text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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The 2005–06 Pittsburgh Penguins season was the team's 39th season in the National Hockey League (NHL). The season was notable for being Sidney Crosby's rookie campaign. However, Crosby's first year didn't help the team, as they suffered another losing season, finishing last place in the Eastern Conference with 58 points, the second worst in the NHL. The Penguins failed to qualify for the Stanley Cup playoffs for the fourth consecutive season. This was the most recent season where they missed the Stanley Cup playoffs. Their 16 years making the playoffs is the longest active streak in the four major North American professional sports.
Off-season
Regular season
The Penguins struggled defensively, finishing 30th overall in goals allowed, with 310 (excluding 6 shootout goals allowed).
Sidney Crosby
Sidney Crosby was selected first overall in the 2005 NHL Entry Draft by the Penguins on July 30, 2005. Due to the labour stoppage in the previous season, the 2005 draft was conducted via a weighted lottery based on each team's playoff appearances and draft lottery victories in the last four years. This lottery system led to the draft being popularly referred to as the "Sidney Crosby Lottery" or the "Sidney Crosby Sweepstakes".
On December 16, 2005, Michel Therrien named Crosby as an alternate captain for the Penguins. The move drew criticism from some hockey pundits, including Don Cherry, who claimed that Crosby had done nothing to earn the position.
Crosby finished his rookie season with the franchise record in assists (63) and points (102) for a rookie, both of which had been previously held by Mario Lemieux. Crosby is the youngest player in the history of the NHL to score 100 points in a single season, and only the seventh rookie ever to hit the benchmark. Overall, Crosby finished sixth in the NHL scoring race and seventh in the NHL in assists. Among Canadian NHL players, he trailed only Joe Thornton and Dany Heatley. While both Crosby and Alexander Ovechkin of the Washington Capitals had impressive rookie campaigns, Crosby finished second behind Ovechkin for the Calder Memorial Trophy for NHL rookie of the year.
Through his first season, Crosby was accused by opposing players and coaches of taking dives and complaining to officials, which has been attributed to his youth.
During his rookie campaign, he was second on his team and fourth among all NHL rookies in penalty minutes, and is the only rookie to accumulate both 100 points and 100 penalty minutes in a single season in NHL history. This magnified his reputation for complaining to NHL officials.
Hockey analyst Kelly Hrudey compared Crosby to Gretzky, who had a similar reputation as a "whiner" in his youth, and suggested that as Crosby matured, he would mellow out and his reputation would fade.
Mario Lemieux
After the lockout concluded, Lemieux returned to the ice for the 2005–06 season. Hopes for the Penguins were high due to the salary cap and revenue sharing, which enabled the team to compete in the market for several star players. Another reason for optimism was the Penguins winning the lottery for the first draft pick, enabling them to select Sidney Crosby. Lemieux opened up his home to Crosby to help the rookie settle in Pittsburgh and Lemieux also served as Crosby's mentor.
On January 24, 2006, Lemieux announced his second and permanent retirement from professional hockey at age 40. This followed a half-season in which he struggled not only with the increased speed of the "new NHL" but also with yet another threatening physical ailment, a heart condition called atrial fibrillation that caused him to experience irregular heartbeats.
Although he had put up points at a pace that most NHL forwards would be perfectly content with (22 points in 26 games) in his last season, Lemieux still remarked, "I can no longer play at a level I was accustomed to in the past."
Season standings
Schedule and results
Preseason
|- style="text-align:center; background:#ffc;"
| 1 || September 19 || Pittsburgh Penguins || 2–3 SO || Columbus Blue Jackets || Nationwide Arena || 0–0–1
|- style="text-align:center; background:#ffc;"
| 2 || September 21 || Boston Bruins || 5–4 OT || Pittsburgh Penguins || Mohegan Sun Arena at Casey Plaza || 0–0–2
|- style="text-align:center; background:#fcf;"
| 3 || September 23 || Ottawa Senators || 3–0 || Pittsburgh Penguins || Visions Veterans Memorial Arena || 0–1–2
|- style="text-align:center; background:#fcf;"
| 4 || September 24 || Ottawa Senators || 6–2 || Pittsburgh Penguins || Mohegan Sun Arena at Casey Plaza || 0–2–2
|- style="text-align:center; background:#fcf;"
| 5 || September 25 || Pittsburgh Penguins || 2–3 || Washington Capitals || Hersheypark Arena || 0–3–2
|- style="text-align:center; background:#cfc;"
| 6 || September 27 || Columbus Blue Jackets || 2–7 || Pittsburgh Penguins || Mellon Arena || 1–3–2
|- style="text-align:center; background:#fcf;"
| 7 || September 29 || Pittsburgh Penguins || 1–4 || Ottawa Senators || Corel Centre || 1–4–2
|- style="text-align:center; background:#fcf;"
| 8 || September 30 || Pittsburgh Penguins || 3–4 || Washington Capitals || Verizon Center || 1–5–2
|- style="text-align:center; background:#cfc;"
| 9 || October 2 || Washington Capitals || 1–7 || Pittsburgh Penguins || Mellon Arena || 2–5–2
|- style="text-align:center;"
| colspan=9 | Legend: = Win = Loss = OT/SO Loss
|-
Regular season
|- style="background:#fcf;"
| 1 || 5 || Pittsburgh Penguins || 1–5 || New Jersey Devils || Mellon Arena (18,101) || 0–1–0 || 0 ||
|- style="background:#ffc;"
| 2 || 7 || Pittsburgh Penguins || 2–3 SO || Carolina Hurricanes || Mellon Arena (18,787) || 0–1–1 || 1 ||
|- style="background:#ffc;"
| 3 || 8 || Boston Bruins || 7–6 OT || Pittsburgh Penguins || TD Banknorth Garden (17,132) || 0–1–2 || 2 ||
|- style="background:#ffc;"
| 4 || 10 || Pittsburgh Penguins || 2–3 OT || Buffalo Sabres || Mellon Arena (12,050) || 0–1–3 || 3 ||
|- style="background:#ffc;"
| 5 || 14 || Pittsburgh Penguins || 5–6 OT || Philadelphia Flyers || Mellon Arena (19,566) || 0–1–4 || 4 ||
|- style="background:#fcf;"
| 6 || 15 || Tampa Bay Lightning || 3–1 || Pittsburgh Penguins || Mellon Arena (17,132) || 0–2–4 || 4 ||
|- style="background:#fcf;"
| 7 || 20 || New Jersey Devils || 6–3 || Pittsburgh Penguins || Mellon Arena (16,082) || 0–3–4 || 4 ||
|- style="background:#fcf;"
| 8 || 22 || Pittsburgh Penguins || 3–6 || Boston Bruins || Mellon Arena (17,565) || 0–4–4 || 4 ||
|- style="background:#ffc;"
| 9 || 25 || Florida Panthers || 4–3 OT || Pittsburgh Penguins || Mellon Arena (14,636) || 0–4–5 || 5 ||
|- style="background:#cfc;"
| 10 || 27 || Atlanta Thrashers || 5–7 || Pittsburgh Penguins || Mellon Arena (14,009) || 1–4–5 || 7 ||
|- style="background:#fcf;"
| 11 || 29 || Carolina Hurricanes || 5–3 || Pittsburgh Penguins || Mellon Arena (16,420) || 1–5–5 || 7 ||
|-
|- style="background:#cfc;"
| 12 || 1 || Pittsburgh Penguins || 4–3 OT || New Jersey Devils || Mellon Arena (10,134) || 2–5–5 || 9 ||
|- style="background:#cfc;"
| 13 || 3 || Pittsburgh Penguins || 5–1 || New York Islanders || Nassau Veterans Memorial Coliseum (14,415) || 3–5–5 || 11 ||
|- style="background:#fcf;"
| 14 || 5 || Pittsburgh Penguins || 3–6 || Boston Bruins || Mellon Arena (17,565) || 3–6–5 || 11 ||
|- style="background:#cfc;"
| 15 || 7 || Pittsburgh Penguins || 3–2 || New York Rangers || Mellon Arena (18,200) || 4–6–5 || 13 ||
|- style="background:#fcf;"
| 16 || 9 || Pittsburgh Penguins || 0–5 || Atlanta Thrashers || Mellon Arena (14,046) || 4–7–5 || 13 ||
|- style="background:#cfc;"
| 17 || 10 || Montreal Canadiens || 2–3 SO || Pittsburgh Penguins || Mellon Arena (16,254) || 5–7–5 || 15 ||
|- style="background:#fcf;"
| 18 || 12 || New York Rangers || 6–1 || Pittsburgh Penguins || Madison Square Garden (IV) (17,132) || 5–8–5 || 15 ||
|- style="background:#ffc;"
| 19 || 14 || New York Islanders || 3–2 SO || Pittsburgh Penguins || Mellon Arena (10,793) || 5–8–6 || 16 ||
|- style="background:#cfc;"
| 20 || 16 || Pittsburgh Penguins || 3–2 OT || Philadelphia Flyers || Mellon Arena (19,687) || 6–8–6 || 18 ||
|- style="background:#fcf;"
| 21 || 19 || Philadelphia Flyers || 6–3 || Pittsburgh Penguins || Wachovia Center (17,132) || 6–9–6 || 18 ||
|- style="background:#cfc;"
| 22 || 22 || Washington Capitals || 4–5 || Pittsburgh Penguins || Verizon Center (16,978) || 7–9–6 || 20 ||
|- style="background:#fcf;"
| 23 || 25 || Pittsburgh Penguins || 3–6 || Florida Panthers || Mellon Arena (18,124) || 7–10–6 || 20 ||
|- style="background:#fcf;"
| 24 || 27 || Pittsburgh Penguins || 1–4 || Tampa Bay Lightning || Mellon Arena (20,218) || 7–11–6 || 20 ||
|- style="background:#fcf;"
| 25 || 29 || Buffalo Sabres || 3–2 || Pittsburgh Penguins || HSBC Arena (15,118) || 7–12–6 || 20 ||
|-
|- style="background:#fcf;"
| 26 || 1 || Pittsburgh Penguins || 1–2 || New York Rangers || Mellon Arena (18,200) || 7–13–6 || 20 ||
|- style="background:#fcf;"
| 27 || 3 || Calgary Flames || 3–2 || Pittsburgh Penguins || Pengrowth Saddledome (16,626) || 7–14–6 || 20 ||
|- style="background:#fcf;"
| 28 || 8 || Minnesota Wild || 5–0 || Pittsburgh Penguins || Xcel Energy Center (14,627) || 7–15–6 || 20 ||
|- style="background:#cfc;"
| 29 || 10 || Colorado Avalanche || 3–4 || Pittsburgh Penguins || Mellon Arena (16,677) || 8–15–6 || 22 ||
|- style="background:#fcf;"
| 30 || 12 || Pittsburgh Penguins || 1–3 || Detroit Red Wings || Mellon Arena (20,066) || 8–16–6 || 22 ||
|- style="background:#fcf;"
| 31 || 13 || Pittsburgh Penguins || 0–3 || St. Louis Blues || Mellon Arena (14,336) || 8–17–6 || 22 ||
|- style="background:#ffc;"
| 32 || 16 || Buffalo Sabres || 4–3 OT || Pittsburgh Penguins || HSBC Arena (16,648) || 8–17–7 || 23 ||
|- style="background:#fcf;"
| 33 || 17 || Pittsburgh Penguins || 3–4 || Buffalo Sabres || Mellon Arena (18,690) || 8–18–7 || 23 ||
|- style="background:#fcf;"
| 34 || 23 || Philadelphia Flyers || 5–4 || Pittsburgh Penguins || Wachovia Center (17,132) || 8–19–7 || 23 ||
|- style="background:#ffc;"
| 35 || 27 || Toronto Maple Leafs || 3–2 OT || Pittsburgh Penguins || Air Canada Centre (17,132) || 8–19–8 || 24 ||
|- style="background:#cfc;"
| 36 || 29 || New Jersey Devils || 2–6 || Pittsburgh Penguins || Izod Center (17,132) || 9–19–8 || 26 ||
|- style="background:#cfc;"
| 37 || 31 || New York Rangers || 3–4 OT || Pittsburgh Penguins || Madison Square Garden (IV) (17,132) || 10–19–8 || 28 ||
|-
|- style="background:#ffc;"
| 38 || 2 || Pittsburgh Penguins || 2–3 OT || Toronto Maple Leafs || Mellon Arena (19,449) || 10–19–9 || 29 ||
|- style="background:#cfc;"
| 39 || 3 || Pittsburgh Penguins || 6–4 || Montreal Canadiens || Mellon Arena (21,273) || 11–19–9 || 31 ||
|- style="background:#fcf;"
| 40 || 6 || Pittsburgh Penguins || 4–6 || Atlanta Thrashers || Mellon Arena (17,209) || 11–20–9 || 31 ||
|- style="background:#fcf;"
| 41 || 7 || Atlanta Thrashers || 4–3 || Pittsburgh Penguins || Philips Arena (17,132) || 11–21–9 || 31 ||
|- style="background:#fcf;"
| 42 || 10 || Edmonton Oilers || 3–1 || Pittsburgh Penguins || Rexall Place (14,905) || 11–22–9 || 31 ||
|- style="background:#fcf;"
| 43 || 11 || Pittsburgh Penguins || 1–6 || Columbus Blue Jackets || Mellon Arena (18,136) || 11–23–9 || 31 ||
|- style="background:#fcf;"
| 44 || 13 || Pittsburgh Penguins || 1–4 || Chicago Blackhawks || Mellon Arena (20,541) || 11–24–9 || 31 ||
|- style="background:#fcf;"
| 45 || 15 || Pittsburgh Penguins || 4–5 || Nashville Predators || Mellon Arena (17,113) || 11–25–9 || 31 ||
|- style="background:#fcf;"
| 46 || 16 || Vancouver Canucks || 4–2 || Pittsburgh Penguins || General Motors Place (15,681) || 11–26–9 || 31 ||
|- style="background:#fcf;"
| 47 || 19 || New York Rangers || 4–2 || Pittsburgh Penguins || Madison Square Garden (IV) (14,272) || 11–27–9 || 31 ||
|- style="background:#fcf;"
| 48 || 21 || Philadelphia Flyers || 2–1 || Pittsburgh Penguins || Wachovia Center (17,132) || 11–28–9 || 31 ||
|- style="background:#fcf;"
| 49 || 23 || Pittsburgh Penguins || 2–4 || Philadelphia Flyers || Mellon Arena (19,726) || 11–29–9 || 31 ||
|- style="background:#cfc;"
| 50 || 25 || Washington Capitals || 1–8 || Pittsburgh Penguins || Verizon Center (14,415) || 12–29–9 || 33 ||
|- style="background:#ffc;"
| 51 || 26 || Pittsburgh Penguins || 3–4 SO || New York Islanders || Nassau Veterans Memorial Coliseum (16,362) || 12–29–10 || 34 ||
|- style="background:#fcf;"
| 52 || 28 || Pittsburgh Penguins || 1–7 || New York Rangers || Mellon Arena (18,200) || 12–30–10 || 34 ||
|-
|- style="background:#fcf;"
| 53 || 1 || Pittsburgh Penguins || 1–3 || New York Rangers || Mellon Arena (18,200) || 12–31–10 || 34 ||
|- style="background:#fcf;"
| 54 || 2 || Ottawa Senators || 7–2 || Pittsburgh Penguins || Corel Centre (14,714) || 12–32–10 || 34 ||
|- style="background:#ffc;"
| 55 || 4 || New York Islanders || 5–4 SO || Pittsburgh Penguins || Mellon Arena (11,218) || 12–32–11 || 35 ||
|- style="background:#fcf;"
| 56 || 6 || Pittsburgh Penguins || 2–5 || Ottawa Senators || Mellon Arena (19,877) || 12–33–11 || 35 ||
|- style="background:#fcf;"
| 57 || 8 || Boston Bruins || 3–1 || Pittsburgh Penguins || TD Banknorth Garden (15,603) || 12–34–11 || 35 ||
|- style="background:#cfc;"
| 58 || 10 || Pittsburgh Penguins || 4–3 || Carolina Hurricanes || Mellon Arena (18,830) || 13–34–11 || 37 ||
|- style="background:#cfc;"
| 59 || 11 || Pittsburgh Penguins || 6–3 || Washington Capitals || Mellon Arena (18,277) || 14–34–11 || 39 ||
|-
|- style="background:#fcf;"
| 60 || 1 || Ottawa Senators || 4–3 || Pittsburgh Penguins || Corel Centre (14,026) || 14–35–11 || 39 ||
|- style="background:#fcf;"
| 61 || 4 || Carolina Hurricanes || 7–5 || Pittsburgh Penguins || RBC Center (16,293) || 14–36–11 || 39 ||
|- style="background:#ffc;"
| 62 || 7 || Tampa Bay Lightning || 5–4 SO || Pittsburgh Penguins || St. Pete Times Forum (15,048) || 14–36–12 || 40 ||
|- style="background:#fcf;"
| 63 || 8 || Pittsburgh Penguins || 3–6 || Washington Capitals || Mellon Arena (14,374) || 14–37–12 || 40 ||
|- style="background:#cfc;"
| 64 || 11 || New Jersey Devils || 3–6 || Pittsburgh Penguins || Izod Center (16,061) || 15–37–12 || 42 ||
|- style="background:#cfc;"
| 65 || 12 || Philadelphia Flyers || 0–2 || Pittsburgh Penguins || Wachovia Center (14,904) || 16–37–12 || 44 ||
|- style="background:#fcf;"
| 66 || 16 || Pittsburgh Penguins || 1–2 || New Jersey Devils || Mellon Arena (11,513) || 16–38–12 || 44 ||
|- style="background:#cfc;"
| 67 || 18 || Pittsburgh Penguins || 5–4 || Montreal Canadiens || Mellon Arena (21,273) || 17–38–12 || 46 ||
|- style="background:#fcf;"
| 68 || 19 || Toronto Maple Leafs || 1–0 || Pittsburgh Penguins || Air Canada Centre (15,174) || 17–39–12 || 46 ||
|- style="background:#fcf;"
| 69 || 21 || Pittsburgh Penguins || 2–5 || Ottawa Senators || Mellon Arena (19,360) || 17–40–12 || 46 ||
|- style="background:#cfc;"
| 70 || 24 || New York Islanders || 3–4 OT || Pittsburgh Penguins || Mellon Arena (11,286) || 18–40–12 || 48 ||
|- style="background:#fcf;"
| 71 || 26 || Montreal Canadiens || 6–5 || Pittsburgh Penguins || Bell Centre (14,807) || 18–41–12 || 48 ||
|- style="background:#fcf;"
| 72 || 29 || Florida Panthers || 5–3 || Pittsburgh Penguins || BankAtlantic Center (14,024) || 18–42–12 || 48 ||
|- style="background:#cfc;"
| 73 || 31 || Pittsburgh Penguins || 4–0 || New York Islanders || Nassau Veterans Memorial Coliseum (15,210) || 19–42–12 || 50 ||
|-
|- style="background:#ffc;"
| 74 || 2 || New Jersey Devils || 3–2 OT || Pittsburgh Penguins || Izod Center (14,427) || 19–42–13 || 51 ||
|- style="background:#fcf;"
| 75 || 5 || Pittsburgh Penguins || 4–6 || New Jersey Devils || Mellon Arena (13,118) || 19–43–13 || 51 ||
|- style="background:#cfc;"
| 76 || 7 || Pittsburgh Penguins || 5–1 || Florida Panthers || Mellon Arena (14,776) || 20–43–13 || 53 ||
|- style="background:#fcf;"
| 77 || 8 || Pittsburgh Penguins || 0–1 || Tampa Bay Lightning || Mellon Arena (20,844) || 20–44–13 || 53 ||
|- style="background:#fcf;"
| 78 || 11 || Pittsburgh Penguins || 3–4 || Philadelphia Flyers || Mellon Arena (19,756) || 20–45–13 || 53 ||
|- style="background:#cfc;"
| 79 || 13 || New York Rangers || 3–5 || Pittsburgh Penguins || Madison Square Garden (IV) (15,155) || 21–45–13 || 55 ||
|- style="background:#ffc;"
| 80 || 15 || Pittsburgh Penguins || 4–5 SO || New York Islanders || Nassau Veterans Memorial Coliseum (17,084) || 21–45–14 || 56 ||
|- style="background:#cfc;"
| 81 || 17 || New York Islanders || 1–6 || Pittsburgh Penguins || Mellon Arena (12,282) || 22–45–14 || 58 ||
|- style="background:#fcf;"
| 82 || 18 || Pittsburgh Penguins || 3–5 || Toronto Maple Leafs || Mellon Arena (19,392) || 22–46–14 || 58 ||
|-
|- style="text-align:center;"
| ''Legend: = Win = Loss = OT/SO Loss
Player statistics
Scoring
Position abbreviations: C = Center; D = Defense; G = Goaltender; LW = Left Wing; RW = Right Wing
= Joined team via a transaction (e.g., trade, waivers, signing) during the season. Stats reflect time with the Penguins only.
= Left team via a transaction (e.g., trade, waivers, release) during the season. Stats reflect time with the Penguins only.
Goaltending
Awards and records
Awards
Milestones
Sidney Crosby played his first professional NHL game on October 5, 2005, against the New Jersey Devils, and registered an assist on the team's first goal of the season, scored by Mark Recchi in a 5–1 loss.
November 11, 2005 – Sidney Crosby beat Jose Theodore of the Montreal Canadiens to win his first career shootout.
November 11, 2005- Mario Lemieux scores his seventh goal of the season, and the last of his career.
November 22, 2005 – Sidney Crosby and Alexander Ovechkin face each other for the first time.
March 12, 2006 – Marc-Andre Fleury gets a shutout by stopping 22 shots against the Philadelphia Flyers.
March 29, 2006 – John LeClair scores the 400th goal of his career.
January 24, 2006 – Mario Lemieux announces his retirement.
April 13, 2006 – Sidney Crosby scores four points in one game, including his 90th point of the season.
April 17, 2006 – Sidney Crosby becomes the youngest player to score 100 points in one season.
Transactions
The Penguins were involved in the following transactions from February 17, 2005, the day after the 2004–05 NHL season was officially cancelled, through June 19, 2006, the day of the deciding game of the 2006 Stanley Cup Finals.
Trades
Players acquired
Players lost
Signings
Other
Draft picks
Notes
References
Pitts
Pitts
Pittsburgh Penguins seasons
Pitts
Pitts | {
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My name is LucilleLluveres-Vivo and for the past 29 years as a physician I have attended manypatients with an incurable disease . My aunt Stellasuffered of a progressive type of cancer and during her illness ( for 5 months) my family and I had the privilege to care forher at home .Together we cried, we laughedand her dignified fight ,in spite of her fears and tears, in spite of herintense suffering inspired me to follow my long dream to start hospice" VillaStella" and be a beacon of light for those facing a terminal illness . Hospice Villa Stella, A placeaway from home but with compassionate professional persons dedicated to the well-being and peace of theterminally ill person. Villa Stella, 24 hours ofhigh quality of care ,comfort and love , not just for the patient but also forthe family. But we cannot do it without your financialsupport; We need your help to take thatsecond step.
Anycontribution is welcome to make this Hospice ,Villa Stella a place where comfort and peace come together.
I kindly would like to ask you, by giving financially, to be part of this journey, so each person at VillaStella can embrace life and face eternity ,not with fear and anxiety but in a peaceful and loving way in the presenceof his or her family . | {
"redpajama_set_name": "RedPajamaC4"
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{"url":"http:\/\/www.arvifox.com\/2016\/08\/","text":"## Shunting-yard algorithm\n\nShunting-yard algorithm\n\nIn\u00a0computer science, the\u00a0shunting-yard algorithm\u00a0is a method for parsing mathematical expressions specified in\u00a0infix notation. It can produce either a postfix notation string, also known as\u00a0Reverse Polish notation\u00a0(RPN), or an\u00a0abstract syntax tree\u00a0(AST). The\u00a0algorithm\u00a0was invented by\u00a0Edsger Dijkstra\u00a0and named the \u201cshunting yard\u201d algorithm because its operation resembles that of a\u00a0railroad shunting yard. Dijkstra first described the Shunting Yard Algorithm in the\u00a0Mathematisch Centrum\u00a0report\u00a0MR 34\/61.","date":"2017-11-20 17:04:54","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8278347253799438, \"perplexity\": 8769.562048723043}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934806086.13\/warc\/CC-MAIN-20171120164823-20171120184823-00244.warc.gz\"}"} | null | null |
Q: Why $http service in AngularJS doesn't find action in controller in APS.Net Web API 2 application I have the application, trying to send request to controller by $http service
getUserPost: function(id) {
var posts = [];
var req = {
method: "POST",
url: "/api/getPost",
data: id
};
I always get the next error:
Possibly unhandled rejection: {"data":{"Message":"No HTTP resource was found that matches the request URI 'http://localhost:57196/api/getPost'.","MessageDetail":"No action was found on the controller 'Post' that matches the request."},"status":404,"config":{"method":"POST","transformRequest":[null],"transformResponse":[null],"jsonpCallbackParam":"callback","url":"/api/getPost","data":0,"headers":{"Accept":"application/json, text/plain, */*","Content-Type":"application/json;charset=utf-8"}},"statusText":"Not Found"}
Controller:
[RoutePrefix("api")]
public class PostController : ApiController
{
private readonly DataContext _db = new DataContext();
[Route("getPost")]
[HttpPost]
public List<Post> GetPost(string id)
{
var posts = new List<Post>();
if (id != null)
{
posts = _db.Posts.Where(x => x.UzytkownikId == int.Parse(id)).ToList();
}
return posts;
}
Does anybody know how can I solve this?
A: You have defined the route for action in wrong way. Try adding FromBody to your action in this way:
[Route("getPost")]
[HttpPost]
public List<Post> GetPost([FromBody]string id)
{
var posts = new List<Post>();
if (id != null)
{
posts = _db.Posts.Where(x => x.UzytkownikId == int.Parse(id)).ToList();
}
return posts;
}
| {
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Tondela es una freguesia portuguesa del concelho de Tondela, con 10,87 km² de superficie y 3.935 habitantes (2001). Su densidad de población es de 362,0 hab/km².
Enlaces externos
Tondela en freguesiasdeportugal.com (en portugués)
Gobierno civil e información administrativa del Distrito de Viseu (en portugués)
Freguesias de Tondela | {
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\subsection{Paper Contribution}
The above literature highlights the breadth of different approaches that are proposed to solve the autonomous intersection management problem.
From this literature, we identify the following key lessons.
First, Kuramoto models and other consensus-based approaches have been successfully used to coordinate traffic lights and centralized intersection coordinators, but they have not been explored as means to coordinate autonomous vehicles themselves. Second, approaches that consider multiple intersections and the coupling between them can yield larger fuel savings compared to localized controls. However, most approaches that solve both intra- and inter-junction problems rely on some sort of centralized agent that couples intersections. This paper proposes an approach to solve the autonomous intersection management problem at both levels using the non-linear consensus equation known as the Kuramoto model. The use of Kuramoto allows vehicles to first agree upon the current state of the intersections (i.e. which flow is being serviced), and then to synchronize with the intersections. The conference version of this paper \cite{Rodriguez2019} introduced the basic idea underlying this strategy; namely, the fact that through mapping phase and frequency to position and velocity and synchronizing using Kuramoto, vehicles can agree on crossing times that solve both the intra- and inter-junction problem. In addition, this archival version of the work adds the following key contributions. First, we pair the synchronizing Kuramoto layer with a more sophisticated optimization-based tracker, as opposed to the back-stepping tracker used in \cite{Rodriguez2019}. Second, we now allow right turns in the network, which puts us closer to adapting the strategy to more realistic traffic scenarios. Third, we evaluate the impact of our strategy on fuel consumption and delay time as compared to human-driven vehicles in the presence of traffic lights. The fuel savings are then explained by correlating them to to changes in vehicle behavior (e.g., reductions in energy loss due to the braking portions of start-stop driving).
\section{Introduction}
\subfile{Introduction.tex}
\section{Proposed Strategy}
\subfile{ProposedStrategy.tex}
\section{Designing for Traffic Flow and Safety}
\subfile{TrafficFlowandSafety.tex}
\section{Simulation Studies}
\subfile{SimulationStudies.tex}
\section{Conclusion}
\subfile{Conclusion.tex}
\section*{Acknowledgment}
The information, data, or work presented herein was funded in part by the Advanced Research Projects Agency-Energy (ARPA-E), U.S. Department of Energy, under Award Number DE-AR-0000801. The authors gratefully acknowledge this support.
\ifCLASSOPTIONcaptionsoff
\newpage
\fi
\bibliographystyle{IEEEtran}
\subsection{Kuramoto Synchronization}
The literature on traffic light synchronization using the Kuramoto equation works by describing the agents (i.e the traffic lights) as oscillators and establishing a mapping between the phase of the agent and a control action (i.e switching from green to yellow, or red). In our proposed strategy, where the vehicles are the agents as opposed to the traffic lights, the mapping relates the phase of the vehicles to a position along the road. For each road segment $p$, we define a mapping $g_p(\theta)$ that relates the phase $\theta$ of a vehicle to the vehicle's desired distance to the intersection along the curvature of the road $s^d$:
\begin{equation} \label{gr}
s^d = g_p(\theta)
\end{equation}
We choose $g_p$ to be an affine function of phase; it can therefore be described by two parameters. We call these parameters the wavelength $\lambda$ and the offset $\phi$, where $\lambda$ is the slope and $\phi$ the zero crossing. For a given road $p$, the mapping is then:
\begin{equation} \label{grDef}
g_p(\theta) = (\theta-\phi_{p})\frac{\lambda_{p}}{2\pi}
\end{equation}
\noindent We can think of this map as having wrapped the length along the road around a circle of radius $\frac{\lambda_p}{2\pi}$, and rotated it by an angle $\phi_p$
Assuming steady state tracking of the desired distance to the intersection $s^d$, it follows from the definition of $g_p$ that a vehicle $i$ on road $p$ will reach the intersection when its phase $\theta_i$ is equal to the corresponding offset $\phi_p$. It also follows that two vehicles on road $p$ with a phase difference of some multiple $k$ of $2\pi$, will be separated by $k\lambda_p$ meters. Mathematically, these two properties of our mapping can be expressed as:
\begin{equation} \label{gr0}
g_p(\phi_p) = 0
\end{equation}
\begin{equation} \label{grLambda}
g_p(\theta+2k\pi) - g_p(\theta) = k\lambda_p
\end{equation}
We have yet to define one of the main descriptors of an oscillator: its natural frequency. Since phase is mapped onto position, frequency will be mapped onto velocity. Indeed, from differentiation in time of Eq. \eqref{grDef}, we have a definition of desired vehicle velocity:
\begin{equation} \label{vdDef}
v^d = \dot{\theta}\frac{\lambda_p}{2\pi}
\end{equation}
Under this definition, it follows that the natural frequency $\omega$ of a vehicle is simply the frequency corresponding to the constant nominal desired velocity the vehicle would like to travel at. In our proposed strategy a key constraint is that all vehicles have the same natural frequency $\omega_n$. As such, a road segment $p$ is characterized not only by its wavelength $\lambda_p$, but also by a nominal speed $v_{n,p}$, such that the following constraint is always satisfied\footnote{The possibility of allowing multiple nominal speeds on multi-lane road segments is not precluded by this problem formulation, since the different lanes can correspond to different wavelengths.}:
\begin{equation} \label{OmegaConstraint}
\omega_n = 2\pi \frac{v_{n,p}}{\lambda_p}
\end{equation}
Now that we have a definition of phase and frequency as they relate to desired position and velocity, we consider the dynamics of this phase variable. Specifically, we impose that these dynamics be governed by the Kuramoto equation. This equation was introduced in 1975 to model the dynamics of populations of weakly coupled oscillators that exhibit self-synchronizing behaviour. Synchronization refers to oscillators with different natural frequencies influencing each other to oscillate at the same frequency and a constant phase difference. This occurs mostly in biological systems like populations of flashing fireflies. The governing equation, as proposed by Kuramoto in \cite{Kuramoto1975}, is as follows:
\begin{equation} \label{KuramotoFull}
\dot{\theta_i}(t) = \omega_{i} + \frac{1}{N}\sum^{N}_{j=1} K_{ij}\sin(\theta_j(t)-\theta_i(t))
\end{equation}
In this formulation, the instantaneous frequency of oscillation $\dot{\theta_i}$ is given by the oscillator's natural frequency $\omega_i$ plus the coupling term to all other oscillators based on the sine of their difference in phase multiplied by a coupling term $K_{ij}$.
For all-to-all symmetric coupling, that is $K_{ij}=K$, and a monotonic and uni-modal distribution of natural frequencies $p(\omega)$, the behaviour and stability of the system is well-understood \cite{Strogatz2000}. To illustrate this behaviour, it is useful to express the model in its mean-field form, by introducing the order parameter:
\begin{equation} \label{r}
r(t) e^{\Psi(t) i} = \frac{1}{N}\sum^{N}_{j=1} e^{\theta_j(t) i}
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[]{Figures/unitCircle.pdf}
\caption{The order parameter has magnitude $r$, the coherence, and phase $\Psi$, the mean phase}
\label{unitCircle}
\end{figure}
If each oscillator is thought of as a particle orbiting around the unit circle, the order parameter is the centroid of all oscillators, as shown in Fig. \ref{unitCircle}. The Kuramoto equation can then be rearranged in terms of $r$ and $\Psi$:
\begin{equation} \label{KuramotoMean}
\dot{\theta_i}(t) = \omega_{i} + r(t) K \sin(\Psi(t)-\theta_i(t))
\end{equation}
In this form, one can see that the $i^{th}$ oscillator is pulled towards the mean phase $\Psi$ with an effective coupling $Kr$. The coherence $r$ takes values from 0 to 1, where 0 represents all oscillators orbiting incoherently and 1 represents all of them sharing the same phase.
When the coupling between oscillators is $K=0$, agents orbit the unit circle in complete incoherence and the value of $r$ fluctuates around $0$. As the coupling strength is increased, incoherent behaviour persists until a critical coupling threshold $K_c$ is exceeded. For these larger values of $K$, a subset of oscillators synchronize and start recruiting more and more oscillators. Indeed, a positive relationship exists between the coherence $r$ and the coupling strength $Kr$. From Eq. \eqref{KuramotoMean} we can see that the stronger the coupling, the more the oscillator is pulled towards the mean phase, and as more oscillators orbit near the mean phase, the coherence $r$ increases. Finally, the value of $r$ saturates at some final value below, but near 1, around which it fluctuates.
For normal distributions of natural frequencies and large enough coupling, the resulting behaviour corresponds to all oscillators orbiting with the mean frequency of the original distribution (this is called frequency entrainment) and maintaining a constant phase difference between each other (this is called phase-locking). In the particular case of all natural frequencies being the same, all vehicles phase lock to the mean phase exactly, with no constant phase difference between them, and $r$ converges to 1 exactly. Using the frequency given in Eq. \eqref{OmegaConstraint}, we can write the dynamics of $\theta_i$ as:
\begin{equation} \label{KuramotoMeanWn}
\dot{\theta_i}(t) = \omega_{n} + r(t) K \sin(\Psi_i(t)-\theta_i(t))
\end{equation}
\noindent where the local mean phase $\Psi_i$ is the closest projection to $\theta_i$ of the overall mean. That is:
\begin{equation} \label{MeanPhaseProjection}
\begin{gathered}
\Psi_i(t) = \min_k \{\Psi(t)+2k\pi \}\\
\text{subject to:}\\
-\pi \leq ||\Psi(t)+2k\pi - \theta_i(t)|| \leq \pi\\
k\in \mathbb{Z}
\end{gathered}
\end{equation}
By collapsing the distribution of natural frequencies of the oscillators into a single point (i.e $p(\omega_n)=1$), we force all the phases of the system to converge to the mean phase plus some multiple of $2\pi$, or, in other words, to its closest mean phase $\Psi_i(t)$.
For a population of oscillators with a random distribution of initial phase, the trajectories of the phase, mean phase, and frequency are shown in Fig. \ref{KuramotoExample}. Note that oscillators are basically pulled towards the closest mean phase; we can then think of the mean phase, and its projections every $2\pi$, as beacons that the vehicles are attracted to.
\begin{figure}[t]
\centering
\subfloat[Phase and Mean phase trajectories]{\includegraphics[width=\columnwidth]{Figures/phaseAndMeanPhase.eps}%
\label{phaseAndMeanPhase}}
\hfil
\subfloat[Evolution of frequencies $\theta_i$]{\includegraphics[width=\columnwidth]{Figures/frequency.eps}%
\label{frequencies}}
\caption{Evolution of phase, mean phase, and frequencies for a population of oscillators with random initial phase}
\label{KuramotoExample}
\end{figure}
Finally, we combine the behavior of a Kuramoto-driven system and the mapping between desired position and phase we have defined. This combination constitutes the coordinating layer of our algorithm. Through Kuramoto the vehicles agree on a mean phase for the entire network, and because of the definition of the mapping given by Eqs. \ref{grDef} and \ref{gr0}, the vehicles then attempt to cross the intersection exactly when the mean phase is equal to the offset of the road. As such, the synchronizing Kuramoto layer allows vehicles to negotiate the crossing state of all intersections in the network, regardless of their distance to those intersections.
\subsection{Phase, Offset and Wavelength Constraints}
Three different types of constraints need to be satisfied so that the behaviour of oscillators shown in Fig. \ref{KuramotoExample}, corresponds to solving both the intra-junction and the inter-junction problem; these are:
\begin{enumerate}
\item No two vehicles in the same road segment are being pulled towards the same beacon; this guarantees \textbf{spacing} between vehicles in the same road.
\item The phase offsets for intersecting roads place the vehicles in the intersection at different times; this guarantees alternate \textbf{servicing} at the intersection.
\item The phase of a vehicle as it goes from one road segment to the next segment of the same road (i.e., as it goes straight through an intersection, without turning) does not change (in the unit circle); this guarantees \textbf{continuity} of the through flow, thereby reducing energy losses due to re-synchronization.
\end{enumerate}
Vehicles can meet the spacing constraint by properly correcting their phase when a conflict is detected, which mostly occurs when entering a new road segment. Recall that Kuramoto feedback pulls an oscillators towards whichever mean-phase attractor is closer to its current phase. If we define $\Psi_i$ as the projection of the mean-phase closest to the phase $\theta_i$ of vehicle $i$, according to Eq. \eqref{MeanPhaseProjection}, we can write a phase resetting condition for the vehicles that guarantees the spacing constraint:
\begin{equation}\label{resetting}
\theta_i = \min(\theta_i,\Psi_j-\pi-\epsilon) \quad \forall j \in \{j|s_j>s_i\}
\end{equation}
\noindent where $\epsilon$ is a very small number. By saturating $\theta_i$ in this way, we make sure that $\Psi_i\neq\Psi_j$, which means that no two agents on the same road segment are pulled towards the same attractor.
\begin{figure}[t]
\centering
\subfloat[Inter-junction diagram]{\includegraphics[width=\columnwidth]{Figures/TwoIntersectionDiagram.png}%
\label{sections}}
\hfil
\subfloat[Intra-junction diagram]{\includegraphics[width=\columnwidth]{Figures/OneIntersectionDiagram.png}%
\label{intersections}}
\caption{Variable definition as seen within and between junctions}
\label{RoadGeometry}
\end{figure}
To write the safe servicing and continuity constraints mathematically, we consider the variable definitions in Fig. \ref{RoadGeometry}, where we draw a representative intersection zone. Points $A_1$ and $B_3$ correspond to the origins of road segments 1 and 3; that is, the phase at those points is the offset of the respective road segment. Point $C$ represents the intersecting point between the trajectories of vehicles going straight through both roads. Here, and for the rest of the paper we consider an intersection of two one-way roads with only two conflicting traffic movements for the sake of simplicity. More practical traffic scenarios can be accounted for by partitioning the wavelength into however many flows are necessary.
The servicing constraint, which directly relates to solving the intra-junction problem, aims to maximally space out vehicles crossing the intersection from different roads. It is a constraint on the offset of each road that guarantees that each traffic flow is serviced during a different portion of the cycle. Considering the scenario drawn in Fig. \ref{RoadGeometry} we can see that maximal spacing for vehicle $i$ from the vehicles that cross the intersection before and after itself occurs if it reaches the intersection (i.e. point $C$) exactly between them. Now vehicles $i+1$ and $i-1$ are separated by a full wavelength $\lambda_1$, or by $2\pi$ radians in the phase domain (as follows from equation \eqref{grLambda}) . It follows that the distance between vehicle $i$ and $i-1$ should be half a wavelength, or $\pi$ radians in the phase domain. We can show that this is achieved if the mappings of roads 1 and 3 satisfy the following constraint, which relates the phases of point C as mapped by the mappings of each road.
\begin{equation} \label{servicingGeneral}
g_1^{-1}(A_1C)=\pi+g_3^{-1}(B_3C)
\end{equation}
\noindent where $g_p^{-1}(s)$ is the inverse of the mapping \eqref{gr} for road segment $p$. The arguments $A_1C$ and $B_3C$ are the distances between each road's entrance to the intersection and the collision point. For our proposed mapping \eqref{grDef}, the above equation can be rearranged as:
\begin{equation} \label{servicing}
\phi_3-\phi_1=2\pi(\frac{A_1C}{\lambda_1}-\frac{B_3C}{\lambda_3})-\pi
\end{equation}
Finally, the inter-junction coordination problem can be solved automatically by ensuring continuity between mappings as vehicles go from one road segment to the next. That is, we guarantee that the phases at points $A_1$ and $B_3$ are the same when mapped by roads 1 and 2, and roads 3 and 4 respectively. Recalling that points $A_1$ and $B_3$ are the origins of the intersection region, and using equations \eqref{grDef} and \eqref{gr0}, this amounts to:
\begin{equation}
\begin{split}
g_2^{-1}(L_2+A_1A_1')&=g_1^{-1}(0)=\phi_1\\
g_4^{-1}(L_4+B_3B_3')&=g_3^{-1}(0)=\phi_3
\end{split}
\end{equation}
Rearranging according to our affine mapping of equation \eqref{gr}, we can express the constraints in terms of the offsets of the roads:
\begin{equation}\label{continuity}
\begin{split}
\phi_1-\phi_2 &= \frac{2\pi}{\lambda_2}(L_2+\bar{A_1A_1'}) \;(\text{mod } 2\pi)\\
\phi_3-\phi_4 &= \frac{2\pi}{\lambda_3}(L_4+\bar{B_3B_3'}) \;(\text{mod } 2\pi)\\
\end{split}
\end{equation}
It is worth noting that Eq. \eqref{continuity} can only partially guarantee continuous flow as vehicles travel along a corridor of intersections. For one, the constraint cannot be imposed to turning flows, since the servicing constraint ensures the destination road segment of a turning vehicle will be $\pi$ radians out of phase with respect to its road of origin. Another scenario where flow is disrupted occurs when another vehicle turns into the destination section of the vehicle going straight. In this situation, because of the spacing constraint, the latter vehicle will be forced to slow down to catch the upstream wave. Finally, while the wavelengths can be thought of as adjustable variables in constraint \eqref{continuity}, wavelengths are also constrained by their relationship with frequency and velocity through Eq. \eqref{OmegaConstraint}. Specifically, a change in wavelength from one section to the next would force a change in desired speed through Eq. \eqref{OmegaConstraint} in order to maintain a constant natural frequency, creating an undesirable acceleration or deceleration event. For the rest of this work, we therefore assume that wavelengths and desired speeds are the same across all roads in the network, and we drop the road identifying index $p$ for $\lambda$ and $v^d$.
In guaranteeing spacing, safety and continuity to solve the coordination problem at both scales, we have introduced two different types of constraints. The spacing constraint \eqref{resetting} is a constraint on the actual phase of the vehicles; it forces vehicles to push their desired phase back, and with it the time at which it will cross the intersection. This constraint needs to be checked for and implemented continuously, although it will mostly become active when vehicles change road segments. The servicing and continuity constraints, on the other hand, are constraints on the constant design variables of the network, namely the offsets and wavelengths of the roads, and they are chosen before any vehicles enter the network. Along with the desired speed $v^d$, these design variables determine the maximum throughput of the network as we will discuss in subsequent sections.
\subsection{Optimal mean-phase tracking}
So far we have discussed the dynamics of a vehicle's desired phase, which is then mapped to a desired position. In previous work \cite{Rodriguez2018}, we propose a linear feed-forward/feedback tracker that uses this signal as reference. Further insight into the behaviour of the system of coupled oscillators allows us to propose here a more sophisticated tracking approach, namely, a model predictive optimal controller that minimizes the jerk of vehicles using predictions of both the arrival time imposed by the phase dynamics and the behaviour of other vehicles.
We can show that the synchronizing layer described above determines the time $T_i(t)$ at which the vehicle $i$ should ideally arrive at the intersection. Indeed, the computation of $T_i(t)$ follows from the properties of Eq. \eqref{KuramotoMean}, where the mean-phase $\Psi$ oscillates with a constant frequency $\omega_n$ \cite{Strogatz2000}.
\begin{equation}
\dot{\Psi}(t)=\omega_n(t)
\end{equation}{}
As described in the previous section, vehicle $i$ should reach the intersection when its phase is already tracking its mean phase beacon, which is in turn equal to the offset of the road:
\begin{equation}
\theta_i(T_i)=\Psi_i(T_i)=\phi_p
\end{equation}
It follows from the previous two equations that for vehicle $i$ at time $t$ the expected time of arrival at the intersection is given by:
\begin{equation}
T_i(t) = \frac{\phi_p-\Psi_i(t)}{\omega_n}
\end{equation}
Since the vehicle enters the intersection at time $T_i$, in synchrony with its mean phase beacon, its desired position, velocity and acceleration are also known:
\begin{equation}\label{finalTimeCondition}
\begin{split}
s_i(t+T_i) &= 0\\
v_i(t+T_i) &= v^d\\
a_i(t+T_i) &= 0
\end{split}
\end{equation}
Assuming vehicles can control their jerk, or their change in acceleration, through accurate lower level powertrain and vehicle dynamics controllers, we model these vehicles as third order dynamical systems. Note that we choose a third order system here, instead of the second order system traditionally used to model vehicles, because it will yield smoother acceleration profiles. With the third order model, the state variables for each vehicle are then: (i) its distance to the intersection, along the path of the road; (ii) its velocity; and (iii) its acceleration. The input is the jerk of the vehicle:
\begin{equation}\label{stateEquations}
\begin{split}
\dot{s_i}(t) &= v_i(t)\\
\dot{v_i}(t) &= a_i(t)\\
\dot{a_i}(t) &= u_i(t)
\end{split}
\end{equation}
The control input $u_i(t)$ that places the vehicle at the intersection at the right time, can be the solution of an optimization problem that minimizes mean square jerk:
\begin{equation}\label{optimalControlProblem}
\min \int_t^{t+T_i(t)}\frac{1}{2}u_i(\tau)^2 d\tau
\end{equation}
Subject to:
\begin{center}
State dynamics \eqref{stateEquations}\\
Terminal time conditions \eqref{finalTimeCondition}
\end{center}
\begin{equation}\label{ineqConstraints}
\begin{split}
s_j(t)-s_i(t)-S \leq 0\\
a_{i,\text{min}}\leq a_i(t) \leq a_{i,\text{max}}\\
v_{i,\text{min}}\leq v_i(t) \leq v_{i,\text{max}}
\end{split}
\end{equation}
The additional inequality constraints guarantee that the vehicle stays a safe distance $S$ from its leading vehicle $j$, and that the acceleration and velocity are bounded.
The solution to the problem without the inequality constraints \eqref{ineqConstraints} can be determined analytically by performing a Hamiltonian analysis. This approach is similar to the work of Malikoupoulos et al. in \cite{Rios-torres2017,Zhao2018}, where the solution to a second order dynamical system, where the input is acceleration rather than jerk, is presented. In our case, the optimal trajectories for the input and the states, denoted with an asterisk, are given by:
\begin{equation}\label{unconstrainedSol}
\begin{split}
u^*(t) &= -\frac{1}{2}c_1 t^2 +c_2 t-c_3\\
a^*(t) &= -\frac{1}{6}c_1 t^3 +\frac{1}{2}c_2 t^2 -c_3 t+c_4\\
v^*(t) &= -\frac{1}{24}c_1 t^4 +\frac{1}{6}c_2 t^3 -\frac{1}{2}c_3 t^2 +c_4 t+c_5\\
s^*(t) &= -\frac{1}{120}c_1 t^5 +\frac{1}{24}c_2 t^4 -\frac{1}{6}c_3 t^3 +\frac{1}{2}c_4 t^2 +c_5 t+c_6\\
\end{split}
\end{equation}
The constants $c_{1,...,6}$ in the above equations are integration constants, and they can be solved for by imposing initial and final time conditions. The initial conditions are given by the current state of the vehicle at time $t$, and the final conditions are given in equation \eqref{finalTimeCondition}. The resulting system of equations is linear, and it is solved by inverting a 6-by-6 matrix and multiplying it by the concatenated vector of initial and final conditions.
The above is the solution to the unconstrained problem; the solution to the constrained problem can be determined numerically by discretizing and using a quadratic programming solver. This type of optimization is well-understood, convex, and not computationally prohibitive. It can therefore be performed online at every time step when constraints are active. As such, our proposed solution method consists of computing the analytical solution to the unconstrained problem, and checking for constraint activity. If no constraint is infringed upon by the analytic unconstrained solution, we execute the computed input trajectory. Otherwise, we use this candidate solution as the initial guess to the quadratic programming solver and implement the constrained solution instead.
\subsection{Summary}
To summarize the workings of our algorithm, let us recount the actions vehicle $i$ takes at any given time $t$, after it receives the phase and mapping information from the rest of network:
\begin{enumerate}
\item If the vehicle has just entered a new road segment, it selects its initial phase $\theta_i(t)$ to match its current position according to the mapping of the road.
\item It computes the order parameter of the system of oscillators (i.e the mean-phase $\Psi$ and coherence $r$ of the network), as well as the projection $\Psi_i$ of $\Psi$ closest to its own phase.
\item If both its phase and that of its leading vehicle $j$ are most proximal to the same mean-phase beacon ($\Psi_i = \Psi_j$), the vehicle pushes its phase backwards by $\pi + \epsilon$ radians from the beacon tracked by its leader ($\theta_i = \Psi_j -\pi-\epsilon$).
\item It computes the time it should arrive at the intersection $T_i(t)$ given the current mean-phase.
\item It determines the optimal trajectory of its state and input that minimizes jerk, according to the analytical solution to the unconstrained optimization problem.
\item If the solution violates constraints, it solves the constrained optimization problem numerically.
\item It updates the value of its phase through the Kuramoto equation.
\item It implements the first input command according to the generated input trajectory.
\end{enumerate}
Fig. \ref{controlArch} summarizes this process in a block diagram. The result of following this protocol is that vehicles cross the intersection at different times, and that acceleration maneuvers as they go from one intersection to next are not very aggressive.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Figures/ControlArchitecture.pdf}
\caption{Control Architecture}
\label{controlArch}
\end{figure}
\subsection{State Trajectories}
Fig. \ref{posTrajectoriesJunction} shows the distance to the intersection as a function of time for a group of vehicles approaching the intersection at the center of the network. In this figure, as in subsequent ones, the color of the curve indicates whether the vehicle is travelling down a horizontal (dashed red) or a vertical (solid
blue) road segment, and, for clarity, we flip the sign of the distance along the horizontal directions. We can see that red and blue lines cross the 0 line at different points, meaning that vehicles enter the intersection at different times. Moreover, the plot illustrates how vehicles space out evenly along the same road.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Figures/posTrajectoriesJunction.eps}
\caption{Example position trajectories for a group of vehicles approaching the same intersection along the horizontal (red dotted-solid line) and vertical (blue solid line) roads.}
\label{posTrajectoriesJunction}
\end{figure}
We can also look at the position, velocity and acceleration of a single vehicle as it travels through the network, which we show in Fig. \ref{StateTrajectories}. Here, we have also plotted in solid blue the vertical segments, and in dashed red the horizontal ones. We can see that as the vehicle goes straight through the intersections its velocity profile stays relatively flat, as promoted by the continuity constraint we impose on the mapping and the fact the consensus occurs at a network level. When the vehicle turns in the third intersection it needs to adjust its speed to match the offset of the new road it travels on. The same thing happens as it turns right again in the next intersection.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Figures/ExampleStateTrajectories.eps}
\caption{Example position, velocity and acceleration trajectories for a single vehicle travelling through the network in horizontal (red solid-dotted) or vertical roads (solid blue), along with the reference mean phase and frequency (dotted black).}
\label{StateTrajectories}
\end{figure}
\subsection{Fuel Consumption and Delay Time Results}
We can evaluate the fuel consumption and delay time of vehicles using our strategy compared to simulated human drivers controlled by traffic lights. The baseline drivers are governed by a modified Gipps car following model \cite{Gipps1981} as implemented in Aimsun, an established traffic simulator. We set the input flow of all entry roads at 750 vehicles per hour, with a turn percentage of 20\%. The arrival process of vehicles into the network is the main source of stochasticity in our simulation, and it is modeled as a Poisson arrival process, as is traditionally done in traffic simulation \cite{barcelo2010}. We choose a traffic light cycle of 60 seconds, with 25 seconds of green time for each flow and 10 seconds of clearing time. Furthermore we offset the green time of the lights in pursuit of the "green wave" effect, which occurs when vehicles catch several green windows in a row as they travel down an arterial corridor. We run the baseline simulation for 10 minutes of simulated time, and we replicate the scenario with the same vehicle injection times and paths, but using our Kuramoto strategy instead. Fig. \ref{posTrajectoriesJunctionBaseline} shows the baseline position trajectories corresponding to the same vehicles shown in Fig. \ref{posTrajectoriesJunction} in the last sub-section.
\begin{figure}[!h]
\centering
\includegraphics[width=\columnwidth]{Figures/posTrajectoriesJunctionBaseline.eps}
\caption{Example baseline position trajectories for vehicles approaching an intersection controlled by a traffic light.}
\label{posTrajectoriesJunctionBaseline}
\end{figure}
The described simulation consists of about 750 vehicles, but for our comparisons we consider only the 100$^{th}$ through 600$^{th}$ vehicles. In this way, we allow for the network to build some capacity, and we don't consider vehicles who don't finish their path before the simulation is stopped.
We are interested in looking at two metrics relevant for traffic performance evaluation: fuel consumption and delay time. The delay time is simply the difference between a vehicle's travel time and its corresponding minimum travel time had it cruised at the desired speed of the road, normalized by the total distance traveled. To calculate fuel consumption, we use a fuel map for a 1.2 liter gasoline engine. The map translates every engine torque and speed pair to a fuel rate $\dot{m_f}$. To use it, we first calculate the wheel power required to meet the acceleration and velocity trajectories imposed by the driver. We then estimate the corresponding engine power by assuming an efficiency ratio for the transmission. Finally, we say the vehicle uses the minimum fuel rate associated with this engine power demand, which assumes the transmission can operate at the required engine torque and speed combination. Fig. \ref{optimalFuel} shows the optimal fuel rate vs. engine power line we get.
\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{Figures/optimalFuelLine.eps}
\caption{Optimal fuel rate vs. engine power for a 1.2 liter gasoline engine}
\label{optimalFuel}
\end{figure}
Fig. \ref{FuelAndDelay} shows the delay time and fuel consumed by each of the vehicles for both the baseline and proposed scenario. When we compare the average of both point clouds, we find that our proposed strategy leads to a 48\% and 57\% reduction in fuel consumption and delay time respectively for this particular scenario. Furthermore, we note a significant reduction in the spread of the point cloud, meaning that there is less variability in the anticipated behaviour of the vehicles. Indeed, in the baseline, a vehicle that encounters a desirable green wave of traffic light sequences can traverse the network quickly without stop-and-go behaviour, whereas vehicles that are less lucky are forced to stop at several intersections in sequence.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{Figures/delayAndFuelConsumed.eps}
\caption{Delay time and fuel consumed for each vehicle in the simulation of the baseline (green) and proposed strategy (blue) }
\label{FuelAndDelay}
\end{figure}
If we compute the work done by negative propulsive forces (i.e. braking), drag forces, and rolling resistance forces in our model for longitudinal vehicle dynamics, we can see where energy losses are incurred. The savings in fuel consumption can then indeed be attributed to a reduction in energy losses due to braking. In other words, our strategy improves performance by reducing stop-and-go behaviour, as expected. Fig. \ref{EnergyExp} shows the result of this energy balance.
\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{Figures/energyExpenditure.eps}
\caption{Energy losses by type (brake, drag, and rolling resistance) for the synchronization strategy and the baseline strategy }
\label{EnergyExp}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,381 |
<?php namespace Gravitano\Routing;
use Closure;
use Gravitano\Routing\Exceptions\InvalidRouteActionException;
class Route {
protected $attributes = [];
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{
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}
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{
return isset($this->attributes[$key]) ? $this->attributes[$key] : $default;
}
public function match($uri)
{
return $this->uri == $uri;
}
public function getResponse()
{
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if($action instanceof Closure)
{
return call_user_func($action, $this);
}
elseif(is_string($action))
{
list($controller, $method) = explode('@', $action);
$controller = 'App\\Http\\Controllers\\' . $controller;
return call_user_func_array([new $controller, $method], array());
}
throw new InvalidRouteActionException;
}
public function __get($key)
{
return $this->getAttribute($key);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
{"url":"https:\/\/openwetware.org\/wiki\/BIOL368\/F14:Week_10","text":"# BIOL368\/F14:Week 10\n\nBIOL368: Bioinformatics Laboratory\n\nLoyola Marymount University\n\nThis journal entry is due on Wednesday, November 5 at midnight PST (Tuesday night\/Wednesday morning). NOTE that the server records the time as Eastern Standard Time (EST). Therefore, midnight will register as 03:00.\n\n## Individual Journal Assignment\n\n\u2022 Store this journal entry as \"username Week 10\" (i.e., this is the text to place between the square brackets when you link to this page).\n\u2022 Create the following set of links. These links should all be in your personal template; then use the template on your journal entry.\n\u2022 Don't forget to add the \"BIOL368\/F14\" category to the end of your wiki page.\n\n### HIV Structure Redux\n\n\u2022 For each of the amino acid substitutions that you talked about in your HIV Structure Project Presentation, use StarBiochem or Cn3D to locate the positions of those amino acids on the structure..\n\u2022 Take a screenshot showing a view that shows the amino acid(s) in question. You may choose to do one substitution at a time or show all of them on the same screenshot. Save your screenshot(s) in a PowerPoint slide and use an arrow or circle to point to the amino acid. Label each with the amino acid it is in the structure and what the substitutions were in the subjects you studied.\n\u2022 Interpret whether the amino acids are in a location that should affect the function of gp120.\n\n### Introduction to DNA Microarrays\n\n#### Answer the following Discovery Questions from Chapter 4\n\nAnswer the following questions related to Chapter 4 of Campbell & Heyer (2003). Note that some of the questions below have been reworded from the Discovery Questions in the book:\n\n1. (Question 5, p. 110) Choose two genes from Figure 4.6b (PDF of figures on MyLMUConnect) and draw a graph to represent the change in transcription over time. You can either create your plot in Excel and put the image up on your wiki page or you can do it in hard copy and turn it in in class.\n2. (Question 6b, p. 110) Look at Figure 4.7, which depicts the loss of oxygen over time and the transcriptional response of three genes. These data are the ratios of transcription for genes X, Y, and Z during the depletion of oxygen. Using the color scale from Figure 4.6, determine the color for each ratio in Figure 4.7b. (Use the nomenclature \"bright green\", \"medium green\", \"dim green\", \"black\", \"dim red\", \"medium red\", or \"bright red\" for your answers.)\n3. (Question 7, p. 110) Were any of the genes in Figure 4.7b transcribed similarly? If so, which ones were transcribed similarly to which ones?\n4. (Question 9, p. 118) Why would most spots be yellow at the first time point? I.e., what is the technical reason that spots show up as yellow - where does the yellow color come from? And, what would be the biological reason that the experiment resulted in most spots being yellow?\n5. (Question 10, p. 118) Go to the Saccharomyces Genome Database and search for the gene TEF4; you will see it is involved in translation. Look at the time point labeled OD 3.7 in Figure 4.12, and find the TEF4 spot. Over the course of this experiment, was TEF4 induced or repressed? Hypothesize why TEF4\u2019s change in expression was part of the cell\u2019s response to a reduction in available glucose (i.e., the only available food).\n6. (Question, 11, p. 120) Why would TCA cycle genes be induced if the glucose supply is running out?\n7. (Question 12, p. 120) What mechanism could the genome use to ensure genes for enzymes in a common pathway are induced or repressed simultaneously?\n8. (Question 13, p. 121) Consider a microarray experiment where cells deleted for the repressor TUP1 were subjected to the same experiment of a timecourse of glucose depletion where cells at t0 (plenty of glucose available) are labeled green and cells at later timepoints (glucose depleted) are labeled red. What color would you expect the spots that represented glucose-repressed genes to be in the later time points of this experiment?\n9. (Question 14, p. 121) Consider a microarray experiment where cells that overexpress the transcription factor Yap1p were subjected to the same experiment of a timecourse of glucose depletion where cells at t0 (plenty of glucose available) are labeled green and cells at later timepoints (glucose depleted) are labeled red. What color would you expect the spots that represented Yap1p target genes to be in the later time points of this experiment?\n10. (Question 15, p. 121) Could the loss of a repressor or the overexpression of a transcription factor result in the repression of a particular gene?\n11. (Question 16, p. 121) Using the microarray data, how could you verify that you had truly deleted TUP1 or overexpressed YAP1 in the experiments described in questions 8 and 9?\n\n#### Finding a Journal Club Article\/Microarray Dataset\n\n\u2022 You may choose to work ahead towards your Week 11 Assignment and Journal Club 3 presentation by finding your article and corresponding microarray dataset with which you will perform your final project in the course. Your task is to find a published microarray dataset that fulfills the following criteria:\n\u2022 The data are from a micraorray experiment that measures changes in gene expression (also known as transcription profiling by array).\n\u2022 A minimum of three biological replicates have been performed for each condition measured.\n\u2022 The experiment performed is a competitive hybridization (also known as a \"two-color\" or \"two-channel\") experiment where one sample was labeled with the Cy3 dye and the other sample was labeled with the Cy5 dye.\n\u2022 The control sample needs to be derived from mRNA and not genomic DNA.\n\u2022 The experiment was performed on one of the following species:\n\u2022 Arabidopsis thaliana\n\u2022 Chlamydia trachomatis\n\u2022 Escherichia coli K12\n\u2022 Helicobacter pylori\n\u2022 Mycobacterium smegmatis\n\u2022 Mycobacterium tuberculosis H37Rv\n\u2022 Plasmodium falciparum\n\u2022 Pseudomonas aerugenosa\n\u2022 Saccharomyces cerevisiae (budding yeast)\n\u2022 Salmonella typhimurium\n\u2022 Sinorhizobium meliloti\n\u2022 Staphylococcus aureus COL\n\u2022 Staphylococcus aureus MRSA252\n\u2022 Streptococcus pneumoniae\n\u2022 Vibrio cholerae\n\u2022 Microarray data are not centrally located on the web. Some major sources are:\n\u2022 You learned how to use PubMed, GoogleScholar, and Web of Science for your Week 3 Assignment. In this case, it will be likely be easier to search for the data first in one of the databases listed above and then retrieve the article corresponding to the data.\n\u2022 You must make a list of five potential articles\/datasets on your Week 11 journal page by the end of class on Wednesday, November 5. The instructor will notify you as to which article(s) are approved for your Journal Club presentation and final project.\n\n## Shared Journal Assignment\n\n\u2022 Store your journal entry in the shared BIOL368\/F14:Class Journal Week 10 page. If this page does not exist yet, go ahead and create it.\n\u2022 Link to the shared journal entry from your user page; this should be part of your template.\n\u2022 Link the shared journal page to this assignment page.\n\u2022 Sign your portion of the journal with the standard wiki signature shortcut (~~~~).\n\u2022 Add the \"BIOL368\/F14\" category to the end of the wiki page (if someone has not already done so).\n\n### Reflection\n\nAfter reading the Brown & Botstein (1999), Campbell & Heyer (2003), and DeRisi et al. (1997) readings, reflect on the following:\n\n1. What was the purpose of these readings?\n2. What did I learn from these readings?","date":"2023-02-06 02:08:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.24238653481006622, \"perplexity\": 3021.9785638185654}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500303.56\/warc\/CC-MAIN-20230206015710-20230206045710-00284.warc.gz\"}"} | null | null |
CXX = g++-4.8
LLVM_CONFIG ?= $(HOME)/code/llvm33/Release+Asserts/bin/llvm-config
LLVM_COMPONENTS= $(shell $(LLVM_CONFIG) --components)
LLVM_VERSION = $(shell $(LLVM_CONFIG) --version | cut -b 1-3)
CLANG ?= $(HOME)/code/llvm33/Release+Asserts/bin/clang
CLANG_VERSION = $(shell $(CLANG) --version)
LLVM_BINDIR = $(shell $(LLVM_CONFIG) --bindir)
LLVM_LIBDIR = $(shell $(LLVM_CONFIG) --libdir)
LLVM_AS = $(LLVM_BINDIR)/llvm-as
LLVM_NM = $(LLVM_BINDIR)/llvm-nm
LLVM_CXX_FLAGS = $(shell $(LLVM_CONFIG) --cppflags)
OPTIMIZE ?= -O3
# This can be set to -m32 to get a 32-bit build of Halide on a 64-bit system.
# (Normally this can be done via pointing to a compiler that defaults to 32-bits,
# but that is difficult in some testing situations because it requires having
# such a compiler handy. One still needs to have 32-bit llvm libraries, etc.)
BUILD_BIT_SIZE ?=
LLVM_VERSION_TIMES_10 = $(shell $(LLVM_CONFIG) --version | cut -b 1,3)
LLVM_CXX_FLAGS += -DLLVM_VERSION=$(LLVM_VERSION_TIMES_10)
WITH_NATIVE_CLIENT ?= $(findstring nacltransforms, $(LLVM_COMPONENTS))
WITH_X86 ?= $(findstring x86, $(LLVM_COMPONENTS))
WITH_ARM ?= $(findstring arm, $(LLVM_COMPONENTS))
WITH_ARM64 ?= $(findstring aarch64, $(LLVM_COMPONENTS))
WITH_OPENCL ?= 1
WITH_SPIR ?= 1
# turn off PTX for llvm 3.2
ifneq ($(LLVM_VERSION), 3.2)
WITH_PTX ?= $(findstring nvptx, $(LLVM_COMPONENTS))
endif
NATIVE_CLIENT_CXX_FLAGS = $(if $(WITH_NATIVE_CLIENT), -DWITH_NATIVE_CLIENT=1, )
NATIVE_CLIENT_LLVM_CONFIG_LIB = $(if $(WITH_NATIVE_CLIENT), nacltransforms, )
X86_CXX_FLAGS=$(if $(WITH_X86), -DWITH_X86=1, )
X86_LLVM_CONFIG_LIB=$(if $(WITH_X86), x86, )
ARM_CXX_FLAGS=$(if $(WITH_ARM), -DWITH_ARM=1, )
ARM_LLVM_CONFIG_LIB=$(if $(WITH_ARM), arm, )
PTX_CXX_FLAGS=$(if $(WITH_PTX), -DWITH_PTX=1, )
PTX_LLVM_CONFIG_LIB=$(if $(WITH_PTX), nvptx, )
PTX_DEVICE_INITIAL_MODULES=$(if $(WITH_PTX), libdevice.compute_20.10.bc libdevice.compute_30.10.bc libdevice.compute_35.10.bc, )
OPENCL_CXX_FLAGS=$(if $(WITH_OPENCL), -DWITH_OPENCL=1, )
OPENCL_LLVM_CONFIG_LIB=$(if $(WITH_OPENCL), , )
SPIR_CXX_FLAGS=$(if $(WITH_SPIR), -DWITH_SPIR=1, )
SPIR_LLVM_CONFIG_LIB=$(if $(WITH_SPIR), , )
ARM64_CXX_FLAGS=$(if $(WITH_ARM64), -DWITH_ARM64=1, )
ARM64_LLVM_CONFIG_LIB=$(if $(WITH_ARM64), aarch64, )
CXX_FLAGS = -Wall -Werror -fno-rtti -Woverloaded-virtual -Wno-unused-function -fPIC $(OPTIMIZE) -DCOMPILING_HALIDE $(BUILD_BIT_SIZE)
CXX_FLAGS += $(LLVM_CXX_FLAGS)
CXX_FLAGS += $(NATIVE_CLIENT_CXX_FLAGS)
CXX_FLAGS += $(PTX_CXX_FLAGS)
CXX_FLAGS += $(ARM_CXX_FLAGS)
CXX_FLAGS += $(ARM64_CXX_FLAGS)
CXX_FLAGS += $(X86_CXX_FLAGS)
CXX_FLAGS += $(OPENCL_CXX_FLAGS)
CXX_FLAGS += $(SPIR_CXX_FLAGS)
LIBS = -L $(LLVM_LIBDIR) $(shell $(LLVM_CONFIG) --libs bitwriter bitreader linker ipo mcjit jit $(X86_LLVM_CONFIG_LIB) $(ARM_LLVM_CONFIG_LIB) $(OPENCL_LLVM_CONFIG_LIB) $(SPIR_LLVM_CONFIG_LIB) $(NATIVE_CLIENT_LLVM_CONFIG_LIB) $(PTX_LLVM_CONFIG_LIB) $(ARM64_LLVM_CONFIG_LIB))
ifneq ($(WITH_PTX), )
ifneq (,$(findstring ptx,$(HL_TARGET)))
TEST_PTX = 1
endif
ifneq (,$(findstring cuda,$(HL_TARGET)))
TEST_PTX = 1
endif
endif
ifneq ($(WITH_OPENCL), )
ifneq (,$(findstring opencl,$(HL_TARGET)))
TEST_OPENCL = 1
endif
endif
ifneq ($(WITH_SPIR), )
ifneq (,$(findstring spir,$(HL_TARGET)))
TEST_OPENCL = 1
endif
ifneq (,$(findstring spir64,$(HL_TARGET)))
TEST_OPENCL = 1
endif
endif
TEST_CXX_FLAGS ?= $(BUILD_BIT_SIZE)
UNAME = $(shell uname)
ifeq ($(UNAME), Linux)
TEST_CXX_FLAGS += -rdynamic
ifneq ($(TEST_PTX), )
STATIC_TEST_LIBS ?= -L/usr/lib/nvidia-current -lcuda
endif
HOST_OS=linux
endif
ifeq ($(UNAME), Darwin)
# Someone with an osx box with cuda installed please fix the line below
ifneq ($(TEST_PTX), )
STATIC_TEST_LIBS ?= -framework CUDA
endif
ifneq ($(TEST_OPENCL), )
STATIC_TEST_LIBS ?= -framework OpenCL
endif
HOST_OS=os_x
else
ifneq ($(TEST_OPENCL), )
STATIC_TEST_LIBS ?= -lOpenCL
endif
endif
# Compiling the tutorials requires libpng
LIBPNG_LIBS_DEFAULT = $(shell libpng-config --ldflags)
LIBPNG_CXX_FLAGS ?= $(shell libpng-config --cflags)
# Workaround for libpng-config pointing to 64-bit versions on linux even when we're building for 32-bit
ifneq (,$(findstring -m32,$(CXX)))
ifneq (,$(findstring x86_64,$(LIBPNG_LIBS_DEFAULT)))
LIBPNG_LIBS ?= -lpng
endif
endif
LIBPNG_LIBS ?= $(LIBPNG_LIBS_DEFAULT)
ifdef BUILD_PREFIX
BUILD_DIR = build/$(BUILD_PREFIX)
BIN_DIR = bin/$(BUILD_PREFIX)
DISTRIB_DIR=distrib/$(BUILD_PREFIX)
else
BUILD_DIR = build
BIN_DIR = bin
DISTRIB_DIR=distrib
endif
SOURCE_FILES = CodeGen.cpp CodeGen_Internal.cpp CodeGen_X86.cpp CodeGen_GPU_Host.cpp CodeGen_PTX_Dev.cpp CodeGen_OpenCL_Dev.cpp CodeGen_SPIR_Dev.cpp CodeGen_GPU_Dev.cpp CodeGen_Posix.cpp CodeGen_ARM.cpp IR.cpp IRMutator.cpp IRPrinter.cpp IRVisitor.cpp CodeGen_C.cpp Substitute.cpp ModulusRemainder.cpp Bounds.cpp Derivative.cpp OneToOne.cpp Func.cpp Simplify.cpp IREquality.cpp Util.cpp Function.cpp IROperator.cpp Lower.cpp Debug.cpp Parameter.cpp Reduction.cpp RDom.cpp Profiling.cpp Tracing.cpp StorageFlattening.cpp VectorizeLoops.cpp UnrollLoops.cpp BoundsInference.cpp IRMatch.cpp StmtCompiler.cpp integer_division_table.cpp SlidingWindow.cpp StorageFolding.cpp InlineReductions.cpp RemoveTrivialForLoops.cpp Deinterleave.cpp DebugToFile.cpp Type.cpp JITCompiledModule.cpp EarlyFree.cpp UniquifyVariableNames.cpp CSE.cpp Tuple.cpp Lerp.cpp Target.cpp SkipStages.cpp SpecializeClampedRamps.cpp RemoveUndef.cpp FastIntegerDivide.cpp
# The externally-visible header files that go into making Halide.h. Don't include anything here that includes llvm headers.
HEADER_FILES = Util.h Type.h Argument.h Bounds.h BoundsInference.h Buffer.h buffer_t.h CodeGen_C.h CodeGen.h CodeGen_X86.h CodeGen_GPU_Host.h CodeGen_PTX_Dev.h CodeGen_OpenCL_Dev.h CodeGen_SPIR_Dev.h CodeGen_GPU_Dev.h Deinterleave.h Derivative.h OneToOne.h Extern.h Func.h Function.h Image.h InlineReductions.h integer_division_table.h IntrusivePtr.h IREquality.h IR.h IRMatch.h IRMutator.h IROperator.h IRPrinter.h IRVisitor.h JITCompiledModule.h Lambda.h Debug.h Lower.h MainPage.h ModulusRemainder.h Parameter.h Param.h RDom.h Reduction.h RemoveTrivialForLoops.h Schedule.h Scope.h Simplify.h SlidingWindow.h StmtCompiler.h StorageFlattening.h StorageFolding.h Substitute.h Profiling.h Tracing.h UnrollLoops.h Var.h VectorizeLoops.h CodeGen_Posix.h CodeGen_ARM.h DebugToFile.h EarlyFree.h UniquifyVariableNames.h CSE.h Tuple.h Lerp.h Target.h SkipStages.h SpecializeClampedRamps.h RemoveUndef.h FastIntegerDivide.h
SOURCES = $(SOURCE_FILES:%.cpp=src/%.cpp)
OBJECTS = $(SOURCE_FILES:%.cpp=$(BUILD_DIR)/%.o)
HEADERS = $(HEADER_FILES:%.h=src/%.h)
RUNTIME_CPP_COMPONENTS = android_io cuda fake_thread_pool gcd_thread_pool ios_io android_clock linux_clock nogpu opencl posix_allocator posix_clock osx_clock windows_clock posix_error_handler posix_io nacl_io osx_io posix_math posix_thread_pool android_host_cpu_count linux_host_cpu_count osx_host_cpu_count tracing write_debug_image cuda_debug opencl_debug windows_io
RUNTIME_LL_COMPONENTS = arm posix_math ptx_dev spir_dev spir64_dev spir_common_dev x86_avx x86 x86_sse41
INITIAL_MODULES = $(RUNTIME_CPP_COMPONENTS:%=$(BUILD_DIR)/initmod.%_32.o) $(RUNTIME_CPP_COMPONENTS:%=$(BUILD_DIR)/initmod.%_64.o) $(RUNTIME_LL_COMPONENTS:%=$(BUILD_DIR)/initmod.%_ll.o) $(PTX_DEVICE_INITIAL_MODULES:libdevice.%.bc=$(BUILD_DIR)/initmod_ptx.%_ll.o)
.PHONY: all
all: $(BIN_DIR)/libHalide.a $(BIN_DIR)/libHalide.so include/Halide.h include/HalideRuntime.h test_internal
$(BIN_DIR)/libHalide.a: $(OBJECTS) $(INITIAL_MODULES)
@-mkdir -p $(BIN_DIR)
$(LD) -r -o $(BUILD_DIR)/Halide.o $(OBJECTS) $(INITIAL_MODULES) $(LIBS)
rm -f $(BIN_DIR)/libHalide.a
ar q $(BIN_DIR)/libHalide.a $(BUILD_DIR)/Halide.o
ranlib $(BIN_DIR)/libHalide.a
$(BIN_DIR)/libHalide.so: $(BIN_DIR)/libHalide.a
$(CXX) $(BUILD_BIT_SIZE) -shared $(OBJECTS) $(INITIAL_MODULES) $(LIBS) -o $(BIN_DIR)/libHalide.so
include/Halide.h: $(HEADERS) $(BIN_DIR)/build_halide_h
mkdir -p include
cd src; ../$(BIN_DIR)/build_halide_h $(HEADER_FILES) > ../include/Halide.h; cd ..
include/HalideRuntime.h: src/runtime/HalideRuntime.h
mkdir -p include
cp src/runtime/HalideRuntime.h include/
$(BIN_DIR)/build_halide_h: src/build_halide_h.cpp
g++ $< -o $@
msvc/initmod.cpp: $(INITIAL_MODULES)
echo "extern \"C\" {" > msvc/initmod.cpp
cat $(BUILD_DIR)/initmod*.cpp >> msvc/initmod.cpp
echo "}" >> msvc/initmod.cpp
-include $(OBJECTS:.o=.d)
$(BUILD_DIR)/initmod.%_64.ll: src/runtime/%.cpp $(BUILD_DIR)/clang_ok
@-mkdir -p $(BUILD_DIR)
$(CLANG) -nobuiltininc -fno-blocks -m64 -DCOMPILING_HALIDE -DBITS_64 -emit-llvm -O3 -S src/runtime/$*.cpp -o $@
$(BUILD_DIR)/initmod.%_32.ll: src/runtime/%.cpp $(BUILD_DIR)/clang_ok
@-mkdir -p $(BUILD_DIR)
$(CLANG) -nobuiltininc -fno-blocks -m32 -DCOMPILING_HALIDE -DBITS_32 -emit-llvm -O3 -S src/runtime/$*.cpp -o $@
$(BUILD_DIR)/initmod.%_ll.ll: src/runtime/%.ll
@-mkdir -p $(BUILD_DIR)
cp src/runtime/$*.ll $(BUILD_DIR)/initmod.$*_ll.ll
$(BUILD_DIR)/initmod.%.bc: $(BUILD_DIR)/initmod.%.ll $(BUILD_DIR)/llvm_ok
$(LLVM_AS) $(BUILD_DIR)/initmod.$*.ll -o $(BUILD_DIR)/initmod.$*.bc
$(BUILD_DIR)/initmod.%.cpp: $(BIN_DIR)/bitcode2cpp $(BUILD_DIR)/initmod.%.bc
./$(BIN_DIR)/bitcode2cpp $* < $(BUILD_DIR)/initmod.$*.bc > $@
$(BUILD_DIR)/initmod_ptx.%_ll.cpp: $(BIN_DIR)/bitcode2cpp src/runtime/nvidia_libdevice_bitcode/libdevice.%.bc
./$(BIN_DIR)/bitcode2cpp ptx_$(basename $*)_ll < src/runtime/nvidia_libdevice_bitcode/libdevice.$*.bc > $@
$(BIN_DIR)/bitcode2cpp: src/bitcode2cpp.cpp
@-mkdir -p $(BIN_DIR)
$(CXX) $< -o $@
$(BUILD_DIR)/initmod_ptx.%_ll.o: $(BUILD_DIR)/initmod_ptx.%_ll.cpp
$(CXX) -c $< -o $@ -MMD -MP -MF $(BUILD_DIR)/$*.d -MT $(BUILD_DIR)/$*.o
$(BUILD_DIR)/initmod.%.o: $(BUILD_DIR)/initmod.%.cpp
$(CXX) $(BUILD_BIT_SIZE) -c $< -o $@ -MMD -MP -MF $(BUILD_DIR)/$*.d -MT $(BUILD_DIR)/$*.o
$(BUILD_DIR)/%.o: src/%.cpp src/%.h $(BUILD_DIR)/llvm_ok
@-mkdir -p $(BUILD_DIR)
$(CXX) $(CXX_FLAGS) -c $< -o $@ -MMD -MP -MF $(BUILD_DIR)/$*.d -MT $(BUILD_DIR)/$*.o
.PHONY: clean
clean:
rm -rf $(BIN_DIR)/*
rm -rf $(BUILD_DIR)/*
rm -rf include/*
rm -rf doc
.SECONDARY:
CORRECTNESS_TESTS = $(shell ls test/correctness/*.cpp)
#Note: cuda_tile() doesn't work with OpenCL
#CORRECTNESS_TESTS_CUDA_TILE = $(shell grep -rnwl -v test/correctness/*.cpp -e 'cuda_tile' )
STATIC_TESTS = $(shell ls test/static/*_generate.cpp)
PERFORMANCE_TESTS = $(shell ls test/performance/*.cpp)
ERROR_TESTS = $(shell ls test/error/*.cpp)
TUTORIALS = $(shell ls tutorial/*.cpp)
test_correctness: $(CORRECTNESS_TESTS:test/correctness/%.cpp=test_%)
test_static: $(STATIC_TESTS:test/static/%_generate.cpp=static_%)
test_performance: $(PERFORMANCE_TESTS:test/performance/%.cpp=performance_%)
test_errors: $(ERROR_TESTS:test/error/%.cpp=error_%)
test_tutorials: $(TUTORIALS:tutorial/%.cpp=tutorial_%)
test_valgrind: $(CORRECTNESS_TESTS:test/correctness/%.cpp=valgrind_%)
run_tests: test_correctness test_errors test_tutorials test_static
make test_performance
build_tests: $(CORRECTNESS_TESTS:test/correctness/%.cpp=$(BIN_DIR)/test_%) \
$(PERFORMANCE_TESTS:test/performance/%.cpp=$(BIN_DIR)/performance_%) \
$(ERROR_TESTS:test/error/%.cpp=$(BIN_DIR)/error_%) \
$(STATIC_TESTS:test/static/%_generate.cpp=$(BIN_DIR)/static_%_generate) \
$(TUTORIALS:tutorial/%.cpp=$(BIN_DIR)/tutorial_%)
$(BIN_DIR)/test_internal: test/internal.cpp $(BIN_DIR)/libHalide.so
$(CXX) $(CXX_FLAGS) $< -Isrc -L$(BIN_DIR) -lHalide -lpthread -ldl -o $@
$(BIN_DIR)/test_%: test/correctness/%.cpp $(BIN_DIR)/libHalide.so include/Halide.h
$(CXX) $(TEST_CXX_FLAGS) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -lHalide -lpthread -ldl -o $@
$(BIN_DIR)/performance_%: test/performance/%.cpp $(BIN_DIR)/libHalide.so include/Halide.h test/performance/clock.h
$(CXX) $(TEST_CXX_FLAGS) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -lHalide -lpthread -ldl -o $@
$(BIN_DIR)/error_%: test/error/%.cpp $(BIN_DIR)/libHalide.so include/Halide.h
$(CXX) $(TEST_CXX_FLAGS) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -lHalide -lpthread -ldl -o $@
$(BIN_DIR)/static_%_generate: test/static/%_generate.cpp $(BIN_DIR)/libHalide.so include/Halide.h
$(CXX) $(TEST_CXX_FLAGS) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -lHalide -lpthread -ldl -o $@
tmp/static/%.o: $(BIN_DIR)/static_%_generate
@-mkdir -p tmp/static
cd tmp/static; DYLD_LIBRARY_PATH=../../$(BIN_DIR) LD_LIBRARY_PATH=../../$(BIN_DIR) ../../$<
@-echo
$(BIN_DIR)/static_%_test: test/static/%_test.cpp $(BIN_DIR)/static_%_generate tmp/static/%.o include/HalideRuntime.h
$(CXX) $(TEST_CXX_FLAGS) $(OPTIMIZE) -I tmp/static -I apps/support tmp/static/$*.o $< -lpthread -ldl $(STATIC_TEST_LIBS) -o $@
$(BIN_DIR)/tutorial_%: tutorial/%.cpp $(BIN_DIR)/libHalide.so include/Halide.h
$(CXX) $(TEST_CXX_FLAGS) $(LIBPNG_CXX_FLAGS) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -lHalide -lpthread -ldl $(LIBPNG_LIBS) -o $@
test_%: $(BIN_DIR)/test_%
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$<
@-echo
static_%: $(BIN_DIR)/static_%_test
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$<
@-echo
valgrind_%: $(BIN_DIR)/test_%
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) valgrind --error-exitcode=-1 ../$<
@-echo
# This test is *supposed* to do an out-of-bounds read, so skip it when testing under valgrind
valgrind_tracing_stack: $(BIN_DIR)/test_tracing_stack
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$(BIN_DIR)/test_tracing_stack
@-echo
performance_%: $(BIN_DIR)/performance_%
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$<
@-echo
error_%: $(BIN_DIR)/error_%
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$< 2>&1 | egrep --q "Assertion.*failed"
@-echo
tutorial_%: $(BIN_DIR)/tutorial_%
@-mkdir -p tmp
cd tmp ; DYLD_LIBRARY_PATH=../$(BIN_DIR) LD_LIBRARY_PATH=../$(BIN_DIR) ../$<
@-echo
.PHONY: test_apps
test_apps: $(BIN_DIR)/libHalide.a include/Halide.h
# make -C apps/bilateral_grid clean
# make -C apps/bilateral_grid out.png
# make -C apps/local_laplacian clean
# make -C apps/local_laplacian out.png
# make -C apps/interpolate clean
# make -C apps/interpolate out.png
make -C apps/blur clean
make -C apps/blur test
./apps/blur/test
make -C apps/wavelet clean
make -C apps/wavelet test
make -C apps/c_backend clean
make -C apps/c_backend test
make -C apps/camera_pipe clean
make -C apps/camera_pipe out.png
ifneq (,$(findstring version 3.,$(CLANG_VERSION)))
ifeq (,$(findstring version 3.0,$(CLANG_VERSION)))
CLANG_OK=yes
endif
endif
ifneq (,$(findstring Apple clang version 4.0,$(CLANG_VERSION)))
CLANG_OK=yes
endif
ifneq (,$(findstring Apple LLVM version 5.0,$(CLANG_VERSION)))
CLANG_OK=yes
endif
ifneq (,$(findstring 3.,$(LLVM_VERSION)))
ifeq (,$(findstring 3.0,$(LLVM_VERSION)))
ifeq (,$(findstring 3.1,$(LLVM_VERSION)))
LLVM_OK=yes
endif
endif
endif
ifneq (,$findstring 3.3.,$(LLVM_VERSION))
LLVM_OK=yes
endif
ifneq (,$findstring 3.2.,$(LLVM_VERSION))
LLVM_OK=yes
endif
ifdef CLANG_OK
$(BUILD_DIR)/clang_ok:
@echo "Found a new enough version of clang"
mkdir -p $(BUILD_DIR)
touch $(BUILD_DIR)/clang_ok
else
$(BUILD_DIR)/clang_ok:
@echo "Can't find clang or version of clang too old (we need 3.1 or greater):"
@echo "You can override this check by setting CLANG_OK=y"
echo '$(CLANG_VERSION)'
echo $(findstring version 3,$(CLANG_VERSION))
echo $(findstring version 3.0,$(CLANG_VERSION))
$(CLANG) --version
@exit 1
endif
ifdef LLVM_OK
$(BUILD_DIR)/llvm_ok:
@echo "Found a new enough version of llvm"
mkdir -p $(BUILD_DIR)
touch $(BUILD_DIR)/llvm_ok
else
$(BUILD_DIR)/llvm_ok:
@echo "Can't find llvm or version of llvm too old (we need 3.2 or greater):"
@echo "You can override this check by setting LLVM_OK=y"
$(LLVM_CONFIG) --version
@exit 1
endif
.PHONY: doc
docs: doc
doc: src test
doxygen
$(DISTRIB_DIR)/halide.tgz: $(BIN_DIR)/libHalide.a $(BIN_DIR)/libHalide.so include/Halide.h include/HalideRuntime.h
mkdir -p $(DISTRIB_DIR)/include $(DISTRIB_DIR)/lib
cp $(BIN_DIR)/libHalide.a $(BIN_DIR)/libHalide.so $(DISTRIB_DIR)/lib
cp include/Halide.h $(DISTRIB_DIR)/include
cp include/HalideRuntime.h $(DISTRIB_DIR)/include
tar -czf $(DISTRIB_DIR)/halide.tgz -C $(DISTRIB_DIR) lib include
distrib: $(DISTRIB_DIR)/halide.tgz
$(BIN_DIR)/HalideProf: util/HalideProf.cpp
$(CXX) $(OPTIMIZE) $< -Iinclude -L$(BIN_DIR) -o $@
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,231 |
author: dema
comments: true
date: 2007-11-04 17:11:01+00:00
layout: post
slug: fon-e-litalico-ntu-o-culu-al-prossimo
title: 'FON e L''italico ''ntu o culu al prossimo '
wordpress_id: 137
categories:
- condivisione
- cultura
- fon
- peer to peer
- sharing
tags:
- condivisione
- cultura
- fon
- peer to peer
- sharing
---
Questo post prende spunto dal [commento di ieri](http://itfonblog.wordpress.com/2007/11/03/luca-sartoni-no-wifi-no-party/#comments) di Francesco D'Elia su questo blog.
Francesco lamenta sostanzialmente **una scarsa partecipazione dei foneros italiani** al mantenimento del network . Dice infatti che nella sua esperienza , moltissimi punti hotspot segnati correttamente attivi sulla mappa Fon , in realtà sono o spenti , oppure non posizionati correttamente per permettere al segnale wifi di raggiungere anche l'esterno.
Francesco ha ragione da vendere , e alla sua stessa conclusione ero già approdato io più di un anno fa .
Purtroppo la scarsa coscienza dei foneros italiani è da imputare al classico attengiamento menfreghista verso i meccanismi di condivisione. Quando si tratta di mettere in atto queste pratiche , il nostro atteggiamento è quello di chiudere la porta in faccia al prossimo.
Emblematico è l'aneddoto del novello internettiano che rivoltosi all'esperto di turno per fargli usare un programma di P2P , quando **si accorge che altri peer si collegano al suo computer** per attingere dai suoi file , dice: " ma che vogliono questi , io non do proprio un cazzo a nessuno".
E' la logica tutta italiana del fare i furbi . Fon all'inizio **regalava router linksys** o li vendeva ad un prezzo scontatissimo . Molti hanno visto un'opportunità per portare a casa un buon router a costo zero .
Poi con l'avvento della fonera il fenomeno si è un po ridimensionato , ma in molti , una volta resisi conto che la loro connettività veniva usata anche dagli altri , hanno deciso di spegnerla.
Occorre un cambio di mentalità e non solo per quanto riguarda gli atteggiamenti su internet , ma anche nel vivere quotidiano verso il prossimo.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 971 |
package teams.api.validations;
import org.junit.Test;
import org.springframework.test.util.ReflectionTestUtils;
import teams.Seed;
import teams.api.ExternalTeamController;
import teams.domain.ExternalTeam;
import teams.domain.FederatedUser;
import teams.domain.Role;
import teams.domain.Team;
import teams.exception.IllegalLinkExternalTeamException;
import java.util.Collections;
import java.util.HashMap;
import static java.util.Collections.singletonList;
import static org.junit.Assert.assertNotNull;
public class ExternalTeamValidatorTest implements Seed {
private ExternalTeamValidator subject = new ExternalTeamController();
@Test
public void externalTeamNotLinked() {
Team team = team();
ExternalTeam externalTeam = new ExternalTeam();
subject.externalTeamNotLinked(team, externalTeam);
}
@Test(expected = IllegalLinkExternalTeamException.class)
public void externalTeamNotLinkedException() {
Team team = teamLinkedWithExternalTeam();
subject.externalTeamNotLinked(team, team.getExternalTeams().iterator().next());
}
public Team teamLinkedWithExternalTeam() {
Team team = team();
ExternalTeam externalTeam = new ExternalTeam();
ReflectionTestUtils.setField(externalTeam, "identifier", "identifier");
team.getExternalTeams().add(externalTeam);
return team;
}
@Test
public void externalTeamLinked() {
Team team = teamLinkedWithExternalTeam();
subject.externalTeamLinked(team, team.getExternalTeams().iterator().next());
}
@Test(expected = IllegalLinkExternalTeamException.class)
public void externalTeamLinkedException() {
Team team = team();
ExternalTeam externalTeam = new ExternalTeam();
subject.externalTeamLinked(team, externalTeam);
}
@Test
public void isAllowedToLinkExternalTeam() throws Exception {
subject.isAllowedToLinkExternalTeam(Role.ADMIN, team(), federatedUser());
}
@Test(expected = IllegalLinkExternalTeamException.class)
public void isAllowedToLinkExternalTeamNotAllowed() throws Exception {
subject.isAllowedToLinkExternalTeam(Role.MEMBER, team(), federatedUser());
}
@Test
public void externalTeamFromFederatedUser() throws Exception {
FederatedUser federatedUser = new FederatedUser(person("urn"), "nope", "OC",
singletonList(externalTeam("identifier")), Collections.emptyMap(), new HashMap<>());
ExternalTeam externalTeam = subject.externalTeamFromFederatedUser(federatedUser, "identifier");
assertNotNull(externalTeam);
}
@Test(expected = IllegalLinkExternalTeamException.class)
public void externalTeamFromFederatedUserNotMember() throws Exception {
FederatedUser federatedUser = new FederatedUser(person("urn"), "nope", "OC",
singletonList(externalTeam("nope")), Collections.emptyMap(), new HashMap<>());
ExternalTeam externalTeam = subject.externalTeamFromFederatedUser(federatedUser, "identifier");
assertNotNull(externalTeam);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,428 |
(BPT) – When it comes to business travel, more women are taking to the road and sky to get to their destinations. In fact, women account for 47 percent of all business travel worldwide, reports the Global Business Travel Association.
"Travel apps are a huge time saver. I use TripIt to organize my itinerary. I use OpenTable and TripAdvisor to find places to dine. And of course, I use airline, car rental and hotel apps during my trip. My airline app will allow me to check in online, serve as a mobile boarding pass and notify me of gate changes and delays. Many hotels now have apps that offer digital check-in and digital keys.
Posted in Travel and tagged business travel, women travelers. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,516 |
Home Tags Elle Fanning
Tag: Elle Fanning
How to Talk to Girls at Parties – Recap/ Review (with Spoilers)
Brilliantly weird, comical and touching, somehow How to Talk to Girls at Parties taps into something absurd without getting lost in its own madness. Director(s) John Cameron Mitchell Screenplay By John Cameron Mitchell, Philippa Goslett Date Released 5/25/2018 Genre(s) Sci-Fi, Comedy, Romance Noted...
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Overview A rather slow film which explores the relationship between a man and daughter. The man being an accomplished jazz musician with a drug addiction, and the daughter seemingly aimless. Review (with Spoilers) - Below Characters &...
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The Boxtrolls – Overview/ Review (with Spoilers)
Overview The Boxtrolls is a kid's movie which shows how foolish, arrogant and gullible adults are. Even when all the evidence needed is right before their eyes. Review (with Spoilers) Though ParaNorman, Laika's last production, didn't really...
Jinn - Summary/ Review (with Spoilers)
Black & Privileged: Volume 1 - Summary, Review (with Spoilers)
2 TV Series Collected Quotes: Anne with an E 96.1% (36)
As one of Myfanwy's pursuers get captured, complications with the Gestalt, Monica, and the relationship between Conrad and Linda continue.
Euphoria: Season 1, Episode 5 "'03 Bonnie and Clyde" – Recap,... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,118 |
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The CNC SUPRA Vertical Mill is a versatile machine that can be used for product development, engraving, and teaching applications for vocation schools or science labs By using this milling machine, any user can create simple and complex. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,337 |
{"url":"https:\/\/math.stackexchange.com\/questions\/3346313\/prove-that-c-0x-is-separable-given-that-x-is-locally-compact-metric-space","text":"# Prove that $C_0(X)$ is separable given that X is locally compact metric space\n\nI'm struggling to prove the following fact:\n\nSuppose that $$X$$ is locally compact metric space. Let us denote with $$C_0(X)$$ the space of functions vanishing at infinity (i.e., $$\\forall f \\in C_0(X)$$ $$\\forall \\varepsilon > 0$$ $$\\exists \\, E\\subset X$$ s.t. $$E$$ is compact and $$|f(x)|<\\varepsilon$$ for $$x \\in X\\setminus E$$). Then $$C_0(X)$$ is separable.\n\nI've proven that $$C_0(X)$$ equipped with a supremum norm is a Banach space, and that $$C_c(X)$$ (functions with compact support) are dense in $$C_0(X)$$, so my guess would be to somehow use those facts to prove that $$C_0(X)$$ is separable. However, I can't exactly see how. I've seen the cases for compact spaces or using the assumption of $$\\sigma$$-compactness. Any help is highly appreciated.\n\n\u2022 Can you use the Stone-Weierstrass theorem and construct a countable subalgebra that separates points? \u2013\u00a0Matthew Leingang Sep 6 '19 at 14:12\n\u2022 @MatthewLeingang wouldn't you need it to contain some non-zero constant function? But it wouldn't vanish at infinity \u2013\u00a0GSofer Sep 6 '19 at 14:18\n\u2022 @GSofer My question wasn't a hint; it was a guess. ;^) \u2013\u00a0Matthew Leingang Sep 6 '19 at 14:52\n\u2022 Are you missing a separability assumption? As written I think this is false. For example, let $X$ be an uncountable set with the discrete metric. Then for each $x \\in X$, define $f_x(x) = 1$ and $f_x(y) = 0$ for $y \\neq x$. Then $\\{f_x: x \\in X\\} \\subseteq C_0(X)$ and $\\|f_x - f_y\\|_\\infty = 1$ for $y \\neq x$ which means that $C_0(X)$ has an uncountable discrete subset and so isn't separable. With the added assumption of separability of $X$ this is true since you can e.g. embed $C_0(X)$ into $C(\\tilde{X})$ where $\\tilde{X}$ is the one-point compactification of $X$. \u2013\u00a0Rhys Steele Sep 6 '19 at 15:26\n\u2022 en.wikipedia.org\/wiki\/\u2026 does not require any constant functions. So you can try to modify the C(X) proof, but that requires that $X$ is separable. Let $D\\subset X$ be dense and countable. Set $d_x(y):=1\/(1+d(x,y))\\ (x\\in D)$. Let $A$ be the subalgebra of $C_0(X)$ generated by the functions $d_x\\in C_0$,* i.e., the set of linear combinations of their products. Let $A'$ be the subset with rational coefficients. Then $A'$ is countable and dense in $A$. But $A$ separates points and vanishes nowhere. So we are done if * is true. Is it? \u2013\u00a0user3810316 Jun 10 at 20:34\n\nAs Rhys Steele mentions, this is not true unless you assume $$X$$ to be second countable (or, equivalently for metric spaces, separable). Rhys gives a counterexample showing the theorem can fail without this assumption, but more is true: it always fails without this assumption.\nProposition. Let $$X$$ be a locally compact Hausdorff space. If $$C_0(X)$$ is separable then $$X$$ is second countable.\nProof. Let $$\\{f_n\\}$$ be a countable dense subset of $$C_0(X)$$, and for each $$n$$ let $$U_n = \\{x \\in X: f_n(x) > 1\/2\\}$$, which is an open subset of $$X$$. I claim that $$\\{U_n\\}$$ is a countable base for the topology of $$X$$. For let $$x \\in X$$ and let $$V$$ be an open neighborhood of $$x$$. Then by Urysohn's lemma for locally compact Hausdorff spaces, there exists a function $$f$$ compactly supported inside $$V$$ with $$f(x) = 1$$. In particular $$f \\in C_c(X) \\subset C_0(X)$$, so by density, we can find some $$f_n$$ with $$\\|f-f_n\\|_\\infty < 1\/2$$. Then we have $$f_n(x) > 1\/2$$ so $$x \\in U_n$$. Moreover, if $$y \\in U_n$$ then $$f_n(y) > 1\/2$$ and so $$f(y) > 0$$, which implies $$y \\in V$$. Therefore $$U_n \\subset V$$. This proves that $$\\{U_n\\}$$ is a base.\nNow, supposing that $$X$$ is second countable, you can proceed in a similar way to the compact case, applying the locally compact version of Stone-Weierstrass. Using the second countability and local compactness, you should be able to construct a countable family $$f_n$$ of compactly supported functions which separates points and vanishes nowhere. Then consider the algebra $$\\mathcal{A}_0$$ generated over $$\\mathbb{Q}$$ by the $$f_n$$; i.e. all functions consisting of finite rational linear combinations of finite products of the $$f_n$$. Show that $$\\mathcal{A}_0$$ is countable, and that the closure of $$\\mathcal{A}_0$$ is a closed algebra over $$\\mathbb{R}$$. Stone-Weierstrass then implies that the closure of $$\\mathcal{A}_0$$ equals $$C_0(X)$$, so $$C_0(X)$$ is separable.\nIn the case of $$X = \\mathbb{R}$$. Consider the (countable) set of function $$G = \\{(P I_n) * \\eta_m: P \\in \\mathbb{Q}(x), m, n \\in \\mathbb{N}\\} \\subset C_0(X),$$ where $$I_n (x) = \\mathbf{1}_{[-n, n]}(x)$$ and $$\\eta_m = \\frac{1}{m} \\eta(\\frac{x}{m})$$, $$\\eta$$ is a mollifier and $$*$$ means convolution. For arbitray $$f \\in C_0(X)$$, assuming it is supported on $$[-N, N]$$, by Weiterstrass theorem we can find a sequence of $$P_n \\in \\mathbb{Q}(x)$$ such that $$P_n(x) I_N (x)$$ approximates $$f(x)$$ uniformly. Using the property of mollifier, we conlcude $$G$$ is dense in $$C_0(X)$$. Therefore, $$C_0(X)$$ is separable.","date":"2020-09-29 11:08:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 68, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9660502076148987, \"perplexity\": 88.84944493130503}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600401641638.83\/warc\/CC-MAIN-20200929091913-20200929121913-00028.warc.gz\"}"} | null | null |
\section{Introduction}
The original $T1$ theorem of David and Journ\'{e} \cite{DaJo}, which
characterized boundedness of a singular integral operator by testing over
indicators $\mathbf{1}_{Q\text{ }}$ of cubes $Q$, was quickly extended to a
Tb$ theorem by David, Journ\'{e} and Semmes \cite{DaJoSe}, in which the
indicators $\mathbf{1}_{Q\text{ }}$ were replaced by testing functions $
\mathbf{1}_{Q\text{ }}$ for an accretive function $b$, i.e. $0<c\leq \func{R
}b\leq \left\vert b\right\vert \leq C<\infty $. Here the accretive function
b$ could be chosen to adapt well to the operator at hand, resulting in
almost immediate verification of the $b$-testing conditions, despite
difficulty in verifying the $1$-testing conditions. One motivating example
of this phenomenon is the boundedness of the Cauchy integral on Lipschitz
curves\footnote
The problem reduces to boundedness on $L^{2}\left( \mathbb{R}\right) $ of
the singular integral operator $C_{A}$ with kernel $K_{A}\left( x,y\right)
\equiv \frac{1}{x-y+i\left( A\left( x\right) -A\left( y\right) \right) }$,
where the curve has graph $\left\{ x+iA\left( x\right) :x\in \mathbb{R
\right\} $. Now $b\left( x\right) \equiv 1+iA^{\prime }\left( x\right) $ is
accretive and the $b$-testing condition $\int_{I}\left\vert C_{A}\left(
\mathbf{1}_{I}b\right) \left( x\right) \right\vert ^{2}dx\leq \mathfrak{T
_{H}^{b}\left\vert I\right\vert $ follows from $\left\vert C_{A}\left(
\mathbf{1}_{I}b\right) \left( x\right) \right\vert ^{2}\approx \ln \frac
x-\alpha }{\beta -x}$, for $x\in I=\left[ \alpha ,\beta \right] $. In the
case of a $C^{1,\delta }$ curve, the kernel $K_{A}$ is $C^{1,\delta }$ and
any $Tb$ theorem applies with $T=C_{A}$ and $\sigma =\omega =dx$, to show
that $C_{A}$ is bounded on $L^{2}\left( \mathbb{R}\right) $.}. See e.g. \cit
[pages 310-316]{Ste}.
Subsequently, M. Christ \cite{Chr} obtained a far more robust \emph{local}
Tb$ theorem in the setting of homogeneous spaces, in which the testing
functions could be further specialized to $b_{Q}\mathbf{1}_{Q\text{ }}$,
where now the accretive functions $b_{Q}$ can be chosen to \emph{differ} for
\emph{each} cube $Q$. Applications of the local $Tb$ theorem included
boundedness of layer potentials, see e.g. \cite{AAAHK} and references there;
and the Kato problem, see \cite{HoMc}, \cite{HoLaMc} and \cite{AuHoLaMcTc}:
and many authors, including G. David \cite{Dav1}; Nazarov, Treil and Volberg
\cite{NTV3}, \cite{NTV2}; Auscher, Hofmann, Muscalu, Tao and Thiele \cit
{AuHoMuTaTh}, Hyt\"{o}nen and Martikainen \cite{HyMa}, and more recently
Lacey and Martikainen \cite{LaMa}, set about proving extensions of the local
$Tb$ theorem, for example to include a single upper doubling weight together
with weaker upper bounds on the function $b$. But these extensions were
modelled on the `nondoubling' methods that arose in connection with upper
doubling measures in the analytic capacity problem, see Mattila, Melnikov
and Verdera \cite{MaMeVe}, G. David \cite{Dav1}, \cite{Dav2}, X. Tolsa \cit
{Tol}, and alsoVolberg \cite{Vol}, and were thus constrained to a single
weight - a setting in which both the Muckenhoupt and energy conditions
follow from the upper doubling condition.
In this paper, we consider only the case of dimension $n=1$, and we adapt
methods from the theory of two weight $T1$ theorems, which arose from \cit
{NTV4}, \cite{Vol}, \cite{LaSaShUr3}, \cite{Lac}, \cite{SaShUr7} and \cit
{SaShUr9}, and were used in \cite{HyMa} as well, to prove a two weight local
$Tb$ theorem. These methods involve the `testing' perspective toward
characterizing two weight norm inequalities for an operator $T$. As
suggested by work originating in \cite{DaJo} and \cite{Saw3}, it is
plausible to conjecture that a given operator $T$ is bounded from one
weighted space to another if and only if both it and its dual are bounded
when tested over a suitable family of functions related geometrically to $T
, e.g. testing over indicators of intervals for fractional integrals $T$ as
in \cite{Saw3}.
\textbf{Muckenhoupt conditions}: However, for even the simplest singular
integral, the Hilbert transform, testing over indicators of intervals no
longer suffices\footnote
consider e.g. $d\omega \left( x\right) =\delta _{0}\left( x\right) $ and
d\sigma \left( x\right) =\left\vert x\right\vert dx$.}, and an additional
`side condition' on the weight pair is required - namely the Muckenhoupt
\mathfrak{A}_{2}$ condition, a simpler form of which was shown by Hunt,
Muckenhoupt and Wheeden \cite{HuMuWh} to characterize the one weight
inequality for the Hilbert transform. This side condition is a size
condition on the weight pair that is typically shown to be necessary by
testing over so-called tails of indicators of intervals, and indeed is known
to be necessary for boundedness of a broad class of fractional singular
integrals that are `strongly elliptic'. Using this side condition of
Muckenhoupt, the solution of the NTV conjecture, due to the authors and M.
Lacey in the two part paper \cite{LaSaShUr3}-\cite{Lac}, shows that the
Hilbert transform $H$ is bounded between weighted $L^{2}$ spaces if and only
if the Muckenhoupt condition and the two testing conditions over indicators
of intervals all hold. However, the testing conditions for singular
integrals, unlike those for positive operators such as fractional integrals,
are extremely unstable and in principle difficult to check \cite{LaSaUr2}.
On the other hand, given a weight pair, it may be possible to produce a
family of testing functions adapted to intervals on which the boundedness of
the operator is evident. In such a case, one would like to conclude that
finding an appropriately \emph{nondegenerate} family of such testing
functions, for which the corresponding testing conditions hold, is enough to
guarantee boundedness of the operator - bringing us back to a local $Tb$
theorem. In any event, one would in general like to understand the weakest
testing conditions that are sufficient for two weight boundedness of a given
operator.
\textbf{Energy conditions}: Our $Tb$ theorem lies in this direction, but the
method of proof requires in addition a second `side condition', namely the
so-called energy condition, introduced in \cite{LaSaUr2}. The energy
condition is necessary for the boundedness of the Hilbert transform, and
actually follows there from testing over indicators of intervals and,
through the Muckenhoupt condition, testing over tails of indicators of
intervals as well. More generally, it is known that the energy condition is
necessary for boundedness of \emph{gradient elliptic} fractional singular
integrals on the real line \cite{SaShUr11}, but fails to be necessary for
certain elliptic singular integrals on the line.
\textbf{Failure of sufficiency of Muckenhoupt and Energy conditions}:
However, the weight pair $\left( \omega ,\ddot{\sigma}\right) $ constructed
in \cite{LaSaUr2} satisfies the Muckenhoupt and energy conditions, yet fails
to satisfy the norm inequality for the Hilbert transform\footnote
The interested reader can easily verify this, or see previous versions of
the current paper where details are included.}. This shows that, even
assuming the necessary conditions of Muckenhoupt and energy, we still need
some sort of testing conditions, and our $Tb$ theorem essentially leaves the
choice of testing conditions at our disposal - subject only to nondegeneracy
and size conditions.
For example, in the case of the Hilbert transform, Theorem \ref{dim one}
below roughly says this. As we are dealing with the case of general locally
finite positive Borel measures, all intervals appearing in this paper should
be assumed to closed on the left and open on the right, except when
otherwise noted.
\begin{theorem}[$Tb$ for Hilbert transform]
Let $H$ denote the Hilbert transform on the real line $\mathbb{R}$, let
\sigma $ and $\omega $ be locally finite positive Borel measures on $\mathbb
R}$. Then $H_{\sigma }$, where $H_{\sigma }f\equiv H\left( f\sigma \right)
, is bounded from \thinspace $L^{2}\left( \sigma \right) $ to $L^{2}\left(
\omega \right) $ \emph{if and only if} the Muckenhoupt and energy side
conditions hold, as well as the $\mathbf{b}$-testing and $\mathbf{b}^{\ast }
-testing condition
\begin{equation*}
\int_{I}\left\vert T_{\sigma }^{\alpha }b_{I}\right\vert ^{2}d\omega \leq
\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert
I\right\vert _{\sigma }\text{ and }\int_{J}\left\vert T_{\omega }^{\alpha
,\ast }b_{J}^{\ast }\right\vert ^{2}d\sigma \leq \left( \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}^{\ast },\ast }\right) ^{2}\left\vert J\right\vert
_{\omega }\ ,
\end{equation*
taken over two families of test functions $\left\{ b_{I}\right\} _{I\in
\mathcal{P}}$ and $\left\{ b_{J}^{\ast }\right\} _{J\in \mathcal{P}}$, where
$b_{I}$ and $b_{J}^{\ast }$ are only required to be nondegenerate in an
average sense, and to be just slightly better than $L^{2}$ functions
themselves, namely $L^{p}$ for some $p>2$.
\end{theorem}
The families of test functions $\left\{ b_{I}\right\} _{I\in \mathcal{P}}$
and $\left\{ b_{J}^{\ast }\right\} _{J\in \mathcal{P}}$ in the $Tb$ theorem
above are nondegenerate and slightly better than $L^{2}$ functions, but
otherwise remain at the disposal of the reader. It is this flexibility in
choosing families of test functions that distinguishes this characterization
as compared to the corresponding $T1$ theorem\footnote
The energy conditions in (\ref{strong b* energy}) and (\ref{strong b energy
) below are relatively simple, stable and checkable conditions on a weight
pair, that are in addition, an almost immediate consequence of the
Muckenhoupt side condition and the testing conditions for $H$ over
indicators of intervals \cite{LaSaUr2}.}. The $Tb$ theorem here generalizes
many of the one-weight $Tb$ theorems in one dimension, since in the upper
doubling case, the Muckenhoupt $\mathfrak{A}_{2}$ condition and the energy
condition easily follow from the upper doubling condition. Recall that in
the one-weight case with doubling and upper doubling measures $\mu $, there
has been a long and sustained effort to relax the integrability conditions
of the testing functions: see e.g. S. Hofmann \cite{Hof} and Alfonseca,
Auscher, Axelsson, Hofmann and Kim \cite{AAAHK}. Subsequently, Hyt\"{o}nen-
Martikainen \cite{HyMa} assumed $Tb$ in $L^{s}\left( \mu \right) $ for some
s>2$, and the one weight theorem with testing functions $b$ in $L^{2}\left(
\mu \right) $ was attained by Lacey-Martikainen \cite{LaMa}, but their
argument strongly uses methods not immediately available in the two weight
setting.
Finally, we point out that the proof of our $Tb$ theorem is mostly
self-contained, but at the expense of considerable length. This is not just
for the convenience of the reader, but mainly because we must repeat much of
the proof strategy from \cite{NTV4}, \cite{LaSaShUr3}, \cite{Lac}, \cit
{SaShUr7}, \cite{SaShUr9} and \cite{SaShUr10}, as the new ideas used here
force redevelopment of many of the previous arguments in these papers. We
now turn to a brief discussion of these new ideas for those readers already
acquainted with the theory of $T1$ and local $Tb$ theorems. See the brief
schematic (\ref{schematic}) below for a picture summary of the
decompositions involved.
\subsection{New ideas}
For those readers already familiar with the theory of local $Tb$ theorems,
we describe here some of the new techniques introduced in this paper to
handle the two weight situation. There are many difficulties to be overcome
in proving a local $Tb$ theorem, even in the one weight setting, as compared
to the corresponding $T1$ theorem, and we indicate four of them now.
\begin{enumerate}
\item \textbf{First difficulty:} In order to control the dual martingale
differences for `breaking' children, i.e. when the testing function
corresponding to a child is \textbf{not} the restriction of the testing
function of the parent, we need to construct coronas in which the
restrictions don't change, and for which the `breaking' intervals satisfy a
Carleson condition. This makes the so-called `nearby' inner products
\left\langle T_{\sigma }^{\alpha }b_{I},b_{J}^{\ast }\right\rangle _{\omega
} $, i.e. those in which the intervals $I$ and $J$ are close in both
position and scale, extremely difficult to estimate due to the fact that the
testing conditions are lost in the corona, except at the tops of coronas,
and are replaced with just a \textbf{weak} testing condition. Ironically,
these nearby inner products are the easiest to estimate in the proof of a
T1 $ theorem since the testing conditions remain in force in the coronas
there. In the one weight setting in \cite{NTV3}, \cite{HyMa} and \cite{LaMa
, special considerations, such as boundedness of Poisson integrals, are
taken into account in handling nearby inner products with random surgery,
and are unavailable to us here.\newline
\textbf{Resolution:} We develop a new recursive method for controlling the
nearby form by the \emph{energy conditions} and testing at the tops of the
coronas, in which we resurrect the original testing functions discarded
during the corona construction. This is presented in Section \ref{Sec nearby
.
\item \textbf{Second difficulty:} Both dual martingale and martingale
differences fail to satisfy two-sided frame-like and Riesz-like inequalities
in the setting of a $Tb$ theorem when $p=2$, complicating the treatment of
paraproducts.\newline
\textbf{Resolution:} We assume $p>2$ in the upper $L^{p}$ control of testing
functions, and then reduce this case to that of \emph{bounded} testing
functions using an absorption and recursion argument. For families of
bounded testing functions, we prove two-sided \emph{Riesz-like} inequalities
for dual martingale differences that are more robust than frame inequalities
(but only one-sided \emph{Riesz-like} inequalities for martingale
differences), and that enable many of the $T1$ two weight techniques to
carry over here in the $Tb$ setting. In particular these are key to
controlling paraproducts here.
\item \textbf{Third difficulty:} Only a weaker form of goodness due to Hy
\"{o}nen and Martikainen \cite{HyMa} is available for use in two weight $Tb$
theorems, complicating Lacey's treatment of the stopping form.\newline
\textbf{Resolution:} We adapt the two weight $T1$ arguments to accommodate
\emph{weak goodness} in two ways, the first highly nontrivial and second
more straightforward:\newline
(\textbf{1}) in bounding the stopping form by Lacey's size functional on
admissible collections and bottom/up corona construction in \cite{Lac}, as
adapted in \cite{SaShUr9} and \cite{SaShUr10}, but using an additional
top/down `indented' corona construction, along with an enlargement of the
skeleton $\limfunc{skel}\left( I\right) $ of an interval $I$ to the body
\limfunc{body}\left( I\right) $ of $I$, in order to deal with the lack of
goodness in telescoping intervals - see Section \ref{Sec stop},\newline
(\textbf{2}) in controlling functional energy as in \cite{SaShUr9} and \cit
{SaShUr10}, but with a different decomposition of the stopping intervals
into `Whitney' intervals, and modified pseudoprojections to accommodate two
independent families of grids - see Appendix B.
\item \textbf{Fourth difficulty:} The dual martingale differences are not in
general projections when some of the children `break', and the Monotonicity
Lemma fails to hold in any of the traditional forms in the setting of $T1$
theorems.\newline
\textbf{Resolution:} We introduce an additional square function bound on the
right hand side involving an infimum of averages, $\inf_{z\in \mathbb{R
}\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega
}\left\vert x-z\right\vert \right) ^{2}$, summed over broken children. We
also use the fact that the corresponding `unbroken' dual martingale
differences form projections, but then we also need to modify the testing
function at the top of a corona, and also refine the triple corona
construction, so that dual martingale differences have controlled averages
on children throughout the corona.
\end{enumerate}
\subsection{Standard fractional singular integrals}
Let $0\leq \alpha <1$. We define a standard $\alpha $-fractional CZ kernel
K^{\alpha }(x,y)$ to be a real-valued function defined on $\mathbb{R}\times
\mathbb{R}$ satisfying the following fractional size and smoothness
conditions of order $1+\delta $ for some $\delta >0$: For $x\neq y$
\begin{eqnarray}
\left\vert K^{\alpha }\left( x,y\right) \right\vert &\leq &C_{CZ}\left\vert
x-y\right\vert ^{\alpha -1}\text{ and }\left\vert \nabla K^{\alpha }\left(
x,y\right) \right\vert \leq C_{CZ}\left\vert x-y\right\vert ^{\alpha -2},
\label{sizeandsmoothness'} \\
\left\vert \nabla K^{\alpha }\left( x,y\right) -\nabla K^{\alpha }\left(
x^{\prime },y\right) \right\vert &\leq &C_{CZ}\left( \frac{\left\vert
x-x^{\prime }\right\vert }{\left\vert x-y\right\vert }\right) ^{\delta
}\left\vert x-y\right\vert ^{\alpha -2},\ \ \ \ \ \frac{\left\vert
x-x^{\prime }\right\vert }{\left\vert x-y\right\vert }\leq \frac{1}{2},
\notag
\end{eqnarray
and the last inequality also holds for the adjoint kernel in which $x$ and
y $ are interchanged. We note that a more general definition of kernel has
only order of smoothness $\delta >0$, rather than $1+\delta $, but the use
of the Monotonicity and Energy Lemmas in arguments below involves first
order Taylor approximations to the kernel functions $K^{\alpha }\left( \cdot
,y\right) $.
\subsubsection{Defining the norm inequality}
We now turn to a precise definition of the weighted norm inequalit
\begin{equation}
\left\Vert T_{\sigma }^{\alpha }f\right\Vert _{L^{2}\left( \omega \right)
}\leq \mathfrak{N}_{T_{\sigma }^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) },\ \ \ \ \ f\in L^{2}\left( \sigma \right) .
\label{two weight'}
\end{equation
For this we follow the lead of \cite{NTV3} and introduce a family $\left\{
\eta _{\delta ,R}^{\alpha }\right\} _{0<\delta <R<\infty }$ of nonnegative
functions on $\left[ 0,\infty \right) $ so that the truncated kernels
K_{\delta ,R}^{\alpha }\left( x,y\right) =\eta _{\delta ,R}^{\alpha }\left(
\left\vert x-y\right\vert \right) K^{\alpha }\left( x,y\right) $ are bounded
with compact support for fixed $x$ or $y$. Then the truncated operators
\begin{equation}
T_{\sigma ,\delta ,R}^{\alpha }f\left( x\right) \equiv \int_{\mathbb{R
}K_{\delta ,R}^{\alpha }\left( x,y\right) f\left( y\right) d\sigma \left(
y\right) ,\ \ \ \ \ x\in \mathbb{R}, \label{def truncation}
\end{equation
are pointwise well-defined, and we will refer to the pair $\left( K^{\alpha
},\left\{ \eta _{\delta ,R}^{\alpha }\right\} _{0<\delta <R<\infty }\right) $
as an $\alpha $-fractional singular integral operator, which we typically
denote by $T^{\alpha }$, suppressing the dependence on the truncations.
\begin{definition}
\label{truncated op}We say that an $\alpha $-fractional singular integral
operator $T^{\alpha }=\left( K^{\alpha },\left\{ \eta _{\delta ,R}^{\alpha
}\right\} _{0<\delta <R<\infty }\right) $ satisfies the norm inequality (\re
{two weight'}) provide
\begin{equation*}
\left\Vert T_{\sigma ,\delta ,R}^{\alpha }f\right\Vert _{L^{2}\left( \omega
\right) }\leq \mathfrak{N}_{T_{\sigma }^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) },\ \ \ \ \ f\in L^{2}\left( \sigma \right)
,0<\delta <R<\infty .
\end{equation*}
\end{definition}
It turns out that, in the presence of the Muckenhoupt conditions (\ref{def
A2}) below, the norm inequality (\ref{two weight'}) is essentially
independent of the choice of truncations used, and this is explained in some
detail in \cite{NTV3}, \cite{LaSaShUr3} and \cite{SaShUr10}. Thus, as in
\cite{SaShUr10}, we are free to use the tangent line truncations described
there throughout the proofs of our results.
\subsection{Weakly accretive $\mathbf{b}$-testing and $\mathbf{b}^{\ast }
-testing conditions}
Denote by $\mathcal{P}$ the collection of intervals in $\mathbb{R}$. Note
that we include an $L^{p}$ upper bound in our definition of `$p$-weakly
accretive family' of functions.
\begin{definition}
Let $p\geq 2$ and let $\mu $ be a locally finite positive Borel measure on
\mathbb{R}$. We say that a family $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in
\mathcal{P}}$ of functions indexed by $\mathcal{P}$ is a $p$\emph{-weakly }
\mu $\emph{-accretive} family of functions on $\mathbb{R}$ i
\begin{eqnarray}
&&\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{P},
\label{local accretive} \\
1 &\leq &\left\vert \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}b_{Q}d\mu \right\vert \leq \left( \frac{1}{\left\vert Q\right\vert
_{\mu }}\int_{Q}\left\vert b_{Q}\right\vert ^{p}d\mu \right) ^{\frac{1}{p
}\leq C_{\mathbf{b}}<\infty ,\ \ \ \ \ Q\in \mathcal{P}\ . \notag
\end{eqnarray}
\end{definition}
Suppose $\sigma $ and $\omega $ are locally finite positive Borel measures
on $\mathbb{R}$. The $\mathbf{b}$-testing conditions for $T^{\alpha }$ and
\mathbf{b}^{\ast }$-testing conditions for the dual $T^{\alpha ,\ast }$ are
given b
\begin{eqnarray}
\int_{Q}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega &\leq
&\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert
Q\right\vert _{\sigma }\ ,\ \ \ \ \ \text{for all intervals }Q,
\label{b testing cond} \\
\int_{Q}\left\vert T_{\omega }^{\alpha ,\ast }b_{Q}^{\ast }\right\vert
^{2}d\sigma &\leq &\left( \mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast
}}\right) ^{2}\left\vert Q\right\vert _{\omega }\ ,\ \ \ \ \ \text{for all
intervals }Q, \notag
\end{eqnarray
where these inequalities are interpreted as holding uniformly over
truncations of $T_{\sigma }^{\alpha }$ and $T^{\alpha ,\ast }$.
\subsection{Poisson integrals and the Muckenhoupt condition $\mathfrak{A
_{2}^{\protect\alpha }$}
Let $\mu $ be a locally finite positive Borel measure on $\mathbb{R}$, and
suppose $Q$ is an interval in $\mathbb{R}$. Recall that $\left\vert
Q\right\vert =\ell \left( Q\right) $ for an interval $Q$. The two $\alpha
-fractional Poisson integrals of $\mu $ on an interval $Q$ are given by the
following expressions
\begin{eqnarray}
\mathrm{P}^{\alpha }\left( Q,\mu \right) &\equiv &\int_{\mathbb{R}}\frac
\left\vert Q\right\vert }{\left( \left\vert Q\right\vert +\left\vert
x-x_{Q}\right\vert \right) ^{2-\alpha }}d\mu \left( x\right) ,
\label{def Poisson} \\
\mathcal{P}^{\alpha }\left( Q,\mu \right) &\equiv &\int_{\mathbb{R}}\left(
\frac{\left\vert Q\right\vert }{\left( \left\vert Q\right\vert +\left\vert
x-x_{Q}\right\vert \right) ^{2}}\right) ^{1-\alpha }d\mu \left( x\right) ,
\notag
\end{eqnarray
where $\left\vert x-x_{Q}\right\vert $ denotes distance between $x$ and
x_{Q}$ and $\left\vert Q\right\vert $ denotes the Lebesgue measure of the
interval $Q$. We refer to $\mathrm{P}^{\alpha }$ as the \emph{standard}
Poisson integral and to $\mathcal{P}^{\alpha }$ as the \emph{reproducing}
Poisson integral. Note that these two kernels satisfy
\begin{equation*}
0\leq \mathrm{P}^{\alpha }\left( Q,\mu \right) \leq C\mathcal{P}^{\alpha
}\left( Q,\mu \right) ,\ \ \ \ \text{for all intervals }Q\text{ and positive
measures }\mu .
\end{equation*}
We now define the \emph{one-tailed Muckenhoupt constant with holes }
\mathcal{A}_{2}^{\alpha }$ using the reproducing Poisson kernel $\mathcal{P
^{\alpha }$. On the other hand, the standard Poisson integral $\mathrm{P
^{\alpha }$ arises naturally throughout the proof of the $Tb$ theorem in
estimating oscillation of the fractional singular integral $T^{\alpha }$,
and in the definition of the energy conditions below.
\begin{definition}
Suppose $\sigma $ and $\omega $ are locally finite positive Borel measures
on $\mathbb{R}$. The one-tailed Muckenhoupt constants $\mathcal{A
_{2}^{\alpha }$ and $\mathcal{A}_{2}^{\alpha ,\ast }$ with holes for the
weight pair $\left( \sigma ,\omega \right) $ are given b
\begin{eqnarray}
\mathcal{A}_{2}^{\alpha } &\equiv &\sup_{Q\in \mathcal{P}}\mathcal{P
^{\alpha }\left( Q,\mathbf{1}_{Q^{c}}\sigma \right) \frac{\left\vert
Q\right\vert _{\omega }}{\left\vert Q\right\vert ^{1-\alpha }}<\infty ,
\label{def call A2} \\
\mathcal{A}_{2}^{\alpha ,\ast } &\equiv &\sup_{Q\in \mathcal{P}}\mathcal{P
^{\alpha }\left( Q,\mathbf{1}_{Q^{c}}\omega \right) \frac{\left\vert
Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{1-\alpha }}<\infty .
\notag
\end{eqnarray}
\end{definition}
Note that these definitions are the conditions with `holes' introduced by Hy
\"{o}nen \cite{Hyt} - the supports of the measures $\mathbf{1}_{Q^{c}}\sigma
$ and $\mathbf{1}_{Q}\omega $ in the definition of $\mathcal{A}_{2}^{\alpha
} $ are disjoint, and so any common point masses of $\sigma $ and $\omega $
do not appear simultaneously in the factors of any of the products $\mathcal
P}^{\alpha }\left( Q,\mathbf{1}_{Q^{c}}\sigma \right) \frac{\left\vert
Q\right\vert _{\omega }}{\left\vert Q\right\vert ^{1-\alpha }}$. We will
also use the smaller `offset' Muckenhoupt conditio
\begin{equation*}
A_{2}^{\alpha }\equiv \sup_{\substack{ Q,Q^{\prime }\in \mathcal{P} \\
\text{ and }Q^{\prime }\text{ are adjacent, }\ell \left( Q\right) =\ell
\left( Q^{\prime }\right) }}\frac{\left\vert Q\right\vert _{\omega }}
\left\vert Q\right\vert ^{1-\alpha }}\frac{\left\vert Q^{\prime }\right\vert
_{\sigma }}{\left\vert Q^{\prime }\right\vert ^{1-\alpha }}<\infty ,
\end{equation*
but the classical Muckenhoupt condition $A_{2}^{\alpha ,\limfunc{class
}\equiv \sup_{Q\in \mathcal{P}}\frac{\left\vert Q\right\vert _{\omega
}\left\vert Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{2-2\alpha }
<\infty $ will find no use in the two weight setting with common point
masses permitted.
\begin{remark}
Initially, these definitions of Muckenhoupt type were given in the following
`one weight' case, $d\omega \left( x\right) =w\left( x\right) dx$ and
d\sigma \left( x\right) =\frac{1}{w\left( x\right) }dx$, where $\mathcal{A
_{2}^{\alpha }\left( \lambda w,\left( \lambda w\right) ^{-1}\right)
\mathcal{A}_{2}^{\alpha }\left( w,w^{-1}\right) $ is homogeneous of degree
0 $. Of course the two weight version is homogeneous of degree $2$ in the
weight pair, $\mathcal{A}_{2}^{\alpha }\left( \lambda \sigma ,\lambda \omega
\right) =\lambda ^{2}\mathcal{A}_{2}^{\alpha }\left( \sigma ,\omega \right)
, while all of the other conditions we consider in connection with two
weight norm inequalities, including the operator norm $\mathfrak{N
_{T^{\alpha }}\left( \sigma ,\omega \right) $ itself, are homogeneous of
degree $1$ in the weight pair. This awkwardness regarding the homogeneity of
Muckenhoupt conditions could be rectified by simply taking the square root
of $\mathcal{A}_{2}^{\alpha }$ and renaming it, but the current definition
is so entrenched in the literature, in particular in connection with the
A_{2}$ conjecture, that we will leave it as is.
\end{remark}
\subsubsection{Punctured $A_{2}^{\protect\alpha }$ conditions}
The \emph{classical} $A_{2}^{\alpha }$ characteristic $\sup_{Q\in \mathcal{P
}\frac{\left\vert Q\right\vert _{\omega }}{\left\vert Q\right\vert
^{1-\alpha }}\frac{\left\vert Q\right\vert _{\sigma }}{\left\vert
Q\right\vert ^{1-\alpha }}$ fails to be finite when the measures $\sigma $
and $\omega $ have a common point mass - simply let $Q$ in the $\sup $ above
shrink to a common mass point. But there is a substitute that is quite
similar in character that is motivated by the fact that for large intervals
Q$, the $\sup $ above is problematic only if just \emph{one} of the measures
is \emph{mostly} a point mass when restricted to $Q$.
Given an at most countable set $\mathfrak{P}=\left\{ p_{k}\right\}
_{k=1}^{\infty }$ in $\mathbb{R}$, an interval $Q\in \mathcal{P}$, and a
positive locally finite Borel measure $\mu $, define
\begin{equation}
\mu \left( Q,\mathfrak{P}\right) \equiv \left\vert Q\right\vert _{\mu }-\sup
\left\{ \mu \left( p_{k}\right) :p_{k}\in Q\cap \mathfrak{P}\right\} ,
\label{puncture}
\end{equation
where the supremum is actually achieved since $\sum_{p_{k}\in Q\cap
\mathfrak{P}}\mu \left( p_{k}\right) <\infty $ as $\mu $ is locally finite.
The quantity $\mu \left( Q,\mathfrak{P}\right) $ is simply the $\widetilde
\mu }$ measure of $Q$ where $\widetilde{\mu }$ is the measure $\mu $ with
its largest point mass from $\mathfrak{P}$ in $Q$ removed. Given a locally
finite positive measure pair $\left( \sigma ,\omega \right) $, let
\begin{equation}
\mathfrak{P}_{\left( \sigma ,\omega \right) }=\left\{ p_{k}\right\}
_{k=1}^{\infty } \label{def common point mass}
\end{equation
be the at most countable set of common point masses of $\sigma $ and $\omega
$. Then the weighted norm inequality (\ref{two weight'}) typically implies
finiteness of the following \emph{punctured} Muckenhoupt conditions
\begin{eqnarray}
A_{2}^{\alpha ,\limfunc{punct}}\left( \sigma ,\omega \right) &\equiv
&\sup_{Q\in \mathcal{P}}\frac{\omega \left( Q,\mathfrak{P}_{\left( \sigma
,\omega \right) }\right) }{\left\vert Q\right\vert ^{1-\alpha }}\frac
\left\vert Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{1-\alpha }},
\label{def punct} \\
A_{2}^{\alpha ,\ast ,\limfunc{punct}}\left( \sigma ,\omega \right) &\equiv
&\sup_{Q\in \mathcal{P}}\frac{\left\vert Q\right\vert _{\omega }}{\left\vert
Q\right\vert ^{1-\alpha }}\frac{\sigma \left( Q,\mathfrak{P}_{\left( \sigma
,\omega \right) }\right) }{\left\vert Q\right\vert ^{1-\alpha }}. \notag
\end{eqnarray}
All of the above Muckenhuopt conditions $\mathcal{A}_{2}^{\alpha }$,
\mathcal{A}_{2}^{\alpha ,\ast }$, $A_{2}^{\alpha ,\limfunc{punct}}$ and
A_{2}^{\alpha ,\ast ,\limfunc{punct}}$ are necessary for boundedness of an
elliptic $\alpha $-fractional singular integral $T_{\sigma }^{\alpha }$ on
the line from\thinspace $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega
\right) $ (see \cite{SaShUr10}). It is convenient to defin
\begin{equation}
\mathfrak{A}_{2}^{\alpha }\equiv \mathcal{A}_{2}^{\alpha }+\mathcal{A
_{2}^{\alpha ,\ast }+A_{2}^{\alpha ,\limfunc{punct}}+A_{2}^{\alpha ,\ast
\limfunc{punct}}\ . \label{def A2}
\end{equation}
\subsection{Energy Conditions}
Here is the definition of the strong energy conditions, which we sometimes
refer to simply as the energy conditions. Let $m_{I}^{\mu }\equiv \frac{1}
\left\vert I\right\vert }\int xd\mu \left( x\right) $ be the average of $x$
over $I$ with respect to the measure $\mu $, which we often abbreviate to
m_{I}$ when the measure $\mu $ is understood.
\begin{definition}
\label{def strong quasienergy}Let $0\leq \alpha <1$. Suppose $\sigma $ and
\omega $ are locally finite positive Borel measures on $\mathbb{R}$. Then
the \emph{strong} energy constant $\mathcal{E}_{2}^{\alpha }$ is defined by
\begin{equation}
\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\equiv \sup_{I=\dot{\cup}I_{r}
\frac{1}{\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\left( \frac
\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1}_{I}\sigma \right) }{\left\vert
I_{r}\right\vert }\right) ^{2}\left\Vert x-m_{I_{r}}^{\omega }\right\Vert
_{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}\ ,
\label{strong b* energy}
\end{equation
where the supremum is taken over arbitrary decompositions of an interval $I$
using a pairwise disjoint union of subintervals $I_{r}$. Similarly, we
define the dual \emph{strong} energy constant $\mathcal{E}_{\alpha }^
\limfunc{strong},\mathbf{b},\ast }$ by switching the roles of $\sigma $ and
\omega $
\begin{equation}
\left( \mathcal{E}_{2}^{\alpha ,\ast }\right) ^{2}\equiv \sup_{I=\dot{\cup
I_{r}}\frac{1}{\left\vert I\right\vert _{\omega }}\sum_{r=1}^{\infty }\left(
\frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1}_{I}\omega \right) }
\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert x-m_{I_{r}}^{\sigma
}\right\Vert _{L^{2}\left( \mathbf{1}_{I_{r}}\sigma \right) }^{2}\ .
\label{strong b energy}
\end{equation}
\end{definition}
These energy conditions are necessary for boundedness of elliptic and
gradient elliptic operators, including the Hilbert transform (but not for
certain elliptic singular operators that fail to be gradient elliptic) - see
\cite{SaShUr11}, and see also (\ref{energy condition is necd}) below for
control of the energy constants $\mathcal{E}_{2}^{\alpha }$ and $\mathcal{E
_{2}^{\alpha ,\ast }$\ by the $\mathbf{1}$-testing and Muckenhoupt constants
$\mathfrak{T}_{T^{\alpha }}^{\mathbf{1}}$, $\mathfrak{T}_{T^{\alpha }}^
\mathbf{1},\ast }$ and $\sqrt{\mathfrak{A}_{2}^{\alpha }}$. It is convenient
to defin
\begin{equation}
\mathfrak{E}_{2}^{\alpha }\equiv \mathcal{E}_{2}^{\alpha }+\mathcal{E
_{2}^{\alpha ,\ast }, \label{def frak energy}
\end{equation
as well a
\begin{equation}
\mathcal{NTV}_{\alpha }\equiv \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}^{\ast },\ast }+\sqrt{\mathfrak{A
_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\ . \label{def NTV}
\end{equation}
\section{The local $Tb$ theorem and proof preliminaries}
We derive a local $Tb$ theorem based in part on our \emph{proof} of the $T1$
theorem in \cite{SaShUr7}, \cite{SaShUr6}, \cite{SaShUr9} and \cite{SaShUr10
, in turn based on prior work in \cite{NTV4}, \cite{LaSaShUr3} and \cite{Lac
, and in part on the \emph{proof} of one weight $Tb$ theorems in Nazarov,
Treil and Volberg \cite{NTV3} and Hyt\"{o}nen and Martikainen \cite{HyMa}.
Recall from \cite{SaShUr11} that an $\alpha $-fractional singular integral
T^{\alpha }$ with kernel $K^{\alpha }$ is said to be \emph{elliptic} if
\left\vert K^{\alpha }\left( x,y\right) \right\vert \geq c\left\vert
x-y\right\vert ^{\alpha -1}$, and \emph{gradient elliptic} if the kernel
K^{\alpha }\left( x,y\right) $ satisfie
\begin{equation}
\frac{d}{dx}K^{\alpha }\left( x,y\right) ,-\frac{d}{dy}K^{\alpha }\left(
x,y\right) \geq c\left\vert x-y\right\vert ^{\alpha -2}.
\label{def grad elliptic}
\end{equation
The Hilbert transform kernel $K\left( x,y\right) =\frac{1}{y-x}$ satisfies
\ref{def grad elliptic}) with $\alpha =0$. In dimension $n=1$ the
Muckenhoupt conditions are necessary for norm boundedness of elliptic
operators by results in \cite{LaSaUr2}, \cite{Hyt2} and \cite{SaShUr9}, and
the energy conditions are necessary for norm boundedness of gradient
elliptic operators by results in \cite{SaShUr11}. Moreover, in dimension
n=1 $, Hyt\"{o}nen \cite[Corollary 3.10]{Hyt2} proves that full testing is
controlled by testing and the Muckenhoupt conditions for the Hilbert
transform, and this is easily extended to $0\leq \alpha <1$ - see (\ref{full
proved}) below. Here is our two weight local $Tb$ theorem.
\begin{theorem}
\label{dim one}Suppose that $\sigma $ and $\omega $ are locally finite
positive Borel measures on the real line $\mathbb{R}$.
\begin{enumerate}
\item Assume that $T^{\alpha }$ is a standard $\alpha $-fractional elliptic
and gradient elliptic singular integral operator on $\mathbb{R}$, and set
T_{\sigma }^{\alpha }f=T^{\alpha }\left( f\sigma \right) $ for any smooth
truncation of $T_{\sigma }^{\alpha }$, so that $T_{\sigma }^{\alpha }$ is
\emph{apriori} bounded from $L^{2}\left( \sigma \right) $ to $L^{2}\left(
\omega \right) $.
\item Let $p>2$ and let $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P
} $ be a $p$-weakly $\sigma $-accretive family of functions on $\mathbb{R}$,
and let $\mathbf{b}^{\ast }=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P
} $ be a $p$-weakly $\omega $-accretive family of functions on $\mathbb{R}$.
\item Then for $0\leq \alpha <1$, the operator $T_{\sigma }^{\alpha }$ is
bounded from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega \right) $
with operator norm $\mathfrak{N}_{T_{\sigma }^{\alpha }}$, i.e.
\begin{equation*}
\left\Vert T_{\sigma }^{\alpha }f\right\Vert _{L^{2}\left( \omega \right)
}\leq \mathfrak{N}_{T_{\sigma }^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) },\ \ \ \ \ f\in L^{2}\left( \sigma \right) ,
\end{equation*
\textbf{uniformly} in smooth truncations of $T^{\alpha }$ \emph{if and only
if} the Muckenhoupt and energy conditions hold, i.e. $\mathcal{A
_{2}^{\alpha },\mathcal{A}_{2}^{\alpha ,\ast },A_{2}^{\alpha ,\limfunc{punct
},A_{2}^{\alpha ,\ast ,\limfunc{punct}},\mathcal{E}_{2}^{\alpha },\mathcal{E
_{2}^{\alpha ,\ast }<\infty $, and the $\mathbf{b}$-testing conditions for
T^{\alpha }$ and the $\mathbf{b}^{\ast }$-testing conditions for the dual
T^{\alpha ,\ast }$ both hold. Moreover, we have the equivalence
\begin{equation*}
\mathfrak{N}_{T^{\alpha }}\approx \mathcal{NTV}_{\alpha }=\mathfrak{T}_
\mathbf{R}^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{\mathbf{R}^{\alpha }}^
\mathbf{b}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E
_{2}^{\alpha }\ .
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{remark}
\label{special}In the special case that $\sigma =\omega =\mu $, the
classical Muckenhoupt $A_{2}^{\alpha }$ condition i
\begin{equation*}
\sup_{Q\in \mathcal{P}}\frac{\left\vert Q\right\vert _{\mu }}{\left\vert
Q\right\vert ^{1-\alpha }}\frac{\left\vert Q\right\vert _{\mu }}{\left\vert
Q\right\vert ^{1-\alpha }}<\infty ,
\end{equation*
which is the upper doubling measure\ condition with exponent $1-\alpha $,
i.e.
\begin{equation*}
\left\vert Q\right\vert _{\mu }\leq C\ell \left( Q\right) ^{1-\alpha },\ \ \
\ \ \text{for all intervals }Q,\text{ }
\end{equation*
which of course prohibits point masses in $\mu $. Both Poisson integrals are
then bounded,
\begin{eqnarray*}
\mathrm{P}^{\alpha }\left( Q,\mu \right) &\lesssim &\sum_{k=0}^{\infty
\frac{\left\vert Q\right\vert }{\left( 2^{k}\left\vert Q\right\vert \right)
^{2-\alpha }}\left\vert 2^{k}Q\right\vert _{\mu }\lesssim \sum_{k=0}^{\infty
}\frac{\left\vert Q\right\vert }{\left( 2^{k}\left\vert Q\right\vert \right)
^{2-\alpha }}\left( 2^{k}\left\vert Q\right\vert \right) ^{1-\alpha }=2, \\
\mathcal{P}^{\alpha }\left( Q,\mu \right) &\lesssim &\sum_{k=0}^{\infty
}\left( \frac{\left\vert Q\right\vert }{\left( 2^{k}\left\vert Q\right\vert
\right) ^{2}}\right) ^{1-\alpha }\left\vert 2^{k}Q\right\vert _{\mu
}\lesssim \sum_{k=0}^{\infty }\left( \frac{\left\vert Q\right\vert }{\left(
2^{k}\left\vert Q\right\vert \right) ^{2}}\right) ^{1-\alpha }\left(
2^{k}\left\vert Q\right\vert \right) ^{1-\alpha }=C_{\alpha }<\infty ,
\end{eqnarray*
and it follows easily that the equal weight pair $\left( \mu ,\mu \right) $
satisfies not only the Muckenhoupt $\mathfrak{A}_{2}^{\alpha }$ condition,
but also the strong energy condition $\mathfrak{E}_{2}^{\alpha }$
\begin{eqnarray*}
&&\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{
}_{I}\sigma \right) }{\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert x
\mathbb{E}_{I_{r}}^{\omega }x\right\Vert _{L^{2}\left( \omega \right)
}^{2}\leq C\sum_{r=1}^{\infty }\left\Vert \frac{x-\mathbb{E}_{I_{r}}^{\omega
}x}{\left\vert I_{r}\right\vert }\right\Vert _{L^{2}\left( \omega \right)
}^{2} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq
C\sum_{r=1}^{\infty }\left\vert I_{r}\right\vert _{\omega }\leq C\left\vert
I\right\vert _{\omega }=C\left\vert I\right\vert _{\sigma }\ ,
\end{eqnarray*
since $\omega =\sigma $. Thus Theorem \ref{dim one}, when restricted to a
single weight $\sigma =\omega $, recovers a weaker version of the one weight
theorem of Lacey and Martikainen \cite[Theorem 1.1]{LaMa} for dimension $n=1$
- weaker due to our assumption that $p>2$. On the other hand, the
possibility of a two weight theorem for a $2$-weakly $\mu $-accretive family
is highly problematic, as one of the key proof strategies used in \cite{LaMa}
in the one weight case is a reduction to testing over $f$ and $g$ with
controlled $L^{\infty }$ norm via interpolation, a strategy that appears to
be unavailable in the two weight setting.
\end{remark}
\begin{problem}
Does Theorem \ref{dim one} remain true in the case $p=2$, i.e. when $\mathbf
b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ is a $2$-weakly $\sigma
-accretive family of functions, and $\mathbf{b}^{\ast }=\left\{ b_{Q}^{\ast
}\right\} _{Q\in \mathcal{P}}$ is a $2$-weakly $\omega $-accretive family of
functions?
\end{problem}
\begin{problem}
Are the energy conditions in Theorem \ref{dim one} already implied by the
Muckenhoupt, $\mathbf{b}$-testing and dual $\mathbf{b}^{\ast }$-testing
conditions for a pair of $p$-weakly accretive families when $p>2$?
\end{problem}
In order to prove Theorem \ref{dim one}, we first establish some improved
properties for a $p$-weakly $\mu $-accretive family, and establish some
improved energy conditions related to the families of testing functions
\mathbf{b}$ and $\mathbf{b}^{\ast }$. We turn to these matters in the next
three subsections.
\subsection{Reduction to the pointwise lower bound property}
Here we show that we may assume without loss of generality that the $p
-weakly accretive families of testing functions $b_{Q}$ and $b_{Q}^{\ast }$
for $Q\in \mathcal{P}$ satisfy the \emph{pointwise lower bound property},
written $PLBP$
\begin{equation}
\left\vert b_{Q}\left( x\right) \right\vert \geq c_{1}>0\ \ \ \ \ \text{for
\ Q\in \mathcal{P}\text{ and }\sigma \text{-a.e. }x\in \mathbb{R},
\label{plb}
\end{equation
for some positive constant $c_{1}$. Of course if $b_{Q}=\mathbf{1}_{Q}b$ for
some globally defined $b$, then the $PLBP$ is immediate from Lebesgue's
dyadic differentiation theorem. We make the following definition of a $p
\emph{-strongly} $\mu $-accretive family in $\mathbb{R}$.
\begin{definition}
\label{def strongly accretive}Let $\mu $ be a positive Borel measure on
\mathbb{R}$. We say that a family $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in
\mathcal{P}}$ of functions indexed by $\mathcal{P}$ is a $p$\emph{-strongly
$\mu $\emph{-accretive} family of functions on $\mathbb{R}$ if the $b_{Q}$
are real-valued and there are positive constants $C_{\mathbf{b}}$, and
c_{1} $ such that
\begin{eqnarray*}
&&\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{P}, \\
&&0<1\leq \left\vert \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}b_{Q}d\mu \right\vert \leq \left( \frac{1}{\left\vert Q\right\vert
_{\mu }}\int_{Q}\left\vert b_{Q}\right\vert ^{p}d\mu \right) ^{\frac{1}{p
}\leq C_{\mathbf{b}}<\infty ,\ \ \ \ \ Q\in \mathcal{P}\ , \\
&&\left\vert b_{Q}\left( x\right) \right\vert \geq c_{1}>0\ \ \ \ \ \text
for }\sigma \text{-a.e. }x\in \mathbb{R}.
\end{eqnarray*}
\end{definition}
To obtain that the families $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in
\mathcal{P}^{n}}$ and $\mathbf{b}^{\ast }=\left\{ b_{Q}^{\ast }\right\}
_{Q\in \mathcal{P}^{n}}$ can be assumed to satisfy the $PLBP$ requires some
effort. But first, let us make a simple observation (essentially in \cit
{HyMa}) under the additional assumption that the \emph{breaking intervals}
Q $, those for which there is a dyadic child $Q^{\prime }$ of $Q$ with
b_{Q^{\prime }}\neq \mathbf{1}_{Q^{\prime }}b_{Q}$, satisfy a $\mu
-Carleson condition. If $G_{k}\equiv \cup \mathcal{G}_{k}$ wher
\begin{equation*}
\mathcal{G}_{k}\equiv \left\{ A\in \mathcal{A}:A\text{ is a }k^{th}\text{
generation breaking interval}\right\} ,
\end{equation*
then $\left\vert \dbigcap\limits_{k=1}^{\infty }G_{k}\right\vert _{\sigma
}=0 $ since $\left\vert G_{k}\right\vert _{\sigma }\lesssim 2^{-\delta k}$
for some $\delta >0$ by the Carleson condition on breaking intervals. Thus
for $\sigma $-almost every $x$, the sequence of test functions $\left\{
b_{Q}\right\} _{Q:\ x\in Q}$, when arranged in order of decreasing length of
$Q$, has the property that all sufficiently small $Q$ with $x\in Q$ belong
to the same corona $\mathcal{C}_{A}$ with $x\in A$, and hence $b_{Q}=\mathbf
1}_{Q}b_{A}$ for sufficiently small intervals $Q$ containing $x$. Suppose
that $A\in \mathcal{G}_{k}$. Then by Lebesgue's dyadic differentiation
theorem, we have
\begin{equation*}
\left\vert b_{A}\left( x\right) \right\vert =\left\vert \lim_{Q\searrow x
\frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}b_{A}d\mu \right\vert
=\lim_{Q\searrow x}\left\vert \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}b_{Q}d\mu \right\vert \geq c>0,
\end{equation*
for $\sigma $-a.e. $x\in A\setminus \left( \cup \mathcal{G}_{k+1}\right) $.
But this misses showing that $\left\vert b_{A}\left( x\right) \right\vert
\geq c>0$ on $A\cap \left( \cup \mathcal{G}_{k+1}\right) $, and for this we
must work harder.
\begin{proposition}
\label{lower bound}Let $p\geq 2$. Suppose that $\mathbf{b}=\left\{
b_{Q}\right\} _{Q\in \mathcal{P}}$ is a $p$-weakly $\sigma $-accretive
family of complex-valued functions on $\mathbb{R}$ that satisfies the
\mathbf{b}$-testing condition (\ref{b testing cond}) for a fractional
singular integral operator $T^{\alpha }$. Then there is a $p$-weakly $\sigma
$-accretive family $\widetilde{\mathbf{b}}=\left\{ \widetilde{b}_{Q}\right\}
_{Q\in \mathcal{P}}$ that satisfies the $PLBP$. Moreover, the full
\widetilde{\mathbf{b}}$-testing condition (\ref{b testing cond}) for
T^{\alpha }$ holds and we have the estimate,
\begin{eqnarray*}
&&\left\{ \widetilde{b}_{Q}\right\} _{Q\in \mathcal{P}^{n}}\text{ is }p\text
-strongly }\sigma \text{-accretive}, \\
\mathfrak{FT}_{T^{\alpha }}^{\widetilde{\mathbf{b}}}\left( \sigma ,\omega
\right) &\leq &C_{\alpha }\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b
}\left( \sigma ,\omega \right) +\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}^{\ast
},\ast }\left( \sigma ,\omega \right) +\sqrt{\mathcal{A}_{2}^{\alpha }\left(
\sigma ,\omega \right) }+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }\left( \sigma
,\omega \right) }\right) .
\end{eqnarray*}
\end{proposition}
\begin{proof}
For every interval $Q\in \mathcal{P}$ defin
\begin{equation*}
E\left( Q\right) \equiv \left\{ x\in Q:\left\vert b_{Q}\left( x\right)
\right\vert <\frac{1}{4}\right\} .
\end{equation*
Momentarily fix an interval $Q\in \mathcal{P}$ and $\delta >0$, and let
\left\{ I_{j}\left( Q\right) \right\} _{j=1}^{\infty }$ be a collection of
pairwise disjoint intervals such tha
\begin{eqnarray*}
&&E\left( Q\right) =\left\{ x\in Q:\left\vert b_{Q}\left( x\right)
\right\vert <\frac{1}{4}\right\} \subset \dbigcup\limits_{j=1}^{\infty
}I_{j}\left( Q\right) \ ; \\
&&\left\vert \dbigcup\limits_{j=1}^{\infty }I_{j}\left( Q\right) \setminus
E\left( Q\right) \right\vert _{\sigma }<\delta \left\vert Q\right\vert
_{\sigma }\ .
\end{eqnarray*
Note tha
\begin{eqnarray*}
\left\vert Q\right\vert _{\sigma } &\leq &\left\vert \int_{Q}b_{Q}d\sigma
\right\vert =\left\vert \int_{E\left( Q\right) }b_{Q}d\sigma \right\vert
+\left\vert \int_{Q\setminus E\left( Q\right) }b_{Q}d\sigma \right\vert \\
&\leq &\frac{1}{4}\left\vert E\left( Q\right) \right\vert _{\sigma }+\eta
\int_{Q\setminus E\left( Q\right) }\left\vert b_{Q}\right\vert ^{2}d\sigma
\frac{1}{\eta }\int_{Q\setminus E\left( Q\right) }d\sigma \\
&\leq &\frac{1}{4}\left\vert E\left( Q\right) \right\vert _{\sigma }+\eta
\int_{Q}\left\vert b_{Q}\right\vert ^{2}d\sigma +\frac{1}{\eta }\left\vert
Q\setminus E\left( Q\right) \right\vert _{\sigma } \\
&\leq &\frac{1}{4}\left\vert E\left( Q\right) \right\vert _{\sigma }+\eta
C\left\vert Q\right\vert _{\sigma }+\frac{1}{\eta }\left\vert Q\setminus
E\left( Q\right) \right\vert _{\sigma }\ .
\end{eqnarray*
Thus taking $\eta =\frac{1}{2C}$, and dividing through by $\left\vert
Q\right\vert _{\sigma }$, we get
\begin{eqnarray*}
1 &\leq &\frac{1}{4}\frac{\left\vert E\left( Q\right) \right\vert _{\sigma
}{\left\vert Q\right\vert _{\sigma }}+\frac{1}{2}+2C\frac{\left\vert
Q\setminus E\left( Q\right) \right\vert _{\sigma }}{\left\vert Q\right\vert
_{\sigma }}\leq \frac{3}{4}+2C\frac{\left\vert Q\setminus E\left( Q\right)
\right\vert _{\sigma }}{\left\vert Q\right\vert _{\sigma }}; \\
&\Longrightarrow &\frac{1}{4}\leq 2C\frac{\left\vert Q\setminus E\left(
Q\right) \right\vert _{\sigma }}{\left\vert Q\right\vert _{\sigma }
\Longrightarrow \left\vert Q\setminus E\left( Q\right) \right\vert _{\sigma
}\geq \frac{1}{8C}\left\vert Q\right\vert _{\sigma } \\
&\Longrightarrow &\left\vert E\left( Q\right) \right\vert _{\sigma }\leq
\left( 1-\frac{1}{8C}\right) \left\vert Q\right\vert _{\sigma }=\beta
\left\vert Q\right\vert _{\sigma }\ ,
\end{eqnarray*
where $\beta =1-\frac{1}{8C}$. Now we note that since $\delta >0$ can be
taken arbitrarily small, we may without loss of generality take $\delta =0
\footnote
A rigorous limiting argument can be modelled after that given here.}.
Altogether then, we have shown that for every $Q\in \mathcal{P}$, there is a
pairwise disjoint collection of intervals $\left\{ I_{j}^{Q}\right\} _{j}$
such tha
\begin{eqnarray*}
E\left( Q\right) &\equiv &\left\{ x\in Q:\left\vert b_{Q}\left( x\right)
\right\vert <\frac{1}{4}\right\} =\overset{\cdot }{\dbigcup }_{j=1}^{\infty
}I_{j}\left( Q\right) \ , \\
\text{and }\left\vert E\left( Q\right) \right\vert _{\sigma }
&=&\sum_{j=1}^{\infty }\left\vert I_{j}\left( Q\right) \right\vert _{\sigma
}\leq \beta \left\vert Q\right\vert _{\sigma }\ ,\text{ where }0<\beta =1
\frac{1}{8C}<1.
\end{eqnarray*}
Now we begin the first step of the construction of a new family $\left\{
\widetilde{b}_{Q}\right\} _{Q\in \mathcal{D}}$ that satisfies both the
accretivity conditions and the testing conditions, as well as the pointwise
lower bound condition. We start by defining for $\mathbf{\epsilon }=\left\{
\epsilon _{j}\right\} _{j=1}^{\infty }$
\begin{equation*}
\widetilde{b}_{Q}^{\mathbf{\epsilon }}\left( x\right) \equiv b_{Q}\left(
x\right) +\sum_{j=1}^{\infty }\epsilon _{j}b_{I_{j}\left( Q\right) }\left(
x\right) \ ,
\end{equation*
where $\epsilon _{j}\in \left\{ -1,1\right\} $ for all $j\geq 1$.
We first note that we can assume that the collection of intervals $\left\{
I_{i}\left( Q\right) \right\} _{i}$ is subordinate to the collection of
children of $Q$, i.e. $I_{i}\left( Q\right) \subset Q^{\prime }$ for some
Q^{\prime }\in \mathfrak{C}\left( Q\right) $ depending on $i$. Then we have
for each $Q^{\prime }\in \mathfrak{C}\left( Q\right) $ tha
\begin{eqnarray*}
\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}^{\mathbf{\epsilon
}\right\vert ^{p}d\sigma &=&\int_{Q^{\prime }}\left\vert
b_{Q}+\sum_{j=1}^{\infty }\epsilon _{j}b_{I_{j}}\right\vert ^{p}d\sigma \\
&\leq &2^{p}\left\{ \int_{Q^{\prime }}\left\vert b_{Q}\right\vert
^{2p}d\sigma +\sum_{j:\ I_{j}\subset Q^{\prime }}\int \left\vert
b_{I_{j}}\right\vert ^{p}d\sigma \right\} \\
&\leq &C_{p}\left\{ \left\vert Q^{\prime }\right\vert _{\sigma }+\sum_{j:\
I_{j}\subset Q^{\prime }}\left\vert I_{j}\right\vert _{\sigma }\right\} \leq
C_{p}\left( 1+\beta \right) \left\vert Q^{\prime }\right\vert _{\sigma }\ .
\end{eqnarray*}
Now with $\mathbb{E}$ denoting expectation with respect to the uniform
probability measure on $\Omega _{\infty }\equiv \left\{ -1,1\right\} ^
\mathbb{N}}$, we hav
\begin{equation*}
\mathbb{E}\int \widetilde{b}_{Q}^{\mathbf{\epsilon }}d\sigma =\int
b_{Q}d\sigma \geq 1>0\ ,
\end{equation*
an
\begin{eqnarray*}
&&\mathbb{E}\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q
\widetilde{b}_{Q}^{\mathbf{\epsilon }}\right) \right\vert ^{2}d\omega
\mathbb{E}\int \left\vert T_{\sigma }^{\alpha }\left(
b_{Q}+\sum_{j=1}^{\infty }\epsilon _{j}b_{I_{j}}\right) \right\vert
^{2}d\omega \\
&=&\mathbb{E}\int \left\{ \left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert
^{2}+2\func{Re}\sum_{j=1}^{\infty }\epsilon _{j}\left( T_{\sigma }^{\alpha
}b_{Q}\right) \overline{\left( T_{\sigma }^{\alpha }b_{I_{j}}\right)
+\sum_{j,k=1}^{\infty }\epsilon _{j}\epsilon _{k}\left( T_{\sigma }^{\alpha
}b_{I_{j}}\right) \overline{\left( T_{\sigma }^{\alpha }b_{I_{k}}\right)
\right\} d\omega \\
&\leq &\int \left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega
+\sum_{j=1}^{\infty }\int \left\vert T_{\sigma }^{\alpha
}b_{I_{j}}\right\vert ^{2}d\omega \\
&\leq &\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}\left\vert Q\right\vert
_{\sigma }+\sum_{j=1}^{\infty }\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b
}\left\vert I_{i}\left( Q\right) \right\vert _{\sigma }\leq \mathfrak{F
_{T^{\alpha }}^{\mathbf{b}}\left[ 1+\beta \right] \left\vert Q\right\vert
_{\sigma }\ .
\end{eqnarray*}
So altogether, at this point in the first step of the construction, we have
for each pair $\left( Q,E\left( Q\right) \right) $ consisting of an interval
$Q$ and a subset $E\left( Q\right) $ having measure at most $\beta
\left\vert Q\right\vert _{\sigma }$
\begin{eqnarray*}
\left\vert E\left( Q\right) \right\vert _{\sigma } &\leq &\beta \left\vert
Q\right\vert _{\sigma }\ , \\
\mathbb{E}\int \widetilde{b}_{Q}^{\mathbf{\epsilon }}d\sigma &=&\int
b_{Q}d\sigma \geq \left\vert Q\right\vert _{\sigma }>0\ , \\
\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}^{\mathbf{\epsilon
}\right\vert ^{p}d\sigma &\leq &C_{p}\left( 1+\beta \right) \left\vert
Q^{\prime }\right\vert _{\sigma }\ ,\ \ \ \ \ Q^{\prime }\in \mathfrak{C
\left( Q\right) \ , \\
\mathbb{E}\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q
\widetilde{b}_{Q}^{\mathbf{\epsilon }}\right) \right\vert ^{2}d\omega &\leq
\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}\left[ 1+\beta \right] \left\vert
Q\right\vert _{\sigma }\ .
\end{eqnarray*}
Now we choose a positive constant $A$ large enough so that with
probabilities $\frac{1}{2}$ for $\int \widetilde{b}_{Q}^{\mathbf{\epsilon
}d\sigma $ and $\frac{3}{4}$ for $\int_{Q}\left\vert T_{\sigma }^{\alpha
}\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^{\mathbf{\epsilon }}\right)
\right\vert ^{2}d\omega $ (note that $\left( 1-\frac{1}{2}\right) +\left( 1
\frac{3}{4}\right) =\frac{3}{4}<1$), there exists $\mathbf{\epsilon }\in
\Omega _{\infty }$ so that $\widetilde{b}_{Q}^{1}\equiv \widetilde{b}_{Q}^
\mathbf{\epsilon }}$ satisfie
\begin{eqnarray}
\int \widetilde{b}_{Q}^{1}d\sigma &\geq &\left\vert Q\right\vert _{\sigma
}>0\ , \label{exists} \\
\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}^{1}\right\vert ^{p}d\sigma
&\leq &AC_{p}\left( 1+\beta \right) \left\vert Q^{\prime }\right\vert
_{\sigma }\ ,\ \ \ \ \ Q^{\prime }\in \mathfrak{C}\left( Q\right) \ , \notag
\\
\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b
_{Q}^{1}\right) \right\vert ^{2}d\omega &\leq &A\mathfrak{FT}_{T^{\alpha }}^
\mathbf{b}}\left[ 1+\beta \right] \beta \left\vert Q\right\vert _{\sigma }\ .
\notag
\end{eqnarray}
To see how big $A$ must be taken to achieve (\ref{exists}), we use
Chebyshev's inequality as follows. Take $N$ large, se
\begin{equation*}
\widetilde{b}_{Q}^{\mathbf{\epsilon },N}\left( x\right) \equiv b_{Q}\left(
x\right) +\sum_{j=1}^{N}\epsilon _{j}b_{I_{j}\left( Q\right) }\left(
x\right) ,
\end{equation*
and equip $\Omega _{N}=\left\{ -1,1\right\} ^{N}$ with the uniform
probability measure that assigns mass $\frac{1}{2^{N}}$ to each $\mathbf
\epsilon }\in \Omega _{N}$. Then for each child $Q^{\prime }$ of $Q$ we have
as above tha
\begin{equation*}
\frac{1}{\#\Omega _{N}}\sum_{\mathbf{\epsilon }\in \Omega _{N}}\int
\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^
\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega =\mathbb{E}\int
\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^
\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega \leq C\left( 1+\beta
\right) \left\vert Q\right\vert _{\sigma }\ ,
\end{equation*
which by Chebyshev's inequality implies tha
\begin{eqnarray*}
&&AC\left( 1+\beta \right) \left\vert Q\right\vert _{\sigma }\ \#\left\{
\mathbf{\epsilon }\in \Omega _{N}:\int \left\vert T_{\sigma }^{\alpha
}\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^{\mathbf{\epsilon },N}\right)
\right\vert ^{2}d\omega >AC\left( 1+\beta \right) \left\vert Q\right\vert
_{\sigma }\right\} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq \sum_{\mathbf{\epsilon }\in \Omega
_{N}}\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b
_{Q}^{\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega \leq \#\Omega
_{N}\ C\left( 1+\beta \right) \left\vert Q\right\vert _{\sigma }\ ,
\end{eqnarray*
which gives in turn tha
\begin{equation*}
\frac{1}{\#\Omega _{N}}\#\left\{ \mathbf{\epsilon }\in \Omega _{N}:\int
\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^
\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega >AC\left( 1+\beta
\right) \left\vert Q\right\vert _{\sigma }\right\} \leq \frac{1}{A}.
\end{equation*
This of course just says that the probability that $\int \left\vert
T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^{\mathbf
\epsilon },N}\right) \right\vert ^{2}d\omega $ exceeds $A$ times $C\left(
1+\beta \right) \left\vert Q\right\vert _{\sigma }$ is less than $\frac{1}{A}
$. So in order to achieve that this latter probability is at most $\frac{1}{
}$, we take $A\geq 4$. Then we get tha
\begin{equation*}
\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^
\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega \leq AC\left( 1+\beta
\right) \left\vert Q\right\vert _{\sigma }\ ,\ \ \ \ \ Q^{\prime }\in
\mathfrak{C}\left( Q\right) \ ,
\end{equation*
holds for a set of $\mathbf{\epsilon }$ in $\Omega _{N}$ of probability at
least $1-\frac{1}{4}=\frac{3}{4}$.
So altogether, provided that we take $A=4$, we obtain that all inequalities,
\begin{eqnarray*}
\int \widetilde{b}_{Q}^{\mathbf{\epsilon },N}d\sigma &\geq &\left\vert
Q\right\vert _{\sigma }>0\ , \\
\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}^{\mathbf{\epsilon
,N}\right\vert ^{p}d\sigma &\leq &C_{p}\left( 1+\beta \right) \left\vert
Q^{\prime }\right\vert _{\sigma }\ ,\ \ \ \ \ Q^{\prime }\in \mathfrak{C
\left( Q\right) \ , \\
\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b}_{Q}^
\mathbf{\epsilon },N}\right) \right\vert ^{2}d\omega &\leq &A\mathfrak{FT
_{T^{\alpha }}^{\mathbf{b}}\left[ 1+\beta \right] \beta \left\vert
Q\right\vert _{\sigma }\ ,
\end{eqnarray*
hold simultaneously for a set of $\mathbf{\epsilon }$ in $\Omega _{N}$ of
probability at leas
\begin{equation*}
1-\left\{ \left( 1-\frac{1}{2}\right) +\left( 1-\frac{3}{4}\right) \right\}
=1-\left\{ \frac{1}{2}+\frac{1}{4}\right\} =\frac{1}{4}.
\end{equation*
Since these estimates are independent of $N$, we can let $N\rightarrow
\infty $ to obtain that there is at least one $\mathbf{\epsilon }\in \Omega
_{\infty }$ for which (\ref{exists}) holds with $\widetilde{b}_{Q}^{1}\equiv
\widetilde{b}_{Q}^{\mathbf{\epsilon },N}$.
We also have $\left\vert \widetilde{b}_{Q}^{1}\right\vert \geq \frac{3}{4}$
in $Q$ except on the exceptional se
\begin{equation*}
G\equiv \overset{\cdot }{\dbigcup }_{j=1}^{\infty }E\left( I_{j}^{Q}\right)
\end{equation*
whose $\sigma $-measure satisfies
\begin{equation*}
\left\vert G\right\vert _{\sigma }=\sum_{j=1}^{\infty }\left\vert E\left(
I_{j}^{Q}\right) \right\vert _{\sigma }\leq \sum_{j=1}^{\infty }\beta
\left\vert I_{j}^{Q}\right\vert _{\sigma }=\beta \left\vert E\left( Q\right)
\right\vert _{\sigma }\leq \beta ^{2}\left\vert Q\right\vert _{\sigma }\ .
\end{equation*
Now we consider the se
\begin{equation*}
F\left( I_{j}\left( Q\right) \right) \equiv \left\{ x\in E\left( I_{j}\left(
Q\right) \right) :\left\vert b_{Q}\left( x\right) +\epsilon
_{j}b_{I_{j}^{Q}}\left( x\right) \right\vert <\frac{1}{4}\right\} .
\end{equation*
We may assume tha
\begin{equation*}
F\left( I_{j}\left( Q\right) \right) =\overset{\cdot }{\dbigcup
_{k=1}^{\infty }I_{k}\left( I_{j}\left( Q\right) \right)
\end{equation*
where $\left\{ I_{k}\left( I_{j}\left( Q\right) \right) \right\}
_{k=1}^{\infty }$ is a pairwise disjoint collection of intervals for each $j
. Now we apply the above step to each pair $\left( I_{j}\left( Q\right)
,F\left( I_{j}\left( Q\right) \right) \right) $ consisting of an interval
I_{j}^{Q}$ and a subset $F\left( I_{j}^{Q}\right) $ having measure at most
\beta \left\vert I_{j}^{Q}\right\vert _{\sigma }$. Then arguing as above for
each such pair, and adding results, we obtain that there exists $\mathbf
\epsilon }_{j}$ so that $\widetilde{b}_{Q}^{2}\equiv \sum_{j=1}^{\infty
\widetilde{b^{1}}_{Q}^{\mathbf{\epsilon }_{j}}$ satisfie
\begin{eqnarray*}
\int \widetilde{b}_{Q}^{2}d\sigma &\geq &\left\vert Q\right\vert _{\sigma
}>0\ , \\
\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}^{2}\right\vert ^{p}d\sigma
&\leq &AC_{p}\left( 1+\beta +\beta ^{2}\right) \left\vert Q^{\prime
}\right\vert _{\sigma }\ ,\ \ \ \ \ Q^{\prime }\in \mathfrak{C}\left(
Q\right) \ , \\
\int \left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}\widetilde{b
_{Q}^{2}\right) \right\vert ^{2}d\omega &\leq &A\mathfrak{FT}_{T^{\alpha }}^
\mathbf{b}}\left[ 1+\beta +\beta ^{2}\right] \beta \left\vert Q\right\vert
_{\sigma }\ ,
\end{eqnarray*
as well as $\left\vert \widetilde{b}_{Q}^{2}\right\vert \geq \frac{3}{4}$ in
$Q$ except on the exceptional se
\begin{equation*}
G\equiv \overset{\cdot }{\dbigcup }_{j,k=1}^{\infty }E\left( I_{k}\left(
I_{j}\left( Q\right) \right) \right) ,
\end{equation*
whose $\sigma $-measure satisfies
\begin{eqnarray*}
\left\vert G\right\vert _{\sigma } &=&\sum_{j,k=1}^{\infty }\left\vert
E\left( I_{k}\left( I_{j}\left( Q\right) \right) \right) \right\vert
_{\sigma }\leq \sum_{j=1}^{\infty }\beta \left\vert E\left( I_{j}\left(
Q\right) \right) \right\vert _{\sigma } \\
&\leq &\sum_{j=1}^{\infty }\beta ^{2}\left\vert I_{j}^{Q}\right\vert
_{\sigma }=\beta ^{2}\left\vert E\left( Q\right) \right\vert _{\sigma }\leq
\beta ^{3}\left\vert Q\right\vert _{\sigma }\ .
\end{eqnarray*
Continuing in this way, we end up with the desired function $\widetilde{b
_{Q}\left( x\right) =\lim_{n\rightarrow \infty }\widetilde{b}_{Q}^{n}\left(
x\right) $, since $\beta <1$ implies that the collection of nested intervals
$\left\{ I_{j_{N}}\left( ...I_{j_{2}}\left( I_{j_{1}}\left( Q\right) \right)
...\right) \right\} $ satisfy a $\sigma $-Carleson condition. We emphasize
that for each interval $Q$, we then have $\left\vert \widetilde{b}_{Q}\left(
x\right) \right\vert \geq \frac{3}{4}$ for $\sigma $-a.e. $x\in Q$, as well
as $\int_{Q^{\prime }}\left\vert \widetilde{b}_{Q}\right\vert ^{2}d\sigma
\leq C^{\prime }\left\vert Q^{\prime }\right\vert _{\sigma }\ $for all
Q^{\prime }\in \mathfrak{C}\left( Q\right) $. The full $\widetilde{\mathbf{b
}$-testing condition constant that we have obtained satisfie
\begin{equation*}
\mathfrak{FT}_{T^{\alpha }}^{\widetilde{\mathbf{b}}}\left( \sigma ,\omega
\right) \leq C_{\alpha }\left( \mathfrak{FT}_{T^{\alpha }}^{\mathbf{b
}\left( \sigma ,\omega \right) +\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b
^{\ast },\ast }\left( \sigma ,\omega \right) +\sqrt{\mathcal{A}_{2}^{\alpha
}\left( \sigma ,\omega \right) }+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }\left(
\sigma ,\omega \right) }\right) ,
\end{equation*
and since the full testing constant is controlled by the testing constant
and the Muckenhoupt constant in (\ref{full proved})\ below, the proof is
complete.
\end{proof}
\subsection{Reduction to real bounded accretive families\label{str acc}}
Recall that a vector of `complex-valued testing functions' $\mathbf{b}\equiv
\left\{ b_{Q}\right\} _{Q\in \mathcal{D}}$ is a $p$-\emph{strongly }$\mu $
\emph{accretive} family if
\begin{equation*}
\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{P},
\end{equation*
\begin{equation}
1\leq \left\vert \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}b_{Q}d\mu
\right\vert \leq \left( \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}\left\vert b_{Q}\right\vert ^{p}d\mu \right) ^{\frac{1}{p}}\leq C_
\mathbf{b}}\left( p\right) <\infty ,\ \ \ \ \ Q\in \mathcal{P}\ ,
\label{acc}
\end{equation
and if $\mathbf{b}$ satisfies the $PLBP$, i.e.
\begin{equation*}
\left\vert b_{Q}\left( x\right) \right\vert \geq c_{1}>0\ \ \ \ \ \text{for
\ Q\in \mathcal{D}\text{ and }\sigma -a.e.x\in \mathbb{R}^{n}.
\end{equation*
We begin by noting that if $b_{Q}$ satisfies (\ref{acc}) with $\mu =\sigma
, and satisfies a given $\mathbf{b}$-testing condition for a weight pair
\left( \sigma ,\omega \right) $, then $\func{Re}b_{Q}$ satisfies $\left(
\frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert \func{Re
b_{Q}\right\vert ^{p}d\mu \right) ^{\frac{1}{p}}\leq C_{\mathbf{b}}\left(
p\right) $ and the given $\mathbf{b}$-testing condition for $\left( \sigma
,\omega \right) $ with $\func{Re}b_{Q}$ in place of $b_{Q}$.
\begin{conclusion}
We may assume throughout the proof of Theorem \ref{dim one} that our $p
-weakly\emph{\ }$\mu $-accretive families $\mathbf{b}\equiv \left\{
b_{Q}\right\} _{Q\in \mathcal{D}}$ and $\mathbf{b}^{\ast }\equiv \left\{
b_{Q}^{\ast }\right\} _{Q\in \mathcal{G}}$ consist of \textbf{real-valued}
functions that in addition satisfy the $PLBP$ and $1\leq \frac{1}{\left\vert
Q\right\vert _{\sigma }}\int_{Q}b_{Q}d\sigma $ and $1\leq \frac{1}
\left\vert Q\right\vert _{\sigma }}\int_{Q}b_{Q}^{\ast }d\sigma $.
\end{conclusion}
Next we show that the assumption of testing conditions for a Calder\'{o
n-Zygmund operator $T$ and $p$-strongly $\mu $-accretive testing functions
\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ and $\mathbf{b}^{\ast
}=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$ with $p>2$ can always
be replaced with real-valued $\infty $-strongly $\mu $-accretive testing
functions, thus reducing the $Tb$ theorem for the case $p>2$ to the case
when $p=\infty $ and the $PLBP$ (\ref{plb}) holds. We now proceed to develop
a precise statement. We extend (\ref{acc}) to $2\leq p\leq \infty $ b
\begin{eqnarray}
&&\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{P},
\label{acc infinity} \\
1 &\leq &\left\vert \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}b_{Q}d\mu \right\vert \leq \left\{
\begin{array}{cc}
\left( \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert
b_{Q}\right\vert ^{p}d\mu \right) ^{\frac{1}{p}}\leq C_{\mathbf{b}}\left(
p\right) <\infty & \text{for }2\leq p<\infty \\
\left\Vert b_{Q}\right\Vert _{L^{\infty }\left( \mu \right) }\leq C_{\mathbf
b}}\left( \infty \right) <\infty & \text{for }p=\inft
\end{array
\right. ,\ \ \ \ \ Q\in \mathcal{P}\ . \notag
\end{eqnarray}
\begin{proposition}
\label{conditional}Let $0\leq \alpha <1$, and let $\sigma $ and $\omega $ be
locally finite positive Borel measures on the real line $\mathbb{R}$, and
let $T^{\alpha }$ be a standard $\alpha $-fractional elliptic and gradient
elliptic singular integral operator on $\mathbb{R}$. Set $T_{\sigma
}^{\alpha }f=T^{\alpha }\left( f\sigma \right) $ for any smooth truncation
of $T_{\sigma }^{\alpha }$, so that $T_{\sigma }^{\alpha }$ is \emph{apriori}
bounded from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega \right) $.
Finally, define the sequence of positive extended real number
\begin{equation*}
\left\{ p_{n}\right\} _{n=0}^{\infty }=\left\{ \frac{2}{1-\left( \frac{2}{3
\right) ^{n}}\right\} _{n=0}^{\infty }=\left\{ \infty ,6,\frac{18}{5},\frac
162}{65},...\right\} .
\end{equation*
Suppose that the following statement is true:\medskip
\begin{description}
\item[$\left( \mathcal{S}_{\infty }\right) $] If $\mathbf{b}=\left\{
b_{Q}\right\} _{Q\in \mathcal{P}}$ is an $\infty $-strongly $\sigma
-accretive family of functions on $\mathbb{R}$, and if $\mathbf{b}^{\ast
}=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$ is an $\infty
-strongly $\omega $-accretive family of functions on $\mathbb{R}$, then the
operator norm $\mathfrak{N}_{T_{\sigma }^{\alpha }}$ of $T_{\sigma }^{\alpha
}$ from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega \right) $, i.e.
the best constant i
\begin{equation*}
\left\Vert T_{\sigma }^{\alpha }f\right\Vert _{L^{2}\left( \omega \right)
}\leq \mathfrak{N}_{T_{\sigma }^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) },\ \ \ \ \ f\in L^{2}\left( \sigma \right) ,
\end{equation*
\textbf{uniformly} in smooth truncations of $T^{\alpha }$, satisfie
\begin{equation*}
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha }}^{\mathbf{
}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\lesssim \mathfrak{N}_{T^{\alpha }}\lesssim \left( C_{\mathbf{b}}\left(
\infty \right) +C_{\mathbf{b}^{\ast }}\left( \infty \right) \right) \left(
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha }}^{\mathbf{
}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right) \ ,
\end{equation*
where $C_{\mathbf{b}}\left( \infty \right) ,C_{\mathbf{b}^{\ast }}\left(
\infty \right) $ are the accretivity constants in (\ref{acc infinity}), and
the constants implied by $\lesssim $ depend on $\alpha $ and the Calder\'{o
n-Zygmund constant $C_{CZ}$ in (\ref{sizeandsmoothness'}).\\[0.15cm]
Then for each $n\geq 0$, the following statements hold:\medskip
\item[$\left( \mathcal{S}_{n}\right) $] Let $p\in \left( p_{n+1},p_{n}\right]
$. If $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ is a $p
-strongly $\sigma $-accretive family of functions on $\mathbb{R}$, and if
\mathbf{b}^{\ast }=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$ is a
p $-strongly $\omega $-accretive family of functions on $\mathbb{R}$, then
the operator norm $\mathfrak{N}_{T_{\sigma }^{\alpha }}$ of $T_{\sigma
}^{\alpha }$ from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega
\right) $, i.e. the best constant i
\begin{equation*}
\left\Vert T_{\sigma }^{\alpha }f\right\Vert _{L^{2}\left( \omega \right)
}\leq \mathfrak{N}_{T_{\sigma }^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) },\ \ \ \ \ f\in L^{2}\left( \sigma \right) ,
\end{equation*
\textbf{uniformly} in smooth truncations of $T^{\alpha }$, satisfie
\begin{equation*}
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha }}^{\mathbf{
}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\lesssim \mathfrak{N}_{T^{\alpha }}\lesssim \left( C_{\mathbf{b}}\left(
p\right) +C_{\mathbf{b}^{\ast }}\left( p\right) \right) ^{3^{n+1}}\left(
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha }}^{\mathbf{
}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right) \ ,
\end{equation*
where $C_{\mathbf{b}}\left( p\right) ,C_{\mathbf{b}^{\ast }}\left( p\right) $
are the accretivity constants in (\ref{acc}), and the constants implied by
\lesssim $ depend on $p$, $\alpha $, and the Calder\'{o}n-Zygmund constant
C_{CZ}$ in (\ref{sizeandsmoothness'}).
\end{description}
\end{proposition}
\begin{proof}[Proof of Proposition \protect\ref{conditional}]
We first prove $\left( \mathcal{S}_{0}\right) $. So fix $p\in \left(
p_{1},p_{0}\right) =\left( 6,\infty \right) $, and let $\mathbf{b}=\left\{
b_{Q}\right\} _{Q\in \mathcal{P}}$ be a $p$-weakly $\sigma $-accretive
family of functions on $\mathbb{R}$, and let $\mathbf{b}^{\ast }=\left\{
b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$ be a $p$-weakly $\omega
-accretive family of functions on $\mathbb{R}$. Let $0<\varepsilon <1$ (to
be chosen differently at various points in the argument below) and defin
\begin{equation}
\lambda =\lambda \left( \varepsilon \right) =\left( \frac{p}{p-2}C_{\mathbf{
}}\left( p\right) ^{p}\frac{1}{\varepsilon }\right) ^{\frac{1}{p-2}},
\label{lambda choice}
\end{equation
and a new collection of test functions
\begin{equation}
\widehat{b}_{Q}\equiv 2b_{Q}\left( \mathbf{1}_{\left\{ \left\vert
b_{Q}\right\vert \leq \lambda \right\} }+\frac{\lambda }{\left\vert
b_{Q}\right\vert }\mathbf{1}_{\left\{ \left\vert b_{Q}\right\vert >\lambda
\right\} }\right) ,\ \ \ \ \ Q\in \mathcal{P}, \label{new}
\end{equation
which continue to satisfy the $PLBP$ (\ref{plb}). We comput
\begin{eqnarray*}
\int_{\left\{ \left\vert b_{Q}\right\vert >\lambda \right\} }\left\vert
b_{Q}\right\vert ^{2}d\sigma &=&\int_{\left\{ \left\vert b_{Q}\right\vert
>\lambda \right\} }\left[ \int_{0}^{\left\vert b_{Q}\right\vert }2tdt\right]
d\sigma \\
&=&\int \int_{\left\{ \left( x,t\right) \in \mathbb{R}\times \left( 0,\infty
\right) :\max \left\{ t,\lambda \right\} <\left\vert b_{Q}\left( x\right)
\right\vert \right\} }2tdtd\sigma \left( x\right) \\
&=&\int_{\left[ 0,\lambda \right] }\int_{\left\{ x\in \mathbb{R}:\lambda
<\left\vert b_{Q}\left( x\right) \right\vert \right\} }d\sigma \left(
x\right) 2tdt+\int_{\left( \lambda ,\infty \right) }\int_{\left\{ x\in
\mathbb{R}:t<\left\vert b_{Q}\left( x\right) \right\vert \right\} }d\sigma
\left( x\right) 2tdt \\
&=&\lambda ^{2}\left\vert \left\{ \left\vert b_{Q}\right\vert >\lambda
\right\} \right\vert _{\sigma }+\int_{\lambda }^{\infty }\left\vert \left\{
\left\vert b_{Q}\right\vert >t\right\} \right\vert _{\sigma }2tdt,
\end{eqnarray*
and henc
\begin{eqnarray}
\int_{\left\{ \left\vert b_{Q}\right\vert >\lambda \right\} }\left\vert
b_{Q}\right\vert ^{2}d\sigma &\leq &\lambda ^{2}\frac{1}{\lambda ^{p}}\left(
\int \left\vert b_{Q}\right\vert ^{p}d\sigma \right) +\int_{\lambda
}^{\infty }\frac{1}{t^{p}}\left( \int \left\vert b_{Q}\right\vert
^{p}d\sigma \right) 2tdt \label{hence} \\
&=&\left\{ \lambda ^{2-p}+\int_{\lambda }^{\infty }2t^{1-p}dt\right\} C_
\mathbf{b}}\left( p\right) ^{p}\left\vert Q\right\vert _{\sigma } \notag \\
&=&\frac{p}{p-2}\lambda ^{2-p}C_{\mathbf{b}}\left( p\right) ^{p}\left\vert
Q\right\vert _{\sigma }=\varepsilon \left\vert Q\right\vert _{\sigma }\ ,
\notag
\end{eqnarray
by (\ref{lambda choice}). Thus we have the lower bound
\begin{eqnarray}
\left\vert \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widehat{b
_{Q}d\sigma \right\vert &=&2\left\vert \frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}b_{Q}d\sigma -\frac{1}{\left\vert Q\right\vert _{\sigma }
\int_{Q}b_{Q}\left( \frac{\lambda }{\left\vert b_{Q}\right\vert }-1\right)
\mathbf{1}_{\left\{ \left\vert b_{Q}\right\vert >\lambda \right\} }d\sigma
\right\vert \label{low} \\
&\geq &2\left\vert \frac{1}{\left\vert Q\right\vert _{\sigma }
\int_{Q}b_{Q}d\mu \right\vert -2\left( \frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\left\vert b_{Q}\right\vert ^{2}\mathbf{1}_{\left\{
\left\vert b_{Q}\right\vert >\lambda \right\} }d\sigma \right) ^{\frac{1}{2}}
\notag \\
&\geq &2-2\left( \frac{1}{\left\vert Q\right\vert _{\sigma }}\varepsilon
\left\vert Q\right\vert _{\sigma }\right) ^{\frac{1}{2}}=2-2\sqrt
\varepsilon }\geq 1>0,\ \ \ \ \ Q\in \mathcal{P}\ , \notag
\end{eqnarray
if we choose $0<\varepsilon \leq \frac{1}{4}$. For an upper bound we have
\begin{equation*}
\left\Vert \widehat{b}_{Q}\right\Vert _{L^{\infty }\left( \sigma \right)
}\leq 2\lambda =2\lambda \left( \varepsilon \right) =2\left( \frac{p}{p-2}C_
\mathbf{b}}\left( p\right) ^{p}\frac{1}{\varepsilon }\right) ^{\frac{1}{p-2
},
\end{equation*
which altogether shows tha
\begin{eqnarray*}
C_{\widehat{\mathbf{b}}}\left( \infty \right) &\leq &2\left( \frac{p}{p-2}C_
\mathbf{b}}\left( p\right) ^{p}\frac{1}{\varepsilon }\right) ^{\frac{1}{p-2
}=2\left( \frac{p}{p-2}\right) ^{\frac{1}{p-2}}C_{\mathbf{b}}\left( p\right)
^{\frac{p}{p-2}}\varepsilon ^{-\frac{1}{p-2}}, \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for }0<\varepsilon \leq
\frac{1}{4}.
\end{eqnarray*
Similarly we hav
\begin{eqnarray*}
C_{\widehat{\mathbf{b}}^{\ast }}\left( \infty \right) &\leq &2\left( \frac{
}{p-2}C_{\mathbf{b}^{\ast }}\left( p\right) ^{p}\frac{1}{\varepsilon ^{\ast
}\right) ^{\frac{1}{p-2}}=2\left( \frac{p}{p-2}\right) ^{\frac{1}{p-2}}C_
\mathbf{b}^{\ast }}\left( p\right) ^{\frac{p}{p-2}}\left( \varepsilon ^{\ast
}\right) ^{-\frac{1}{p-2}} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{for }0<\varepsilon ^{\ast
}\leq \frac{1}{4}.
\end{eqnarray*}
Moreover, we also hav
\begin{eqnarray*}
\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }\widehat{b}_{Q}\right\vert
^{2}d\omega } &\leq &2\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha
}b_{Q}\right\vert ^{2}d\omega }+2\sqrt{\int_{Q}\left\vert T_{\sigma
}^{\alpha }\mathbf{1}_{\left\{ \left\vert b_{Q}\right\vert >\lambda \right\}
}\left( \frac{\lambda }{\left\vert b_{Q}\right\vert }-1\right)
b_{Q}\right\vert ^{2}d\omega } \\
&\leq &2\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\sqrt{\left\vert Q\right\vert
_{\sigma }}+2\mathfrak{N}_{T^{\alpha }}\sqrt{\int_{\left\{ \left\vert
b_{Q}\right\vert >\lambda \right\} }\left\vert b_{Q}\right\vert ^{2}d\sigma }
\\
&=&2\left\{ \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\sqrt{\varepsilon
\mathfrak{N}_{T^{\alpha }}\right\} \sqrt{\left\vert Q\right\vert _{\sigma }
\ ,\ \ \ \ \ \text{for all intervals }Q,
\end{eqnarray*
which shows tha
\begin{equation}
\mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}}}\leq 2\mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+2\sqrt{\varepsilon }\mathfrak{N}_{T^{\alpha }}\ .
\label{test}
\end{equation}
Now we take $\varepsilon =\varepsilon ^{\ast }$ and apply the fact that
\left( \mathcal{S}_{\infty }\right) $ holds to obtai
\begin{eqnarray*}
\mathfrak{N}_{T^{\alpha }} &\lesssim &\left( C_{\widehat{\mathbf{b}}}\left(
\infty \right) +C_{\widehat{\mathbf{b}}^{\ast }}\left( \infty \right)
\right) \left\{ \mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}}}+\mathfrak{
}_{T^{\alpha ,\ast }}^{\widehat{\mathbf{b}}^{\ast }}+\sqrt{\mathfrak{A
_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\frac{p}{p-2}}\varepsilon ^{-\frac{1}{p-2
}\left\{ \left[ \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\sqrt{\varepsilon
\mathfrak{N}_{T^{\alpha }}\right] +\left[ \mathfrak{T}_{T^{\alpha ,\ast }}^
\mathbf{b}^{\ast }}+\sqrt{\varepsilon }\mathfrak{N}_{T^{\alpha }}\right]
\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\frac{p}{p-2}}\varepsilon ^{-\frac{1}{p-2
}\left\{ \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha
,\ast }}^{\mathbf{b}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E
_{2}^{\alpha }\right\} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\left( C_{\mathbf{b}}\left(
p\right) +C_{\mathbf{b}^{\ast }}\left( p\right) \right) ^{\frac{p}{p-2
}\varepsilon ^{\frac{1}{2}-\frac{1}{p-2}}\mathfrak{N}_{T^{\alpha }}\ .
\end{eqnarray*
Now we choose
\begin{equation*}
\varepsilon =\frac{1}{\Gamma }\left( C_{\mathbf{b}}\left( p\right) +C_
\mathbf{b}^{\ast }}\left( p\right) \right) ^{-\frac{\frac{p}{p-2}}{\frac{1}{
}-\frac{1}{p-2}}}
\end{equation*
with $\Gamma $ large enough, depending only on the implied constant $C_
\limfunc{implied}}$ (where $\lesssim $ is written $\leq C_{\limfunc{implied
} $), so that the final term on the right satisfie
\begin{eqnarray*}
&&C_{\limfunc{implied}}\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b
^{\ast }}\left( p\right) \right) ^{\frac{p}{p-2}}\ \varepsilon ^{\frac{1}{2}
\frac{1}{p-2}}\ \mathfrak{N}_{T^{\alpha }} \\
&=&C_{\limfunc{implied}}\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b
^{\ast }}\left( p\right) \right) ^{\frac{p}{p-2}}\ \left( \frac{1}{\Gamma
\right) ^{\frac{1}{2}-\frac{1}{p-2}}\left( C_{\mathbf{b}}\left( p\right) +C_
\mathbf{b}^{\ast }}\left( p\right) \right) ^{-\frac{p}{p-2}}\ \mathfrak{N
_{T^{\alpha }} \\
&\leq &C_{\limfunc{implied}}\ \left( \frac{1}{\Gamma }\right) ^{\frac{1}{4
}\ \mathfrak{N}_{T^{\alpha }}=\frac{1}{2}\mathfrak{N}_{T^{\alpha }}\ ,
\end{eqnarray*
i.e, we choose $\Gamma =\left( 2C_{\limfunc{implied}}\right) ^{4}$, and
where we have used $\frac{1}{2}-\frac{1}{p-2}\geq \frac{1}{4}$ for $p>6$.
This term can then be absorbed into the left hand side to obtai
\begin{eqnarray*}
\mathfrak{N}_{T^{\alpha }} &\lesssim &\left( C_{\mathbf{b}}\left( p\right)
+C_{\mathbf{b}^{\ast }}\left( p\right) \right) ^{\frac{p}{p-2}}\left( \left(
C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast }}\left( p\right) \right)
^{-\frac{\frac{p}{p-2}}{\frac{1}{2}-\frac{1}{p-2}}}\right) ^{-\frac{1}{p-2}}
\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \left\{ \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b
^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\frac{p}{p-2}\left\{ 1+\frac{\frac{1}{p-2}}
\frac{1}{2}-\frac{1}{p-2}}\right\} }\left\{ \mathfrak{T}_{T^{\alpha }}^
\mathbf{b}}+\mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}+\sqrt
\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\right\}
\end{eqnarray*
Since
\begin{equation*}
\frac{p}{p-2}\left\{ 1+\frac{\frac{1}{p-2}}{\frac{1}{2}-\frac{1}{p-2}
\right\} =\left( 1+\frac{2}{p-2}\right) \left( 1+\frac{2}{p-4}\right) \leq
\text{ for }p>6,
\end{equation*
we ge
\begin{equation*}
\mathfrak{N}_{T^{\alpha }}\lesssim \left( C_{\mathbf{b}}\left( p\right) +C_
\mathbf{b}^{\ast }}\left( p\right) \right) ^{3}\left\{ \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b
^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right\} ,
\end{equation*
which completes the proof of $\left( \mathcal{S}_{0}\right) $.
Suppose now, in order to derive a contradiction, that $\left( \mathcal{S
_{n}\right) $ fails for some $n\geq 1$. Then there exists an integer $n\geq
1 $ and an exponent $r\in \left( p_{n+1},p_{n}\right] $ such tha
\begin{equation*}
\left( \mathcal{S}_{r}\right) \text{ fails and }\left( \mathcal{S
_{q}\right) \text{ holds\ for }q>p_{n}.
\end{equation*
We now show that $\left( \mathcal{S}_{p}\right) $ holds for all $p\in \left(
p_{n+1},p_{n}\right] $, contradicting the fact that $\left( \mathcal{S
_{r}\right) $ fails. So fix $n\geq 1$, $p\in \left( p_{n+1},p_{n}\right] $,
and suppose that $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ is a
$p$-weakly $\sigma $-accretive family of functions on $\mathbb{R}$, and that
$\mathbf{b}^{\ast }=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$ is a
p$-weakly $\omega $-accretive family of functions on $\mathbb{R}$. Note that
the sequence $\left\{ p_{n}\right\} _{n=0}^{\infty }=\left\{ \frac{2}
1-\left( \frac{2}{3}\right) ^{n}}\right\} _{n=0}^{\infty }$ satisfies the
recursion relatio
\begin{equation*}
p_{n+1}=\frac{6}{1+\frac{4}{p_{n}}},\text{ equivalently }p_{n}=\frac{4}
\frac{6}{p_{n+1}}-1},\text{\ \ \ \ \ }n\geq 0.
\end{equation*
Choose $q\in \left( p_{n},p_{n-1}\right] $ so tha
\begin{equation}
p>\frac{6}{1+\frac{4}{q}},\text{ i.e. }q<\frac{4}{\frac{6}{p}-1},
\label{restrict}
\end{equation
which can be done since $p>p_{n+1}=\frac{2}{1-\left( \frac{2}{3}\right)
^{n+1}}$ is equivalent to $p_{n}=\frac{2}{1-\left( \frac{2}{3}\right) ^{n}}
\frac{4}{\frac{6}{p}-1}$, which leaves room to choose $q$ satisfying
p_{n}<q<\frac{4}{\frac{6}{p}-1}$.
Now let $0<\varepsilon <1$ (to be fixed later), define $\lambda =\lambda
\left( \varepsilon \right) $ as in (\ref{lambda choice}), and define
\widehat{b}_{Q}$ as in (\ref{new}). Recall from (\ref{hence}) and (\ref{low
) that we then hav
\begin{equation*}
\int_{\left\{ \left\vert b_{Q}\right\vert >\lambda \right\} }\left\vert
b_{Q}\right\vert ^{2}d\sigma \leq \varepsilon \left\vert Q\right\vert
_{\sigma }\ ,
\end{equation*
an
\begin{equation*}
\left\vert \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widehat{b
_{Q}d\sigma \right\vert \geq 1,\ \ \ \ \ Q\in \mathcal{P}\ ,
\end{equation*
if we choose $0<\varepsilon \leq \frac{1}{4}$. We of course have the
previous upper bound
\begin{equation*}
\left\Vert \widehat{b}_{Q}\right\Vert _{L^{\infty }\left( \sigma \right)
}\leq 2\lambda =2\lambda \left( \varepsilon \right) =2\left( \frac{p}{p-2}C_
\mathbf{b}}\left( p\right) ^{p}\frac{1}{\varepsilon }\right) ^{\frac{1}{p-2
},
\end{equation*
and while this turned out to be sufficient in the case $n=0$, we must do
better than $O\left( \frac{1}{\varepsilon }\right) ^{\frac{1}{p-2}}$ in the
case $n\geq 1$. In fact we compute the $L^{q}$ norm instead, recalling that
q>p$
\begin{eqnarray*}
&&\left( \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert
\widehat{b}_{Q}\right\vert ^{q}d\mu \right) ^{\frac{1}{q}}=2\left( \frac{1}
\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert b_{Q}\left( \mathbf{1
_{\left\{ \left\vert b_{Q}\right\vert \leq \lambda \right\} }+\frac{\lambda
}{\left\vert b_{Q}\right\vert }\mathbf{1}_{\left\{ \left\vert
b_{Q}\right\vert >\lambda \right\} }\right) \right\vert ^{q}d\mu \right) ^
\frac{1}{q}} \\
&=&2\left( \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{\left\{ \left\vert
b_{Q}\right\vert \leq \lambda \right\} }\left[ \int_{0}^{\left\vert
b_{Q}\right\vert }qt^{q-1}dt\right] d\sigma +\frac{\lambda ^{q}\left\vert
\left\{ \left\vert b_{Q}\right\vert >\lambda \right\} \right\vert _{\mu }}
\left\vert Q\right\vert _{\mu }}\right) ^{\frac{1}{q}} \\
&\leq &2\left( \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{0}^{\lambda
\left[ \int_{\left\{ t<\left\vert b_{Q}\right\vert \leq \lambda \right\}
}d\sigma \right] qt^{q-1}dt+C_{\mathbf{b}}\left( p\right) ^{p}\lambda
^{q-p}\right) ^{\frac{1}{q}} \\
&\leq &2\left( \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{0}^{\lambda
\left[ \frac{1}{t^{p}}\int \left\vert b_{Q}\right\vert ^{p}d\sigma \right]
qt^{q-1}dt+C_{\mathbf{b}}\left( p\right) ^{p}\lambda ^{q-p}\right) ^{\frac{
}{q}} \\
&\leq &2C_{\mathbf{b}}\left( p\right) ^{\frac{p}{q}}\left( \int_{0}^{\lambda
}qt^{q-p-1}dt+\lambda ^{q-p}\right) ^{\frac{1}{q}}=2\left( C_{\mathbf{b
}\left( p\right) ^{p}\frac{2q-p}{q-p}\lambda ^{q-p}\right) ^{\frac{1}{q}},
\end{eqnarray*
which shows that $C_{\widehat{\mathbf{b}}}\left( q\right) $ satisfies the
estimat
\begin{eqnarray*}
C_{\widehat{\mathbf{b}}}\left( q\right) &\leq &2C_{\mathbf{b}}\left(
p\right) ^{\frac{p}{q}}\left( \frac{2q-p}{q-p}\right) ^{\frac{1}{q}}\left[
\left( \frac{p}{p-2}C_{\mathbf{b}}\left( p\right) ^{p}\frac{1}{\varepsilon
\right) ^{\frac{1}{p-2}}\right] ^{1-\frac{p}{q}} \\
&\lesssim &C_{\mathbf{b}}\left( p\right) ^{\frac{p}{q}\left( 1+\frac{q-p}{p-
}\right) }\varepsilon ^{-\frac{1-\frac{p}{q}}{p-2}}\lesssim C_{\mathbf{b
}\left( p\right) ^{\frac{3}{2}}\varepsilon ^{-\frac{1-\frac{p}{q}}{p-2}},
\end{eqnarray*
a significant improvement over the bound $O\left( \varepsilon ^{-\frac{1}{p-
}}\right) $. Here we have used that if $p>\frac{6}{1+\frac{4}{q}}=\frac{6q}
q+4}$, the
\begin{eqnarray*}
\frac{p}{q}\left( 1+\frac{q-p}{p-2}\right) &=&\frac{p}{p-2}\frac{q-2}{q}
\frac{\frac{6q}{q+4}}{\frac{6q}{q+4}-2}\frac{q-2}{q} \\
&=&\frac{6q}{6q-\left( 2q+8\right) }\frac{q-2}{q}=\frac{6\left( q-2\right) }
4q-8}=\frac{3}{2}.
\end{eqnarray*}
Moreover, from (\ref{test}) we also hav
\begin{equation*}
\mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}}}\leq 2\mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+2\sqrt{\varepsilon }\mathfrak{N}_{T^{\alpha }}\ .
\end{equation*
We can do the same for the dual testing functions $\mathbf{b}^{\ast
}=\left\{ b_{Q}^{\ast }\right\} _{Q\in \mathcal{P}}$, and then altogether we
have bot
\begin{eqnarray*}
1 &\leq &\left\vert \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q
\widehat{b}_{Q}d\sigma \right\vert \leq \left\Vert \widehat{b
_{Q}\right\Vert _{L^{q}\left( \sigma \right) }\leq C_{\mathbf{b}}\left(
p\right) ^{\frac{3}{2}}\varepsilon ^{-\frac{1-\frac{p}{q}}{p-2}},\ \ \ \ \
Q\in \mathcal{P}\ , \\
&&\ \ \ \ \ \ \ \ \ \ \mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}}}\leq
\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+2\sqrt{\varepsilon }\mathfrak{N
_{T^{\alpha }}\ ,
\end{eqnarray*
as well a
\begin{eqnarray*}
1 &\leq &\left\vert \frac{1}{\left\vert Q\right\vert _{\omega }}\int_{Q
\widehat{b^{\ast }}_{Q}d\omega \right\vert \leq \left\Vert \widehat{b^{\ast
}_{Q}\right\Vert _{L^{q}\left( \omega \right) }\leq C_{\mathbf{b}^{\ast
}}\left( p\right) ^{\frac{3}{2}}\varepsilon ^{-\frac{1-\frac{p}{q}}{p-2}},\
\ \ \ \ Q\in \mathcal{P}\ , \\
&&\ \ \ \ \ \ \ \ \ \ \mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}^{\ast
}}\leq 2\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}^{\ast }}+2\sqrt{\varepsilon
^{\ast }}\mathfrak{N}_{T^{\alpha }}\ ,
\end{eqnarray*
provide
\begin{equation}
0<\varepsilon =\varepsilon ^{\ast }\leq \frac{1}{4}\text{ }. \label{eps}
\end{equation}
We now use these estimates, together with the fact that $\left( \mathcal{S
_{n-1}\right) $ holds, to obtai
\begin{eqnarray*}
\mathfrak{N}_{T^{\alpha }} &\lesssim &\left( C_{\widehat{\mathbf{b}}}\left(
q\right) +C_{\widehat{\mathbf{b}}^{\ast }}\left( q\right) \right)
^{3^{n}}\left\{ \mathfrak{T}_{T^{\alpha }}^{\widehat{\mathbf{b}}}+\mathfrak{
}_{T^{\alpha ,\ast }}^{\widehat{\mathbf{b}}^{\ast }}+\sqrt{\mathfrak{A
_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\frac{3}{2}3^{n}}\varepsilon ^{-\frac{1-\frac{p}
q}}{p-2}} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \left\{ \left[ \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\sqrt{\varepsilon }\mathfrak{N}_{T^{\alpha }
\right] +\left[ \mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}+\sqrt
\varepsilon }\mathfrak{N}_{T^{\alpha }}\right] +\sqrt{\mathfrak{A
_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha }\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\frac{3}{2}3^{n}}\varepsilon ^{-\frac{1-\frac{p}
q}}{p-2}}\left\{ \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T
_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}
\mathfrak{E}_{2}^{\alpha }\right\} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\left( C_{\mathbf{b}}\left(
p\right) +C_{\mathbf{b}^{\ast }}\left( p\right) \right) ^{\frac{3}{2}3^{n}
\sqrt{\varepsilon }\varepsilon ^{-\frac{1-\frac{p}{q}}{p-2}}\mathfrak{N
_{T^{\alpha }}\ .
\end{eqnarray*
We can absorb the last term on the right hand side above into the left hand
side for $\varepsilon >0$ sufficiently small, since (\ref{restrict}) gives
\frac{\frac{6}{p}-1}{2}<\frac{2}{q}$, and hence
\begin{equation}
\frac{1}{2}-\frac{1-\frac{p}{q}}{p-2}=\frac{p\left( 1+\frac{2}{q}\right) -4}
2p-4}>\frac{p\left( 1+\frac{\frac{6}{p}-1}{2}\right) -4}{2p-4}=\frac{1}{4}.
\label{quarter}
\end{equation
In fact, we choose
\begin{equation*}
\varepsilon =\frac{1}{\Gamma }\left( C_{\mathbf{b}}\left( p\right) +C_
\mathbf{b}^{\ast }}\left( p\right) \right) ^{\left[ \frac{3}{2}3^{n}\right]
\left[ \frac{1}{\frac{1-\frac{p}{q}}{p-2}-\frac{1}{2}}\right] }
\end{equation*
with $\Gamma $ sufficiently large, depending only on the implied constant,
to ge
\begin{eqnarray*}
\mathfrak{N}_{T^{\alpha }} &\lesssim &\left( C_{\mathbf{b}}\left( p\right)
+C_{\mathbf{b}^{\ast }}\left( p\right) \right) ^{\left[ \frac{3}{2}3^{n
\right] }\left( \left( \left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b
^{\ast }}\left( p\right) \right) ^{\left[ \frac{3}{2}3^{n}\right] \left[
\frac{1}{\frac{1-\frac{p}{q}}{p-2}-\frac{1}{2}}\right] }\right) ^{-\frac{1
\frac{p}{q}}{p-2}}\right) \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \left\{ \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b
^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right\} \\
&\lesssim &\left( C_{\mathbf{b}}\left( p\right) +C_{\mathbf{b}^{\ast
}}\left( p\right) \right) ^{\left[ \frac{3}{2}3^{n}\right] \left[ 1+1\right]
}\left\{ \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha
,\ast }}^{\mathbf{b}^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E
_{2}^{\alpha }\right\} .
\end{eqnarray*
Here we have used that (\ref{quarter}) applied twice implie
\begin{equation*}
\frac{\frac{1-\frac{p}{q}}{p-2}}{\frac{1}{2}-\frac{1-\frac{p}{q}}{p-2}}<
\frac{1-\frac{p}{q}}{p-2}\leq 1.
\end{equation*
So we finally hav
\begin{equation*}
\mathfrak{N}_{T^{\alpha }}\lesssim \left( C_{\mathbf{b}}\left( p\right) +C_
\mathbf{b}^{\ast }}\left( p\right) \right) ^{3^{n+1}}\left\{ \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b
^{\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha
}\right\} ,
\end{equation*
which completes the proof of Proposition \ref{conditional}.
\end{proof}
\begin{remark}
Propositions \ref{lower bound}\ and \ref{conditional} extend to higher
dimensions with analogous proofs.
\end{remark}
\begin{conclusion}
\label{bounded PLBP}We may assume for the proof of Theorem \ref{dim one}
given below that $p=\infty $ and that the testing functions are real-valued,
satisfy the $PLBP$ and satisf
\begin{eqnarray}
&&\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{P},
\label{acc infinity'} \\
1 &\leq &\frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}b_{Q}d\mu \leq
\left\Vert b_{Q}\right\Vert _{L^{\infty }\left( \mu \right) }\leq C_{\mathbf
b}}\left( \infty \right) <\infty ,\ \ \ \ \ Q\in \mathcal{P}\ . \notag
\end{eqnarray}
\end{conclusion}
\subsection{Reverse H\"{o}lder control of children}
Here we begin to further reduce the proof of Theorem \ref{dim one} to the
case of bounded real testing functions $\mathbf{b}=\left\{ b_{Q}\right\}
_{Q\in \mathcal{P}}$ having reverse H\"{o}lder control
\begin{equation}
\left\vert \frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}b_{Q}d\sigma \right\vert \geq c\left\Vert \mathbf{1
_{Q^{\prime }}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }>0,
\label{rev Hol con}
\end{equation
for all children $Q^{\prime }\in \mathfrak{C}\left( Q\right) $ with
\left\vert Q^{\prime }\right\vert _{\sigma }>0$ and $Q\in \mathcal{P}$.
\subsubsection{Control of averages over children}
Here we address the case of a single interval $Q$.
\begin{lemma}
\label{further red}Suppose that $\sigma $ and $\omega $ are locally finite
positive Borel measures on the real line $\mathbb{R}$. Assume that
T^{\alpha }$ is a standard $\alpha $-fractional elliptic and gradient
elliptic singular integral operator on $\mathbb{R}$, and set $T_{\sigma
}^{\alpha }f=T^{\alpha }\left( f\sigma \right) $ for any smooth truncation
of $T_{\sigma }^{\alpha }$, so that $T_{\sigma }^{\alpha }$ is \emph{apriori}
bounded from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega \right) $.
Let $Q\in \mathcal{P}$ and let $\mathfrak{N}_{T^{\alpha }}\left( Q\right) $
be the best constant in the local inequalit
\begin{equation*}
\sqrt{\int_{Q^{\prime }}\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1
_{Q}f\right) \right\vert ^{2}d\omega }\leq \mathfrak{N}_{T^{\alpha }}\left(
Q\right) \sqrt{\int_{Q}\left\vert f\right\vert ^{2}d\sigma }\ ,\ \ \ \ \
f\in L^{2}\left( \mathbf{1}_{Q}\sigma \right) .
\end{equation*
Suppose that $b_{Q}$ is a real-valued function supported in $Q$ such that
\begin{eqnarray*}
&&1\leq \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}b_{Q}d\sigma
\leq \left\Vert \mathbf{1}_{Q}b_{Q}\right\Vert _{L^{\infty }\left( \sigma
\right) }\leq C_{b_{Q}}\ , \\
&&\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega
}\leq \mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) \sqrt{\left\vert
Q\right\vert _{\sigma }}\ .
\end{eqnarray*
Then for every $0<\delta <\frac{1}{4C_{\mathbf{b}}^{3}}$, there exists a
real-valued function $\widetilde{b}_{Q}$ supported in $Q$ such that
\begin{eqnarray*}
&&1\leq \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widetilde{b
_{Q}d\sigma \leq \left\Vert \mathbf{1}_{Q}\widetilde{b}_{Q}\right\Vert
_{L^{\infty }\left( \sigma \right) }\leq 2\left( 1+\sqrt{C_{b_{Q}}}\right)
C_{b_{Q}}\ , \\
&&\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert
^{2}d\omega }\leq \left[ 2\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right)
+4C_{b_{Q}}^{\frac{3}{2}}\delta ^{\frac{1}{4}}\mathfrak{N}_{T^{\alpha
}}\left( Q\right) \right] \sqrt{\left\vert Q\right\vert _{\sigma }}\ , \\
&&0<\left\Vert \mathbf{1}_{Q_{i}}\widetilde{b}_{Q}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq \frac{16C_{b_{Q}}}{\delta }\left\vert \frac{1}
\left\vert Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}\widetilde{b}_{Q}d\sigma
\right\vert \ ,\ \ \ \ \ Q_{i}\in \mathfrak{C}\left( Q\right) .
\end{eqnarray*
Moreover, if $\left\vert b_{Q}\right\vert \geq c_{1}>0$, then we may take
\left\vert \widetilde{b}_{Q}\right\vert \geq c_{1}$ as well.
\end{lemma}
\begin{proof}
Let $0<\delta <1$ and fix $Q\in \mathcal{P}$. By assumption we hav
\begin{equation*}
1\leq \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}b_{Q}d\sigma \leq
\left\Vert \mathbf{1}_{Q}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right)
}\leq C_{b_{Q}}.
\end{equation*
Let $Q_{\limfunc{left}}$ and $Q_{\limfunc{right}}$ be the children of $Q$.
We now define $\widetilde{b}_{Q}$. First we note that the inequalit
\begin{equation}
\left\vert \frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}b_{Q}d\sigma \right\vert <\frac{\delta }{C_{b_{Q}}
\left\Vert \mathbf{1}_{Q^{\prime }}b_{Q}\right\Vert _{L^{\infty }\left(
\sigma \right) } \label{Q' big}
\end{equation
cannot hold for $Q^{\prime }$ equal to both $Q_{\limfunc{left}}$ and $Q_
\limfunc{right}}$, since otherwise we obtain the contradiction
\begin{eqnarray*}
\left\vert \int_{Q}b_{Q}d\sigma \right\vert &\leq &\left\vert \int_{Q_
\limfunc{left}}}b_{Q}d\sigma \right\vert +\left\vert \int_{Q_{\limfunc{right
}}b_{Q}d\sigma \right\vert \\
&<&\frac{\delta }{C_{\mathbf{b}}}\left( \left\vert Q_{\limfunc{left
}\right\vert _{\sigma }\left\Vert \mathbf{1}_{Q_{\limfunc{left
}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }+\left\vert Q_
\limfunc{right}}\right\vert _{\sigma }\left\Vert \mathbf{1}_{Q_{\limfunc
right}}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }\right) \\
&\leq &\frac{\delta }{C_{b_{Q}}}\left\vert Q\right\vert _{\sigma }\left\Vert
\mathbf{1}_{Q}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }\leq
\delta \left\vert \int_{Q}b_{Q}d\sigma \right\vert <\left\vert
\int_{Q}b_{Q}d\sigma \right\vert .
\end{eqnarray*
If (\ref{Q' big}) holds for neither $Q_{\limfunc{left}}$ nor $Q_{\limfunc
right}}$, then we simply define $\widetilde{b}_{Q}=b_{Q}$. If (\ref{Q' big})
holds for just one of the children, say $Q_{\limfunc{left}}$, then we define
$\widetilde{b}_{Q}$ differently according to how large the $L^{1}\left(
\sigma \right) $-average $\frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma $ is.
\textbf{Case (0)} $\frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma =0$:
In this case we defin
\begin{equation*}
\widetilde{b}_{Q}\equiv \delta \mathbf{1}_{Q_{\limfunc{left}}}+b_{Q}\mathbf{
}_{Q_{\limfunc{right}}}\ ,
\end{equation*
and the reader can easily verify that the conclusions of Lemma \ref{further
red} hold.
\textbf{Case (1)} $0<\frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma \leq
\sqrt{C_{b_{Q}}\delta }$: In this case we define
\begin{equation*}
\widetilde{b}_{Q}\equiv \left( \frac{1}{\left\vert Q_{\limfunc{left
}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert
d\sigma \right) \mathbf{1}_{Q_{\limfunc{left}}}+b_{Q}\mathbf{1}_{Q_{\limfunc
right}}}\ .
\end{equation*}
With this definition we then hav
\begin{eqnarray*}
1 &\leq &\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}b_{Q}d\sigma
\leq \frac{1}{\left\vert Q\right\vert _{\sigma }}\left( \int_{Q_{\limfunc
left}}}\left\vert b_{Q}\right\vert d\sigma +\int_{Q_{\limfunc{right
}}b_{Q}d\sigma \right) \\
&=&\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widetilde{b
_{Q}d\sigma \leq \left\Vert \widetilde{b}_{Q}\right\Vert _{L^{\infty }\left(
\sigma \right) }\leq \left\Vert b_{Q}\right\Vert _{L^{\infty }\left( \sigma
\right) }\leq C_{b_{Q}}\ ,
\end{eqnarray*
and bot
\begin{eqnarray*}
\frac{\left\Vert \mathbf{1}_{Q_{\limfunc{left}}}\widetilde{b}_{Q}\right\Vert
_{L^{\infty }\left( \sigma \right) }}{\left\vert \frac{1}{\left\vert Q_
\limfunc{left}}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\widetilde{b
_{Q}d\sigma \right\vert } &=&\frac{\frac{1}{\left\vert Q_{\limfunc{left
}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert
d\sigma }{\left\vert \frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma
\right\vert }=1<\frac{1}{\delta }C_{b_{Q}}\text{\ }, \\
\frac{\left\Vert \mathbf{1}_{Q_{\limfunc{right}}}\widetilde{b
_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }}{\left\vert \frac{1}
\left\vert Q_{\limfunc{right}}\right\vert _{\sigma }}\int_{Q_{\limfunc{right
}}\widetilde{b}_{Q}d\sigma \right\vert } &=&\frac{\left\Vert \mathbf{1}_{Q_
\limfunc{right}}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }}
\left\vert \frac{1}{\left\vert Q_{\limfunc{right}}\right\vert _{\sigma }
\int_{Q_{\limfunc{right}}}b_{Q}d\sigma \right\vert }<\frac{1}{\delta
C_{b_{Q}}\ ,
\end{eqnarray*
where the second line follows since (\ref{Q' big}) fails for $Q^{\prime }=Q_
\limfunc{right}}$.
Finally we check the testing condition in this case. We have from
Minkowski's inequality,
\begin{eqnarray*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega } &\leq &\sqrt
\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma
}^{\alpha }b_{Q}\right\vert ^{2}d\omega }+\sqrt{\frac{1}{\left\vert
Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma }^{\alpha }\left(
\widetilde{b}_{Q}-b_{Q}\right) \right\vert ^{2}d\omega } \\
&\leq &\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) +\mathfrak{N
_{T^{\alpha }}\left( Q\right) \sqrt{\frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\left\vert \widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\sigma
}.
\end{eqnarray*
In the case (\ref{Q' big}) holds for neither $Q_{\limfunc{left}}$ nor $Q_
\limfunc{right}}$, then $\widetilde{b}_{Q}-b_{Q}=0$ and s
\begin{equation*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega }\leq
\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) \ .
\end{equation*
In the case (\ref{Q' big}) holds for just one child, say $Q_{\limfunc{left}}
, the
\begin{eqnarray*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
\widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\omega } &=&\sqrt{\frac{1}
\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert
\left( \frac{1}{\left\vert Q_{\limfunc{left}}\right\vert _{\sigma }}\int_{Q_
\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma \right)
-b_{Q}\right\vert ^{2}d\sigma } \\
&\leq &\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc
left}}}\left\vert \frac{1}{\left\vert Q_{\limfunc{left}}\right\vert _{\sigma
}}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma \right\vert
^{2}d\sigma }+\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q_
\limfunc{left}}}\left\vert b_{Q}\right\vert ^{2}d\sigma } \\
&\leq &\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc
left}}}C_{\mathbf{b}}\delta d\sigma }+\sqrt{C_{\mathbf{b}}\frac{1}
\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert
b_{Q}\right\vert d\sigma } \\
&\leq &\sqrt{C_{b_{Q}}\delta }+\sqrt{C_{b_{Q}}\sqrt{C_{b_{Q}}\delta }}\leq
2C_{b_{Q}}^{\frac{3}{4}}\delta ^{\frac{1}{4}}.
\end{eqnarray*}
\textbf{Case (2)} $\frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert d\sigma
\sqrt{C_{b_{Q}}\delta }$: Let
\begin{eqnarray*}
\mathbf{1}_{Q_{\limfunc{left}}}\left( x\right) b_{Q}\left( x\right)
&=&p\left( x\right) -n\left( x\right) , \\
\mathbf{1}_{Q_{\limfunc{left}}}\left( x\right) \left\vert b_{Q}\left(
x\right) \right\vert &=&p\left( x\right) +n\left( x\right) ,
\end{eqnarray*
where $p\left( x\right) $ and $n\left( x\right) $ are the positive and
negative parts respectively of $b_{Q}$ on $Q_{\limfunc{left}}$. Then define
\widetilde{b}_{Q}$ b
\begin{equation*}
\widetilde{b}_{Q}\equiv \left\{
\begin{array}{ccc}
\left( \frac{1}{\left\vert Q_{\limfunc{left}}\right\vert _{\sigma }}\int_{Q_
\limfunc{left}}}\left[ p-n\left( 1+\sqrt{C_{\mathbf{b}}\delta }\right)
\right] d\sigma \right) \mathbf{1}_{Q_{\limfunc{left}}}+b_{Q}\mathbf{1}_{Q_
\limfunc{right}}} & \text{ if } & \int_{Q_{\limfunc{left}}}pd\sigma
<\int_{Q_{\limfunc{left}}}nd\sigma \\
\left( \frac{1}{\left\vert Q_{\limfunc{left}}\right\vert _{\sigma }}\int_{Q_
\limfunc{left}}}\left[ \left( 1+\sqrt{C_{\mathbf{b}}\delta }\right) p-
\right] d\sigma \right) \mathbf{1}_{Q_{\limfunc{left}}}+b_{Q}\mathbf{1}_{Q_
\limfunc{right}}} & \text{ if } & \int_{Q_{\limfunc{left}}}pd\sigma \geq
\int_{Q_{\limfunc{left}}}nd\sigm
\end{array
\right. .
\end{equation*}
\textbf{Subcase (2a)} $\int_{Q_{\limfunc{left}}}pd\sigma <\int_{Q_{\limfunc
left}}}nd\sigma $: In this case we hav
\begin{eqnarray*}
1 &\leq &\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}b_{Q}d\sigma
\frac{1}{\left\vert Q\right\vert _{\sigma }}\left( \int_{Q_{\limfunc{left
}}\left( p-n\right) d\sigma +\int_{Q_{\limfunc{right}}}b_{Q}d\sigma \right)
\\
&&-\frac{\sqrt{C_{b_{Q}}\delta }}{\left\vert Q\right\vert _{\sigma }
\int_{Q_{\limfunc{left}}}nd\sigma +\frac{\sqrt{C_{b_{Q}}\delta }}{\left\vert
Q\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}nd\sigma \\
&=&\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widetilde{b
_{Q}d\sigma +\frac{\sqrt{C_{b_{Q}}\delta }}{\left\vert Q\right\vert _{\sigma
}}\int_{Q_{\limfunc{left}}}nd\sigma \leq \frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\widetilde{b}_{Q}d\sigma +\frac{\sqrt{C_{b_{Q}}\delta }}
\left\vert Q\right\vert _{\sigma }}C_{b_{Q}}\left\vert Q_{\limfunc{left
}\right\vert _{\sigma } \\
&\leq &\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widetilde{b
_{Q}d\sigma +\sqrt{C_{b_{Q}}\delta }C_{b_{Q}}\leq \frac{1}{\left\vert
Q\right\vert _{\sigma }}\int_{Q}\widetilde{b}_{Q}d\sigma +\frac{1}{2},
\end{eqnarray*
if we choose $0<\delta <\frac{1}{4C_{\mathbf{b}}^{3}}$. This gives the lower
bound $\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\widetilde{b
_{Q}d\sigma \geq \frac{1}{2}$, and for an upper bound we hav
\begin{equation*}
\left\Vert \widetilde{b}_{Q}\right\Vert _{L^{\infty }\left( \sigma \right)
}\leq \left( 1+\sqrt{C_{b_{Q}}}\right) C_{b_{Q}}\ .
\end{equation*
Since we are taking $\delta <\frac{1}{4C_{\mathbf{b}}^{3}}$, we have $1
\sqrt{C_{b_{Q}}\delta }\leq 2$, and so we also have both
\begin{eqnarray*}
\frac{\left\Vert \mathbf{1}_{Q_{\limfunc{left}}}\widetilde{b}_{Q}\right\Vert
_{L^{\infty }\left( \sigma \right) }}{\left\vert \frac{1}{\left\vert Q_
\limfunc{left}}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\widetilde{b
_{Q}d\sigma \right\vert } &\leq &\frac{\left( 1+\sqrt{C_{b_{Q}}\delta
\right) C_{b_{Q}}}{\left\vert \frac{1}{\left\vert Q_{\limfunc{left
}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left[ p-n\left( 1+\sqrt
C_{b_{Q}}\delta }\right) \right] d\sigma \right\vert } \\
&\leq &\frac{\left( 1+\sqrt{C_{b_{Q}}\delta }\right) C_{b_{Q}}}{\left\vert
\sqrt{C_{b_{Q}}\delta }\frac{1}{\left\vert Q_{\limfunc{left}}\right\vert
_{\sigma }}\int_{Q_{\limfunc{left}}}nd\sigma \right\vert }\leq \frac
4C_{b_{Q}}}{\sqrt{C_{b_{Q}}\delta }\frac{1}{\left\vert Q_{\limfunc{left
}\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert b_{Q}\right\vert
d\sigma } \\
&\leq &\frac{4C_{b_{Q}}}{C_{b_{Q}}\delta }=\frac{4}{\delta }\text{\ }, \\
\frac{\left\Vert \mathbf{1}_{Q_{\limfunc{right}}}\widetilde{b
_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }}{\left\vert \frac{1}
\left\vert Q_{\limfunc{right}}\right\vert _{\sigma }}\int_{Q_{\limfunc{right
}}\widetilde{b}_{Q}d\sigma \right\vert } &=&\frac{\left\Vert \mathbf{1}_{Q_
\limfunc{right}}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }}
\left\vert \frac{1}{\left\vert Q_{\limfunc{right}}\right\vert _{\sigma }
\int_{Q_{\limfunc{right}}}b_{Q}d\sigma \right\vert }<\frac{1}{\delta
C_{b_{Q}}\ ,
\end{eqnarray*
where the second line follows since (\ref{Q' big}) fails for $Q^{\prime }=Q_
\limfunc{right}}$.
Finally we check the testing condition in this case. We have from
Minkowski's inequality,
\begin{eqnarray*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega } &\leq &\sqrt
\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma
}^{\alpha }b_{Q}\right\vert ^{2}d\omega }+\sqrt{\frac{1}{\left\vert
Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma }^{\alpha }\left(
\widetilde{b}_{Q}-b_{Q}\right) \right\vert ^{2}d\omega } \\
&\leq &\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) +\mathfrak{N
_{T^{\alpha }}\left( Q\right) \sqrt{\frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\left\vert \widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\sigma
}.
\end{eqnarray*
Now recall we are assuming $\int_{Q_{\limfunc{left}}}nd\sigma >\int_{Q_
\limfunc{left}}}pd\sigma $, so tha
\begin{eqnarray*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
\widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\sigma } &=&\sqrt{\frac{1}
\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc{left}}}\left\vert \sqrt
C_{\mathbf{b}}\delta }n\right\vert ^{2}d\sigma }\leq \sqrt{C_{b_{Q}}\delta
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q_{\limfunc{left
}}\left\vert n\right\vert ^{2}d\sigma } \\
&\lesssim &\sqrt{C_{b_{Q}}\delta }C_{b_{Q}}=C_{b_{Q}}^{\frac{3}{2}}\sqrt
\delta }.
\end{eqnarray*
Thus in any case we hav
\begin{equation*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega }\leq
\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) +2C_{b_{Q}}^{\frac{3}{2
}\delta ^{\frac{1}{2}}\mathfrak{N}_{T^{\alpha }}\left( Q\right) \ .
\end{equation*
\textbf{Subcase (2b)} $\int_{Q_{\limfunc{left}}}pd\sigma \geq \int_{Q_
\limfunc{left}}}nd\sigma $: The same estimates arise in this case, except
that we get the better lower bound $\frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\widetilde{b}_{Q}d\sigma \geq 1$.
Collecting all of our estimates for $\widetilde{b}_{Q}$ in the various cases
above we hav
\begin{eqnarray*}
&&\frac{1}{2}\leq \frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q
\widetilde{b}_{Q}d\sigma \leq \left\Vert \mathbf{1}_{Q}\widetilde{b
_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }\leq \left( 1+\sqrt
C_{b_{Q}}}\right) C_{b_{Q}}\ ,\ \ \ \ \ Q\in \mathcal{P}\ , \\
&&\frac{\left\Vert \mathbf{1}_{Q^{\prime }}\widetilde{b}_{Q}\right\Vert
_{L^{\infty }\left( \sigma \right) }}{\left\vert \frac{1}{\left\vert
Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime }}\widetilde{b}_{Q}d\sigma
\right\vert },\frac{\left\Vert \mathbf{1}_{Q^{\prime }}\widetilde{b
_{Q}^{\ast }\right\Vert _{L^{\infty }\left( \omega \right) }}{\left\vert
\frac{1}{\left\vert Q^{\prime }\right\vert _{\omega }}\int_{Q^{\prime }
\widetilde{b}_{Q}^{\ast }d\omega \right\vert }<\frac{4}{\delta }C_{b_{Q}}\
,\ \ \ \ \ Q\in \mathcal{P}\text{ and }Q^{\prime }\in \mathfrak{C}\left(
Q\right) \ , \\
&&\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega }\leq
\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) +2C_{b_{Q}}^{\frac{3}{2
}\delta ^{\frac{1}{4}}\mathfrak{N}_{T^{\alpha }}\left( Q\right) \ \ \ \ \
Q\in \mathcal{P}\ .
\end{eqnarray*
In order to obtain the inequalities for $\widetilde{b}_{Q}$ in the
conclusion of Lemma \ref{further red}, we simply multiply the above function
$\widetilde{b}_{Q}$ by a factor of $2$.
Finally, if $\left\vert b_{Q}\right\vert \geq c_{1}>0$, we need only
consider \textbf{Case (2)} above, in which case we have $\left\vert
b_{Q}\right\vert \geq \left\vert \widetilde{b}_{Q}\right\vert $. This
completes the proof of Lemma \ref{further red}.
\end{proof}
\subsubsection{Control of averages in coronas}
Let $\mathcal{D}_{Q}$ be the grid of dyadic subintervals of $Q$. In the
construction of the triple corona below, we will need to repeat the
construction in the previous subsubsection for a subdecomposition $\left\{
Q_{i}\right\} _{i=1}^{\infty }$ of dyadic subintervals $Q_{i}\in \mathcal{D
_{Q}$ of an interval $Q$. Define the corona corresponding to the
subdecomposition $\left\{ Q_{i}\right\} _{i=1}^{\infty }$ by
\begin{equation*}
\mathcal{C}_{Q}\equiv \mathcal{D}_{Q}\setminus \bigcup_{i=1}^{\infty
\mathcal{D}_{Q_{i}}\ .
\end{equation*}
\begin{lemma}
\label{prelim control of corona}Suppose that $\sigma $ and $\omega $ are
locally finite positive Borel measures on the real line $\mathbb{R}$. Assume
that $T^{\alpha }$ is a standard $\alpha $-fractional elliptic and gradient
elliptic singular integral operator on $\mathbb{R}$, and set $T_{\sigma
}^{\alpha }f=T^{\alpha }\left( f\sigma \right) $ for any smooth truncation
of $T_{\sigma }^{\alpha }$, so that $T_{\sigma }^{\alpha }$ is \emph{apriori}
bounded from $L^{2}\left( \sigma \right) $ to $L^{2}\left( \omega \right) $.
Let $Q\in \mathcal{P}$ and let $\mathfrak{N}_{T^{\alpha }}\left( Q\right) $
be the best constant in the local inequalit
\begin{equation*}
\sqrt{\int_{Q^{\prime }}\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1
_{Q}f\right) \right\vert ^{2}d\omega }\leq \mathfrak{N}_{T^{\alpha }}\left(
Q\right) \sqrt{\int_{Q}\left\vert f\right\vert ^{2}d\sigma }\ ,\ \ \ \ \
f\in L^{2}\left( \mathbf{1}_{Q}\sigma \right) .
\end{equation*
Let $\left\{ Q_{i}\right\} _{i=1}^{\infty }\subset \mathcal{D}_{Q}$\ be a
collection of pairwise disjoint dyadic subintervals of $Q$. Suppose that
b_{Q}$ is a real-valued function supported in $Q$ such that
\begin{eqnarray*}
&&1\leq \frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}b_{Q}d\sigma \leq \left\Vert \mathbf{1}_{Q^{\prime
}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }\leq C_{\mathbf{b}}\
,\ \ \ \ \ Q^{\prime }\in \mathcal{C}_{Q}\ , \\
&&\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega
}\leq \mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) \sqrt{\left\vert
Q\right\vert _{\sigma }}\ .
\end{eqnarray*
Then for every $0<\delta <\frac{1}{4C_{\mathbf{b}}^{3}}$, there exists a
real-valued function $\widetilde{b}_{Q}$ supported in $Q$ such tha
\begin{eqnarray*}
&&1\leq \frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}\widetilde{b}_{Q}d\sigma \leq \left\Vert \mathbf{1
_{Q^{\prime }}\widetilde{b}_{Q}\right\Vert _{L^{\infty }\left( \sigma
\right) }\leq 2\left( 1+\sqrt{C_{\mathbf{b}}}\right) C_{\mathbf{b}}\ ,\ \ \
\ \ Q^{\prime }\in \mathcal{C}_{Q}\ , \\
&&\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert
^{2}d\omega }\leq \left[ 2\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right)
+4C_{\mathbf{b}}^{\frac{3}{2}}\delta ^{\frac{1}{4}}\mathfrak{N}_{T^{\alpha
}}\left( Q\right) \right] \sqrt{\left\vert Q\right\vert _{\sigma }}\ , \\
&&0<\left\Vert \mathbf{1}_{Q_{i}}\widetilde{b}_{Q}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq \frac{16C_{\mathbf{b}}}{\delta }\left\vert
\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}\widetilde{b
_{Q}d\sigma \right\vert \ ,\ \ \ \ \ 1\leq i<\infty .
\end{eqnarray*
Moreover, if $\left\vert b_{Q}\right\vert \geq c_{1}>0$, then we may take
\left\vert \widetilde{b}_{Q}\right\vert \geq c_{1}$ as well.
\end{lemma}
The additional gain in the lemma is in the final line that controls the
degeneracy of $\widetilde{b}_{Q}$ at the `bottom' of the corona $\mathcal{C
_{Q}$ by establishing a reverse H\"{o}lder control. Note that if we combine
this control with the accretivity control in the corona $\mathcal{C}_{Q}$,
namely
\begin{equation*}
\left\Vert \mathbf{1}_{Q^{\prime }}\widetilde{b}_{Q}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq 2\left( 1+\sqrt{C_{\mathbf{b}}}\right) C_
\mathbf{b}}\leq 2\left( 1+\sqrt{C_{\mathbf{b}}}\right) C_{\mathbf{b}}\frac{
}{\left\vert Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime }}\widetilde{
}_{Q}d\sigma ,
\end{equation*
we obtain reverse H\"{o}lder control throughout the entire collection
\mathcal{C}_{Q}\cup \left\{ Q_{i}\right\} _{i=1}^{\infty }$:
\begin{equation*}
\left\Vert \mathbf{1}_{I}\widetilde{b}_{Q^{\prime }}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq C_{\delta ,\mathbf{b}}\left\vert \frac{1}
\left\vert I\right\vert _{\sigma }}\int_{I}\widetilde{b}_{Q^{\prime
}}d\sigma \right\vert ,\ \ \ \ \ I\in \mathfrak{C}\left( Q^{\prime }\right)
,Q^{\prime }\in \mathcal{C}_{Q}\text{ }.
\end{equation*
This has the crucial consequence that the martingale and dual martingale
differences $\bigtriangleup _{Q^{\prime }}^{\sigma ,\mathbf{b}}$ and
\square _{Q^{\prime }}^{\sigma ,\mathbf{b}}$ associated with these functions
as defined in (\ref{def diff}) of Appendix A, satisf
\begin{equation}
\left\vert \bigtriangleup _{Q^{\prime }}^{\sigma ,\mathbf{b}}h\right\vert
,\left\vert \square _{Q^{\prime }}^{\sigma ,\mathbf{b}}h\right\vert \leq
C_{\delta ,\mathbf{b}}\sum_{I\in \mathfrak{C}\left( Q^{\prime }\right)
}\left( \frac{1}{\left\vert I\right\vert _{\sigma }}\int_{I}\left\vert
h\right\vert d\sigma +\frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}\left\vert h\right\vert d\sigma \right) \mathbf{1}_{I}\ .
\label{cruc conseq}
\end{equation
See Appendix A for more detail on this. However, the defect in this lemma is
that we lose the weak testing condition for $\widetilde{b}_{Q}$ in the
corona even if we had assumed it at the outset for $b_{Q}$.
\begin{proof}
The proof of Lemma \ref{prelim control of corona} is similar to that of the
special case given by Lemma \ref{further red}. Indeed, we defin
\begin{eqnarray*}
\widetilde{b}_{Q} &\equiv &\sum_{i\in G_{0}}\delta \mathbf{1
_{Q_{i}}+\sum_{i\in G_{+}}\left( \frac{1}{\left\vert Q_{i}\right\vert
_{\sigma }}\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma \right) \mathbf{1
_{Q_{i}} \\
&&+\sum_{i\in B_{-}}\left( \frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left[ p_{i}-n_{i}\left( 1+\sqrt{C_{\mathbf{b}}\delta }\right)
\right] d\sigma \right) \mathbf{1}_{Q_{i}} \\
&&+\sum_{i\in B_{+}}\left( \frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left[ \left( 1+\sqrt{C_{\mathbf{b}}\delta }\right) p_{i}-n_{i
\right] d\sigma \right) \mathbf{1}_{Q_{i}} \\
&&+b_{Q}\mathbf{1}_{Q\setminus \cup _{i=1}^{\infty }Q_{i}}\ ,
\end{eqnarray*
wher
\begin{eqnarray*}
G_{0} &\equiv &\left\{ i:\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma =0\right\} \\
G_{+} &\equiv &\left\{ i:0<\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma \leq \sqrt{C_{\mathbf{b
}\delta }\right\} , \\
B_{-} &\equiv &\left\{ i:\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma >\sqrt{C_{\mathbf{b}}\delta
\text{ and }\int_{Q_{i}}nd\sigma >\int_{Q_{i}}pd\sigma \right\} , \\
B_{+} &\equiv &\left\{ i:\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }
\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma >\sqrt{C_{\mathbf{b}}\delta
\text{ and }\int_{Q_{i}}pd\sigma \geq \int_{Q_{i}}nd\sigma \right\} .
\end{eqnarray*
First we note tha
\begin{eqnarray*}
1 &\leq &\frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}b_{Q}d\sigma \leq \frac{1}{\left\vert Q^{\prime
}\right\vert _{\sigma }}\int_{Q^{\prime }}\widetilde{b}_{Q}d\sigma +\frac{1}
\left\vert Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime }}\left( b_{Q}
\widetilde{b}_{Q}\right) d\sigma \\
&\leq &\frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime
}}\widetilde{b}_{Q}d\sigma +\sum_{i:\ Q_{i}\subset Q^{\prime }}\frac{1}
\left\vert Q^{\prime }\right\vert _{\sigma }}\left\{
\begin{array}{ccc}
\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma -\delta & \text{ in } &
\text{\textbf{Case (0)}} \\
2\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma & \text{ in } & \text
\textbf{Case (1)}} \\
\sqrt{C_{\mathbf{b}}\delta }\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma
& \text{ in } & \text{\textbf{Case (2)}
\end{array
\right. \\
&\leq &\frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime
}}\widetilde{b}_{Q}d\sigma +2\sqrt{C_{\mathbf{b}}\delta }C_{\mathbf{b
}\sum_{i:\ Q_{i}\subset Q^{\prime }}\frac{\left\vert Q_{i}\right\vert
_{\sigma }}{\left\vert Q^{\prime }\right\vert _{\sigma }}\leq \frac{1}
\left\vert Q^{\prime }\right\vert _{\sigma }}\int_{Q^{\prime }}\widetilde{b
_{Q}d\sigma +\frac{1}{2},
\end{eqnarray*
if $0<\delta <\frac{1}{4C_{\mathbf{b}}^{3}}$.
Then we estimate the testing condition for the interval $Q$ b
\begin{eqnarray*}
\sqrt{\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
T_{\sigma }^{\alpha }\widetilde{b}_{Q}\right\vert ^{2}d\omega } &\leq &\sqrt
\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma
}^{\alpha }b_{Q}\right\vert ^{2}d\omega }+\sqrt{\frac{1}{\left\vert
Q\right\vert _{\sigma }}\int_{Q}\left\vert T_{\sigma }^{\alpha }\left(
\widetilde{b}_{Q}-b_{Q}\right) \right\vert ^{2}d\omega } \\
&\leq &\mathfrak{T}_{T^{\alpha }}^{b_{Q}}\left( Q\right) +\mathfrak{N
_{T^{\alpha }}\left( Q\right) \sqrt{\frac{1}{\left\vert Q\right\vert
_{\sigma }}\int_{Q}\left\vert \widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\sigma
},
\end{eqnarray*
and note that the arguments above show that
\begin{eqnarray*}
\widetilde{b}_{Q}-b_{Q} &=&\sum_{i\in G}\left( \frac{1}{\left\vert
Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}\left\vert b_{Q}\right\vert d\sigma
\right) \mathbf{1}_{Q_{i}}-\sum_{i\in G}b_{Q}\mathbf{1}_{Q_{i}} \\
&&-\sum_{i\in B_{-}}\left( \frac{\sqrt{C_{\mathbf{b}}\delta }}{\left\vert
Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}n_{i}d\sigma \right) \mathbf{1
_{Q_{i}}+\sum_{i\in B_{+}}\left( \frac{\sqrt{C_{\mathbf{b}}\delta }}
\left\vert Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}p_{i}d\sigma \right)
\mathbf{1}_{Q_{i}},
\end{eqnarray*
satisfies an inequality of the for
\begin{equation*}
\int_{Q}\left\vert \widetilde{b}_{Q}-b_{Q}\right\vert ^{2}d\sigma \leq
C\left( C_{\mathbf{b}}\right) \delta ^{\frac{1}{4}}\sum_{i=1}^{\infty
}\left\vert Q_{i}\right\vert _{\sigma }\leq C\left( C_{\mathbf{b}}\right)
\delta ^{\frac{1}{4}}\left\vert Q\right\vert _{\sigma }\ .
\end{equation*}
\end{proof}
\begin{remark}
The estimate $\int_{Q}\left\vert \widetilde{b}_{Q}-b_{Q}\right\vert
^{2}d\sigma \leq C\left( C_{\mathbf{b}}\right) \delta ^{\frac{1}{4
}\sum_{i=1}^{\infty }\left\vert Q_{i}\right\vert _{\sigma }$ in the last
line of the above proof is of course too large in general to be dominated by
a fixed multiple of $\left\vert Q^{\prime }\right\vert _{\sigma }$ for
Q^{\prime }\in \mathcal{C}_{Q}$, and this is the reason we have no control
of weak testing for $\widetilde{b}_{Q}$ in the rest of the corona even if we
assume weak testing for $b_{Q}$ in the corona $\mathcal{C}_{Q}$. This defect
is addressed in the next subsection below.
\end{remark}
\subsection{Three corona decompositions}
We will use multiple corona constructions, namely a Calder\'{o}n-Zygmund
decomposition, an accretive decomposition, a weak testing decomposition, and
an energy decomposition, in order to reduce matters to the stopping form,
which is treated in Section \ref{Sec stop} by adapting the bottom/up
stopping time in the argument of M. Lacey in \cite{Lac}, and using an
additional `indented' top/down corona construction, in order to accommodate
weak goodness. We will then iterate these corona decompositions into a
single corona decomposition, which we refer to as the \emph{triple corona}.
More precisely, we iterate the first generation of common stopping times
with an infusion of the reverse H\"{o}lder condition on children, followed
by another iteration of the first generation of weak testing stopping times.
Recall that we must show the bilinear inequalit
\begin{equation*}
\left\vert \int \left( T_{\sigma }^{\alpha }f\right) gd\omega \right\vert
\leq \mathfrak{N}_{T^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }\left\Vert g\right\Vert _{L^{2}\left( \omega \right) },\ \ \ \ \
f\in L^{2}\left( \sigma \right) \text{ and }g\in L^{2}\left( \omega \right) .
\end{equation*}
\subsubsection{The Calder\'{o}n-Zygmund corona decomposition}
We first introduce the Calder\'{o}n-Zygmund stopping times $\mathcal{F}$ for
a function $\phi \in L^{2}\left( \mu \right) $ relative to an interval
S_{0} $ and a positive constant $C_{0}\geq 4$. Let $\mathcal{F}=\left\{
F\right\} _{F\in \mathcal{F}}$ be the collection of Calder\'{o}n-Zygmund
stopping intervals for $\phi $ defined so that $F\subset S_{0}$, $S_{0}\in
\mathcal{F} $, and for all $F\in \mathcal{F}$ with $F\subsetneqq S_{0}$ we
hav
\begin{eqnarray*}
\frac{1}{\left\vert F\right\vert _{\mu }}\int_{F}\left\vert \phi \right\vert
d\mu &>&C_{0}\frac{1}{\left\vert \pi _{\mathcal{F}}F\right\vert _{\mu }
\int_{\pi _{\mathcal{F}}F}\left\vert \phi \right\vert d\mu ; \\
\frac{1}{\left\vert F^{\prime }\right\vert _{\mu }}\int_{F^{\prime
}}\left\vert \phi \right\vert d\mu &\leq &C_{0}\frac{1}{\left\vert \pi _
\mathcal{F}}F\right\vert _{\mu }}\int_{\pi _{\mathcal{F}}F}\left\vert \phi
\right\vert d\mu \ \ \ \ \text{ for }F\subsetneqq F^{\prime }\subset \pi _
\mathcal{F}}F.
\end{eqnarray*
To achieve this construction we use the following definition.
\begin{definition}
\label{CZ stopping times}Let $C_{0}\geq 4$. Given a dyadic grid $\mathcal{D}$
an interval $S_{0}\in \mathcal{D}$, define $\mathcal{S}\left( S_{0}\right) $
to be the \emph{maximal} $\mathcal{D}$-subintervals $I\subset S_{0}$ such
tha
\begin{equation*}
\frac{1}{\left\vert I\right\vert _{\mu }}\int_{I}\left\vert \phi \right\vert
d\mu >C_{0}\frac{1}{\left\vert S_{0}\right\vert _{\mu }}\int_{S_{0}}\lef
\vert \phi \right\vert d\mu \ ,
\end{equation*
and then define the $CZ$ stopping intervals of $S_{0}$ to be the collection
\begin{equation*}
\mathcal{S}=\left\{ S_{0}\right\} \cup \dbigcup\limits_{n=0}^{\infty
\mathcal{S}_{n}
\end{equation*
where $\mathcal{S}_{0}=\mathcal{S}\left( S_{0}\right) $ and $\mathcal{S
_{n+1}=\dbigcup\limits_{S\in \mathcal{S}_{n}}\mathcal{S}\left( S\right) $
for $n\geq 0$.
\end{definition}
Let $\mathcal{D}=\dbigcup\limits_{F\in \mathcal{F}}\mathcal{C}_{F}$ be the
associated corona decomposition of the dyadic grid $\mathcal{D}$ wher
\begin{equation*}
\mathcal{C}_{F}\equiv \left\{ F^{\prime }\in \mathcal{D}:F\supset F^{\prime
}\supsetneqq H\text{ for some }H\in \mathfrak{C}_{\mathcal{F}}\left(
F\right) \right\} .
\end{equation*
We now recall some of the definitions just used above. See \cite{SaShUr7}
and/or \cite{SaShUr6} for more detail. For an interval $I\in \mathcal{D}$
let $\pi _{\mathcal{D}}I$ be the $\mathcal{D}$-parent of $I$ in the grid
\mathcal{D}$, and let $\pi _{\mathcal{F}}I$ be the smallest member of
\mathcal{F}$ that \emph{strictly} contains $I$. For $F,F^{\prime }\in
\mathcal{F}$, we say that $F^{\prime }$ is an $\mathcal{F}$-child of $F$ if
\pi _{\mathcal{F}}\left( F^{\prime }\right) =F$ (it could be that $F=\pi _
\mathcal{D}}F^{\prime }$), and we denote by $\mathfrak{C}_{\mathcal{F
}\left( F\right) $ the set of $\mathcal{F}$-children of $F$. We call $\pi _
\mathcal{F}}\left( F^{\prime }\right) $ the $\mathcal{F}$-parent of
F^{\prime }\in \mathcal{F}$.
The stopping intervals $\mathcal{F}$ above satisfy a Carleson condition
\begin{equation}
\sum_{F\in \mathcal{F}:\ F\subset \Omega }\left\vert F\right\vert _{\mu
}\leq C\left\vert \Omega \right\vert _{\mu }\ ,\ \ \ \ \ \text{for all open
sets }\Omega . \label{CZ Car}
\end{equation
Indeed,
\begin{equation*}
\sum_{F^{\prime }\in \mathfrak{C}_{\mathcal{F}}\left( F\right) }\left\vert
F^{\prime }\right\vert _{\mu }\leq \sum_{F^{\prime }\in \mathfrak{C}_
\mathcal{F}}\left( F\right) }\frac{\int_{F^{\prime }}\left\vert \phi
\right\vert d\mu }{C_{0}\frac{1}{\left\vert F\right\vert _{\mu }
\int_{F}\left\vert \phi \right\vert d\mu }\leq \frac{1}{C_{0}},
\end{equation*
and standard arguments now complete the proof of the Carleson condition.
We emphasize that accretive functions $b$ play no role in the Calder\'{o
n-Zygmund corona decomposition.
\subsubsection{The $\mathbf{b}$-accretive / weak testing corona decompositio
}
Recall that we are assuming $p=\infty $, and that our testing functions
\mathbf{b}$ and $\mathbf{b}^{\ast }$ are real-valued, in the proof of
Theorem \ref{dim one}. We use a corona construction modelled after that of
Hyt\"{o}nen and Martikainen \cite{HyMa}, that delivers a \emph{weak corona
testing condition} that coincides with the testing condition itself \textbf
only} at the tops of the coronas. This corona decomposition is developed to
optimize the choice of a new family of testing functions $\left\{ \widehat{b
_{Q}\right\} _{Q\in \mathcal{D}}$ taken from the vector $\mathbf{b}\equiv
\left\{ b_{Q}\right\} _{Q\in \mathcal{D}}$ so that we have
\begin{enumerate}
\item the telescoping property at our disposal in each accretive corona,
\item a weak corona testing condition remains in force for the new testing
functions $\widehat{b}_{Q}$ that coincides with the usual testing condition
at the tops of the coronas,
\item the tops of the coronas, i.e. the stopping intervals, enjoy a Carleson
condition.
\end{enumerate}
We will sometimes refer to the old family as the \emph{original} family, and
denote it by $\left\{ b_{Q}^{\limfunc{orig}}\right\} _{Q\in \mathcal{D}}$.
The original family will reappear later in helping to estimate the nearby
form in Section \ref{Sec nearby}.
Let $\sigma $ and $\omega $ be locally finite Borel measures on $\mathbb{R}
. We assume that the vector of `testing functions' $\mathbf{b}\equiv \left\{
b_{Q}\right\} _{Q\in \mathcal{D}}$ is an $\infty $-strongly $\sigma
-accretive real-valued family, i.e.
\begin{equation*}
\limfunc{support}b_{Q}\subset Q\ ,\ \ \ \ \ Q\in \mathcal{D},
\end{equation*
an
\begin{equation*}
1\leq \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}b_{Q}d\sigma \leq
\left\Vert b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }\leq C_
\mathbf{b}}<\infty ,\ \ \ \ \ Q\in \mathcal{D}\ ,
\end{equation*
and also that $\mathbf{b}^{\ast }\equiv \left\{ b_{Q}\right\} _{Q\in
\mathcal{D}}$ is an $\infty $-strongly $\omega $-accretive real-valued
family, and we assume in addition the testing condition
\begin{eqnarray*}
\int_{Q}\left\vert T_{\sigma }^{\alpha }\left( \mathbf{1}_{Q}b_{Q}\right)
\right\vert ^{2}d\omega &\leq &\mathfrak{T}_{T^{\alpha }}^{\mathbf{b
}\left\vert Q\right\vert _{\sigma }\ ,\ \ \ \ \ \text{for all intervals }Q,
\\
\int_{Q}\left\vert T_{\omega }^{\alpha ,\ast }\left( \mathbf{1
_{Q}b_{Q}^{\ast }\right) \right\vert ^{2}d\sigma &\leq &\mathfrak{T
_{T^{\alpha }}^{\mathbf{b}^{\ast },\ast }\left\vert Q\right\vert _{\omega }\
,\ \ \ \ \ \text{for all intervals }Q.
\end{eqnarray*}
\begin{definition}
\label{accretive stopping times gen}Given a dyadic grid $\mathcal{D}$ an
interval $S_{0}\in \mathcal{D}$, define $\mathcal{S}\left( S_{0}\right) $ to
be the \emph{maximal} $\mathcal{D}$-subintervals $I\subset S_{0}$ such tha
\begin{eqnarray*}
\text{\textbf{either} }\left\vert \frac{1}{\left\vert I\right\vert _{\sigma
}\int_{I}b_{S_{0}}d\sigma \right\vert &<&\gamma \ , \\
\ \text{\textbf{or} }\int_{I}\left\vert T_{\sigma }^{\alpha }\left(
b_{S_{0}}\right) \right\vert ^{2}d\omega &>&\Gamma \left( \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert I\right\vert _{\sigma
\text{ },
\end{eqnarray*
where the positive constants $\gamma ,\Gamma $ satisfy $0<\gamma <1<\Gamma
<\infty $. Then define the $\mathbf{b}$-accretive stopping intervals of
S_{0}$ to be the collection
\begin{equation*}
\mathcal{S}=\left\{ S_{0}\right\} \cup \dbigcup\limits_{n=0}^{\infty
\mathcal{S}_{n}
\end{equation*
where $\mathcal{S}_{0}=\mathcal{S}\left( S_{0}\right) $ and $\mathcal{S
_{n+1}=\dbigcup\limits_{S\in \mathcal{S}_{n}}\mathcal{S}\left( S\right) $
for $n\geq 0$.
\end{definition}
For $\gamma <1$ chosen small enough and $\Gamma >1$ chosen large enough, the
$\mathbf{b}$-accretive stopping intervals satisfy a $\sigma $-Carleson
condition relative to the measure $\sigma $, and the corresponding stopping
functions $b_{S_{0}}$ satisfy \emph{weak} testing inequalities in the
corona. The following lemma is essentially in \cite{HyMa}, but we include a
proof for completeness.
\begin{lemma}[\protect\cite{HyMa}]
\label{Car and Test gen}For $\gamma <1$ small enough and $\Gamma >1$ large
enough we have the following:
\begin{enumerate}
\item For every open set $\Omega $ we have we have the inequality
\begin{equation}
\sum_{S\in \mathcal{S}:\ S\subset \Omega }\left\vert S\right\vert _{\sigma
}\leq C\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert
\Omega \right\vert _{\sigma }\ . \label{Car gen}
\end{equation}
\item For every interval $S\in \mathcal{C}_{S_{0}}$ we have the weak corona
testing inequality
\begin{equation}
\int_{S}\left\vert T_{\sigma }^{\alpha }b_{S_{0}}\right\vert ^{2}d\omega
\leq C\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert
S\right\vert _{\sigma }\ . \label{Test gen}
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
We first address the Carleson condition (\ref{Car gen}). A standard argument
reduces matters to the case where $\Omega $ is an interval $Q\in \mathcal{S}$
with $\left\vert Q\right\vert _{\sigma }>0$. It suffices to consider each of
the two stopping criteria separately. We first address the stopping
condition $\left\vert \frac{1}{\left\vert I\right\vert _{\sigma }
\int_{I}b_{S_{0}}d\sigma \right\vert <\gamma $. Throughout this proof we
will denote the union of these children $\mathcal{S}\left( Q\right) $ of $Q$
by $E\left( Q\right) \equiv \dbigcup\limits_{S\in \mathcal{S}\left( Q\right)
}S$. Then we hav
\begin{equation*}
\left\vert \int_{E\left( Q\right) }b_{Q}d\sigma \right\vert \leq \sum_{S\in
\mathcal{S}\left( Q\right) }\left\vert \int_{S}b_{Q}d\sigma \right\vert
<\gamma \sum_{S\in \mathcal{S}\left( Q\right) }\left\vert S\right\vert
_{\sigma }\leq \gamma \left\vert Q\right\vert _{\sigma }\ ,
\end{equation*
which together with our hypotheses on $b_{Q}$ give
\begin{eqnarray*}
\left\vert Q\right\vert _{\sigma } &<&\left\vert \int_{Q}b_{Q}d\sigma
\right\vert =\left\vert \int_{E\left( Q\right) }b_{Q}d\sigma \right\vert
+\left\vert \int_{Q\setminus E\left( Q\right) }b_{Q}d\sigma \right\vert \\
&\leq &\gamma \left\vert Q\right\vert _{\sigma }+\sqrt{\int_{Q\setminus
E\left( Q\right) }\left\vert b_{Q}\right\vert ^{2}d\sigma }\sqrt{\left\vert
Q\setminus E\left( Q\right) \right\vert _{\sigma }} \\
&\leq &\gamma \left\vert Q\right\vert _{\sigma }+\Gamma C_{\mathbf{b}}\sqrt
\left\vert Q\right\vert _{\sigma }}\sqrt{\left\vert Q\setminus E\left(
Q\right) \right\vert _{\sigma }}.
\end{eqnarray*
Rearranging the inequality yields successivel
\begin{eqnarray*}
\left( 1-\gamma \right) \left\vert Q\right\vert _{\sigma } &\leq &\Gamma C_
\mathbf{b}}\sqrt{\left\vert Q\right\vert _{\sigma }}\sqrt{\left\vert
Q\setminus E\left( Q\right) \right\vert _{\sigma }}; \\
\left( 1-\gamma \right) ^{2}\left\vert Q\right\vert _{\sigma }^{2} &\leq
&\Gamma ^{2}C_{\mathbf{b}}^{2}\left\vert Q\right\vert _{\sigma }\left\vert
Q\setminus E\left( Q\right) \right\vert _{\sigma }; \\
\frac{\left( 1-\gamma \right) ^{2}}{\Gamma ^{2}C_{\mathbf{b}}^{2}}\left\vert
Q\right\vert _{\sigma } &\leq &\left\vert Q\setminus E\left( Q\right)
\right\vert _{\sigma }\ ,
\end{eqnarray*
which in turn give
\begin{eqnarray*}
\sum_{S\in \mathcal{S}\left( Q\right) }\left\vert S\right\vert _{\sigma }
&=&\left\vert Q\right\vert _{\sigma }-\left\vert Q\setminus E\left( Q\right)
\right\vert _{\sigma } \\
&\leq &\left\vert Q\right\vert _{\sigma }-\frac{\left( 1-\gamma \right) ^{2
}{\Gamma ^{2}C_{\mathbf{b}}^{2}}\left\vert Q\right\vert _{\sigma }=\left( 1
\frac{\left( 1-\gamma \right) ^{2}}{\Gamma ^{2}C_{\mathbf{b}}^{2}}\right)
\left\vert Q\right\vert _{\sigma }\equiv \beta \left\vert Q\right\vert
_{\sigma }\ ,
\end{eqnarray*
where $0<\beta <1$ since $1\leq C_{\mathbf{b}}$. If we now iterate this
inequality, we obtain for each $k\geq 1$
\begin{eqnarray*}
\sum_{\substack{ S\in \mathcal{S}:\ S\subset Q \\ \pi _{\mathcal{S
}^{\left( k\right) }\left( S\right) =Q}}\left\vert S\right\vert _{\sigma }
&=&\sum _{\substack{ S\in \mathcal{S}:\ S\subset Q \\ \pi _{\mathcal{S
}^{\left( k-1\right) }\left( S\right) =Q}}\sum_{S^{\prime }\in \mathcal{S
\left( S\right) }\left\vert S^{\prime }\right\vert _{\sigma }\leq \sum
_{\substack{ S\in \mathcal{S}:\ S\subset Q \\ \pi _{\mathcal{S}}^{\left(
k-1\right) }\left( S\right) =Q}}\beta \left\vert S\right\vert _{\sigma } \\
&&\vdots \\
&\leq &\sum_{\substack{ S\in \mathcal{S}:\ S\subset Q \\ \pi _{\mathcal{S
}^{\left( 1\right) }\left( S\right) =Q}}\left( 1-\gamma ^{2}\right)
^{k-1}\left\vert S\right\vert _{\sigma }\leq \beta ^{k}\left\vert
Q\right\vert _{\sigma }\ .
\end{eqnarray*
Finally the
\begin{equation*}
\sum_{S\in \mathcal{S}:\ S\subset Q}\left\vert S\right\vert _{\sigma }\leq
\left\vert Q\right\vert _{\sigma }+\sum_{k=1}^{\infty }\sum_{\substack{ S\in
\mathcal{S}:\ S\subset Q \\ \pi _{\mathcal{S}}^{\left( k\right) }\left(
S\right) =Q}}\left\vert S\right\vert _{\sigma }\leq \sum_{k=0}^{\infty
}\beta ^{k}\left\vert Q\right\vert _{\sigma }=\frac{1}{1-\beta }\left\vert
Q\right\vert _{\sigma }=\frac{\Gamma ^{2}C_{\mathbf{b}}^{2}}{\left( 1-\gamma
\right) ^{2}}\left\vert Q\right\vert _{\sigma }\ .
\end{equation*}
Now we turn to the second stopping criterion
\begin{equation*}
\int_{I}\left\vert T_{\sigma }^{\alpha }\left( b_{S_{0}}\right) \right\vert
^{2}d\omega >\Gamma \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}\left\vert I\right\vert _{\sigma }.
\end{equation*
We hav
\begin{eqnarray*}
\sum_{S\in \mathfrak{C}_{\mathcal{S}}\left( S_{0}\right) }\left\vert
S\right\vert _{\sigma } &\leq &\frac{1}{\Gamma \left( \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}}\sum_{S\in \mathfrak{C}_{\mathcal{S
}\left( S_{0}\right) }\int_{S}\left\vert T_{\sigma }^{\alpha }\left(
b_{S_{0}}\right) \right\vert ^{2}d\omega \\
&\leq &\frac{1}{\Gamma \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}}\sum_{S_{0}}\int_{S}\left\vert T_{\sigma }^{\alpha }\left(
b_{S_{0}}\right) \right\vert ^{2}d\omega \leq \frac{1}{\Gamma }\left\vert
S_{0}\right\vert _{\sigma }.
\end{eqnarray*
Iterating this inequality give
\begin{equation*}
\sum_{\substack{ S\in \mathcal{S} \\ S\subset S_{0}}}\left\vert
S\right\vert _{\sigma }\leq \sum_{k=0}^{\infty }\frac{1}{\Gamma ^{k}
\left\vert S_{0}\right\vert _{\sigma }=\frac{\Gamma }{\Gamma -1}\left\vert
S_{0}\right\vert _{\sigma },
\end{equation*
and the
\begin{equation*}
\sum_{\substack{ S\in \mathcal{S} \\ S\subset \Omega }}\left\vert
S\right\vert _{\sigma }=\sum_{\substack{ \text{maximal }S_{0}\in \mathcal{S}
\\ S_{0}\subset \Omega }}\sum_{\substack{ S\in \mathcal{S} \\ S\subset
S_{0} }}\left\vert S\right\vert _{\sigma }\leq \frac{\Gamma }{\Gamma -1}\sum
_{\substack{ \text{maximal }S_{0}\in \mathcal{S} \\ S_{0}\subset \Omega }
\left\vert S_{0}\right\vert _{\sigma }=\frac{\Gamma }{\Gamma -1}\left\vert
\Omega \right\vert _{\sigma }\ .
\end{equation*}
Finally, for $I\in \mathcal{C}_{S_{0}}$ we have the weak testing inequalit
\begin{equation*}
\int_{I}\left\vert T_{\sigma }^{\alpha }\left( b_{S_{0}}\right) \right\vert
^{2}d\omega \leq \Gamma \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b
}\right) ^{2}\left\vert I\right\vert _{\sigma }\ ,
\end{equation*}
and this completes the proof of Lemma \ref{Car and Test gen}.
\end{proof}
\subsubsection{The energy corona decompositions}
Given a weight pair $\left( \sigma ,\omega \right) $, we now construct an
energy corona decomposition for $\sigma $, and an energy corona
decomposition for $\omega $, that uniformize estimates (c.f. \cite{NTV3},
\cite{LaSaUr2}, \cite{SaShUr6} and \cite{SaShUr7}). In order to define these
constructions, we recall that the energy condition constant $\mathcal{E
_{2}^{\alpha }$ in Definition \ref{def strong quasienergy} is given b
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\equiv \sup_{\substack{ Q\in
\mathcal{P} \\ Q\supset \dot{\cup}J_{r}}}\frac{1}{\left\vert Q\right\vert
_{\sigma }}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( J_{r}
\mathbf{1}_{Q}\sigma \right) }{\left\vert J_{r}\right\vert }\right)
^{2}\left\Vert x-m_{J_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{J_{r}}\omega
\right) }^{2}\ ,
\end{equation*
where $\dot{\cup}J_{r}$ is an arbitary subdecomposition of $Q$ into
intervals $J_{r}\in \mathcal{P}$. In the next definition we restrict the
intervals $Q$ to a dyadic grid $\mathcal{D}$, but keep the subintervals
J_{r}$ unrestricted.
\begin{definition}
\label{def energy corona 3}Given a dyadic grid $\mathcal{D}$ and an interval
$S_{0}\in \mathcal{D}$, define $\mathcal{S}\left( S_{0}\right) $ to be the
\emph{maximal} $\mathcal{D}$-subintervals $I\subset S_{0}$ such tha
\begin{equation}
\sup_{I\supset \dot{\cup}J_{r}}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P
^{\alpha }\left( J_{r},\mathbf{1}_{S_{0}}\sigma \right) }{\left\vert
J_{r}\right\vert }\right) ^{2}\left\Vert x-m_{J_{r}}\right\Vert
_{L^{2}\left( \mathbf{1}_{J_{r}}\omega \right) }^{2}\geq C_{\limfunc{energy}
\left[ \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+\mathfrak{A
_{2}^{\alpha }\right] \ \left\vert I\right\vert _{\sigma },
\label{def stop 3}
\end{equation
where the intervals $J_{r}\in \mathcal{P}$ are pairwise disjoint in $I$,
\mathcal{E}_{2}^{\alpha }$ is the energy condition constant, and $C_
\limfunc{energy}}$ is a sufficiently large positive constant depending only
on $\alpha $. Then define the $\sigma $-energy stopping intervals of $S_{0}$
to be the collection
\begin{equation*}
\mathcal{S}=\left\{ S_{0}\right\} \cup \dbigcup\limits_{n=0}^{\infty
\mathcal{S}_{n}
\end{equation*
where $\mathcal{S}_{0}=\mathcal{S}\left( S_{0}\right) $ and $\mathcal{S
_{n+1}=\dbigcup\limits_{S\in \mathcal{S}_{n}}\mathcal{S}\left( S\right) $
for $n\geq 0$.
\end{definition}
We now claim that from the energy condition $\mathcal{E}_{2}^{\alpha
\mathcal{<\infty }$, we obtain the $\sigma $-Carleson estimate
\begin{equation}
\sum_{S\in \mathcal{S}:\ S\subset I}\left\vert S\right\vert _{\sigma }\leq
2\left\vert I\right\vert _{\sigma },\ \ \ \ \ I\in \mathcal{D}^{\sigma }.
\label{sigma Carleson 3}
\end{equation
Indeed, for any $S_{1}\in \mathcal{S}$ we hav
\begin{eqnarray*}
\sum_{S\in \mathfrak{C}_{\mathcal{S}}\left( S_{1}\right) }\left\vert
S\right\vert _{\sigma } &\leq &\frac{1}{C_{\limfunc{energy}}\left( \mathcal{
}_{2}^{\alpha }\right) ^{2}}\sum_{S\in \mathfrak{C}_{\mathcal{S}}\left(
S_{1}\right) }\sup_{S\supset \dot{\cup}J_{r}}\sum_{r=1}^{\infty }\left(
\frac{\mathrm{P}^{\alpha }\left( J_{r},\mathbf{1}_{S_{1}}\sigma \right) }
\left\vert J_{r}\right\vert }\right) ^{2}\left\Vert x-m_{J_{r}}\right\Vert
_{L^{2}\left( \mathbf{1}_{J_{r}}\omega \right) }^{2} \\
&\leq &\frac{1}{C_{\limfunc{energy}}\left( \mathcal{E}_{2}^{\alpha }\right)
^{2}}\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\left\vert S_{1}\right\vert
_{\sigma }=\frac{1}{C_{\limfunc{energy}}}\left\vert S_{1}\right\vert
_{\sigma }\ ,
\end{eqnarray*
upon noting that the union of the subdecompositions $\dot{\cup}J_{r}\subset
S $ over $S\in \mathfrak{C}_{\mathcal{S}}\left( S_{1}\right) $ is a
subdecomposition of $S_{1}$, and the proof of the Carleson estimate is now
finished by iteration in the standard way.
Finally, we record the reason for introducing energy stopping times. If
\begin{equation}
\mathbf{X}_{\alpha }\left( \mathcal{C}_{S}\right) ^{2}\equiv \sup_{I\in
\mathcal{C}_{S}}\frac{1}{\left\vert I\right\vert _{\sigma }}\sup_{I\supset
\dot{\cup}J_{r}}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left(
J_{r},\mathbf{1}_{S}\sigma \right) }{\left\vert J_{r}\right\vert }\right)
^{2}\left\Vert x-m_{J_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{J_{r}}\omega
\right) }^{2} \label{def stopping energy 3}
\end{equation
is (the square of) the $\alpha $\emph{-stopping energy} of the weight pair
\left( \sigma ,\omega \right) $ with respect to the corona $\mathcal{C}_{S}
, then we have the \emph{stopping energy bounds
\begin{equation}
\mathbf{X}_{\alpha }\left( \mathcal{C}_{S}\right) \leq \sqrt{C_{\limfunc
energy}}}\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+\mathfrak{A
_{2}^{\alpha }},\ \ \ \ \ S\in \mathcal{S}, \label{def stopping bounds 3}
\end{equation
where $\mathfrak{A}_{2}^{\alpha }$ and the energy constant $\mathfrak{E
_{2}^{\alpha }$ are controlled by the assumptions in Theorem \ref{dim one}.
\subsection{Iterated coronas and general stopping data}
We will use a construction that permits \emph{iteration} of the above three
corona decompositions by combining Definitions \ref{CZ stopping times}, \re
{accretive stopping times gen} and \ref{def energy corona 3} into a single
stopping condition. However, there is one remaining difficulty with the
triple corona constructed in this way, namely if a stopping interval $I\in
\mathcal{A}$ is a child of an interval $Q$ in the corona $\mathcal{C}_{A}$,
then the modulus of the average $\left\vert \frac{1}{\left\vert I\right\vert
_{\sigma }}\int_{I}b_{Q}d\sigma \right\vert $ of $b_{Q}$ on $I$ may be far
smaller than the sup norm of $\left\vert b_{Q}\right\vert $ on the child $I
, indeed it may be that $\frac{1}{\left\vert I\right\vert _{\sigma }
\int_{I}b_{Q}d\sigma =0$. This of course destroys any reasonable estimation
of the martingale and dual martingale differences $\bigtriangleup
_{Q}^{\sigma ,\mathbf{b}}f$ and $\square _{Q}^{\sigma ,\mathbf{b}}f$ used in
the proof of Theorem \ref{dim one}, and so we will use Lemma \ref{prelim
control of corona} on the function $b_{A}$ to obtain a new function
\widetilde{b}_{A}$ for which this problem is circumvented at the `bottom' of
the corona, i.e. for those $A^{\prime }\in C_{\mathcal{A}}\left( A\right) $.
We then refer to the stopping times $A^{\prime }\in C_{\mathcal{A}}\left(
A\right) $ as `shadow' stopping times since we have lost control of the weak
testing condition relative to the new function $\widetilde{b}_{A}$. Thus we
must redo the weak testing stopping times for the new function $\widetilde{b
_{A}$, but also stopping if we hit one of the shadow stopping times. Here
are the details.
\begin{definition}
\label{def shadow}Let $C_{0}\geq 4$, $0<\gamma <1$ and $1<\Gamma <\infty $.
Suppose that $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ is an
\infty $-strongly $\sigma $-accretive family on $\mathbb{R}$. Given a dyadic
grid $\mathcal{D}$ and an interval $Q\in \mathcal{D}$, define the collection
of `shadow' stopping times $\mathcal{S}_{\limfunc{shadow}}\left( Q\right) $
to be the \emph{maximal} $\mathcal{D}$-subintervals $I\subset Q$ such that
eithe
\begin{equation*}
\frac{1}{\left\vert I\right\vert _{\sigma }}\int_{I}\left\vert f\right\vert
d\sigma >C_{0}\frac{1}{\left\vert Q\right\vert _{\sigma }}\int_{Q}\left\vert
f\right\vert d\sigma \ ,
\end{equation*
o
\begin{equation*}
\left\vert \frac{1}{\left\vert I\right\vert _{\mu }}\int_{I}b_{Q}d\sigma
\right\vert <\gamma \ \text{or }\int_{I}\left\vert T_{\sigma }^{\alpha
}\left( b_{Q}\right) \right\vert ^{2}d\omega >\Gamma \left( \mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert I\right\vert _{\sigma
\text{ },
\end{equation*
o
\begin{equation*}
\sup_{I\supset \dot{\cup}J_{r}}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P
^{\alpha }\left( J_{r},\sigma \right) }{\left\vert J_{r}\right\vert }\right)
^{2}\left\Vert x-m_{J_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{J_{r}}\omega
\right) }^{2}\geq C_{\limfunc{energy}}\left[ \left( \mathcal{E}_{2}^{\alpha
\mathbf{b},\mathbf{b}^{\ast }}\right) ^{2}+\mathfrak{A}_{2}^{\alpha }\right]
\ \left\vert I\right\vert _{\sigma }\ .
\end{equation*}
\end{definition}
Now we apply Lemma \ref{prelim control of corona}\ to the function $b_{Q}$
with the subdecomposition $\mathcal{S}_{\limfunc{shadow}}\left( Q\right)
\equiv \left\{ Q_{i}\right\} _{i=1}^{\infty }$ to obtain a new function
\widetilde{b}_{Q}$ satisfying the propertie
\begin{eqnarray}
&&\limfunc{support}\widetilde{b}_{Q}\subset Q\ , \label{props} \\
&&1\leq \frac{1}{\left\vert Q^{\prime }\right\vert _{\sigma }
\int_{Q^{\prime }}\widetilde{b}_{Q}d\sigma \leq \left\Vert \mathbf{1
_{Q^{\prime }}\widetilde{b}_{Q}\right\Vert _{L^{\infty }\left( \sigma
\right) }\leq 2\left( 1+\sqrt{C_{\mathbf{b}}}\right) C_{\mathbf{b}}\ ,\ \ \
\ \ Q^{\prime }\in \mathcal{C}_{Q}\ , \notag \\
&&\sqrt{\int_{Q}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega
}\leq \left[ 2\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\left( Q\right) +4C_
\mathbf{b}}^{\frac{3}{2}}\delta ^{\frac{1}{4}}\mathfrak{N}_{T^{\alpha
}}\left( Q\right) \right] \sqrt{\left\vert Q\right\vert _{\sigma }}\ ,
\notag \\
&&\left\Vert \mathbf{1}_{Q_{i}}\widetilde{b}_{Q}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq \frac{16C_{\mathbf{b}}}{\delta }\left\vert
\frac{1}{\left\vert Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}\widetilde{b
_{Q}d\sigma \right\vert \ ,\ \ \ \ \ 1\leq i<\infty . \notag
\end{eqnarray
Note that each of the functions $\widetilde{b}_{Q^{\prime }}\equiv \mathbf{1
_{Q^{\prime }}\widetilde{b}_{Q}$, for $Q^{\prime }\in \mathcal{C}_{Q}$, now
satisfies the crucial reverse H\"{o}lder propert
\begin{equation*}
\left\Vert \mathbf{1}_{I}\widetilde{b}_{Q^{\prime }}\right\Vert _{L^{\infty
}\left( \sigma \right) }\leq C_{\delta ,\mathbf{b}}\left\vert \frac{1}
\left\vert I\right\vert _{\sigma }}\int_{I}\widetilde{b}_{Q^{\prime
}}d\sigma \right\vert \ ,\ \ \ \ \ \text{for all }I\in \mathfrak{C}\left(
Q^{\prime }\right) ,\ Q^{\prime }\in \mathcal{C}_{Q}.
\end{equation*
Indeed, if $I$ equals one of the $Q_{i}$ then the reverse H\"{o}lder
condition in the last line of (\ref{props}) applies, while if $I\in \mathcal
C}_{Q}$ then the accretivity in the second line of (\ref{props}) applies.
Since we have lost the weak testing condition in the corona for this new
function $\widetilde{b}_{Q}$, the next step is to run again the weak testing
construction of stopping times, but this time starting with the new function
$\widetilde{b}_{Q}$, and also stopping if we hit one of the `shadow'
stopping times $Q_{i}$. Here is the new stopping criterion.
\begin{definition}
\label{def iterated}Let $C_{0}\geq 4$ and $1<\Gamma <\infty $. Let $\mathcal
S}_{\limfunc{shadow}}\left( Q\right) \equiv \left\{ Q_{i}\right\}
_{i=1}^{\infty }$ be as in Definition \ref{def shadow}. Define $\mathcal{S}_
\limfunc{iterated}}\left( Q\right) $ to be the \emph{maximal} $\mathcal{D}
-subintervals $I\subset Q$ such that eithe
\begin{equation*}
\int_{I}\left\vert T_{\sigma }^{\alpha }\left( \widetilde{b}_{Q}\right)
\right\vert ^{2}d\omega >\Gamma \left( \mathfrak{T}_{T^{\alpha }}^
\widetilde{\mathbf{b}}}\right) ^{2}\left\vert I\right\vert _{\sigma }\text{
,
\end{equation*
o
\begin{equation*}
I=Q_{i}\text{ for some }1\leq i<\infty .
\end{equation*}
\end{definition}
Thus for each interval $Q$ we have now constructed \emph{iterated stopping
children} $\mathcal{S}_{\limfunc{iterated}}\left( Q\right) $ by first
constructing shadow stopping times $\mathcal{S}_{\limfunc{shadow}}\left(
Q\right) $ using one step of the triple corona construction, then modifying
the testing function to have reverse H\"{o}lder controlled children, and
finally running again the weak testing stopping time construction to get
\mathcal{S}_{\limfunc{iterated}}\left( Q\right) $. These iterated stopping
times $\mathcal{S}_{\limfunc{iterated}}\left( Q\right) $ have control of CZ
averages of $f$ and energy control of $\sigma $ and $\omega $, simply
because these controls were achieved in the shadow construction, and were
unaffected by either the application of Lemma \ref{prelim control of corona}
or the rerunning of the weak testing stopping criterion for $\widetilde{b
_{Q}$. And of course we now have weak testing within the corona determined
by $Q$ and $\mathcal{S}_{\limfunc{iterated}}\left( Q\right) $, and we also
have the crucial reverse H\"{o}lder condition on all the children of
intervals in the corona. With all of this in hand, here then is the
definition of the construction of iterated coronas.
\begin{definition}
\label{iterated stopping times}Let $C_{0}\geq 4$, $0<\gamma <1$ and
1<\Gamma <\infty $. Suppose that $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in
\mathcal{P}}$ is an $\infty $-strongly $\sigma $-accretive family on
\mathbb{R}$. Given a dyadic grid $\mathcal{D}$ and an interval $S_{0}$ in
\mathcal{D}$, define the iterated stopping intervals of $S_{0}$ to be the
collection
\begin{equation*}
\mathcal{S}=\left\{ S_{0}\right\} \cup \dbigcup\limits_{n=0}^{\infty
\mathcal{S}_{n}
\end{equation*
where $\mathcal{S}_{0}=\mathcal{S}_{\limfunc{iterated}}\left( S_{0}\right) $
and $\mathcal{S}_{n+1}=\dbigcup\limits_{S\in \mathcal{S}_{n}}\mathcal{S}_
\limfunc{iterated}}\left( S\right) $ for $n\geq 0$, and where $\mathcal{S}_
\limfunc{iterated}}\left( Q\right) $ is defined in Definition \ref{def
iterated}.
\end{definition}
It is useful to append to the notion of stopping times $\mathcal{S}$ in the
above $\sigma $-iterated corona decomposition a positive constant $A_{0}$
and an additional structure $\alpha _{\mathcal{S}}$ called stopping bounds
for a function $f$. We will refer to the resulting\ triple $\left( A_{0}
\mathcal{F},\alpha _{\mathcal{F}}\right) $ as constituting stopping data for
$f$. If $\mathcal{F}$ is a grid, we define $F^{\prime }\prec F$ if
F^{\prime }\subsetneqq F$ and $F^{\prime },F\in \mathcal{F}$. Recall that
\pi _{\mathcal{F}}F^{\prime }$ is the smallest $F\in \mathcal{F}$ such that
F^{\prime }\prec F$.
\begin{definition}
\label{general stopping data}Suppose we are given a positive constant
A_{0}\geq 4$, a subset $\mathcal{F}$ of the dyadic grid $\mathcal{D}$
(called the stopping times), and a corresponding sequence $\alpha _{\mathcal
F}}\equiv \left\{ \alpha _{\mathcal{F}}\left( F\right) \right\} _{F\in
\mathcal{F}}$ of nonnegative numbers $\alpha _{\mathcal{F}}\left( F\right)
\geq 0$ (called the stopping bounds). Let $\left( \mathcal{F},\prec ,\pi _
\mathcal{F}}\right) $ be the tree structure on $\mathcal{F}$ inherited from
\mathcal{D}$, and for each $F\in \mathcal{F}$ denote by $\mathcal{C
_{F}=\left\{ I\in \mathcal{D}:\pi _{\mathcal{F}}I=F\right\} $ the corona
associated with $F$:
\begin{equation*}
\mathcal{C}_{F}=\left\{ I\in \mathcal{D}:I\subset F\text{ and }I\not\subset
F^{\prime }\text{ for any }F^{\prime }\prec F\right\} .
\end{equation*
We say the triple $\left( A_{0},\mathcal{F},\alpha _{\mathcal{F}}\right) $
constitutes \emph{stopping data} for a function $f\in L_{loc}^{1}\left(
\sigma \right) $ if
\begin{enumerate}
\item $\mathbb{E}_{I}^{\sigma }\left\vert f\right\vert \leq \alpha _
\mathcal{F}}\left( F\right) $ for all $I\in \mathcal{C}_{F}$ and $F\in
\mathcal{F}$,
\item $\sum_{F^{\prime }\preceq F}\left\vert F^{\prime }\right\vert _{\sigma
}\leq A_{0}\left\vert F\right\vert _{\sigma }$ for all $F\in \mathcal{F}$,
\item $\sum_{F\in \mathcal{F}}\alpha _{\mathcal{F}}\left( F\right)
^{2}\left\vert F\right\vert _{\sigma }\mathbf{\leq }A_{0}^{2}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }^{2}$,
\item $\alpha _{\mathcal{F}}\left( F\right) \leq \alpha _{\mathcal{F}}\left(
F^{\prime }\right) $ whenever $F^{\prime },F\in \mathcal{F}$ with $F^{\prime
}\subset F$.
\end{enumerate}
\end{definition}
Property (1) says that $\alpha _{\mathcal{F}}\left( F\right) $ bounds the
averages of $f$ in the corona $\mathcal{C}_{F}$, and property (2) says that
the intervals at the tops of the coronas satisfy a Carleson condition
relative to the weight $\sigma $. Note that a standard `maximal interval'
argument extends the Carleson condition in property (2) to the inequalit
\begin{equation}
\sum_{F^{\prime }\in \mathcal{F}:\ F^{\prime }\subset A}\left\vert F^{\prime
}\right\vert _{\sigma }\leq A_{0}\left\vert A\right\vert _{\sigma }\text{
for all open sets }A\subset \mathbb{R}. \label{Car ext}
\end{equation
Property (3) is the quasi-orthogonality condition that says the sequence of
functions $\left\{ \alpha _{\mathcal{F}}\left( F\right) \mathbf{1
_{F}\right\} _{F\in \mathcal{F}}$ is in the vector-valued space $L^{2}\left(
\ell ^{2};\sigma \right) $ with control, and is often referred to as a
Carleson embedding theorem, and property (4) says that the control on
stopping data is nondecreasing on the stopping tree $\mathcal{F}$. We
emphasize that we are \emph{not} assuming in this definition the stronger
property that there is $C>1$ such that $\alpha _{\mathcal{F}}\left(
F^{\prime }\right) >C\alpha _{\mathcal{F}}\left( F\right) $ whenever
F^{\prime },F\in \mathcal{F}$ with $F^{\prime }\subsetneqq F$. Instead, the
properties (2) and (3) substitute for this lack. Of course the stronger
property \emph{does} hold for the familiar \emph{Calder\'{o}n-Zygmund}
stopping data determined by the following requirements for $C>1$
\begin{equation*}
\mathbb{E}_{F^{\prime }}^{\sigma }\left\vert f\right\vert >C\mathbb{E
_{F}^{\sigma }\left\vert f\right\vert \text{ whenever }F^{\prime },F\in
\mathcal{F}\text{ with }F^{\prime }\subsetneqq F,\ \ \ \ \ \mathbb{E
_{I}^{\sigma }\left\vert f\right\vert \leq C\mathbb{E}_{F}^{\sigma
}\left\vert f\right\vert \text{ for }I\in \mathcal{C}_{F},
\end{equation*
which are themselves sufficiently strong to automatically force properties
(2) and (3) with $\alpha _{\mathcal{F}}\left( F\right) =\mathbb{E
_{F}^{\sigma }\left\vert f\right\vert $.
We have the following useful consequence of (2) and (3) that says the
sequence $\left\{ \alpha _{\mathcal{F}}\left( F\right) \mathbf{1
_{F}\right\} _{F\in \mathcal{F}}$ has a \emph{quasi-orthogonal} property
relative to $f$ with a constant $A_{0}^{\prime }$ depending only on $A_{0}$
(see e.g. \cite{SaShUr7})
\begin{equation}
\left\Vert \sum_{F\in \mathcal{F}}\alpha _{\mathcal{F}}\left( F\right)
\mathbf{1}_{F}\right\Vert _{L^{2}\left( \sigma \right) }^{2}\leq
A_{0}^{\prime }\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }^{2}.
\label{q orth}
\end{equation}
\begin{proposition}
\label{data}Let $f\in L^{2}\left( \sigma \right) $, let $\mathcal{F}$ be the
iterated corona $\mathcal{S}\left( S_{0}\right) $ in Definition \re
{iterated stopping times}, and define stopping data $\alpha _{\mathcal{F}}$
by $\alpha _{F}=\frac{1}{\left\vert F\right\vert _{\sigma }
\int_{F}\left\vert f\right\vert d\sigma $. Then there is $A_{0}\geq 4$,
depending only on the constant $C_{0}$ in Definition \ref{CZ stopping times
, such that the triple $\left( A_{0},\mathcal{F},\alpha _{\mathcal{F
}\right) $ constitutes \emph{stopping data} for the function $f$.
\end{proposition}
\begin{proof}
This is an easy exercise using (\ref{CZ Car}), (\ref{Car gen}) and (\re
{sigma Carleson 3}), and is left for the reader.
\end{proof}
\subsection{Grid parameterizations}
It is important to use \emph{two} independent random grids, one for each
function $f$ and $g$ simultaneously, as this is necessary in order to apply
probabilistic methods to the dual martingale averages $\square _{I}^{\mu
\mathbf{b}}$ that depend, not only on $I$, but also on the underlying grid
in which $I$ lives. The proof methods for functional energy from \cit
{SaShUr7} and \cite{SaShUr6} relied heavily on the use of a single grid, and
this must now be modified to accommodate two independent grids.
Now we recall the construction from our paper \cite{SaShUr10}. We
momentarily fix a large positive integer $M\in \mathbb{N}$, and consider the
tiling of $\mathbb{R}$ by the family of intervals $\mathbb{D}_{M}\equiv
\left\{ I_{\alpha }^{M}\right\} _{\alpha \in \mathbb{Z}}$ having side length
$2^{-M}$ and given by $I_{\alpha }^{M}\equiv I_{0}^{M}+2^{-M}\alpha $ where
I_{0}^{M}=\left[ 0,2^{-M}\right) $. A \emph{dyadic grid} $\mathcal{D}$ built
on $\mathbb{D}_{M}$ is\ defined to be a family of intervals $\mathcal{D}$
satisfying:
\begin{enumerate}
\item Each $I\in \mathcal{D}$ has side length $2^{-\ell }$ for some $\ell
\in \mathbb{Z}$ with $\ell \leq M$, and $I$ is a union of $2^{M-\ell }$
intervals from the tiling $\mathbb{D}_{M}$,
\item For $\ell \leq M$, the collection $\mathcal{D}_{\ell }$ of intervals
in $\mathcal{D}$ having side length $2^{-\ell }$ forms a pairwise disjoint
decomposition of the space $\mathbb{R}$,
\item Given $I\in \mathcal{D}_{i}$ and $J\in \mathcal{D}_{j}$ with $j\leq
i\leq M$, it is the case that either $I\cap J=\emptyset $ or $I\subset J$.
\end{enumerate}
We now momentarily fix a \emph{negative} integer $N\in -\mathbb{N}$, and
restrict the above grids to intervals of side length at most $2^{-N}$
\begin{equation*}
\mathcal{D}^{N}\equiv \left\{ I\in \mathcal{D}:\text{side length of }I\text{
is at most }2^{-N}\right\} \text{.}
\end{equation*
We refer to such grids $\mathcal{D}^{N}$ as a (truncated) dyadic grid
\mathcal{D}$ built on $\mathbb{D}_{M}$ of size $2^{-N}$. There are now two
traditional means of constructing probability measures on collections of
such dyadic grids, namely parameterization by choice of parent, and
parameterization by translation.
\textbf{Construction \#1}: For any
\begin{equation*}
\beta =\{\beta _{i}\}_{i\in _{M}^{N}}\in \omega _{M}^{N}\equiv \left\{
0,1\right\} ^{\mathbb{Z}_{M}^{N}},
\end{equation*
where $\mathbb{Z}_{M}^{N}\equiv \left\{ \ell \in \mathbb{Z}:N\leq \ell \leq
M\right\} $, define the dyadic grid $\mathcal{D}_{\beta }$ built on $\mathbb
D}_{m}$ of size $2^{-N}$ by
\begin{equation}
\mathcal{D}_{\beta }=\left\{ 2^{-\ell }\left( [0,1)+k+\sum_{i:\ \ell <i\leq
m}2^{-i+\ell }\beta _{i}\right) \right\} _{N\leq \ell \leq m,\,k\in {\mathbb
Z}}}\ . \label{def dyadic grid}
\end{equation
Place the uniform probability measure $\rho _{M}^{N}$ on the finite index
space $\omega _{M}^{N}=\left\{ 0,1\right\} ^{\mathbb{Z}_{M}^{N}}$, namely
that which charges each $\beta \in \omega _{M}^{N}$ equally.
\textbf{Construction \#2}: Momentarily fix a (truncated) dyadic grid
\mathcal{D}$ built on $\mathbb{D}_{M}$ of size $2^{-N}$. For any
\begin{equation*}
\gamma \in \Gamma _{M}^{N}\equiv \left\{ \gamma \in 2^{-M}\mathbb{Z
_{+}:\left\vert \gamma \right\vert <2^{-N}\right\} ,
\end{equation*
define the dyadic grid $\mathcal{D}^{\gamma }$ built on $\mathbb{D}_{M}$ of
size $2^{-N}$ b
\begin{equation*}
\mathcal{D}^{\gamma }\equiv \mathcal{D}+\gamma .
\end{equation*
Place the uniform probability measure $\nu _{M}^{N}$ on the finite index set
$\Gamma _{M}^{N}$, namely that which charges each multiindex $\gamma $ in
\Gamma _{M}^{N}$ equally.
The two probability spaces $\left( \left\{ \mathcal{D}_{\beta }\right\}
_{\beta \in \Omega _{M}^{N}},\mu _{M}^{N}\right) $ and $\left( \left\{
\mathcal{D}^{\gamma }\right\} _{\gamma \in \Gamma _{M}^{N}},\nu
_{M}^{N}\right) $ are isomorphic since both collections $\left\{ \mathcal{D
_{\beta }\right\} _{\beta \in \Omega _{M}^{N}}$ and $\left\{ \mathcal{D
^{\gamma }\right\} _{\gamma \in \Gamma _{M}^{N}}$ describe the set
\boldsymbol{A}_{M}^{N}$ of \textbf{all} (truncated) dyadic grids $\mathcal{D
^{\gamma }$ built on $\mathbb{D}_{m}$ of size $2^{-N}$, and since both
measures $\mu _{M}^{N}$ and $\nu _{M}^{N}$ are the uniform measure on this
space. The first construction may be thought of as being \emph{parameterized
by scales} - each component $\beta _{i}$ in $\beta =\{\beta _{i}\}_{i\in
_{M}^{N}}\in \omega _{M}^{N}$ amounting to a choice of the two possible
tilings at level $i$ that respect the choice of tiling at the level below -
and since any grid in $\boldsymbol{A}_{M}^{N}$ is determined by a choice of
scales , we see that $\left\{ \mathcal{D}_{\beta }\right\} _{\beta \in
\Omega _{M}^{N}}=\boldsymbol{A}_{M}^{N}$. The second construction may be
thought of as being \emph{parameterized by translation} - each $\gamma \in
\Gamma _{M}^{N}$ amounting to a choice of translation of the grid $\mathcal{
}$ fixed in construction \#2\ - and since any grid in $\boldsymbol{A
_{M}^{N} $ is determined by any of the intervals at the top level, i.e. with
side length $2^{-N}$, we see that $\left\{ \mathcal{D}^{\gamma }\right\}
_{\gamma \in \Gamma _{M}^{N}}=\boldsymbol{A}_{M}^{N}$ as well, since every
interval at the top level in $\boldsymbol{A}_{M}^{N}$ has the form $Q+\gamma
$ for some $\gamma \in \Gamma _{M}^{N}$ and $Q\in \mathcal{D}$ at the top
level in $\boldsymbol{A}_{M}^{N}$ (i.e. every interval at the top level in
\boldsymbol{A}_{M}^{N}$ is a union of small intervals in $\mathbb{D}_{m}$,
and so must be a translate of some $Q\in \mathcal{D}$ by an amount $2^{-M}$
times an element of $\mathbb{Z}_{+}$). Note also that $\#\Omega
_{M}^{N}=\#\Gamma _{M}^{N}=2^{M-N}$. We will use $\boldsymbol{E}_{\Omega
_{M}^{N}}$ to denote expectation with respect to this common probability
measure on $\boldsymbol{A}_{M}^{N}$.
\begin{notation}
\label{suppress M and N}For purposes of notation and clarity, we now
suppress all reference to $M$ and $N$ in our families of grids, and in the
notations $\Omega $ and $\Gamma $ for the parameter sets, and we use
\boldsymbol{P}_{\Omega }$ and $\boldsymbol{E}_{\Omega }$ to denote
probability and expectation with respect to families of grids, and instead
proceed as if all grids considered are unrestricted. The careful reader can
supply the modifications necessary to handle the assumptions made above on
the grids $\mathcal{D}$ and the functions $f$ and $g$ regarding $M$ and $N$.
\end{notation}
\subsection{The Monotonicity Lemma}
As in virtually all proofs of a two weight $T1$ theorem (see e.g. \cit
{LaSaShUr3}, \cite{Lac}, \cite{SaShUr7} and/or \cite{SaShUr6}), the key to
starting an estimate for any of the forms we consider below, is the
Monotonicity Lemma and the Energy Lemma, to which we now turn. In dimension
n=1$ (\cite{LaSaShUr3}, \cite{Lac}) the Haar functions have opposite sign on
their children, and this was exploited in a simple but powerful monotonicity
argument. In higher dimensions, this simple argument no longer holds and
that Monotonicity Lemma is replaced with the Lacey-Wick formulation of the
Monotonicity Lemma (see \cite{LaWi}, and also \cite{SaShUr6}) involving the
smaller Poisson operator. As the martingale differences with test functions
b_{Q\,}$ here are no longer of one sign on children, we will adapt the
Lacey-Wick formulation of the Monotonicity Lemma to the operator $T^{\alpha
} $ and the dual martingale differences $\left\{ \square _{J}^{\omega
\mathbf{b}^{\ast }}\right\} _{J\in \mathcal{G}}$, bearing in mind that the
operators $\square _{J}^{\omega ,\mathbf{b}^{\ast }}$ are no longer
projections, which results in only a one-sided estimate with additional
terms on the right hand side. It is here that we need the crucial property
that $\limfunc{Range}\square _{J}^{\omega ,\mathbf{b}^{\ast }}$ is
orthogonal to constants, $\int \left( \square _{J}^{\omega ,\mathbf{b}^{\ast
}}\Psi \right) d\sigma =\int \left( \triangle _{J}^{\sigma ,\mathbf{b}^{\ast
}}1\right) \Psi d\omega =\int \left( 0\right) \Psi d\omega =0$ (see Appendix
A). See Definition \ref{controlled accretive} in Appendix A for the
terminology `$p$-weakly $\mu $-controlled accretive family' along with more
detail on martingale and dual martingale expansions.
Recall from Appendix A tha
\begin{eqnarray*}
\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\mathbf{1
_{Q}\left( x\right) \frac{1}{\int_{Q}b_{Q}d\mu }\int_{Q}fb_{Q}d\mu ,\ \ \ \
\ Q\in \mathcal{P}\ , \\
\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\mathbf{1
_{Q}\left( x\right) b_{Q}\left( x\right) \frac{1}{\int_{Q}b_{Q}d\mu
\int_{Q}fd\mu ,\ \ \ \ \ Q\in \mathcal{P}\ ,
\end{eqnarray*
and
\begin{eqnarray*}
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left(
\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E}_{Q^{\prime
}}^{\mu ,\mathbf{b}}f\left( x\right) \right) -\mathbb{E}_{Q}^{\mu ,\mathbf{b
}f\left( x\right) =\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbf
1}_{Q^{\prime }}\left( x\right) \left( \mathbb{E}_{Q^{\prime }}^{\mu
\mathbf{b}}f\left( x\right) -\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left(
x\right) \right) , \\
\square _{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left(
\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{F}_{Q^{\prime
}}^{\mu ,\mathbf{b}}f\left( x\right) \right) -\mathbb{F}_{Q}^{\mu ,\mathbf{b
}f\left( x\right) =\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbf
1}_{Q^{\prime }}\left( x\right) \left( \mathbb{F}_{Q^{\prime }}^{\mu
\mathbf{b}}f\left( x\right) -\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left(
x\right) \right) ,
\end{eqnarray*
and from (\ref{Carleson avg op})
\begin{equation*}
\nabla _{Q}^{\mu }h=\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken
}\left( Q\right) }\left( E_{Q^{\prime }}^{\mu }\left\vert h\right\vert
\right) ^{2}\mathbf{1}_{Q^{\prime }}\ .
\end{equation*}
We will also need the smaller Poisson integral used in the Lacey-Wick
formulation of the Monontonicity Lemma
\begin{equation*}
\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mu \right) \equiv \int \frac
\left\vert J\right\vert ^{\frac{1+\delta }{n}}}{\left( \left\vert
J\right\vert +\left\vert y-c_{J}\right\vert \right) ^{n+1+\delta -\alpha }
d\mu \left( y\right) ,
\end{equation*
which is discussed in more detail below.
\begin{lemma}[Monotonicity Lemma]
\label{mono}Suppose that$\ I$ and $J$ are intervals in $\mathbb{R}$ such
that $J\subset \gamma J\subset I$ for some $\gamma >1$, and that $\mu $ is a
signed measure on $\mathbb{R}$ supported outside $I$. Let $0<\delta <1$ and
let $\Psi \in L^{2}\left( \omega \right) $. Finally suppose that $T^{\alpha
} $ is a standard fractional singular integral on $\mathbb{R}$ as in \cit
{SaShUr6}, \cite{SaShUr7} and \cite{SaShUr9} with $0\leq \alpha <1$, and
suppose that $\mathbf{b}^{\ast }$ is an $\infty $-weakly $\mu $-controlled
accretive family on $\mathbb{R}$. Then we have the estimat
\begin{equation}
\left\vert \left\langle T^{\alpha }\mu ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}\Psi \right\rangle _{\omega }\right\vert \lesssim C_{\mathbf{b
^{\ast }}C_{CZ}\ \Phi ^{\alpha }\left( J,\left\vert \mu \right\vert \right)
\ \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }, \label{estimate}
\end{equation
wher
\begin{eqnarray*}
\Phi ^{\alpha }\left( J,\left\vert \mu \right\vert \right) &\equiv &\frac
\mathrm{P}^{\alpha }\left( J,\left\vert \mu \right\vert \right) }{\left\vert
J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }+\frac{\mathrm{P
_{1+\delta }^{\alpha }\left( J,\left\vert \mu \right\vert \right) }
\left\vert J\right\vert }\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf
1}_{J}\omega \right) }, \\
\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &\equiv &\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{2}+\inf_{z\in \mathbb{R}}\sum_{J^{\prime }\in \mathfrak{C
_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime }\right\vert
_{\omega }\left( E_{J^{\prime }}^{\omega }\left\vert x-z\right\vert \right)
^{2}, \\
\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert
_{L^{2}\left( \mu \right) }^{\bigstar 2} &\equiv &\left\Vert \square
_{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert _{L^{2}\left( \mu \right)
}^{2}+\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left[ E_{J^{\prime }}^{\omega
}\left\vert \Psi \right\vert \right] ^{2}.
\end{eqnarray*
All of the implied constants above depend only on $\gamma >1$, $0<\delta <1$
and $0<\alpha <1$.
\end{lemma}
Using $\bigtriangledown _{J}^{\omega }h=\sum_{J^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( JQ\right) }\left( E_{J^{\prime }}^{\omega
}\left\vert h\right\vert \right) ^{2}\mathbf{1}_{J^{\prime }}$ defined in
\ref{Carleson avg op}) in Appendix A, we can rewrite the expressions
\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}$ and $\left\Vert \square
_{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert _{L^{2}\left( \mu \right)
}^{\bigstar 2}$ a
\begin{eqnarray*}
\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &\equiv &\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{2}+\inf_{z\in \mathbb{R}}\left\Vert \bigtriangledown
_{J}^{\omega }\left( x-z\right) \right\Vert _{L^{2}\left( \omega \right)
}^{2}, \\
\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert
_{L^{2}\left( \mu \right) }^{\bigstar 2} &\equiv &\left\Vert \square
_{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert _{L^{2}\left( \mu \right)
}^{2}+\left\Vert \bigtriangledown _{J}^{\omega }\Psi \right\Vert
_{L^{2}\left( \omega \right) }^{2}.
\end{eqnarray*}
\begin{proof}
We also use formulas (\ref{def pi box}), (\ref{square of delta}) and the
estimate (\ref{F est}) from Appendix A
\begin{eqnarray*}
\square _{Q}^{\mu ,\pi ,\mathbf{b}}f &=&\left[ \sum_{Q^{\prime }\in
\mathfrak{C}\left( Q\right) }\mathbb{F}_{Q^{\prime }}^{\mu ,\pi ,\mathbf{b}}
\right] -\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f=\sum_{Q^{\prime }\in \mathfrak{C
\left( Q\right) }\mathbb{F}_{Q^{\prime }}^{\mu ,b_{Q}}f-\mathbb{F}_{Q}^{\mu
,b_{Q}}f, \\
\mathbb{F}_{Q}^{\mu ,\pi ,\mathbf{b}}f &=&\mathbf{1}_{Q}\frac{b_{\pi Q}}
\int_{Q}b_{\pi Q}d\mu }\int_{Q}fd\mu , \\
\square _{Q}^{\mu ,\mathbf{b}} &=&\square _{Q}^{\mu ,\pi ,\mathbf{b}}\square
_{Q}^{\mu ,\pi ,\mathbf{b}}+\square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b}
\text{ and }\square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b}}f=\sum_{Q^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right) }\mathbb{F}_{Q^{\prime
}}^{\mu ,b_{Q^{\prime }}}f-\mathbb{F}_{Q^{\prime }}^{\mu ,b_{Q}}f, \\
\left\vert \square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b}}f\right\vert
&\lesssim &\left\vert \bigtriangledown _{Q}^{\mu }f\right\vert ,
\end{eqnarray*
with similar equalities and inequalities for $\bigtriangleup $ and $\mathbb{
}$. Here $\mathfrak{C}_{\limfunc{broken}}\left( Q\right) $ denotes the set
of broken children, i.e. those $Q^{\prime }\in \mathfrak{C}\left( Q\right) $
for which $b_{Q^{\prime }}\neq \mathbf{1}_{Q^{\prime }}b_{Q}$, and more
generally and typically, $\mathfrak{C}_{\limfunc{broken}}\left( Q\right)
\mathfrak{C}\left( Q\right) \cap \mathcal{A}$ where $\mathcal{A}$ is a
collection of stopping intervals that includes the broken children and
satisfies a $\sigma $-Carleson condition. Using $\square _{J}^{\omega
\mathbf{b}^{\ast }}=\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\square
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}+\square _{J,\limfunc{broken}}^{\omega
,\mathbf{b}^{\ast }}$, we writ
\begin{eqnarray*}
\left\vert \left\langle T^{\alpha }\mu ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}\Psi \right\rangle _{\omega }\right\vert &=&\left\vert \left\langle
T^{\alpha }\mu ,\left( \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\square
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}+\square _{J,\limfunc{broken}}^{\omega
,\mathbf{b}^{\ast }}\right) \Psi \right\rangle _{\omega }\right\vert \\
&\leq &\left\vert \left\langle T^{\alpha }\mu ,\square _{J}^{\omega ,\pi
\mathbf{b}^{\ast }}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi
\right\rangle _{\omega }\right\vert +\left\vert \left\langle T^{\alpha }\mu
,\square _{J,\limfunc{broken}}^{\omega ,\mathbf{b}^{\ast }}\Psi
\right\rangle _{\omega }\right\vert \equiv I+II.
\end{eqnarray*
Since $\left\langle 1,\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast
}}h\right\rangle _{\omega }=0$, we use $m_{J}=\frac{1}{\left\vert
J\right\vert _{\omega }}\int_{J}xd\omega \left( x\right) $ to obtai
\begin{equation*}
T^{\alpha }\mu \left( x\right) -T^{\alpha }\mu \left( m_{J}\right) =\int
\left[ \left( K^{\alpha }\right) \left( x,y\right) -\left( K^{\alpha
}\right) \left( m_{J},y\right) \right] d\mu \left( y\right) =\int \left[
\left( K_{y}^{\alpha }\right) ^{\prime }\left( \theta \left( x,m_{J}\right)
\right) \left( x-m_{J}\right) \right] d\mu \left( y\right)
\end{equation*
for some $\theta \left( x,m_{J}\right) \in J$ to obtai
\begin{eqnarray*}
I &=&\left\vert \int \left[ T^{\alpha }\mu \left( x\right) -T^{\alpha }\mu
\left( m_{J}\right) \right] \ \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast
}}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right) d\omega
\left( x\right) \right\vert \\
&=&\left\vert \int \left\{ \int \left[ \left( K_{y}^{\alpha }\right)
^{\prime }\left( \theta \left( x,m_{J}\right) \right) \right] d\mu \left(
y\right) \right\} \ \left( x-m_{J}\right) \ \square _{J}^{\omega ,\pi
\mathbf{b}^{\ast }}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left(
x\right) d\omega \left( x\right) \right\vert \\
&\leq &\left\vert \int \left\{ \int \left[ \left( K_{y}^{\alpha }\right)
^{\prime }\left( m_{J}\right) \right] d\mu \left( y\right) \right\} \ \left(
x-m_{J}\right) \ \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\square
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right) d\omega \left(
x\right) \right\vert \\
&&+\left\vert \int \left\{ \int \left[ \left( K_{y}^{\alpha }\right)
^{\prime }\left( \theta \left( x,m_{J}\right) \right) -\left( K_{y}^{\alpha
}\right) ^{\prime }\left( m_{J}\right) \right] d\mu \left( y\right) \right\}
\ \left( x-m_{J}\right) \ \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast
}}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right) d\omega
\left( x\right) \right\vert \\
&\equiv &I_{1}+I_{2}.
\end{eqnarray*
Now we estimat
\begin{eqnarray*}
I_{1} &=&\left\vert \int \left[ \left( K_{y}^{\alpha }\right) ^{\prime
}\left( m_{J}\right) \right] d\mu \left( y\right) \right\vert \ \left\vert
\int \left( x-m_{J}\right) \ \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast
}}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right) d\omega
\left( x\right) \right\vert \\
&=&\left\vert \int \left[ \left( K_{y}^{\alpha }\right) ^{\prime }\left(
m_{J}\right) \right] d\mu \left( y\right) \right\vert \ \left\vert \int
\left( \bigtriangleup _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}x\right) \
\left( \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right)
\right) \ d\omega \left( x\right) \right\vert \\
&\lesssim &C_{CZ}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \mu
\right\vert \right) }{\left\vert J\right\vert }\ \left\Vert \bigtriangleup
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }\left\Vert \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi
\right\Vert _{L^{2}\left( \omega \right) }\ ,
\end{eqnarray*
an
\begin{eqnarray*}
I_{2} &\lesssim &C_{CZ}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left(
J,\left\vert \mu \right\vert \right) }{\left\vert J\right\vert }\int
\left\vert x-m_{J}\right\vert \left\vert \square _{J}^{\omega ,\pi ,\mathbf{
}^{\ast }}\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \left( x\right)
\right\vert d\omega \left( x\right) \\
&\lesssim &C_{CZ}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\left\vert
\mu \right\vert \right) }{\left\vert J\right\vert }\sqrt{\int_{J}\left\vert
x-m_{J}\right\vert ^{2}d\omega \left( x\right) }\left\Vert \square
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\square _{J}^{\omega ,\pi ,\mathbf{b
^{\ast }}\Psi \right\Vert _{L^{2}\left( \omega \right) } \\
&\lesssim &C_{CZ}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\left\vert
\mu \right\vert \right) }{\left\vert J\right\vert }\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }\left\Vert
\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi \right\Vert _{L^{2}\left(
\omega \right) }\ .
\end{eqnarray*
For term $II$ we fix $z\in \overline{J}$ for the moment. Then since
\left\langle 1,\square _{J,\limfunc{broken}}^{\omega ,\mathbf{b}^{\ast
}}h\right\rangle _{\omega }=\left\langle 1,\square _{J}^{\omega ,\mathbf{b
^{\ast }}h-\square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}h\right\rangle
_{\omega }=0$, we hav
\begin{equation*}
II=\left\vert \left\langle T^{\alpha }\mu ,\square _{J,\limfunc{broken
}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\rangle _{\omega }\right\vert
=\left\vert \int \left\{ \int \left[ \left( K_{y}^{\alpha }\right) ^{\prime
}\left( \theta \left( x,z\right) \right) \right] d\mu \left( y\right)
\right\} \left( x-z\right) \ \square _{J,\limfunc{broken}}^{\omega ,\mathbf{
}^{\ast }}\Psi \left( x\right) d\omega \left( x\right) \right\vert .
\end{equation*}
Using reverse H\"{o}lder control of children (\ref{rev Hol con}), we obtain
the estimate (\ref{F est}) from Appendix A
\begin{equation*}
\left\vert \square _{J,\limfunc{broken}}^{\omega ,\mathbf{b}^{\ast }}\Psi
\right\vert =\left\vert \sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken
}\left( JQ\right) }\left( \mathbb{F}_{J^{\prime }}^{\omega ,b_{J^{\prime }}}
\mathbb{F}_{J^{\prime }}^{\omega ,b_{J}}\right) \Psi \right\vert \lesssim
\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right) }\mathbf
1}_{J^{\prime }}E_{J^{\prime }}^{\omega }\left\vert \Psi \right\vert ,
\end{equation*
and s
\begin{equation*}
II\lesssim C_{CZ}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \mu
\right\vert \right) }{\left\vert J\right\vert }\sqrt{\sum_{J^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega }\left\vert
x-z\right\vert \right) ^{2}}\sqrt{\sum_{J^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( J\right) }\left\vert J^{\prime }\right\vert _{\omega
}\left[ E_{J^{\prime }}^{\omega }\left\vert \Psi \right\vert \right] ^{2}}.
\end{equation*}
Combining the estimates for terms $I$ and $II$, we obtai
\begin{eqnarray*}
&&\left\vert \left\langle T^{\alpha }\mu ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}\Psi \right\rangle _{\omega }\right\vert \\
&\lesssim &C_{CZ}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \mu
\right\vert \right) }{\left\vert J\right\vert }\ \left\Vert \bigtriangleup
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }\left\Vert \square _{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}\Psi
\right\Vert _{L^{2}\left( \omega \right) } \\
&&+C_{CZ}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\left\vert \mu
\right\vert \right) }{\left\vert J\right\vert }\left\Vert x-m_{J}\right\Vert
_{L^{2}\left( \mathbf{1}_{J}\omega \right) }\left\Vert \square _{J}^{\omega
,\pi ,\mathbf{b}^{\ast }}\Psi \right\Vert _{L^{2}\left( \omega \right) } \\
&&+C_{CZ}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \mu \right\vert
\right) }{\left\vert J\right\vert }\inf_{z\in \overline{J}}\sqrt
\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega
}\left\vert x-z_{2}\right\vert \right) ^{2}}\sqrt{\sum_{J^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left[ E_{J^{\prime }}^{\omega }\left\vert \Psi
\right\vert +E_{J}^{\omega }\left\vert \Psi \right\vert \right] ^{2}},
\end{eqnarray*
and then noting that the infimum over $z\in \mathbb{R}$ is achieved for
z\in \overline{J}$, and using the triangle inequality on $\square
_{J}^{\omega ,\pi ,\mathbf{b}^{\ast }}=\square _{J}^{\omega ,\mathbf{b
^{\ast }}-\square _{J,\limfunc{broken}}^{\omega ,\mathbf{b}^{\ast }}$ we get
(\ref{estimate}).
\end{proof}
The right hand side of (\ref{estimate}) in the Monotonicity Lemma will be
typically estimated in what follows using the frame inequalities (see
Appendix A) for any interval $K$
\begin{eqnarray*}
\sum_{J\subset K}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi
\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2} &\lesssim
&\left\Vert \Psi \right\Vert _{L^{2}\left( \omega \right) }^{2}\ , \\
\sum_{J\subset K}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} &\lesssim
&\int_{K}\left\vert x-m_{K}\right\vert ^{2}d\omega \left( x\right) \ ,
\end{eqnarray*
together with these inequalities for the square function expressions.
\begin{lemma}
For any interval $K$ we hav
\begin{eqnarray}
\sum_{J\subset K}\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
J\right) }\left\vert J^{\prime }\right\vert _{\omega }\left[ E_{J^{\prime
}}^{\omega }\left\vert \Psi \right\vert \left( x\right) \right] ^{2}
&\lesssim &\int_{K}\left\vert \Psi \left( x\right) \right\vert ^{2}d\omega
\left( x\right) , \label{with both} \\
\text{and }\sum_{J\subset K}\inf_{z\in \mathbb{R}}\sum_{J^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega }\left\vert
x-z\right\vert \right) ^{2} &\lesssim &\int_{K}\left\vert x-m_{K}\right\vert
^{2}d\omega \left( x\right) . \notag
\end{eqnarray}
\end{lemma}
\begin{proof}
The first inequality in (\ref{with both}) is just the Carleson embedding
theorem since the intervals $\left\{ J^{\prime }\in \mathfrak{C}_{\limfunc
broken}}\left( J\right) :J\subset K\right\} $ satisfy an $\omega $-Carleson
condition, and the second inequality in (\ref{with both}) follows by
choosing $z=m_{K}$ to obtai
\begin{equation*}
\inf_{z\in \mathbb{R}}\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken
}\left( J\right) }\left\vert J^{\prime }\right\vert _{\omega }\left(
E_{J^{\prime }}^{\omega }\left\vert x-z\right\vert \right) ^{2}\leq
\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega
}\left\vert x-m_{K}\right\vert \right) ^{2},
\end{equation*
and then applying the Carleson embedding theorem again
\begin{equation*}
\sum_{J\subset K}\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
J\right) }\left\vert J^{\prime }\right\vert _{\omega }\left( E_{J^{\prime
}}^{\omega }\left\vert x-m_{K}\right\vert \right) ^{2}\lesssim
\int_{K}\left\vert x-m_{K}\right\vert ^{2}d\omega \left( x\right) .
\end{equation*}
\end{proof}
\subsubsection{The smaller Poisson integral}
The expressions $\inf_{z\in \mathbb{R}}\frac{\mathrm{P}_{1+\delta }^{\alpha
}\left( J,\left\vert \mu \right\vert \right) }{\left\vert J\right\vert
\left\Vert x-z\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }$ are typically easier to sum due
to the small Poisson operator $\mathrm{P}_{1+\delta }^{\alpha }\left(
J,\left\vert \mu \right\vert \right) $. To illlustrate, we show here one way
in which we can exploit the additional decay in the Poisson integral
\mathrm{P}_{1+\delta }^{\alpha }$. Suppose that $J$ is good in $I$ with
\ell \left( J\right) =2^{-s}\ell \left( I\right) $ (see Definition \ref{good
arb} below for `goodness'). We then comput
\begin{eqnarray*}
\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{A\setminus
I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}{n}}} &\approx
&\int_{A\setminus I}\frac{\left\vert J\right\vert ^{\frac{\delta }{n}}}
\left\vert y-c_{J}\right\vert ^{n+1+\delta -\alpha }}d\sigma \left( y\right)
\\
&\leq &\int_{A\setminus I}\left( \frac{\left\vert J\right\vert ^{\frac{1}{n}
}{\limfunc{qdist}\left( c_{J},I^{c}\right) }\right) ^{\delta }\frac{1}
\left\vert y-c_{J}\right\vert ^{n+1-\alpha }}d\sigma \left( y\right) \\
&\lesssim &\left( \frac{\left\vert J\right\vert ^{\frac{1}{n}}}{\limfunc
qdist}\left( c_{J},I^{c}\right) }\right) ^{\delta }\frac{\mathrm{P}^{\alpha
}\left( J,\mathbf{1}_{A\setminus I}\sigma \right) }{\left\vert J\right\vert
^{\frac{1}{n}}},
\end{eqnarray*
and use the goodness inequality
\begin{equation*}
d\left( c_{J},I^{c}\right) \geq \frac{1}{2}\ell \left( I\right)
^{1-\varepsilon }\ell \left( J\right) ^{\varepsilon }\geq \frac{1}{2
2^{s\left( 1-\varepsilon \right) }\ell \left( J\right) ,
\end{equation*
to conclude tha
\begin{equation}
\left( \frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{A\setminus
I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}{n}}}\right) \lesssim
2^{-s\delta \left( 1-\varepsilon \right) }\frac{\mathrm{P}^{\alpha }\left( J
\mathbf{1}_{A\setminus I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}
n}}}. \label{Poisson decay}
\end{equation
Now we can estimat
\begin{eqnarray*}
&&\sum_{J\subset K:\ J\text{ is }\limfunc{good}\text{ in }K}\inf_{z\in
\mathbb{R}}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1
_{K^{c}}\left\vert \mu \right\vert \right) }{\left\vert J\right\vert
\left\Vert x-z\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}\Psi \right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\sqrt{\sum_{J\subset K:\ J\text{ is }\limfunc{good}\text{ in
K}\left( \frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1
_{K^{c}}\left\vert \mu \right\vert \right) }{\left\vert J\right\vert
\right) ^{2}\inf_{z\in \mathbb{R}}\left\Vert x-z\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }^{2}}\sqrt{\sum_{J\subset K:\ J\text{ is
\limfunc{good}\text{ in }K}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}\Psi \right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}},
\end{eqnarray*
wher
\begin{eqnarray*}
&&\sum_{J\subset K:\ J\text{ is }\limfunc{good}\text{ in }K}\left( \frac
\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{K^{c}}\left\vert \mu
\right\vert \right) }{\left\vert J\right\vert }\right) ^{2}\inf_{z\in
\mathbb{R}}\left\Vert x-z\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega
\right) }^{2} \\
&=&\sum_{s=0}^{\infty }\sum_{\substack{ J\subset K:\ J\text{ is }\limfunc
good}\text{ in }K \\ \ell \left( J\right) =2^{-s}\ell \left( I\right) }
\left( \frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1
_{K^{c}}\left\vert \mu \right\vert \right) }{\left\vert J\right\vert
\right) ^{2}\inf_{z\in \mathbb{R}}\left\Vert x-z\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }^{2} \\
&\leq &\sum_{s=0}^{\infty }\sum_{\substack{ J\subset K:\ J\text{ is
\limfunc{good}\text{ in }K \\ \ell \left( J\right) =2^{-s}\ell \left(
I\right) }}\left( 2^{-s\delta \left( 1-\varepsilon \right) }\frac{\mathrm{P
^{\alpha }\left( J,\mathbf{1}_{K^{c}}\sigma \right) }{\left\vert
J\right\vert ^{\frac{1}{n}}}\right) ^{2}\inf_{z\in \mathbb{R}}\left\Vert
x-z\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2} \\
&\leq &\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{K^{c}}\sigma
\right) }{\left\vert K\right\vert ^{\frac{1}{n}}}\right)
^{2}\sum_{s=0}^{\infty }\sum_{\substack{ J\subset K:\ J\text{ is }\limfunc
good}\text{ in }K \\ \ell \left( J\right) =2^{-s}\ell \left( I\right) }
2^{-2s\delta \left( 1-\varepsilon \right) }\inf_{z\in \mathbb{R}}\left\Vert
x-z\right\Vert _{L^{2}\left( \mathbf{1}_{K}\omega \right) }^{2} \\
&\lesssim &\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1
_{K^{c}}\sigma \right) }{\left\vert K\right\vert ^{\frac{1}{n}}}\right)
^{2}\inf_{z\in \mathbb{R}}\left\Vert x-z\right\Vert _{L^{2}\left( \mathbf{1
_{K}\omega \right) }^{2}\ ,
\end{eqnarray*
and where we have used (\ref{Poisson inequality}), which gives in particula
\begin{equation*}
\mathrm{P}^{\alpha }(J,\mu \mathbf{1}_{I^{c}})\lesssim \left( \frac{\ell
\left( J\right) }{\ell \left( I\right) }\right) ^{1-\varepsilon \left(
2-\alpha \right) }\mathrm{P}^{\alpha }(I,\mu \mathbf{1}_{I^{c}}).
\end{equation*
for $J\subset I$ and $d\left( J,\partial I\right) >\tfrac{1}{2}\ell \left(
J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }$. We will use
such arguments repeatedly in the sequel.
Armed with the Monotonicity Lemma and the lower frame inequalit
\begin{equation*}
\sum_{I\in \mathcal{D}}\left\Vert \square _{I}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \mu \right) }^{\bigstar 2}\lesssim \left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }^{2}\ ,
\end{equation*
we can obtain a $\mathbf{b}^{\ast }$-analogue of the Energy Lemma as in \cit
{SaShUr7} and/or \cite{SaShUr6}.
\subsubsection{The Energy Lemma}
Suppose now we are given a subset $\mathcal{H}$ of the dyadic grid $\mathcal
G}$.
\begin{notation}
\label{nonstandard norm}Due to the failure of both martingale and dual
martingale pseudoprojections $\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b
^{\ast }}x$ and $\mathsf{P}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}g$, as
in Definition \ref{Psi op} in Appendix A, to satisfy inequalities of the
form $\left\Vert \mathsf{P}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }\lesssim \left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }$ when the children `break', it
is convenient to define the `square function norms' $\left\Vert \mathsf{Q}_
\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit }$ and $\left\Vert \mathsf{P}_{\mathcal{H}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }$
of the pseudoprojections
\begin{equation*}
\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x=\sum_{J\in \mathcal{H
}\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\text{ and }\mathsf{P}_
\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}g=\sum_{J\in \mathcal{H}}\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\ ,
\end{equation*
b
\begin{eqnarray*}
\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &\equiv &\sum_{J\in \mathcal{H
}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}=\sum_{J\in \mathcal{H
}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\sum_{J\in \mathcal{H}}\inf_{z\in \mathbb
R}}\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega
}\left\vert x-z\right\vert \right) ^{2}, \\
\left\Vert \mathsf{P}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2} &\equiv &\sum_{J\in \mathcal{H
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2}=\sum_{J\in \mathcal{H
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\sum_{J\in \mathcal{H}}\sum_{J^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left[ E_{J^{\prime }}^{\omega }\left\vert
g\right\vert +E_{J}^{\omega }\left\vert g\right\vert \right] ^{2},
\end{eqnarray*
for any subset $\mathcal{H}\subset \mathcal{G}$. The average $E_{J}^{\omega
}\left\vert x-z\right\vert $ above is taken with respect to the variable $x
, i.e. $E_{J}^{\omega }\left\vert x-z\right\vert =\frac{1}{\left\vert
J\right\vert _{\omega }}\int \left\vert x-z\right\vert d\omega \left(
x\right) $, and it is important that the infimum $\inf_{z\in \mathbb{R}}$ is
taken \emph{inside} the sum $\sum_{J\in \mathcal{H}}$.
\end{notation}
Note that we are defining here square function expressions related to
pseudoprojections, which depend not only on the functions $\mathsf{Q}_
\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x$ and $\mathsf{P}_{\mathcal{H
}^{\omega ,\mathbf{b}^{\ast }}g$, but also on the particular representations
$\sum_{J\in \mathcal{H}}\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x$
and $\sum_{J\in \mathcal{H}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}g$.
This slight abuse of notation should not cause confusion, and it provides a
useful way of bookkeeping the sums of squares of norms of martingale and
dual martingale differences $\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf
b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{2}$ and $\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{2}$, along with the norms of the associated Carleson square
function expressions
\begin{eqnarray*}
\sum_{J\in \mathcal{H}}\inf_{z\in \mathbb{R}}\left\Vert \nabla _{J}^{\omega
}\left( x-z\right) \right\Vert _{L^{2}\left( \omega \right) }^{2}
&=&\sum_{J\in \mathcal{H}}\inf_{z\in \mathbb{R}}\sum_{J^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega }\left\vert
x-z\right\vert \right) ^{2} \\
\sum_{J\in \mathcal{H}}\left\Vert \nabla _{J}^{\omega }\Psi \right\Vert
_{L^{2}\left( \omega \right) }^{2} &=&\sum_{J\in \mathcal{H}}\sum_{J^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left[ E_{J^{\prime }}^{\omega }\left\vert \Psi
\right\vert \right] ^{2}.
\end{eqnarray*
Note also that the upper weak Riesz inequalities in Appendix A below yield
the inequalitie
\begin{eqnarray*}
\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2} &\lesssim &\sum_{J\in \mathcal{H
}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}\leq \left\Vert \mathsf{Q}_{\mathcal{H
}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\ , \\
\left\Vert \mathsf{P}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2} &\lesssim &\sum_{J\in \mathcal{H
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}\leq \left\Vert \mathsf{P}_{\mathcal{H
}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{\bigstar 2}\ .
\end{eqnarray*
We will exclusively use $\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} $ in connection with energy terms, and use $\left\Vert \mathsf{P}_
\mathcal{H}}^{\sigma ,\mathbf{b}^{\ast }}f\right\Vert _{L^{2}\left( \sigma
\right) }^{\bigstar 2}$ and $\left\Vert \mathsf{P}_{\mathcal{H}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}
\ in connection with functions $f\in L^{2}\left( \sigma \right) $ and $g\in
L^{2}\left( \omega \right) $. Finally, note that $\mathsf{Q}_{\mathcal{H
}^{\omega ,\mathbf{b}^{\ast }}x=\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b
^{\ast }}\left( x-m\right) $ for any constant $m$. We also define the `large
$\mathcal{G}$-pseudoprojections
\begin{equation*}
\mathsf{Q}_{L}^{\omega ,\mathbf{b}^{\ast }}\equiv \sum_{J^{\prime }\in
\mathcal{G}:\ J^{\prime }\subset L}\bigtriangleup _{J^{\prime }}^{\omega
\mathbf{b}^{\ast }},\ \ \ \ \ \text{for any interval }L\text{.}
\end{equation*}
Recall tha
\begin{equation*}
\Phi ^{\alpha }\left( J,\nu \right) \equiv \frac{\mathrm{P}^{\alpha }\left(
J,\nu \right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup
_{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit }+\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\nu \right) }
\left\vert J\right\vert }\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf
1}_{J}\omega \right) }\ .
\end{equation*}
\begin{lemma}[\textbf{Energy Lemma}]
\label{ener}Let $J\ $be an interval in $\mathcal{G}$. Let $\Psi _{J}$ be an
L^{2}\left( \omega \right) $ function supported in $J$ with vanishing
\omega $-mean, and let $\mathcal{H}\subset \mathcal{G}$ be such that
J^{\prime }\subset J$ for every $J^{\prime }\in \mathcal{H}$. Let $\nu $ be
a positive measure supported in $\mathbb{R}\setminus \gamma J$ with $\gamma
>1$, and for each $J^{\prime }\in \mathcal{H}$, let $d\nu _{J^{\prime
}}=\varphi _{J^{\prime }}d\nu $ with $\left\vert \varphi _{J^{\prime
}}\right\vert \leq 1$. Suppose that $\mathbf{b}^{\ast }$ is an $\infty
-weakly $\mu $-controlled accretive family on $\mathbb{R}$. Let $T^{\alpha }$
be a standard $\alpha $-fractional singular integral operator with $0\leq
\alpha <1$. Then we hav
\begin{eqnarray*}
&&\left\vert \sum_{J^{\prime }\in \mathcal{H}}\left\langle T^{\alpha }\left(
\nu _{J^{\prime }}\right) ,\square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}\Psi _{J}\right\rangle _{\omega }\right\vert \lesssim C_{\gamma
}\sum_{J^{\prime }\in \mathcal{H}}\Phi ^{\alpha }\left( J^{\prime },\nu
\right) \left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi
_{J}\right\Vert _{L^{2}\left( \mu \right) }^{\bigstar } \\
&\lesssim &C_{\gamma }\sqrt{\sum_{J^{\prime }\in \mathcal{H}}\Phi ^{\alpha
}\left( J^{\prime },\nu \right) ^{2}}\sqrt{\sum_{J^{\prime }\in \mathcal{H
}\left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi
_{J}\right\Vert _{L^{2}\left( \mu \right) }^{\bigstar 2}} \\
&\leq &C_{\gamma }\left( \frac{\mathrm{P}^{\alpha }\left( J,\nu \right) }
\left\vert J\right\vert }\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }
\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\nu \right) }{\left\vert
J\right\vert }\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1
_{J}\omega \right) }\right) \left\Vert \mathsf{P}_{\mathcal{H}}^{\omega
\mathbf{b}^{\ast }}\Psi _{J}\right\Vert _{L^{2}\left( \mu \right)
}^{\bigstar },
\end{eqnarray*
and in particular the `energy' estimat
\begin{equation*}
\left\vert \left\langle T^{\alpha }\varphi \nu ,\Psi _{J}\right\rangle
_{\omega }\right\vert \lesssim C_{\gamma }\left( \frac{\mathrm{P}^{\alpha
}\left( J,\nu \right) }{\left\vert J\right\vert }\left\Vert \mathsf{Q
_{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit }+\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\nu \right) }
\left\vert J\right\vert }\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf
1}_{J}\omega \right) }\right) \left\Vert \sum_{J^{\prime }\subset J}\square
_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi _{J}\right\Vert
_{L^{2}\left( \mu \right) }^{\bigstar }\ ,
\end{equation*
where $\left\Vert \sum_{J^{\prime }\subset J}\square _{J^{\prime }}^{\omega
\mathbf{b}^{\ast }}\Psi _{J}\right\Vert _{L^{2}\left( \mu \right)
}^{\bigstar }\lesssim \left\Vert \Psi _{J}\right\Vert _{L^{2}\left( \mu
\right) }$, and the `pivotal' boun
\begin{equation*}
\left\vert \left\langle T^{\alpha }\left( \varphi \nu \right) ,\Psi
_{J}\right\rangle _{\omega }\right\vert \lesssim C_{\gamma }\mathrm{P
^{\alpha }\left( J,\left\vert \nu \right\vert \right) \sqrt{\left\vert
J\right\vert _{\omega }}\left\Vert \Psi _{J}\right\Vert _{L^{2}\left( \omega
\right) }\ ,
\end{equation*
for any function $\varphi $ with $\left\vert \varphi \right\vert \leq 1$.
\end{lemma}
\begin{proof}
Beginning with the first display in the conclusion of the Energy Lemma, we
need only prove the first line since the next two follow from Poisson
inequalities and the definitions in Notation \ref{nonstandard norm}. Using
the Monotonicity Lemma \ref{mono}, followed by $\left\vert \nu _{J^{\prime
}}\right\vert \leq \nu $, the Poisson equivalence
\begin{equation}
\frac{\mathrm{P}^{\alpha }\left( J^{\prime },\nu \right) }{\left\vert
J^{\prime }\right\vert }\approx \frac{\mathrm{P}^{\alpha }\left( J,\nu
\right) }{\left\vert J\right\vert },\ \ \ \ \ J^{\prime }\subset J\subset
\gamma J,\ \ \ \limfunc{supp}\nu \cap \gamma J=\emptyset ,
\label{Poisson equiv}
\end{equation
and the weak frame inequalities for dual martingale differences in Appendix
A, we hav
\begin{eqnarray*}
&&\left\vert \sum_{J^{\prime }\in \mathcal{H}}\left\langle T^{\alpha }\left(
\nu _{J^{\prime }}\right) ,\square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}\Psi _{J}\right\rangle _{\omega }\right\vert \lesssim \sum_{J^{\prime }\in
\mathcal{H}}\Phi ^{\alpha }\left( J^{\prime },\left\vert \mu \right\vert
\right) \left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi
_{J}\right\Vert _{L^{2}\left( \mu \right) }^{\bigstar } \\
&\leq &\sum_{J^{\prime }\in \mathcal{H}}\left( \frac{\mathrm{P}^{\alpha
}\left( J^{\prime },\nu \right) }{\left\vert J^{\prime }\right\vert
\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) ,z}^{\spadesuit }+\frac{\mathrm{
}_{1+\delta }^{\alpha }\left( J^{\prime },\nu \right) }{\left\vert J^{\prime
}\right\vert }\left\Vert x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \mathbf{
}_{J^{\prime }}\omega \right) }\right) \left\Vert \square _{J^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}\Psi _{J}\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \\
&\lesssim &\left( \sum_{J^{\prime }\in \mathcal{H}}\left( \frac{\mathrm{P
^{\alpha }\left( J^{\prime },\nu \right) }{\left\vert J^{\prime }\right\vert
}\right) ^{2}\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) ,z}^{\spadesuit 2}\right)
^{\frac{1}{2}}\left( \sum_{J^{\prime }\in \mathcal{H}}\left\Vert \square
_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi _{J}\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2}\right) ^{\frac{1}{2}} \\
&&+\left( \sum_{J^{\prime }\in \mathcal{H}}\left( \frac{\mathrm{P}_{1+\delta
}^{\alpha }\left( J^{\prime },\left\vert \mu \right\vert \right) }
\left\vert J^{\prime }\right\vert }\right) ^{2}\left\Vert x-m_{J^{\prime
}}\right\Vert _{L^{2}\left( \mathbf{1}_{J^{\prime }}\omega \right)
}^{2}\right) ^{\frac{1}{2}}\left( \sum_{J^{\prime }\in \mathcal{H
}\left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\Psi
_{J}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}\right) ^{\frac{
}{2}} \\
&\lesssim &\left( \frac{\mathrm{P}^{\alpha }\left( J,\nu \right) }
\left\vert J\right\vert }\right) \left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega
,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \Psi _{J}\right\Vert _{L^{2}\left( \omega \right) }+\frac{1}
\gamma ^{\delta ^{\prime }}}\left( \frac{\mathrm{P}_{1+\delta ^{\prime
}}^{\alpha }\left( J,\nu \right) }{\left\vert J\right\vert }\right)
\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}\left\Vert \Psi _{J}\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{eqnarray*
The last inequality follows from the following calculation using Haar
projections $\bigtriangleup _{K}^{\omega }$:
\begin{eqnarray}
&&\sum_{J^{\prime }\in \mathcal{H}}\left( \frac{\mathrm{P}_{1+\delta
}^{\alpha }\left( J^{\prime },\nu \right) }{\left\vert J^{\prime
}\right\vert }\right) ^{2}\left\Vert x-m_{J^{\prime }}\right\Vert
_{L^{2}\left( \mathbf{1}_{J^{\prime }}\omega \right) }^{2}
\label{Haar trick} \\
&=&\sum_{J^{\prime }\in \mathcal{H}}\left( \frac{\mathrm{P}_{1+\delta
}^{\alpha }\left( J^{\prime },\nu \right) }{\left\vert J^{\prime
}\right\vert }\right) ^{2}\sum_{J^{\prime \prime }\subset J^{\prime
}}\left\Vert \bigtriangleup _{J^{\prime \prime }}^{\omega }x\right\Vert
_{L^{2}\left( \omega \right) }^{2}=\sum_{J^{\prime \prime }\subset J}\left\{
\sum_{J^{\prime }:\ J^{\prime \prime }\subset J^{\prime }\subset J}\left(
\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime },\nu \right) }
\left\vert J^{\prime }\right\vert }\right) ^{2}\right\} \left\Vert
\bigtriangleup _{J^{\prime \prime }}^{\omega }x\right\Vert _{L^{2}\left(
\omega \right) }^{2} \notag \\
&\lesssim &\frac{1}{\gamma ^{2\delta ^{\prime }}}\sum_{J^{\prime \prime
}\subset J}\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left(
J^{\prime \prime },\nu \right) }{\left\vert J^{\prime \prime }\right\vert
\right) ^{2}\left\Vert \bigtriangleup _{J^{\prime \prime }}^{\omega
}x\right\Vert _{L^{2}\left( \omega \right) }^{2}\leq \frac{1}{\gamma
^{2\delta ^{\prime }}}\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha
}\left( J,\nu \right) }{\left\vert J\right\vert }\right) ^{2}\sum_{J^{\prime
\prime }\subset J}\left\Vert \bigtriangleup _{J^{\prime \prime }}^{\omega
}x\right\Vert _{L^{2}\left( \omega \right) }^{2}\ , \notag
\end{eqnarray
which in turn follows from (recalling $\delta =2\delta ^{\prime }$ and using
$\left\vert J^{\prime }\right\vert +\left\vert y-c_{J^{\prime }}\right\vert
\approx \left\vert J\right\vert +\left\vert y-c_{J}\right\vert $ and $\frac
\left\vert J\right\vert }{\left\vert J\right\vert +\left\vert
y-c_{J}\right\vert }\leq \frac{1}{\gamma }$ for $y\in \mathbb{R}\setminus
\gamma J$
\begin{eqnarray*}
&&\sum_{J^{\prime }:\ J^{\prime \prime }\subset J^{\prime }\subset J}\left(
\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime },\nu \right) }
\left\vert J^{\prime }\right\vert }\right) ^{2}=\sum_{J^{\prime }:\
J^{\prime \prime }\subset J^{\prime }\subset J}\left\vert J^{\prime
}\right\vert ^{2\delta }\left( \int_{\mathbb{R}\setminus \gamma J}\frac{1}
\left( \left\vert J^{\prime }\right\vert +\left\vert y-c_{J^{\prime
}}\right\vert \right) ^{2+\delta -\alpha }}d\nu \left( y\right) \right) ^{2}
\\
&\lesssim &\sum_{J^{\prime }:\ J^{\prime \prime }\subset J^{\prime }\subset
J}\frac{1}{\gamma ^{2\delta ^{\prime }}}\frac{\left\vert J^{\prime
}\right\vert ^{2\delta }}{\left\vert J\right\vert ^{2\delta }}\left( \int_
\mathbb{R}\setminus \gamma J}\frac{\left\vert J\right\vert ^{\delta ^{\prime
}}}{\left( \left\vert J\right\vert +\left\vert y-c_{J}\right\vert \right)
^{2+\delta ^{\prime }-\alpha }}d\nu \left( y\right) \right) ^{2} \\
&=&\frac{1}{\gamma ^{2\delta ^{\prime }}}\left( \sum_{J^{\prime }:\
J^{\prime \prime }\subset J^{\prime }\subset J}\frac{\left\vert J^{\prime
}\right\vert ^{2\delta }}{\left\vert J\right\vert ^{2\delta }}\right) \left(
\frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left( J,\nu \right) }
\left\vert J\right\vert }\right) ^{2}\lesssim \frac{1}{\gamma ^{2\delta
^{\prime }}}\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left(
J,\nu \right) }{\left\vert J\right\vert }\right) ^{2}.
\end{eqnarray*
Finally we obtain the `energy' estimate from the equalit
\begin{equation*}
\Psi _{J}=\sum_{J^{\prime }\subset J}\square _{J^{\prime }}^{\omega ,\mathbf
b}^{\ast }}\Psi _{J}\ ,\ \ \ \ \ (\text{since }\Psi _{J}\text{ has vanishing
}\omega \text{-mean)},\text{ }
\end{equation*
and we obtain the `pivotal' bound from the inequalit
\begin{equation*}
\sum_{J^{\prime \prime }\subset J}\left\Vert \bigtriangleup _{J^{\prime
\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\lesssim \left\Vert \left( x-m_{J}\right)
\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2}\leq \left\vert
J\right\vert ^{2}\left\vert J\right\vert _{\omega }\ .
\end{equation*}
\end{proof}
\subsection{Organization of the proof}
We adapt the proof of the main theorem in \cite{SaShUr7}, \cite{SaShUr9} and
\cite{SaShUr10}, but beginning instead with the decomposition of Hyt\"{o}nen
and Martikainen \cite{HyMa}, to obtain the norm inequalit
\begin{equation*}
\mathfrak{N}_{T^{\alpha }}\lesssim \mathcal{NTV}_{\alpha }=\mathfrak{T
_{T^{\alpha }}^{\mathbf{b}}+\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}^{\ast }}
\sqrt{\mathfrak{A}_{2}^{\alpha }}+\mathfrak{E}_{2}^{\alpha },
\end{equation*
under the \emph{apriori} assumption $\mathfrak{N}_{T^{\alpha }}<\infty $,
which is achieved by considering one of the truncations $T_{\sigma ,\delta
,R}^{\alpha }$ defined in (\ref{def truncation}) above. This will be carried
out in the next five sections of this paper. In the next section we consider
the various form splittings and reduce matters to the \emph{disjoint} form,
the \emph{nearby} form and the \emph{main below} form. Then these latter
three forms are taken up in the subsequent three sections, using material
from the appendices. Finally, the stopping form is treated in the section
following these three.
A major source of difficulty will arise in the infusion of goodness for the
intervals $J$ into the main below form where the sum is taken over all pairs
$\left( I,J\right) $ such that $\ell \left( J\right) \leq \ell \left(
I\right) $. We will infuse goodness in a weak way pioneered by Hyt\"{o}nen
and Martikainen in a one weight setting. This weak form of goodness is then
exploited in all subsequent constructions by typically replacing $J$ with
J^{\maltese }$ in defining relations, where $J^{\maltese }$ is the smallest
interval $K$ for which $J$ is good in $K$ and beyond (see the next section
for terminology, in particular Definition \ref{def sharp cross}).
Another source of difficulty arises in the treatment of the nearby form in
the setting of two weights. The one weight proofs in \cite{HyMa} and \cit
{LaMa} relied strongly on a property peculiar to the one weight setting -
namely the fact already pointed out in Remark \ref{special}\ above, that
both of the Poisson integrals are bounded, namely $\mathrm{P}^{\alpha
}\left( Q,\mu \right) \lesssim 1$ and $\mathcal{P}^{\alpha }\left( Q,\mu
\right) \lesssim 1$. We will circumvent this difficulty by combining a
recursive energy argument with the full testing conditions assumed for the
\infty $-weakly accretive family of \emph{original} testing functions
b_{Q}^{\limfunc{orig}}$, before these conditions were suppressed by corona
constructions that delivered only weak testing conditions for the new family
of testing functions $b_{Q}$.
In Section \ref{Sec stop} we bound the stopping form using the arguments
from \cite{SaShUr7}, \cite{SaShUr9} and \cite{SaShUr10}, which were in turn
based on the bottom/up stopping time and recursion of M. Lacey in \cite{Lac
. Here we introduce an additional top/down `indented' corona construction to
handle the lack of goodness in size testing intervals, and we use an
absorption in place of recursion. Finally, the treatment of various
`straddling lemmas' is complicated by weak goodness, and we use a stronger
form of weak goodness defined with the three point `skeleton' of an interval
replaced by an infinite `body', coupled with two geometric `Key Facts' to
establish these lemmas.
Of particular importance will be two independent results proved in
Appendices A and B that follow from known work with some new twists. In
Appendix A we establish convergence and frame inequalities for martingale
and dual martingale differences, and derive certain weak Riesz inequalities
associated with $\infty $-weakly $\mu $-accretive families of testing
functions, which will find application in treating the paraproduct form
below. The boundedness of testing functions, and the reverse H\"{o}lder
condition on their children, is important here.
In Appendix B we show that the functional energy for an arbitrary pair of
grids is controlled by the Muckenhoupt and energy side conditions. The
somewhat lengthy proof of this latter assertion is similar to the
corresponding proof in the $T1$ setting - see e.g. \cite{SaShUr9} - but
requires a different decomposition of the stopping intervals into `Whitney
intervals' in order to accommodate the weaker notion of goodness used here,
as well as the usual decomposition into maximal deeply embedded intervals
that is used to control expressions involving the `small' Poisson integral.
Finally, we include in Appendix C an up-to-date list of errata for our most
often referred to paper \cite{SaShUr7}.
\section{Form splittings}
\begin{notation}
Fix grids $\mathcal{D}$ and $\mathcal{G}$. We will use $\mathcal{D}$ to
denote the grid associated with $f\in L^{2}\left( \sigma \right) $, and we
will use $\mathcal{G}$ to denote the grid associated with $g\in L^{2}\left(
\omega \right) $.
\end{notation}
We have defined corona decompositions of $f$ and $g$ in the $\sigma
-iterated triple corona construction above, but in order to start these
corona decompositions for $f$ and $g$ respectively within the dyadic grids
\mathcal{D}$ and $\mathcal{G}$, we need to first restrict $f$ and $g$ to be
supported in a large common interval $Q_{\infty }$. Then we cover $Q_{\infty
}$ with two pairwise disjoint intervals $I_{\infty }\in \mathcal{D}$ with
\ell \left( I_{\infty }\right) =\ell \left( Q_{\infty }\right) $, and
similarly cover $Q_{\infty }$ with two pairwise disjoint intervals
J_{\infty }\in \mathcal{G}$ with $\ell \left( J_{\infty }\right) =\ell
\left( Q_{\infty }\right) $. We can now use the broken martingale
decompositions from Appendix A, together with full $\mathbf{b}$-testing (see
(\ref{full b testing}) and (\ref{full proved}) below), to reduce matters to
consideration of the four form
\begin{equation*}
\sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\sum_{J\in \mathcal{G}:\
J\subset J_{\infty }}\int \left( T_{\sigma }^{\alpha }\square _{I}^{\sigma
\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega ,
\end{equation*
with $I_{\infty }$ and $J_{\infty }$ as above, and where we can then use the
intervals $I_{\infty }$ and $J_{\infty }$ as the starting intervals in our
corona constructions below. Indeed, the identities in Lemma \ref{conv prop}
from Appendix A below, giv
\begin{eqnarray*}
f &=&\sum_{I\in \mathcal{D}_{N}}\mathbb{F}_{I}^{\sigma ,\mathbf{b
}f+\sum_{I\in \mathcal{D}:\ I\subset I_{\infty },\ \ell \left( I\right) \geq
N+1}\square _{I}^{\sigma ,\mathbf{b}}f+\mathbb{F}_{I_{\infty }}^{\sigma
\mathbf{b}}f, \\
g &=&\sum_{J\in \mathcal{G}_{N}}\mathbb{F}_{J}^{\omega ,\mathbf{b}^{\ast
}}g+\sum_{J\in \mathcal{G}:\ J\subset J_{\infty },\ \ell \left( J\right)
\geq N+1}\square _{J}^{\omega ,\mathbf{b}^{\ast }}g+\mathbb{F}_{J_{\infty
}}^{\omega ,\mathbf{b}^{\ast }}g,
\end{eqnarray*
which can then be used to write the bilinear form $\int \left( T_{\sigma
}f\right) gd\omega $ as a sum of the form
\begin{eqnarray}
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \int \left( T_{\sigma }f\right) gd\omega
=\sum_{\substack{ \text{four pairs} \\ \left( I_{\infty },J_{\infty
}\right) }}\left\{ \sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\sum_{J\in
\mathcal{G}:\ J\subset J_{\infty }}\int \left( T_{\sigma }^{\alpha }\square
_{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast
}}gd\omega \right. \label{sum of forms} \\
&&\left. +\sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\int \left(
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \mathbb{F
_{J_{\infty }}^{\omega ,\mathbf{b}^{\ast }}gd\omega +\sum_{J\in \mathcal{G
:\ J\subset J_{\infty }}\int \left( T_{\sigma }^{\alpha }\mathbb{F
_{I_{\infty }}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b
^{\ast }}gd\omega +\int \left( T_{\sigma }^{\alpha }\mathbb{F}_{I_{\infty
}}^{\sigma ,\mathbf{b}}f\right) \mathbb{F}_{J_{\infty }}^{\omega ,\mathbf{b
^{\ast }}gd\omega \right\} , \notag
\end{eqnarray
taken over the four pairs of intervals $\left( I_{\infty },J_{\infty
}\right) $ above, plus the limit of the sum of terms involving $\sum_{I\in
\mathcal{D}_{N}}\mathbb{F}_{I}^{\sigma ,\mathbf{b}}f$ and $\sum_{J\in
\mathcal{G}_{N}}\mathbb{F}_{J}^{\omega ,\mathbf{b}^{\ast }}g$. This latter
limit is easily shown to vanish due to the strong convergence of the dual
martingale differences $\square _{I}^{\sigma ,\mathbf{b}}f$ and $\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g$ in $L^{2}\left( \sigma \right) $ and
L^{2}\left( \omega \right) $ respectively. More precisely, we hav
\begin{eqnarray*}
&&\left\vert \int \left( T_{\sigma }^{\alpha }\sum_{I\in \mathcal{D}_{N}
\mathbb{F}_{I}^{\sigma ,\mathbf{b}}f\right) \sum_{J\in \mathcal{G}_{N}
\mathbb{F}_{J}^{\omega ,\mathbf{b}^{\ast }}g\ d\omega \right\vert \lesssim
\mathfrak{N}_{T^{\alpha }}\left\Vert \sum_{I\in \mathcal{D}_{N}}\mathbb{F
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert \sum_{J\in \mathcal{G}_{N}}\mathbb{F}_{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) } \\
&=&\mathfrak{N}_{T^{\alpha }}\left\Vert \sum_{I\in \mathcal{D}:\ \ell \left(
I\right) \geq 2^{N}}\square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert \sum_{J\in \mathcal{G}:\ \ell
\left( J\right) \geq 2^{N}}\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) } \\
&\lesssim &\mathfrak{N}_{T^{\alpha }}\left( \sum_{I\in \mathcal{D}:\ \ell
\left( I\right) \geq 2^{N}}\left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}\right) ^{\frac{1}{2}}\left(
\sum_{J\in \mathcal{G}:\ \ell \left( J\right) \geq 2^{N}}\left\Vert \square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}\right) ^{\frac{1}{2}},
\end{eqnarray*
which tends to $0$ as $N\rightarrow \infty $ sinc
\begin{equation*}
\left( \sum_{I\in \mathcal{D}}\left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}\right) ^{\frac{1}{2}}\left(
\sum_{J\in \mathcal{G}}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right) ^{\frac{1}{2
}\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{equation*}
\begin{remark}
In particular,
\begin{equation*}
\lim_{N\rightarrow \infty }\sup_{I\in \mathcal{D}_{N}}\left\Vert \mathbb{F
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}=0=\lim_{N\rightarrow \infty }\sup_{J\in \mathcal{G}_{N}}\left\Vert \mathbb
F}_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \sigma \right)
}
\end{equation*
and so we can use the pointwise telescoping identitie
\begin{equation*}
\mathbb{F}_{I}^{\sigma ,\mathbf{b}}f\left( x\right) =\sum_{I^{\prime }\in
\mathcal{D}:\ I\subset I^{\prime }}\square _{I^{\prime }}^{\sigma ,\mathbf{b
}f\left( x\right) \text{ and }\mathbb{F}_{J}^{\omega ,\mathbf{b}^{\ast
}}g\left( x\right) =\sum_{J\in \mathcal{G}:\ J\subset J^{\prime }}\square
_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\left( x\right) .
\end{equation*}
\end{remark}
The second, third and fourth sums in (\ref{sum of forms}) can be controlled
by the full testing conditions, e.g
\begin{eqnarray}
&&\left\vert \sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\int \left(
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \mathbb{F
_{J_{\infty }}^{\omega ,\mathbf{b}^{\ast }}gd\omega \right\vert =\left\vert
\int \left( \sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\square
_{I}^{\sigma ,\mathbf{b}}f\right) T_{\omega }^{\alpha ,\ast }\left( \mathbb{
}_{J_{\infty }}^{\omega ,\mathbf{b}^{\ast }}g\right) d\sigma \right\vert
\label{top control} \\
&\leq &\left\Vert \sum_{I\in \mathcal{D}:\ I\subset I_{\infty }}\square
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert T_{\omega }^{\alpha ,\ast }\left( \mathbb{F}_{J_{\infty
}}^{\omega ,\mathbf{b}^{\ast }}g\right) \right\Vert _{L^{2}\left( \sigma
\right) }\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left( \mathfrak{FT}_{T_{\omega }^{\alpha ,\ast }}+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \left\Vert g\right\Vert _{L^{2}\left( \omega \right) }
\notag
\end{eqnarray
since $\mathbb{F}_{J_{\infty }}^{\omega ,\mathbf{b}^{\ast }}g=b_{J_{\infty
}}^{\ast }\frac{E_{J_{\infty }}^{\omega }g}{E_{J_{\infty }}^{\omega
}b_{J_{\infty }}^{\ast }}$ is $b_{J_{\infty }}^{\ast }$ times an `accretive'
average of $g$ on $J_{\infty }$, and similarly for the third and fourth sum.
Finally, the full testing conditions $\mathfrak{FT}_{T_{\sigma }^{\alpha }}^
\mathbf{b}}$ and $\mathfrak{FT}_{T_{\omega }^{\alpha ,\ast }}^{\mathbf{b
^{\ast }}$ are controlled by the usual testing conditions $\mathfrak{T
_{T_{\sigma }^{\alpha }}^{\mathbf{b}}$ and $\mathfrak{T}_{T_{\omega
}^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}$ together with the Muckenhoupt
condition $\mathfrak{A}_{2}^{\alpha }$, by (\ref{full proved}) below.
\begin{description}
\item[Important] In the $\sigma $-iterated triple corona construction we
redefined the family $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{D}}$
so that the new functions $b_{Q}^{\limfunc{new}}$ are given in terms of the
original functions $b_{Q}^{\limfunc{orig}}$ by $b_{Q}^{\limfunc{new}}
\mathbf{1}_{Q}b_{A}^{\limfunc{orig}}$ for $Q\in \mathcal{C}_{A}^{\sigma }$,
and of course we then dropped the superscript `$\limfunc{new}$'. We continue
to refer to the triple stopping intervals $A$ as `breaking' intervals even
if $b_{A}$ happens to equal $\mathbf{1}_{A}b_{\pi A}$. The results of
Appendix A apply with this more inclusive definition of `breaking'
intervals, and the associated definition of `broken' children, since only
the Carleson condition\ on stopping intervals is relevant here.
\end{description}
Altogether this and Proposition \ref{data} give us the \emph{triple corona
decomposition} of $f=\sum_{A\in \mathcal{A}}\mathsf{P}_{\mathcal{C
_{A}}^{\sigma ,\mathbf{b}}f$, where the pseudoprojection $\mathsf{P}_
\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}$ is defined in Appendix A
\begin{equation*}
\mathsf{P}_{\mathcal{C}_{A}}^{\mu ,\mathbf{b}}f=\sum_{I\in \mathcal{C
_{A}}\square _{I}^{\mu ,\mathbf{b}}f\ .
\end{equation*
We now record the main facts proved above, and in Appendix A below, for the
triple corona.
\begin{lemma}
Let $f\in L^{2}\left( \sigma \right) $. We hav
\begin{eqnarray*}
f &=&\sum_{A\in \mathcal{A}}\mathsf{P}_{\mathcal{C}_{A}}^{\sigma }f,\ \ \ \
\ \text{both in the sense of norm convergence in }L^{2}\left( \sigma \right)
\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ and pointwise }\sigma \text
-a.e}.
\end{eqnarray*
The corona tops $\mathcal{A}$ and stopping bounds $\left\{ \alpha _{\mathcal
A}}\left( A\right) \right\} _{A\in \mathcal{A}}$ satisfy properties (1),
(2), (3) and (4) in Definition \ref{general stopping data}, hence constitute
stopping data for $f$. Moreover, $\mathbf{b}=\left\{ b_{I}\right\} _{I\in
\mathcal{D}}$ is an $\infty $-strongly $\sigma $-controlled accretive family
on $\mathcal{D}$ with corona tops $\mathcal{A\subset D}$, where $b_{I}
\mathbf{1}_{I}b_{A}$ has reverse H\"{o}lder control on children for all
I\in \mathcal{C}_{A}$, and the weak corona forward testing condition holds
uniformly in coronas, i.e
\begin{equation*}
\frac{1}{\left\vert I\right\vert _{\sigma }}\int_{I}\left\vert T_{\sigma
}^{\alpha }b_{A}\right\vert ^{2}d\sigma \leq C,\ \ \ \ \ I\in \mathcal{C
_{A}^{\sigma }\ .
\end{equation*
Similar statements hold for $g\in L^{2}\left( \omega \right) $.
\end{lemma}
Now we turn to the various splittings of forms, beginning with the two
weight analogue of the decomposition of Hyt\"{o}nen and Martikainen \cit
{HyMa}. Fix the stopping data $\mathcal{A}$ and $\left\{ \alpha _{\mathcal{A
}\left( A\right) \right\} _{A\in \mathcal{A}}$ and dual martingale
differences $\square _{I}^{\sigma ,\mathbf{b}}$ constructed above with the
triple iterated coronas, as well as the corresponding data for $g$. Here is
a brief schematic diagram of the splittings and decompositions we will
describe below, with associated bounds given in $\fbox{}$. We split the form
$\left\langle T_{\sigma }^{\alpha }f,g\right\rangle _{\omega }$ into the sum
of two essentially symmetric forms by interval size
\begin{equation}
\int \left( T_{\sigma }f\right) gd\omega =\left\{ \sum_{\substack{ I\in
\mathcal{D}:\ J\in \mathcal{G} \\ \ell \left( J\right) \leq \ell \left(
I\right) }}+\sum_{\substack{ I\in \mathcal{D}:\ J\in \mathcal{G} \\ \ell
\left( J\right) >\ell \left( I\right) }}\right\} \int \left( T_{\sigma
}\square _{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b
^{\ast }}gd\omega , \label{ess symm}
\end{equation
and focus on the first sum
\begin{equation*}
\Theta \left( f,g\right) =\sum_{I\in \mathcal{D}\text{ and }J\in \mathcal{G
:\ \ell \left( J\right) \leq \ell \left( I\right) }\left\langle T_{\sigma
}^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f,\square _{J}^{\omega ,\mathbf{
}^{\ast }}\right\rangle _{\omega },
\end{equation*
since the second sum is handled dually, but is easier due to the missing
diagonal
\begin{equation}
\fbox{
\begin{array}{ccccccc}
\Theta \left( f,g\right) & & & & & & \\
\downarrow & & & & & & \\
\Theta _{2}^{\limfunc{good}}\left( =\mathsf{B}_{\Subset _{\mathbf{r
}}\right) \left( f,g\right) & + & \Theta _{1}\left( =\mathsf{B}_{\cap
}\right) \left( f,g\right) & + & \Theta _{3}\left( =\mathsf{B}_{\diagup
}\right) \left( f,g\right) & + & \Theta _{2}^{\limfunc{bad}}\left( f,g\right)
\\
\downarrow & & \fbox{$\mathcal{NTV}_{\alpha }$} & & \fbox{$\mathcal{NTV
_{\alpha }+\sqrt{\theta }\mathfrak{N}_{T^{\alpha }}$} & & \fbox{$2^{
\mathbf{r}\varepsilon }\mathfrak{N}_{T^{\alpha }}$} \\
\downarrow & & & & & & \\
\mathsf{T}_{\limfunc{diagonal}}\left( f,g\right) & + & \mathsf{T}_{\limfunc
far}\limfunc{below}}\left( f,g\right) & + & \mathsf{T}_{\limfunc{far
\limfunc{above}}\left( f,g\right) & + & \mathsf{T}_{\limfunc{disjoint
}\left( f,g\right) \\
\downarrow & & \fbox{$\mathcal{NTV}_{\alpha }$} & & \fbox{$\emptyset $} &
& \fbox{$\emptyset $} \\
\mathsf{B}_{\Subset _{\mathbf{r}}}^{A}\left( f,g\right) & & & & & & \\
\downarrow & & & & & & \\
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc
paraproduct}}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc{neighbour
}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc{broken}}^{A}\left(
f,g\right) \\
\fbox{$\mathcal{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}+\sqrt{A_{2}^{\alpha
\limfunc{punct}}}$} & & \fbox{$\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}$} &
& \fbox{$\sqrt{A_{2}^{\alpha }}$} & & \fbox{$\mathfrak{T}_{T^{\alpha }}^
\mathbf{b}}$
\end{array
$} \label{schematic}
\end{equation}
For the reader's convenience we now collect the various martingale and
probability estimates that will be used in the proof that follows. First we
summarize the martingale identities and estimates from Appendix A that we
will use in our proof, noting in particular that \emph{lower weak Riesz}
inequalities are \textbf{not} used in the proof of our $Tb$ theorem. Suppose
$\mu $ is a positive locally finite Borel measure, and that $\mathbf{b}$ is
an $\infty $-strongly $\mu $-controlled accretive family. Then the following
martingale identities and estimates hold:
\begin{description}
\item[Martingale identities] Both of the following identities hold pointwise
$\mu $-almost everywhere, as well as in the sense of strong convergence in
L^{2}\left( \mu \right) $
\begin{eqnarray*}
f &=&\sum_{I\in \mathcal{D}_{N}}\mathbb{F}_{I}^{\mu ,\mathbf{b}}f+\sum_{I\in
\mathcal{D}:\ \ell \left( I\right) \geq N+1}\square _{I}^{\mu ,\mathbf{b}}f\
, \\
f &=&\sum_{I\in \mathcal{D}_{N}}\mathbb{E}_{I}^{\mu ,\mathbf{b}}f+\sum_{I\in
\mathcal{D}:\ \ell \left( I\right) \geq N+1}\bigtriangleup _{I}^{\mu
\mathbf{b}}f\ .
\end{eqnarray*}
\item[Frame estimates] Both of the following frame estimates hold
\begin{eqnarray}
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\approx &\sum_{Q\in
\mathcal{D}}\left\{ \left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}+\left\Vert \bigtriangledown _{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right\} \label{FRAME} \\
&\approx &\sum_{Q\in \mathcal{D}}\left\{ \left\Vert \bigtriangleup _{Q}^{\mu
,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right) }^{2}+\left\Vert
\bigtriangledown _{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}\right\} \ . \notag
\end{eqnarray}
\item[Weak upper Riesz estimates] Define the pseudoprojections,
\begin{eqnarray*}
\Psi _{\mathcal{B}}^{\mu ,\mathbf{b}}f &\equiv &\sum_{I\in \mathcal{B
}\square _{I}^{\mu ,\mathbf{b}}f, \\
\left( \Psi _{\mathcal{B}}^{\mu ,\mathbf{b}}\right) ^{\ast }f &\equiv
&\sum_{I\in \mathcal{B}}\left( \square _{I}^{\mu ,\mathbf{b}}\right) ^{\ast
}f=\sum_{I\in \mathcal{B}}\bigtriangleup _{I}^{\mu ,\mathbf{b}}f.
\end{eqnarray*
We have the `upper Riesz' inequalities for pseudoprojections $\Psi _
\mathcal{B}}^{\mu ,\mathbf{b}}$ and $\left( \Psi _{\mathcal{B}}^{\mu
\mathbf{b}}\right) ^{\ast }$
\begin{eqnarray}
\left\Vert \Psi _{\mathcal{B}}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\mu \right) }^{2} &\leq &C\sum_{I\in \mathcal{B}}\left\Vert \square
_{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right)
}^{2}+\sum_{I\in \mathcal{B}}\left\Vert \nabla _{I}^{\mu }f\right\Vert
_{L^{2}\left( \mu \right) }^{2}, \label{UPPER RIESZ} \\
\left\Vert \left( \Psi _{\mathcal{B}}^{\mu ,\mathbf{b}}\right) ^{\ast
}f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\leq &C\sum_{I\in \mathcal{B
}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\mu \right) }^{2}+\sum_{I\in \mathcal{B}}\left\Vert \nabla _{I}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}, \notag
\end{eqnarray
for all $f\in L^{2}\left( \mu \right) $ and all subsets $\mathcal{B}$ of the
grid $\mathcal{D}$, and where the positive constant $C$ is independent of
the subset $\mathcal{B}$. Here $\nabla _{I}^{\mu }$ is the Carleson
averaging operator defined in (\ref{Carleson avg op}) in Appendix A.
\end{description}
Now we turn to the probability estimates for martingale differences and
halos that we will use. Recall that given $0<\lambda <\frac{1}{2}$, the
\lambda $-halo of $J$ is defined to be
\begin{equation*}
\partial _{\lambda }J\equiv \left( 1+\lambda \right) J\setminus \left(
1-\lambda \right) J.
\end{equation*
Suppose $\mu $ is a positive locally finite Borel measure, and that $\mathbf
b}$ is an $\infty $-weakly $\mu $-controlled accretive family. Then the
following probability estimates hold. See Definition \ref{def Gbad} below
for the notation $\mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}$.
\begin{description}
\item[Bad cube probability estimates] Suppose that $\mathcal{D}$ and
\mathcal{G}$ are independent random dyadic grids. With $\Psi _{\mathcal{G
_{k-\limfunc{bad}}^{\mathcal{D}}}^{\mu ,\mathbf{b}^{\ast }}g\equiv
\sum_{J\in \mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}}\square _{J}^{\mu
\mathbf{b}^{\ast }}g$ equal to the pseudoprojection of $g$ onto $k$-bad
\mathcal{G}$-intervals, we hav
\begin{equation*}
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\left( \left\Vert \Psi _{\mathcal{G
_{k-\limfunc{bad}}^{\mathcal{D}}}^{\mu ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \mu \right) }^{2}\right) \lesssim \boldsymbol{E}_{\Omega }^
\mathcal{D}}\left( \sum_{J\in \mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}
\left[ \left\Vert \square _{J,\mathcal{G}}^{\mu ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \mu \right) }^{2}+\left\Vert \nabla _{J
\mathcal{G}}^{\mu }g\right\Vert _{L^{2}\left( \mu \right) }^{2}\right]
\right) \leq Ce^{-k\varepsilon }\left\Vert g\right\Vert _{L^{2}\left( \mu
\right) }^{2}\ ,
\end{equation*
where the first inequality is the `weak upper half Riesz' inequality for the
pseudoprojection $\Psi _{\mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}}^{\mu
\mathbf{b}^{\ast }}$, and the second inequality is proved using the frame
inequality in (\ref{main bad prob}) below.
\item[Halo probability estimates] Suppose that $\mathcal{D}$ and $\mathcal{G}
$ are independent random grids. Using the \emph{parameterization by
translations}\ of grids and taking the average over certain translates $\tau
+\mathcal{D}$ of the grid $\mathcal{D}$ we hav
\begin{eqnarray}
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\sum_{I^{\prime }\in \mathcal{D}:\
\ell \left( I^{\prime }\right) \approx \ell \left( J^{\prime }\right)
}\int_{J^{\prime }\cap \partial _{\delta }I^{\prime }}d\omega &\lesssim
\mathbb{\delta }\int_{J^{\prime }}d\omega ,\ \ \ \ \ J^{\prime }\in
\mathfrak{C}\left( J\right) ,J\in \mathcal{G}, \label{hand'} \\
\boldsymbol{E}_{\Omega }^{\mathcal{G}}\sum_{J^{\prime }\in \mathcal{G}:\
\ell \left( J^{\prime }\right) \approx \ell \left( I^{\prime }\right)
}\int_{I^{\prime }\cap \partial _{\delta }J^{\prime }}d\sigma &\lesssim
\mathbb{\delta }\int_{I^{\prime }}d\sigma ,\ \ \ \ \ I^{\prime }\in
\mathfrak{C}\left( I\right) ,I\in \mathcal{D}, \notag
\end{eqnarray
and where the expectations $\boldsymbol{E}_{\Omega }^{\mathcal{D}}$ and
\boldsymbol{E}_{\Omega }^{\mathcal{G}}$ are taken over grids $\mathcal{D}$
and $\mathcal{G}$ respectively. Indeed, it is geometrically evident that for
any fixed pair of side lengths $\ell _{1}\approx \ell _{2}$, the average of
the measure $\left\vert J^{\prime }\cap \partial _{\delta }I^{\prime
}\right\vert _{\omega }$ of the set $J^{\prime }\cap \partial _{\delta
}I^{\prime }$, as an interval $I^{\prime }\in \mathcal{D}$ with side length
\ell \left( I^{\prime }\right) =\ell _{1}$ is translated across an interval
J^{\prime }\in \mathcal{G}$ of side length $\ell \left( J^{\prime }\right)
=\ell _{2}$, is at most $C\left\vert J^{\prime }\right\vert _{\omega }$.
Using this observation it is now easy to see that (\ref{hand'}) holds.
\end{description}
\subsection{The Hyt\"{o}nen-Martikainen decomposition and a weak variant of
NTV goodness\label{Subsec HM}}
Let $\mathbf{b}$ (respectively $\mathbf{b}^{\ast }$) be $\infty $-weakly
\sigma $-controlled (respectively $\omega $-controlled) accretive families.
At the beginning of this section, we reduced the estimation of the bilinear
form $\int_{\mathbb{R}}\left( T_{\sigma }f\right) gd\omega $ to that of the
su
\begin{equation*}
\sum_{I\in \mathcal{D}}\sum_{J\in \mathcal{G}}\int \left( T_{\sigma
}^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega
\mathbf{b}^{\ast }}gd\omega ,
\end{equation*
and then we decomposed this sum by interval side length
\begin{eqnarray*}
\sum_{I\in \mathcal{D}}\sum_{J\in \mathcal{G}}\int \left( T_{\sigma
}^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega
\mathbf{b}^{\ast }}gd\omega &=&\left\{ \sum_{\substack{ I\in \mathcal{D}:\
J\in \mathcal{G} \\ \ell \left( J\right) \leq \ell \left( I\right) }}+\sum
_{\substack{ I\in \mathcal{D}:\ J\in \mathcal{G} \\ \ell \left( J\right)
>\ell \left( I\right) }}\right\} \int \left( T_{\sigma }^{\alpha }\square
_{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast
}}gd\omega \\
&\equiv &\Theta \left( f,g\right) +\Theta ^{\ast }\left( f,g\right) ,
\end{eqnarray*
and noted that by symmetry, it suffices to estimate the first form $\Theta
\left( f,g\right) $. Before introducing goodness into the sum, we follow
\cite{HyMa} and split the form $\Theta \left( f,g\right) $ into 3 pieces
\begin{eqnarray*}
&&\Theta \left( f,g\right) \equiv \sum_{\substack{ I\in \mathcal{D}:\ J\in
\mathcal{G} \\ \ell \left( J\right) \leq \ell \left( I\right) }}\int \left(
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \square
_{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega \\
&=&\sum_{I\in \mathcal{D}}\left\{ \sum_{\substack{ J\in \mathcal{G}:\ \ell
\left( J\right) \leq \ell \left( I\right) \\ d\left( J,I\right) >2\ell
\left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}+\sum
_{\substack{ J\in \mathcal{G}:\ \ell \left( J\right) \leq 2^{-\mathbf{r
}\ell \left( I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right)
^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}+\sum_{\substack{
J\in \mathcal{G}:\ 2^{-\mathbf{r}}\ell \left( I\right) <\ell \left( J\right)
\leq \ell \left( I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right)
^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}\right\} \int \left(
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \square
_{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega \\
&\equiv &\Theta _{1}\left( f,g\right) +\Theta _{2}\left( f,g\right) +\Theta
_{3}\left( f,g\right) \ ,
\end{eqnarray*
where $\varepsilon >0$ will be chosen to satisfy $0<\varepsilon <\frac{1}{2
\leq \frac{1}{2-\alpha }$ later, and the goodness parameter $\mathbf{r}$ is
then determined in (\ref{choice of r}) below. Now the disjoint form $\Theta
_{1}\left( f,g\right) $ can be handled by `long-range' and `short-range'
arguments which we give in the next section below, and the nearby form
\Theta _{3}\left( f,g\right) $ will be handled in the subsequent section
using probabilistic surgery methods and a new deterministic surgery
involving energy conditions and the `original' testing functions discarded
in the corona construction. The remaining form $\Theta _{2}\left( f,g\right)
$ will be treated further in this section after introducing weak goodness.
\subsubsection{Good intervals with `$\limfunc{body}$'}
We begin with the weaker extension of goodness introduced in \cite{HyMa},
except that we will make it a bit stronger by replacing the skeleton `
\limfunc{skel}K$' of an interval $K$, as used in \cite{HyMa}, by a larger
collection of points `$\limfunc{body}K$', which we call the dyadic body of
K $. This modification will prove useful in establishing the Straddling
Lemma in the treatment of the stopping form in Section \ref{Sec stop} below.
Let $\mathcal{P}$ denote the collection of all intervals in $\mathbb{R}$.
The content of the next four definitions is inspired by, or sometimes
identical with, that already appearing in the work of Nazarov, Treil and
Volberg in \cite{NTV1} and \cite{NTV3}.
\begin{definition}
\label{skel spray body}Let $K\in \mathcal{P}$.
\begin{enumerate}
\item Define the \emph{skeleton} `$\limfunc{skel}K$' of $K$ to consist of
its two endpoints and its midpoint.
\item For a point $x$ in $\mathbb{R}$, define the \emph{dyadic spray}
\mathbb{S}_{x}^{\limfunc{dy}}\equiv \left\{ x\right\} \cup \left\{ x\pm
\frac{1}{2^{j}}\right\} _{k\in \mathbb{Z}}$ of $x$ to consist of $x$ and all
points $y$ in $\mathbb{R}$ that have distance $\frac{1}{2^{j}}$ from $x$ for
some $j\in \mathbb{Z}$.
\item Then define the \emph{dyadic body} `$\limfunc{body}K$' of an interval
K\in \mathcal{P}$ to be the intersection of $\overline{K}$ with the union of
the dyadic sprays of its two endpoints, i.e. if $K=\left[ a,b\right) $, the
\begin{equation*}
\limfunc{body}K=\overline{K}\cap \left( \mathbb{S}_{a}^{\limfunc{dy}}\cup
\mathbb{S}_{b}^{\limfunc{dy}}\right) .
\end{equation*}
\end{enumerate}
\end{definition}
Thus the body of the unit interval $\left[ 0,1\right) $ consists of the
points
\begin{equation*}
\left\{ 0\right\} \dot{\cup}\left\{ \frac{1}{2^{j}}\right\} _{j=1}^{\infty
\dot{\cup}\left\{ 1-\frac{1}{2^{j}}\right\} _{j=2}^{\infty }\dot{\cup
\left\{ 1\right\} \ ,
\end{equation*
which have the endpoints of $\left[ 0,1\right) $ as cluster points.
\begin{definition}
\label{good arb}Let $0<\varepsilon <1$ (to be chosen later). For intervals
J,K\in \mathcal{P}$ with $\ell \left( J\right) \leq \ell \left( K\right) $,
we define $J$ to be $\varepsilon -\limfunc{good}$ \emph{with respect to} an
interval $K$ if
\begin{equation}
d\left( J,\limfunc{body}K\right) >2\left\vert J\right\vert ^{\varepsilon
}\left\vert K\right\vert ^{1-\varepsilon }, \label{eps far}
\end{equation
and we say $J$ is $\varepsilon -\limfunc{bad}$ \emph{with respect to} $K$ if
(\ref{eps far}) fails. We also say that $J$ is $\varepsilon -\limfunc{good}$
\emph{inside} an interval $K$ if $J$ is $\varepsilon -\limfunc{good}$ with
respect to $K$ and $J\subset K$.
\end{definition}
A key consequence of an interval $J$ being $\varepsilon -\limfunc{good}$
inside an interval $S$, is that $J$ must then be contained in some \emph
dyadic subinterval} $K$ of $S$ with $3K\subset S$
\begin{equation}
\text{If }J\text{ is }\varepsilon -\limfunc{good}\text{inside }S\text{, then
}J\subset K\text{ for some }K\in \mathcal{W}\left( S\right) ,
\label{key contain}
\end{equation
where $\mathcal{W}\left( S\right) $ is the collection of maximal dyadic
subintervals of $S$ whose triples are contained in $S$. Indeed, the
endpoints of the intervals in $\mathcal{W}\left( S\right) $ are precisely
the $\limfunc{body}$ of $S$. Note that this property can fail if we use the
smaller set $\limfunc{skel}S$ in place of $\limfunc{body}S$ in Definition
\ref{good arb}, since then an $\varepsilon -\limfunc{good}$ interval $J$
could intersect one of the sprays. Of course we will also need to know that
\limfunc{body}S$ is not so much larger than $\limfunc{skel}S$ that the
crucial probability estimate for good intervals fails - namely we need to
know that given $k\gg 1$, an interval $J\subset S$ of side length $\ell
\left( J\right) =2^{-k}\ell \left( S\right) $ is $\varepsilon -\limfunc{good}
$ inside $S$ with `large probability'. This will be made precise below using
random dyadic grids.
\begin{definition}
\label{good two grids}Let $\mathcal{D}$ and $\mathcal{G}$ be dyadic grids.
Define $\mathcal{G}_{\left( k,\varepsilon \right) -\limfunc{good}}^{\mathcal
D}}$ to consist of those $J\in \mathcal{G}$ such that $J$ is $\varepsilon
\limfunc{good}$ \textbf{inside} every interval $K\in \mathcal{D}$ with
K\cap J\neq \emptyset $ that lies at least $k$ levels `above' $J$, i.e.
\ell \left( K\right) \geq 2^{k}\ell \left( J\right) $ (note that the use of
\textbf{inside} forces such $K$ with $K\cap J\neq \emptyset $ to actually
contain $J$). We also define $J$ to be `$\varepsilon -\limfunc{good}$ inside
an interval $K$ and beyond' if $J\in \mathcal{G}_{\left( k,\varepsilon
\right) -\limfunc{good}}^{\mathcal{D}}$ where $k=\log _{2}\frac{\ell \left(
K\right) }{\ell \left( J\right) }$ and where $K\cap J\neq \emptyset $,
equivalently in this situation $K\supset J$. As the goodness parameter
\varepsilon $ will eventually be fixed throughout the proof, we sometimes
suppress it, and simply say `$J$ is $\limfunc{good}$ inside an interval $K$
and beyond' instead of `$J$ is $\varepsilon -\limfunc{good}$ inside an
interval $K$ and beyond'. When $\varepsilon >0$ is understood, we will often
write $\mathcal{G}_{k-\limfunc{good}}^{\mathcal{D}}=\mathcal{G}_{\left(
k,\varepsilon \right) -\limfunc{good}}^{\mathcal{D}}$.
\end{definition}
\begin{remark}
Note tha
\begin{equation*}
\mathcal{G}_{\left( k,\varepsilon \right) -\limfunc{good}}^{\mathcal{D
}\equiv \left\{ J\in \mathcal{G}:J\text{ is }\varepsilon -\limfunc{good
\text{ with respect to every }K\in \mathcal{D}\text{ with }\ell \left(
K\right) \geq 2^{k}\ell \left( J\right) .\right\}
\end{equation*
Indeed, if $J$ is $\varepsilon -\limfunc{bad}$ with respect to some $K\in
\mathcal{D}$ with $K\cap J=\emptyset $, then $J$ is also $\varepsilon
\limfunc{bad}$ with respect to one of the two neighbours (of the same side
length) of $K$ in $\mathcal{D}$.
\end{remark}
\subsubsection{Grid probability}
As pointed out on page 14 of \cite{HyMa} by Hyt\"{o}nen and Martikainen,
there are subtle difficulties associated in using dual martingale
decompositions of functions which depend on the entire dyadic grid, rather
than on just the local interval in the grid. We will proceed at first in the
spirit of \cite{HyMa}, and the goodness that we will infuse below into the
main `below' form $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) $
will be the Hyt\"{o}nen-Martikainen `weak' version of NTV goodness, but
using the body `$\limfunc{body}I$' of an interval rather than\ its skeleton
$\limfunc{skel}I$': every pair $\left( I,J\right) \in \mathcal{D}\times
\mathcal{G}$ that arises in the form $\mathsf{B}_{\Subset _{\mathbf{r
}}\left( f,g\right) $ will satisfy $J\in \mathcal{G}_{\left( k,\varepsilon
\right) -\limfunc{good}}^{\mathcal{D}}$ where $\ell \left( I\right)
=2^{k}\ell \left( J\right) $.
Now we return to the martingale differences $\square _{I}^{\sigma ,\mathbf{b
}$ and $\square _{J}^{\omega ,\mathbf{b}^{\ast }}$ with controlled families
\mathbf{b}$ and $\mathbf{b}^{\ast }$ in the real line $\mathbb{R}$. When we
want to emphasize that the grid in use is $\mathcal{D}$ or $\mathcal{G}$, we
will denote the martingale difference by $\square _{I,\mathcal{D}}^{\sigma
\mathbf{b}}$, and similarly for $\square _{J,\mathcal{G}}^{\omega ,\mathbf{b
^{\ast }}$. Recall Definition \ref{good arb} for the meaning of when an
interval $J$ is $\varepsilon $-$\limfunc{bad}$ with respect to another
interval $K$.
\begin{definition}
\label{bad in grid}We say that $J\in \mathcal{P}$ is $k$-$\limfunc{bad}$ in
a grid $\mathcal{D}$ if there is an interval $K\in \mathcal{D}$ with $\ell
\left( K\right) =2^{k}\ell \left( J\right) $ such that $J$ is $\varepsilon $
$\limfunc{bad}$ with respect to $K$ (context should eliminate any ambiguity
between the different use of $k$-$\limfunc{bad}$ when $k\in \mathbb{N}$ and
\varepsilon $-$\limfunc{bad}$ when $0<\varepsilon <\frac{1}{2}$).
\end{definition}
A key observation here (see \cite{NTV1}, \cite{NTV2}, \cite{NTV3} or \cit
{NTV4} for the case when goodness is defined using the skeleton instead of
the body) is that for any $J\in \mathcal{G}$ where $\mathcal{D}$ and
\mathcal{G}$ are independent random grids
\begin{equation}
\boldsymbol{P}_{\Omega }^{\mathcal{D}}\left( \mathcal{D}:J\text{ is }k\text{
}\limfunc{bad}\text{ in }\mathcal{D}\right) \equiv \int_{\Omega }\mathbf{1
_{\left\{ \mathcal{D}:\ J\text{ is }k\text{-}\limfunc{bad}\text{ in
\mathcal{D}\right\} }d\mu _{\Omega }\left( \mathcal{D}\right) \leq
C\varepsilon k2^{-\varepsilon k}. \label{key prob}
\end{equation
Indeed, it suffices to consider the case when $J\in \mathcal{G}$ with
J\subset \left[ 0,1\right) $ and $\ell \left( J\right) =2^{-k}$. So fix such
an interval $J$. For each $m\in \mathbb{Z}_{2^{k}}\equiv \left\{ \ell \in
\mathbb{Z}:0\leq \ell \leq 2^{k}-1\right\} $, consider the collection
\mathfrak{D}_{m}$ of all grids $\mathcal{D}$ that contain the interval
I_{m}\equiv \left[ 0,1\right) +\frac{m}{2^{k}}=\left[ \frac{m}{2^{k}},1
\frac{m}{2^{k}}\right) $. Then for every $m$, it is the case that
\begin{enumerate}
\item \textbf{either} $J$ is $\varepsilon $-$\limfunc{bad}$ in $\mathcal{D}$
for all $\mathcal{D}\in \mathfrak{D}_{m}$,
\item \textbf{or} $J$ is $\varepsilon $-$\limfunc{good}$ in $\mathcal{D}$
for all $\mathcal{D}\in \mathfrak{D}_{m}$.
\end{enumerate}
We will say that the \emph{collection} $\mathfrak{D}_{m}$ is $k$-$\limfunc
bad}$ if the first case holds. We have the same dichotomy for $\mathfrak{D
_{m}+s$ if we replace $\left[ 0,1\right) $ with the translate $\left[
0,1\right) +s=\left[ s,1+s\right) $ where $0\leq s<2^{-k}$. We now claim
that for any fixed $0\leq s<2^{-k}$, the number of $k$-$\limfunc{bad}$
collections $\mathfrak{D}_{m}+s$ is at most $C\varepsilon k2^{\left(
1-\varepsilon \right) k}$, hence the proportion of $k$-$\limfunc{bad}$
collections is $\frac{C\varepsilon k2^{\left( 1-\varepsilon \right) k}}{2^{k
}=C\varepsilon k2^{-\varepsilon k}$, from which we obtain the estimate (\re
{key prob}) as follows. Every grid $\mathcal{D}\in \Omega $ is contained in
exactly one of the collections $\left\{ \mathfrak{D}_{m}+s\right\} _{m\in
\mathbb{Z}_{2^{k}}\text{ and }s\in \left[ 0,2^{-k}\right) }$, and s
\begin{eqnarray*}
\boldsymbol{P}_{\Omega }^{\mathcal{D}}\left( \mathcal{D}:J\text{ is }k\text{
}\limfunc{bad}\text{ in }\mathcal{D}\right) &=&\frac{1}{2^{-k}}\int_{\left[
0,2^{-k}\right) }\left\{ \frac{\#\left\{ m\in \mathbb{Z}_{2^{k}}:\ \mathfrak
D}_{m}+s\text{ is }k-\limfunc{bad}\right\} }{\#\mathbb{Z}_{2^{k}}}\right\} ds
\\
&\leq &\frac{1}{2^{-k}}\int_{\left[ 0,2^{-k}\right) }\left\{ \frac
C\varepsilon k2^{\left( 1-\varepsilon \right) k}}{2^{k}}\right\}
ds=C\varepsilon k2^{-\varepsilon k}.
\end{eqnarray*}
To see our claim, it suffices to consider the case $s=0$, to keep the
interval $I_{0}=\left[ 0,1\right) $ fixed, and consider instead the
translates $J_{m}\equiv J+\frac{m}{2^{k}}$ of the interval $J$ for $0\leq
m\leq 2^{k}-1$. Moreover we can assume without loss of generality that $J$
intersects the point $\frac{1}{2^{k}}$ so that all of the intervals $J_{m}$
in the collection $\left\{ J_{m}\right\} _{m=0}^{2^{k}-1}$ lie in $I_{0}$
except for the last one $J_{2^{k}-1}=J+1-\frac{1}{2^{k}}$, which intersects
the point $1$. In this situation our claim become
\begin{equation}
\#\left\{ m\in \mathbb{Z}_{2^{k}}:J_{m}\text{ is }\varepsilon -\limfunc{bad
\text{ in }\left[ 0,1\right) \right\} \leq C\varepsilon k2^{\left(
1-\varepsilon \right) k}. \label{claim becomes}
\end{equation}
To prove (\ref{claim becomes}), we begin by definin
\begin{equation*}
d\equiv \varepsilon k-1\text{ and }L\equiv 2\ell \left( J\right)
^{\varepsilon }\ell \left( I_{0}\right) ^{1-\varepsilon }=2^{1-\varepsilon
k}=2^{-d},
\end{equation*
where we may assume $k>\frac{1}{\varepsilon }$ so that $d>0$. Then if $J_{m}$
is $\varepsilon -\limfunc{bad}$ in $\left[ 0,1\right) $, at least one of the
following two inequalities must hold
\begin{equation*}
\limfunc{dist}\left( J_{m},\mathbb{S}_{0}^{\limfunc{dy}}\right) \leq L,\
\limfunc{dist}\left( J_{m},\mathbb{S}_{1}^{\limfunc{dy}}\right) \leq L,
\end{equation*
where we recall that $\mathbb{S}_{a}^{\limfunc{dy}}$ is the dyadic spray of
a$. Now if $\limfunc{dist}\left( J_{m},\mathbb{S}_{0}^{\limfunc{dy}}\right)
\leq L$, the
\begin{eqnarray*}
\text{either }\limfunc{dist}\left( J_{m},\left\{ 0\right\} \cup \left\{
\frac{1}{2^{j}}\right\} _{j>d}\right) &\leq &L, \\
\text{or }\limfunc{dist}\left( J_{m},\frac{1}{2^{j}}\right) &\leq &L,\ \ \ \
\ \text{for some }0\leq j\leq d.
\end{eqnarray*
However, if
\begin{equation}
\limfunc{dist}\left( J_{m},\left\{ 0\right\} \cup \left\{ \frac{1}{2^{j}
\right\} _{j>d}\right) \leq L, \label{first case}
\end{equation
then we must have
\begin{equation*}
m\ell \left( J\right) \leq \frac{1}{2^{d}}+L=2L,
\end{equation*
and if $\limfunc{dist}\left( J_{m},\frac{1}{2^{j}}\right) \leq L$ for some
0\leq j\leq d$, then we must hav
\begin{equation*}
\left\vert \frac{m}{2^{k}}-\frac{1}{2^{j}}\right\vert \leq 2L.
\end{equation*
So altogether the number of indices $m\in \mathbb{Z}_{2^{k}}$ for which
\limfunc{dist}\left( J_{m},\mathbb{S}_{0}^{\limfunc{dy}}\right) \leq L$
holds is at mos
\begin{eqnarray*}
2\frac{L}{\ell \left( J\right) }+1+\left( d+1\right) \left(
2^{k+2}L+1\right) &=&\left( 2d+3\right) 2^{k+1}L+d+2 \\
&=&\left( 2\varepsilon k+1\right) \cdot 2^{k+1}\cdot 2^{1-\varepsilon
k}+\varepsilon k+1\leq 20\varepsilon k2^{\left( 1-\varepsilon \right) k}.
\end{eqnarray*
Similarly the number of indices $m\in \mathbb{Z}_{2^{k}}$ for which
\limfunc{dist}\left( J_{m},\mathbb{S}_{1}^{\limfunc{dy}}\right) \leq L$
holds is at most $20\varepsilon k2^{\left( 1-\varepsilon \right) k}$. Thus
we conclude that (\ref{claim becomes}) holds with $C=40$.
Then we obtain from (\ref{key prob}), using the lower frame inequality, the
expectation estimat
\begin{eqnarray*}
&&\int_{\Omega }\sum_{J\in \mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}}\left[
\left\Vert \square _{J,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla _{J,\mathcal{G
}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right] d\mu
_{\Omega }\left( \mathcal{D}\right) \\
&=&\sum_{J\in \mathcal{G}}\left[ \left\Vert \square _{J,\mathcal{G}}^{\omega
,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}+\left\Vert \nabla _{J,\mathcal{G}}^{\omega }g\right\Vert _{L^{2}\left(
\omega \right) }^{2}\right] \int_{\Omega }\mathbf{1}_{\left\{ \mathcal{D}:\
\text{ is }k\text{-}\limfunc{bad}\text{ in }\mathcal{D}\right\} }d\mu
_{\Omega }\left( \mathcal{D}\right) \\
&\leq &Ck\varepsilon 2^{-k\varepsilon }\sum_{J\in \mathcal{G}}\left[
\left\Vert \square _{J,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla _{J,\mathcal{G
}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right] \leq
Ck\varepsilon 2^{-k\varepsilon }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }^{2}\ ,
\end{eqnarray*
where $\nabla _{J,\mathcal{G}}^{\omega }$ denotes the `broken' Carleson
averaging operator in (\ref{Carleson avg op}) that depends on the broken
children in the grid $\mathcal{G}$. Altogether then it follows easily tha
\begin{equation}
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\left( \sum_{J\in \bigcup_{\ell
=k}^{\infty }\mathcal{G}_{\ell -\limfunc{bad}}^{\mathcal{D}}}\left[
\left\Vert \square _{J,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla _{J,\mathcal{G
}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right] \right)
\leq Ck\varepsilon 2^{-k\varepsilon }\left\Vert g\right\Vert _{L^{2}\left(
\omega \right) }^{2}\ , \label{main bad prob}
\end{equation
for some large positive constant $C$.
From such inequalities summed for $k\geq \mathbf{r}$, it can be concluded as
in \cite{NTV3} that there is an absolute choice of $\mathbf{r}$ depending on
$0<\varepsilon <\frac{1}{2}$ so that the following holds. Let
T\;:\;L^{2}(\sigma )\rightarrow L^{2}(\omega )$ be a bounded linear
operator. We then have the following traditional inequality for two random
grids in the case that $\mathbf{b}$ is an $\infty $-strongly $\mu
-controlled accretive family:
\begin{equation}
\left\Vert T\right\Vert _{L^{2}(\sigma )\rightarrow L^{2}(\omega )}\leq
2\sup_{\left\Vert f\right\Vert _{L^{2}(\sigma )}=1}\sup_{\left\Vert
g\right\Vert _{L^{2}(\omega )}=1}\boldsymbol{E}_{\Omega }\boldsymbol{E
_{\Omega ^{\prime }}\left\vert \left\langle \sum_{I,J\in \mathcal{D}_
\mathbf{r}-\limfunc{good}}^{\mathcal{G}}}T\left( \square _{I,\mathcal{D
}^{\sigma ,\mathbf{b}}f\right) f,\square _{J,\mathcal{D}}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }\right\vert \,. \label{e.Tgood'}
\end{equation}
However, this traditional method of introducing goodness is flawed here in
the general setting of dual martingale differences, since these differences
are no longer orthogonal projections, and as emphasized in \cite{HyMa}, we
cannot simply add back in bad intervals whenever we want telescoping
identities to hold - but these are needed in order to control the right hand
side of (\ref{e.Tgood'}). In fact, in the analysis of the form $\Theta
\left( f,g\right) $ above, it is necessary to have goodness for the
intervals $J$ and telescoping for the intervals $I$. On the other hand, in
the analysis of the form $\Theta ^{\ast }\left( f,g\right) $ above, it is
necessary to have just the opposite - namely goodness for the intervals $I$
and telescoping for the intervals $J$.
Thus, because in this unfortunate set of circumstances we can no longer `add
back in' bad cubes to achieve telescoping, we are prevented from introducing
goodness in the \emph{full} sum (\ref{ess symm})\ over all $I$ and $J$,
prior to splitting according to side lengths of $I$ and $J$. Thus the
infusion of goodness must come \emph{after} the splitting by side length,
but one must work much harder to introduce goodness directly into the form
\Theta \left( f,g\right) $ \emph{after} we have restricted the sum to
intervals $J$ that have smaller side length than $I$. This is accomplished
in the next subsubsection using the \emph{weaker form of NTV goodness}
introduced by Hyt\"{o}nen and Martikainen in \cite{HyMa} (that permits
certain additional pairs $\left( I,J\right) $ in the good forms where $\ell
\left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) $ and yet $J$ is
\limfunc{bad}$ in the traditional sense), and that will prevail later in the
treatment of the far below forms $\mathsf{T}_{\limfunc{far}\limfunc{below
}^{1}\left( f,g\right) $, and of the local forms $\mathsf{B}_{\Subset _
\mathbf{r}}}^{A}\left( f,g\right) $ (see Subsection \ref{Sub wrapup}) where
the need for using the `body' of an interval will become apparent in dealing
with the stopping form, and also in the treatment of the functional energy
in Appendix B.
\subsubsection{Weak goodness}
Let $\mathcal{D}$ and $\mathcal{G}$ be dyadic grids. It remains to estimate
the form $\Theta _{2}\left( f,g\right) $ which, following \cite{HyMa}, we
will split into a `bad' part and a `good' part. For this we introduce our
main definition associated with the above modification of the weak goodness
of Hyt\"{o}nen and Martikainen, namely the definition of the interval
R^{\maltese }$ in a grid $\mathcal{D}$, given an arbitrary interval $R\in
\mathcal{P}$.
\begin{definition}
\label{def sharp cross}Let $\mathcal{D}$ be a dyadic grid. Given $R\in
\mathcal{P}$, let $R^{\maltese }$ be the smallest (if any such exist)
\mathcal{D}$-dyadic superinterval $Q$ of $R$ such that $R$ is good inside
\textbf{all} $\mathcal{D}$-dyadic superintervals $K$ of $Q$. Of course
R^{\maltese }$ will not exist if there is no $\mathcal{D}$-dyadic interval
Q $ containing $R$ in which $R$ is good. For intervals $R,Q\in \mathcal{P}$
let $\kappa \left( Q,R\right) =\log _{2}\frac{\ell \left( Q\right) }{\ell
\left( R\right) }$. For $R\in \mathcal{P}$ for which $R^{\maltese }$ exists,
let $\kappa \left( R\right) \equiv \kappa \left( R^{\maltese },R\right) $.
\end{definition}
Note that we typically suppress the dependence of $R^{\maltese }$ on the
grid $\mathcal{D}$, since the grid is usually understood from context. If
R^{\maltese }$ exists, we thus have that $R$ is good inside all $\mathcal{D}
-dyadic superintervals $K$ of $R$ with $\ell \left( K\right) \geq \ell
\left( R^{\maltese }\right) $. Note in particular the monotonicity property
for $J^{\prime },J\in \mathcal{P}$
\begin{equation*}
J^{\prime }\subset J\Longrightarrow \left( J^{\prime }\right) ^{\maltese
}\subset J^{\maltese }.
\end{equation*
Here now is the decomposition
\begin{eqnarray*}
\Theta _{2}\left( f,g\right) &=&\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in
\mathcal{G}:\ J^{\maltese }\not\subsetneqq I\text{, }\ell \left( J\right)
\leq 2^{-\mathbf{r}}\ell \left( I\right) \\ d\left( J,I\right) \leq 2\ell
\left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}\int
\left( T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right)
\square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega \\
&&+\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ J^{\maltese
}\subsetneqq I\text{, }\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }}}\int \left( T_{\sigma }^{\alpha
}\square _{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b
^{\ast }}gd\omega \\
&\equiv &\Theta _{2}^{\limfunc{bad}}\left( f,g\right) +\Theta _{2}^{\limfunc
good}}\left( f,g\right) \ ,
\end{eqnarray*
and where if $J^{\maltese }$ fails to exist, we assume by convention that
J^{\maltese }\not\subsetneqq I$, i.e. $J^{\maltese }$ is \emph{not} strictly
contained in $I$, so that the pair $\left( I,J\right) $ is then included in
the bad form $\Theta _{2}^{\limfunc{bad}}\left( f,g\right) $. We will in
fact estimate a larger quantity corresponding to the bad form, namel
\begin{equation}
\Theta _{2}^{\limfunc{bad}\natural }\left( f,g\right) \equiv \sum_{I\in
\mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ J^{\maltese }\not\subsetneqq
I\text{, }\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \int \left( T_{\sigma }^{\alpha
}\square _{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b
^{\ast }}gd\omega \right\vert \label{Theta_2^bad sharp}
\end{equation
with absolute value signs \emph{inside} the sum.
\begin{remark}
We now make some general comments on where we now stand and where we are
going.
\begin{enumerate}
\item In the first sum $\Theta _{2}^{\limfunc{bad}}\left( f,g\right) $
above, we are roughly keeping the pairs of intervals $\left( I,J\right) $
such that $J$ is $\limfunc{bad}$ with respect to some `nearby' interval
having side length larger than that of $I$.
\item We have defined energy and dual energy conditions that are independent
of the testing families (because the definition of $\mathsf{E}\left(
J,\omega \right) =\mathbb{E}_{J}^{\omega ,x}\mathbb{E}_{J}^{\omega
,x^{\prime }}\left( \left\vert \frac{x-x^{\prime }}{\ell \left( J\right)
\right\vert ^{2}\right) $ does not involve pseudoprojections $\square _{J
\mathcal{D}}^{\omega ,\mathbf{b}^{\ast }}$), but the functional energy
condition defined below \emph{does} involve the dual martingale
pseudoprojections $\square _{J,\mathcal{D}}^{\omega ,\mathbf{b}^{\ast }}$.
\item Using the notion of weak goodness above, we will be able to eliminate
all pairs of intervals with $J$ bad in $I$, which then permits control of
the short range form in Section \ref{Sec disj form} and the neighbour form
in Section \ref{Sec Main below} provided $0<\varepsilon <\frac{1}{2-\alpha }
. Defining shifted coronas in terms of $J^{\maltese }$ will then allow
existing arguments to prove the Intertwining Proposition and obtain control
of the functional energy in Appendix B, as well as permitting control of the
stopping form in Section \ref{Sec stop}, but all of this with some new
twists, for example the introduction of a top/down `indented corona' in the
analysis of the stopping form.
\item The nearby form $\Theta _{3}\left( f,g\right) $ is handled in Section
\ref{Sec nearby} using the energy condition assumption along with the
original testing functions $b_{Q}^{\limfunc{orig}}$ discarded during the
construction of the testing/accretive corona.
\end{enumerate}
\end{remark}
These remarks will become clear in this and the following sections. Recall
that we earlier defined in Definition \ref{good two grids}, the set
\mathcal{G}_{k-\limfunc{good}}^{\mathcal{D}}=\mathcal{G}_{\left(
k,\varepsilon \right) -\limfunc{good}}^{\mathcal{D}}$ to consist of those
J\in \mathcal{G}$ such that $J$ is $\varepsilon -\limfunc{good}$ inside
every interval $K\in \mathcal{D}$ with $K\cap J\neq \emptyset $ that lies at
least $k$ levels `above' $J$, i.e. $\ell \left( K\right) \geq 2^{k}\ell
\left( J\right) $. We now define an analogous notion of $\mathcal{G}_{k
\limfunc{bad}}^{\mathcal{D}}$.
\begin{definition}
\label{def Gbad}Let $\varepsilon >0$. Define the set $\mathcal{G}_{k
\limfunc{bad}}^{\mathcal{D}}=\mathcal{G}_{\left( k,\varepsilon \right)
\limfunc{bad}}^{\mathcal{D}}$ to consist of all $J\in \mathcal{G}$ such that
there is a $\mathcal{D}$-interval $K$ with sidelength $\ell \left( K\right)
=2^{k}\ell \left( J\right) $ for which $J$ is $\varepsilon -\limfunc{bad}$
with respect to $K$.
\end{definition}
Note that for grids $\mathcal{D}$ and $\mathcal{G}$, the complement of
\mathcal{G}_{k-\limfunc{good}}^{\mathcal{D}}$ is the union of $\mathcal{G
_{\ell -\limfunc{bad}}^{\mathcal{D}}$ for $\ell \geq k$, i.e
\begin{equation*}
\mathcal{G\setminus G}_{k-\limfunc{good}}^{\mathcal{D}}=\bigcup_{\ell \geq k
\mathcal{G}_{\ell -\limfunc{bad}}^{\mathcal{D}}\ .
\end{equation*
Now assume $\varepsilon >0$. We then have the following important property,
namely for all intervals $R$, and all $k\geq \mathbf{r}$ (where the goodness
parameter $\mathbf{r}$ will be fixed given $\varepsilon >0$ in (\ref{choice
of r}) below):
\begin{equation}
\#\left\{ Q:\kappa \left( Q,R\right) =k\text{ and }d\left( R,Q\right) \leq
2\ell \left( R\right) ^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon
}\right\} \lesssim 1. \label{imp}
\end{equation
As in \cite{HyMa}, se
\begin{equation*}
\mathcal{G}_{\limfunc{bad},n}^{\mathcal{D}}\equiv \left\{ J\in \mathcal{G}:
\text{ is }\varepsilon -\limfunc{bad}\ \text{with respect to some }K\in
\mathcal{D}\text{ with }\ell \left( K\right) \geq n\right\} .
\end{equation*
We will now use the set equalit
\begin{eqnarray}
&&\left\{ J\in \mathcal{G}:\ J^{\maltese }\not\subset I,\text{ }\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) ,\ d\left( J,I\right) \leq
2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon
}\right\} \label{set equ} \\
&=&\left\{ R\in \mathcal{G}_{\limfunc{bad},\ell \left( Q\right) }^{\mathcal{
}}:\ \mathbf{r}\leq \kappa \left( Q,R\right) <\kappa \left( R\right) ,\
d\left( R,Q\right) \leq 2\ell \left( R\right) ^{\varepsilon }\ell \left(
Q\right) ^{1-\varepsilon }\right\} , \notag
\end{eqnarray
which the careful reader can prove by painstakingly verifying both
containments.
Assuming only that $\mathbf{b}$ is $2$-weakly $\mu $-controlled accretive
(recall we are assuming the stronger condition that $\mathbf{b}$ is $\infty
-strongly $\mu $-controlled accretive in our proof here), and following the
proof in \cite{HyMa}, we use (\ref{set equ}) to show that for any fixed
grids $\mathcal{D}$ and $\mathcal{G}$, and any bounded linear operator
T_{\sigma }^{\alpha }$ we have the following inequality for the form $\Theta
_{2}^{\limfunc{bad}\natural ,\limfunc{strict}}\left( f,g\right) $, defined
to be $\Theta _{2}^{\limfunc{bad}\natural }\left( f,g\right) $ as in (\re
{Theta_2^bad sharp}) with the pairs $\left( I,J\right) $ removed when
J^{\maltese }=I$. We use $\varepsilon _{Q,R}=\pm 1$ to obtain
\begin{eqnarray*}
&&\Theta _{2}^{\limfunc{bad}\natural ,\limfunc{strict}}\left( f,g\right)
=\sum_{Q\in \mathcal{D}}\sum_{\substack{ R\in \mathcal{G}_{\limfunc{bad
,\ell \left( Q\right) }^{\mathcal{D}}:\ \mathbf{r}\leq \kappa \left(
Q,R\right) <\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell \left(
R\right) ^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}}\left\vert
\left\langle T_{\sigma }^{\alpha }\left( \square _{Q,\mathcal{D}}^{\sigma
\mathbf{b}}f\right) ,\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle \right\vert \\
&=&\sum_{Q\in \mathcal{D}}\sum_{\substack{ R\in \mathcal{G}_{\limfunc{bad
,\ell \left( Q\right) }^{\mathcal{D}}:\ \mathbf{r}\leq \kappa \left(
Q,R\right) <\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell \left(
R\right) ^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}}\varepsilon
_{Q,R}\left\langle T_{\sigma }^{\alpha }\left( \square _{Q,\mathcal{D
}^{\sigma ,\mathbf{b}}f\right) ,\square _{R,\mathcal{G}}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle \\
&\leq &\sum_{Q\in \mathcal{D}}\left\vert \left\langle T_{\sigma }^{\alpha
}\left( \square _{Q,\mathcal{D}}^{\sigma ,\mathbf{b}}f\right) ,\sum
_{\substack{ R\in \mathcal{G}_{\limfunc{bad},\ell \left( Q\right) }^
\mathcal{D}}:\ \mathbf{r}\leq \kappa \left( Q,R\right) <\kappa \left(
R\right) \\ d\left( R,Q\right) \leq 2\ell \left( R\right) ^{\varepsilon
}\ell \left( Q\right) ^{1-\varepsilon }}}\varepsilon _{Q,R}\square _{R
\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle \right\vert \\
&\leq &\mathfrak{N}_{T^{\alpha }}\sum_{Q\in \mathcal{D}}\left\Vert \square
_{Q,\mathcal{D}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }\left\Vert \sum_{\substack{ R\in \mathcal{G}_{\limfunc{bad},\ell
\left( Q\right) }^{\mathcal{D}}:\ \mathbf{r}\leq \kappa \left( Q,R\right)
<\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell \left( R\right)
^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}}\varepsilon
_{Q,R}\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) } \\
&\leq &\mathfrak{N}_{T^{\alpha }}\sum_{Q\in \mathcal{D}}\left\Vert \square
_{Q,\mathcal{D}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }\sum_{k=\mathbf{r}}^{\infty }\left\Vert \sum_{\substack{ R\in
\mathcal{G}_{\limfunc{bad},\ell \left( Q\right) }^{\mathcal{D}}:k=\kappa
\left( Q,R\right) <\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell
\left( R\right) ^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}
\varepsilon _{Q,R}\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }\ ,
\end{eqnarray*
by Minkowski's inequality, and we continue wit
\begin{eqnarray*}
&\leq &2\mathfrak{N}_{T^{\alpha }}\sum_{k=\mathbf{r}}^{\infty }\left(
\sum_{Q\in \mathcal{D}}\left\Vert \square _{Q,\mathcal{D}}^{\sigma ,\mathbf{
}}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}\right) ^{\frac{1}{2
}\left( \sum_{Q\in \mathcal{D}}\sum_{\substack{ R\in \mathcal{G}_{\limfunc
bad},\ell \left( Q\right) }^{\mathcal{D}}:\ k=\kappa \left( Q,R\right)
<\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell \left( R\right)
^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}}\left( \left\Vert
\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla _{R,\mathcal{G
}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right) \right) ^
\frac{1}{2}} \\
&\lesssim &\mathfrak{N}_{T^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left(
\sigma \right) }\sum_{k=\mathbf{r}}^{\infty }\left( \sum_{R\in \mathcal{G}_
\limfunc{bad},2^{k}\ell \left( R\right) }^{\mathcal{D}}}\left( \left\Vert
\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla _{R,\mathcal{G
}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right) \right) ^
\frac{1}{2}},
\end{eqnarray*
where $\nabla _{R,\mathcal{G}}^{\omega }$ denotes the `broken' Carleson
averaging operator in (\ref{Carleson avg op}) that depends on the grid
\mathcal{G}$, and
\begin{enumerate}
\item the penultimate inequality uses Cauchy-Schwarz in $Q$ and the weak
upper Riesz inequalities (\ref{UPPER RIESZ}) for $\sum_{\substack{ R\in
\mathcal{G}_{\limfunc{bad},\ell \left( Q\right) }^{\mathcal{D}}:\ k=\kappa
\left( Q,R\right) <\kappa \left( R\right) \\ d\left( R,Q\right) \leq 2\ell
\left( R\right) ^{\varepsilon }\ell \left( Q\right) ^{1-\varepsilon }}
\varepsilon _{Q,R}\square _{R,\mathcal{G}}^{\omega ,\mathbf{b}^{\ast }}$,
once for the sum when $\varepsilon _{Q,R}=1$, and again for the sum when
\varepsilon _{Q,R}=-1$. However, we note that since the sum in $R$ is
pigeonholed by $k=\kappa \left( Q,R\right) $, the $R$'s are pairwise
disjoint intervals and the pseudoprojections $\square _{R,\mathcal{G
}^{\omega ,\mathbf{b}^{\ast }}g$ are pairwise orthogonal. Thus we could
instead apply Cauchy-Schwarz first in $R$, and then in $Q$ as was done in
\cite{HyMa}, but we must still apply weak upper Riesz inequalities as above.
\item and the final inequality uses the frame inequality (\ref{FRAME})
together with (\ref{imp}), namely the fact that there are at most $C$
intervals $Q$ such that $\kappa \left( Q,R\right) \geq \mathbf{r}$ is fixed
and $d\left( R,Q\right) \leq 2\ell \left( R\right) ^{\varepsilon }\ell
\left( Q\right) ^{1-\varepsilon }$.
\end{enumerate}
Now it is easy to verify that we have the same inequality for the pairs
\left( J^{\maltese },J\right) $ that were removed, and then we take grid
expectations and use the probability estimate (\ref{main bad prob}) to
obtain for $\varepsilon ^{\prime }=\frac{1}{2}\varepsilon $ that
\begin{eqnarray}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\left( \Theta _{2}^{\limfunc{bad
\natural }\left( f,g\right) \right) \label{HM bad} \\
&\leq &\boldsymbol{E}_{\Omega }^{\mathcal{D}}\mathfrak{N}_{T^{\alpha
}}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\sum_{k=\mathbf{r
}^{\infty }\left( \sum_{R\in \mathcal{G}_{\limfunc{bad},2^{k}\ell \left(
R\right) }^{\mathcal{D}}}\left( \left\Vert \square _{R,\mathcal{G}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}+\left\Vert \nabla _{R,\mathcal{G}}^{\omega }g\right\Vert _{L^{2}\left(
\omega \right) }^{2}\right) \right) ^{\frac{1}{2}} \notag \\
&\leq &\mathfrak{N}_{T^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left(
\sigma \right) }\sum_{k=\mathbf{r}}^{\infty }\left( \boldsymbol{E}_{\Omega
}^{\mathcal{D}}\sum_{R\in \mathcal{G}_{\limfunc{bad},2^{k}\ell \left(
R\right) }^{\mathcal{D}}}\left( \left\Vert \square _{R,\mathcal{G}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}+\left\Vert \nabla _{R,\mathcal{G}}^{\omega }g\right\Vert _{L^{2}\left(
\omega \right) }^{2}\right) \right) ^{\frac{1}{2}} \notag \\
&\lesssim &2^{-\frac{1}{2}\varepsilon ^{\prime }\mathbf{r}}\mathfrak{N
_{T^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\sum_{k
\mathbf{r}}^{\infty }\left( C_{1}2^{-\varepsilon k}\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }^{2}\right) ^{\frac{1}{2}}\leq C_{\limfunc{goo
}}2^{-\frac{1}{2}\varepsilon \mathbf{r}}\mathfrak{N}_{T^{\alpha }}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }\ . \notag
\end{eqnarray
Clearly we can now fix $\mathbf{r}$ sufficiently large depending on
\varepsilon >0$ so tha
\begin{equation}
C_{\limfunc{good}}2^{-\frac{1}{2}\varepsilon \mathbf{r}}<\frac{1}{100},
\label{choice of r}
\end{equation
and then the final term above, namely $C_{\limfunc{good}}2^{-\frac{1}{2
\varepsilon \mathbf{r}}\mathfrak{N}_{T^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }$, can be absorbed at the end of the proof in Subsection \ref{Sub
wrapup}. Note that (\ref{choice of r}) fixes our choice of the parameter
\mathbf{r}$ for any given $\varepsilon >0$. Later we will choose
0<\varepsilon <\frac{1}{2}\leq \frac{1}{2-\alpha }$. It is this type of weak
goodness that we will exploit in the local forms $\mathsf{B}_{\Subset _
\mathbf{r}}}^{A}\left( f,g\right) $ treated below in Section \ref{Sec Main
below}.
We are now left with the following `good' form to control
\begin{equation*}
\Theta _{2}^{\limfunc{good}}\left( f,g\right) =\sum_{I\in \mathcal{D}}\sum
_{\substack{ J^{\maltese }\subsetneqq I:\ \ell \left( J\right) \leq 2^{
\mathbf{r}}\ell \left( I\right) \\ d\left( J,I\right) \leq 2\ell \left(
J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}\int \left(
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right) \square
_{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega .
\end{equation*
The first thing we observe regarding this form is that the intervals $J$
which arise in the sum for $\Theta _{2}^{\limfunc{good}}\left( f,g\right) $
must lie entirely inside $I$ since $J\subset J^{\maltese }\subsetneqq I$.
Then in the remainder of the paper, we proceed to analyze
\begin{equation}
\Theta _{2}^{\limfunc{good}}\left( f,g\right) =\sum_{I\in \mathcal{D
}\sum_{J^{\maltese }\subsetneqq I:\ \ell \left( J\right) \leq 2^{-\mathbf{r
}\ell \left( I\right) }\int \left( T_{\sigma }^{\alpha }\square _{I}^{\sigma
,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega ,
\label{def Theta 2 good}
\end{equation
in the same way we analyzed the below term $\mathsf{B}_{\Subset _{\mathbf{r
}}\left( f,g\right) $ in \cite{SaShUr6}; namely, by implementing the
canonical corona splitting and the decomposition into paraproduct, neighbour
and stopping forms, but now with an additional broken form. We have $\left(
\kappa ,\varepsilon \right) $-goodness available for all the intervals $J\in
\mathcal{G}$ arising in the form $\Theta _{2}^{\limfunc{good}}\left(
f,g\right) $, and moreover, the intervals $I\in \mathcal{D}$ arising in the
form $\Theta _{2}^{\limfunc{good}}\left( f,g\right) $ for a fixed $J$ are
tree-connected, so that telescoping identities hold for these intervals $I$.
This will prove decisive in the following three sections of the paper.
The forms $\Theta _{1}\left( f,g\right) $ and $\Theta _{3}\left( f,g\right) $
are analogous to the disjoint and nearby forms $\mathsf{B}_{\cap }\left(
f,g\right) $ and $\mathsf{B}_{/}\left( f,g\right) $ in \cite{SaShUr6}
respectively. In the next\ two sections, we control the disjoint form
\Theta _{1}\left( f,g\right) $ in essentially the same way that the disjoint
form $\mathsf{B}_{\cap }\left( f,g\right) $ was treated in \cite{SaShUr6}
and in earlier papers of many authors beginning with Nazarov, Treil and
Volberg (see e.g. \cite{Vol}), and we control the nearby form $\Theta
_{3}\left( f,g\right) $ using the probabilistic surgery of Hyt\"{o}nen and
Martikainen building on that of NTV, together with a new deterministic
surgery involving the energy condition and the original testing functions.
But first we recall, in the following subsection, the characterization of
boundedness of one-dimensional forms supported on disjoint intervals \cit
{Hyt2}.
\subsection{A characterization of bilinear forms supported on disjoint
intervals \label{disjoint sets}}
Matters here in the one-dimensional setting are greatly simplified by a
generalization to fractional integrals of Hyt\"{o}nen's characterization of
the \emph{restricted bilinear inequality,
\begin{equation}
\left\vert \int_{\mathbb{R}\setminus I}\left( \int_{I}\frac{f\left( y\right)
}{\left\vert x-y\right\vert }d\sigma \left( y\right) \right) g\left(
x\right) d\omega \left( x\right) \right\vert \lesssim \mathfrak{D}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }\ , \label{bilin disjoint}
\end{equation
for all intervals $I$, in terms of the Muckenhoupt conditions, namel
\begin{equation*}
\mathfrak{D}\approx \sqrt{\mathcal{A}_{2}}+\sqrt{\mathcal{A}_{2}^{\ast }},
\end{equation*
where $\mathfrak{D}$ is the best constant in (\ref{bilin disjoint}). In \cit
{HyMa} this inequality was proved for complementary half-lines, where it was
pointed out that the passage to an interval and its complement is then
routine.
We claim that Hyt\"{o}nen's characterization extends immediately to
fractional integrals on the line with the same proof. Namely, we have,
\begin{eqnarray}
\left\vert \int_{\left( -\infty ,a\right) }\left( \int_{\left( a,\infty
\right) }\frac{f\left( y\right) }{\left\vert x-y\right\vert ^{1-\alpha }
d\sigma \left( y\right) \right) g\left( x\right) d\omega \left( x\right)
\right\vert &\lesssim &\left( \sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt
\mathcal{A}_{2}^{\alpha ,\ast }}\right) \left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\ , \label{disj supp} \\
\left\vert \int_{\mathbb{R}\setminus I}\left( \int_{I}\frac{f\left( y\right)
}{\left\vert x-y\right\vert ^{1-\alpha }}d\sigma \left( y\right) \right)
g\left( x\right) d\omega \left( x\right) \right\vert &\lesssim &\left( \sqrt
\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}\right)
\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }\ , \notag
\end{eqnarray
and that $\sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha
,\ast }}\approx \mathfrak{D}^{\alpha }$ where $\mathfrak{D}^{\alpha }$ is
the best constant in the inequality above (a limiting argument shows that we
may take one of the half-lines to be closed in (\ref{disj supp})). First,
the proof that $\sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A
_{2}^{\alpha ,\ast }}\lesssim \mathfrak{D}^{\alpha }$ is the standard proof
of necessity of the one-tailed Muckenhoupt conditions. In the other
direction, we use the general two weight Hardy inequality of Muckenhoupt as
presented in \cite[Theorem 3.3]{Hyt2}, see also \cite{LaSaUr2}: if $\sigma $
and $\omega $ are locally finite positive Borel measures on the interval
\left( 0,\infty \right) $, the
\begin{equation*}
\int_{0}^{\infty }\left( \int_{\left( 0,x\right] }fd\sigma \right)
^{2}d\omega \left( x\right) \leq C\int_{0}^{\infty }f\left( y\right)
^{2}d\sigma \left( y\right) ,
\end{equation*
holds for all $f\in L^{2}\left( \sigma \right) $ if and only i
\begin{equation*}
A\equiv \sup_{t>0}\left( \int_{\left( 0,t\right] }d\sigma \right) \left(
\int_{\left[ t,\infty \right) }d\omega \right) <\infty .
\end{equation*
Moreover, if $C$ is the best constant above, the
\begin{equation*}
A\leq C\leq 4A.
\end{equation*
We easily obtain the following characterization of an intermediate
inequality:
\begin{equation*}
\int_{0}^{\infty }\int_{0}^{\infty }\frac{f\left( y\right) g\left( x\right)
}{\left( x+y\right) ^{1-\alpha }}d\sigma \left( y\right) d\omega \left(
x\right) \leq C\left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) },
\end{equation*
if and only i
\begin{equation*}
A\equiv \sup_{t>0}\sqrt{\left( \int_{\left( 0,t\right] }d\sigma \right)
\left( \int_{\left[ t,\infty \right) }\frac{d\omega \left( x\right) }
x^{2-2\alpha }}\right) }+\sup_{t>0}\sqrt{\left( \int_{\left( 0,t\right]
}d\omega \right) \left( \int_{\left[ t,\infty \right) }\frac{d\sigma \left(
y\right) }{y^{2-2\alpha }}\right) }<\infty ,
\end{equation*
and moreover the best constant $C$ satisfies $\frac{1}{4}A\leq C\leq 2A$. To
see this we simply use the estimates
\begin{equation*}
\frac{1}{2}\max \left\{ \frac{1}{x^{1-\alpha }}\mathbf{1}_{\left( 0,x\right]
}\left( y\right) ,\frac{1}{y^{1-\alpha }}\mathbf{1}_{\left( 0,y\right]
}\left( x\right) \right\} \leq \frac{1}{\left( x+y\right) ^{1-\alpha }}\leq
\frac{1}{x^{1-\alpha }}\mathbf{1}_{\left( 0,x\right] }\left( y\right) +\frac
1}{y^{1-\alpha }}\mathbf{1}_{\left( 0,y\right] }\left( x\right) ,
\end{equation*
together with Hardy's inequality. From this and duality, we immediately
obtain (\ref{disj supp}).
\subsubsection{Control of triple testing and triple energy}
We also define the \emph{triple} $\mathbf{b}$-testing conditions for
T^{\alpha }$ and \emph{triple} $\mathbf{b}^{\ast }$-testing conditions for
the dual $T^{\alpha ,\ast }$ given b
\begin{eqnarray}
\int_{3Q}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert ^{2}d\omega &\leq
&\left( \mathfrak{3T}_{T^{\alpha }}^{\mathbf{b}}\right) ^{2}\left\vert
Q\right\vert _{\sigma }\ ,\ \ \ \ \ \text{for all intervals }Q,
\label{triple b testing cond} \\
\int_{3Q}\left\vert T_{\omega }^{\alpha ,\ast }b_{Q}^{\ast }\right\vert
^{2}d\sigma &\leq &\left( \mathfrak{3T}_{T^{\alpha }}^{\mathbf{b}^{\ast
},\ast }\right) ^{2}\left\vert Q\right\vert _{\omega }\ ,\ \ \ \ \ \text{for
all intervals }Q, \notag
\end{eqnarray
as well as the \emph{full} $\mathbf{b}$-testing conditions for $T^{\alpha }$
and \emph{full} $\mathbf{b}^{\ast }$-testing conditions for the dual
T^{\alpha ,\ast }$ given b
\begin{eqnarray}
\int_{\mathbb{R}}\left\vert T_{\sigma }^{\alpha }b_{Q}\right\vert
^{2}d\omega &\leq &\left( \mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}\left\vert Q\right\vert _{\sigma }\ ,\ \ \ \ \ \text{for all intervals
Q, \label{full b testing} \\
\int_{\mathbb{R}}\left\vert T_{\omega }^{\alpha ,\ast }b_{Q}^{\ast
}\right\vert ^{2}d\sigma &\leq &\left( \mathfrak{FT}_{T^{\alpha }}^{\mathbf{
}^{\ast },\ast }\right) ^{2}\left\vert Q\right\vert _{\omega }\ ,\ \ \ \ \
\text{for all intervals }Q. \notag
\end{eqnarray
Note that the full testing conditions are implied by the triple testing
conditions and the Muckenhoupt conditions (e.g. use the above
characterization on complementary half-lines)
\begin{equation*}
\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}\lesssim \mathfrak{3T}_{T^{\alpha
}}^{\mathbf{b}}+\sqrt{\mathcal{A}_{2}^{\alpha }}\text{ and }\mathfrak{FT
_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}\lesssim \mathfrak{3T}_{T^{\alpha
,\ast }}^{\mathbf{b}^{\ast }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}.
\end{equation*
Since dimension $n=1$, the full testing conditions are controlled by testing
and Muckenhoupt, as we now show. Indeed, if we now set $f=b_{I}$ in the
second line of (\ref{disj supp}), and take the supremum over all $g\in
L^{2}\left( \omega \right) $ with $\left\Vert g\right\Vert _{L^{2}\left(
\omega \right) }=1$, we obtai
\begin{eqnarray*}
\sqrt{\int_{\mathbb{R}\setminus I}\left( \int_{I}\frac{f\left( y\right) }
\left\vert x-y\right\vert ^{1-\alpha }}d\sigma \left( y\right) \right)
^{2}d\omega \left( x\right) } &\lesssim &\left( \sqrt{\mathcal{A
_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}\right) \left\Vert
b_{I}\right\Vert _{L^{2}\left( \sigma \right) } \\
&\lesssim &\left( \sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A
_{2}^{\alpha ,\ast }}\right) \sqrt{\left\vert I\right\vert _{\sigma }},
\end{eqnarray*
which give
\begin{eqnarray*}
\int_{\mathbb{R}}\left\vert T_{\sigma }^{\alpha }b_{I}\right\vert
^{2}d\omega \left( x\right) &=&\int_{I}\left\vert T_{\sigma }^{\alpha
}b_{I}\right\vert ^{2}d\omega \left( x\right) +\int_{\mathbb{R}\setminus
I}\left\vert T_{\sigma }^{\alpha }b_{I}\right\vert ^{2}d\omega \left(
x\right) \\
&\lesssim &\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}\left\vert I\right\vert _{\sigma }+\int_{\mathbb{R}\setminus I}\left(
\int_{I}\frac{f\left( y\right) }{\left\vert x-y\right\vert ^{1-\alpha }
d\sigma \left( y\right) \right) ^{2}d\omega \left( x\right) \\
&\lesssim &\left\{ \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}+\mathcal{A}_{2}^{\alpha }+\mathcal{A}_{2}^{\alpha ,\ast }\right\}
\left\vert I\right\vert _{\sigma }\ .
\end{eqnarray*
Thus we have obtained control of full $\mathbf{b}$-testing by just $\mathbf{
}$-testing and the Muckenhoupt conditions in dimension $n=1$
\begin{equation}
\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}\lesssim \mathfrak{T}_{T^{\alpha }}^
\mathbf{b}}+\sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha
,\ast }}\text{ and }\mathfrak{FT}_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast
}}\lesssim \mathfrak{T}_{T^{\alpha ,\ast }}^{\mathbf{b}^{\ast }}+\sqrt
\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}\text{ }.
\label{full proved}
\end{equation}
Now we turn to the analogous notion of \emph{triple} energy conditions
defined in analogy with the triple testing condtions. Namely, the sum over
the intervals $I_{r}$ in the energy condition in Definition \ref{def strong
quasienergy} is permitted to extend to the triple $3I$ of the interval $I$,
but with the additional proviso that the distance of $I_{r}$ from the
boundary of $I$ is at least a positive multiple $\delta $ (the exact value
of which is immaterial) of the side length of $I_{r}$.
\begin{definition}
\label{def triple energy}Let $0\leq \alpha <1$ and $0<\delta \leq \frac{1}{2}
$. Suppose $\sigma $ and $\omega $ are locally finite positive Borel
measures on $\mathbb{R}$. Then the \emph{triple} energy constant $\mathcal{E
_{2}^{\alpha ,\limfunc{triple}}$ is defined by
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha ,\limfunc{triple}}\right) ^{2}\equiv \sup
_{\substack{ 3I=\dot{\cup}I_{r} \\ d\left( I_{r},\partial I\right) \geq
\delta \ell \left( I_{r}\right) \text{ when }I_{r}\cap I^{c}=\emptyset }
\frac{1}{\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\left( \frac
\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1}_{I}\sigma \right) }{\left\vert
I_{r}\right\vert }\right) ^{2}\left\Vert x-m_{I_{r}}\right\Vert
_{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}\ ,
\end{equation*
where the supremum is taken over arbitrary decompositions of the triple $3I$
of an interval $I$ using a pairwise disjoint union of subintervals $I_{r}$
whose distance to the boundary of $I$ is at least a positive multiple of
\ell \left( I_{r}\right) $ when $I_{r}$ is not contained in $I$. Similarly,
we define the \emph{dual triple} energy constant $\mathcal{E}_{2}^{\alpha
\limfunc{triple},\ast }$ by switching the roles of $\sigma $ and $\omega $
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha ,\limfunc{triple},\ast }\right) ^{2}\equiv
\sup_{\substack{ 3I=\dot{\cup}I_{r} \\ d\left( I_{r},\partial I\right) \geq
\delta \ell \left( I_{r}\right) \text{ when }I_{r}\cap I^{c}=\emptyset }
\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1
_{I}\omega \right) }{\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert
x-m_{I_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{I_{r}}\sigma \right) }^{2}\
.
\end{equation*}
\end{definition}
We now show that in dimension $n=1$, the triple energy conditions are
controlled by the energy and Muckenhoupt conditions, namel
\begin{equation}
\mathcal{E}_{2}^{\alpha ,\limfunc{triple}}+\mathcal{E}_{2}^{\alpha ,\limfunc
triple},\ast }\lesssim \mathcal{E}_{2}^{\alpha }+\mathcal{E}_{2}^{\alpha
,\ast }+\sqrt{\mathfrak{A}_{2}^{\alpha }} \label{triple energy control}
\end{equation
Indeed, assuming for convenience that $\delta =1$, we need only control by
C\left\vert I\right\vert _{\sigma }$ the sum
\begin{equation*}
\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1
_{I}\sigma \right) }{\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert
x-m_{I_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}\
,
\end{equation*}
over adjacent intervals $J$ and $I$ of equal length where $\left\{
I_{r}\right\} _{r=1}^{\infty }$ is a disjoint decomposition of $J=\dot{\cup
I_{r}$ with $d\left( I_{r},\partial I\right) \geq \ell \left( I_{r}\right) $
for all $r\geq 1$. However, using reversal of energy for the standard
gradient elliptic operator $T^{\alpha }$ with convolution kernel $K^{\alpha
}\left( x\right) =\frac{x}{\left\vert x\right\vert ^{2-\alpha }}$ (see e.g.
\cite{SaShUr10}), we have since $2I_{r}\cap I=\emptyset $,
\begin{eqnarray*}
&&\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{
}_{I}\sigma \right) }{\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert
x-m_{I_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}
\\
&=&\frac{1}{2}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left(
I_{r},\mathbf{1}_{I}\sigma \right) }{\left\vert I_{r}\right\vert }\right)
^{2}\frac{1}{\left\vert I_{r}\right\vert _{\omega }}\int_{I_{r}}\int_{I_{r}
\left\vert x-z\right\vert ^{2}d\omega \left( x\right) d\omega \left( z\right)
\\
&\lesssim &\sum_{r=1}^{\infty }\frac{1}{\left\vert I_{r}\right\vert _{\omega
}}\int_{I_{r}}\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I}\left( x\right) -T_{\sigma }^{\alpha }\mathbf{1}_{I}\left( z\right)
\right\vert ^{2}d\omega \left( x\right) d\omega \left( z\right) \\
&\lesssim &\sum_{r=1}^{\infty }\frac{1}{\left\vert I_{r}\right\vert _{\omega
}}\int_{I_{r}}\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I}\left( x\right) \right\vert ^{2}d\omega \left( x\right) d\omega \left(
z\right) +\sum_{r=1}^{\infty }\frac{1}{\left\vert I_{r}\right\vert _{\omega
}\int_{I_{r}}\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I}\left( z\right) \right\vert ^{2}d\omega \left( x\right) d\omega \left(
z\right) \\
&\lesssim &\sum_{r=1}^{\infty }\int_{I_{r}}\left\vert T_{\sigma }^{\alpha
\mathbf{1}_{I}\right\vert ^{2}d\omega \lesssim \int_{J}\left\vert T_{\sigma
}^{\alpha }\mathbf{1}_{I}\right\vert ^{2}d\omega \lesssim \mathfrak{A
_{2}^{\alpha }\left\vert I\right\vert _{\sigma }\ ,
\end{eqnarray*
which completes the proof of (\ref{triple energy control}).
Finally, we note that a modification of this last argument also shows that
the energy condition itself is controlled by the $\mathbf{1}$-testing
condition and the Muckenhoupt condition. Indeed, as shown in \cite{SaShUr11
,
\begin{eqnarray*}
&&\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r},\mathbf{
}_{I}\sigma \right) }{\left\vert I_{r}\right\vert }\right) ^{2}\left\Vert
x-m_{I_{r}}\right\Vert _{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}
\\
&=&\frac{1}{2}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left(
I_{r},\left[ \mathbf{1}_{I\setminus I_{r}}+\mathbf{1}_{I_{r}}\right] \sigma
\right) }{\left\vert I_{r}\right\vert }\right) ^{2}\frac{1}{\left\vert
I_{r}\right\vert _{\omega }}\int_{I_{r}}\int_{I_{r}}\left\vert
x-z\right\vert ^{2}d\omega \left( x\right) d\omega \left( z\right) \\
&\lesssim &\sum_{r=1}^{\infty }\left( \frac{\mathrm{P}^{\alpha }\left( I_{r}
\mathbf{1}_{I\setminus I_{r}}\sigma \right) }{\left\vert I_{r}\right\vert
\right) ^{2}\frac{1}{\left\vert I_{r}\right\vert _{\omega }
\int_{I_{r}}\int_{I_{r}}\left\vert x-z\right\vert ^{2}d\omega \left(
x\right) d\omega \left( z\right) +\sum_{r=1}^{\infty }A_{2}^{\alpha
\limfunc{energy}}\left\vert I_{r}\right\vert _{\sigma } \\
&\lesssim &\sum_{r=1}^{\infty }\frac{1}{\left\vert I_{r}\right\vert _{\omega
}}\int_{I_{r}}\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I\setminus I_{r}}\left( x\right) -T_{\sigma }^{\alpha }\mathbf{1
_{I\setminus I_{r}}\left( z\right) \right\vert ^{2}d\omega \left( x\right)
d\omega \left( z\right) +\mathfrak{A}_{2}^{\alpha }\left\vert I\right\vert
_{\sigma } \\
&\lesssim &\sum_{r=1}^{\infty }\frac{1}{\left\vert I_{r}\right\vert _{\omega
}}\int_{I_{r}}\int_{I_{r}}\left( \left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I\setminus I_{r}}\left( x\right) \right\vert ^{2}+\left\vert T_{\sigma
}^{\alpha }\mathbf{1}_{I\setminus I_{r}}\left( z\right) \right\vert
^{2}\right) d\omega \left( x\right) d\omega \left( z\right) +\mathfrak{A
_{2}^{\alpha }\left\vert I\right\vert _{\sigma } \\
&\lesssim &\sum_{r=1}^{\infty }\int_{I_{r}}\left\vert T_{\sigma }^{\alpha
\mathbf{1}_{I\setminus I_{r}}\left( x\right) \right\vert ^{2}d\omega \left(
x\right) +\mathfrak{A}_{2}^{\alpha }\left\vert I\right\vert _{\sigma }\ ,
\end{eqnarray*
and now we `plug the hole' to continue wit
\begin{eqnarray*}
\sum_{r=1}^{\infty }\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I\setminus I_{r}}\left( x\right) \right\vert ^{2}d\omega \left( x\right)
&\lesssim &\sum_{r=1}^{\infty }\int_{I_{r}}\left\vert T_{\sigma }^{\alpha
\mathbf{1}_{I}\left( x\right) \right\vert ^{2}d\omega \left( x\right)
+\sum_{r=1}^{\infty }\int_{I_{r}}\left\vert T_{\sigma }^{\alpha }\mathbf{1
_{I_{r}}\left( x\right) \right\vert ^{2}d\omega \left( x\right) \\
&\lesssim &\int_{I}\left\vert T_{\sigma }^{\alpha }\mathbf{1}_{I}\right\vert
^{2}d\omega +\left( \mathfrak{T}_{T^{\alpha }}\right) ^{2}\sum_{r=1}^{\infty
}\left\vert I_{r}\right\vert _{\sigma }\lesssim \left( \mathfrak{T
_{T^{\alpha }}\right) ^{2}\left\vert I\right\vert _{\sigma }\ .
\end{eqnarray*
Altogether this give
\begin{equation}
\mathcal{E}_{2}^{\alpha }+\mathcal{E}_{2}^{\alpha ,\ast }\lesssim \mathfrak{
}_{T^{\alpha }}+\mathfrak{T}_{T^{\alpha }}^{\ast }+\mathfrak{A}_{2}^{\alpha
}\ . \label{energy condition is necd}
\end{equation}
\section{Disjoint form\label{Sec disj form}}
Here we control the disjoint form $\Theta _{1}\left( f,g\right) $ by further
decomposing it as follows
\begin{eqnarray}
\Theta _{1}\left( f,g\right) &=&\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in
\mathcal{G}:\ \ell \left( J\right) \leq \ell \left( I\right) \\ d\left(
J,I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right)
^{1-\varepsilon }}}\int \left( T_{\sigma }\square _{I}^{\sigma ,\mathbf{b
}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega
\label{decomp long short} \\
&=&\sum_{I\in \mathcal{D}}\left\{ \sum_{\substack{ J\in \mathcal{G}:\ \ell
\left( J\right) \leq \ell \left( I\right) \\ d\left( J,I\right) >\max
\left( \ell \left( I\right) ,2\ell \left( J\right) ^{\varepsilon }\ell
\left( I\right) ^{1-\varepsilon }\right) }}+\sum_{\substack{ J\in \mathcal{G
:\ \ell \left( J\right) \leq \ell \left( I\right) \\ \ell \left( I\right)
\geq d\left( J,I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\right\} \int \left( T_{\sigma }\square
_{I}^{\sigma ,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast
}}gd\omega \notag \\
&\equiv &\Theta _{1}^{\limfunc{long}}\left( f,g\right) +\Theta _{1}^
\limfunc{short}}\left( f,g\right) , \notag
\end{eqnarray
where $\Theta _{1}^{\limfunc{long}}\left( f,g\right) $ is a `long range'
form in which $J$ is far from $I$, and where $\Theta _{1}^{\limfunc{short
}\left( f,g\right) $ is a short range form. It should be noted that weak
goodness plays no role in treating the disjoint form.
\subsection{Long range form}
\begin{lemma}
\label{delta long}We hav
\begin{equation*}
\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ \ell \left(
J\right) \leq \ell \left( I\right) \\ d\left( J,I\right) >\ell \left(
I\right) }}\left\vert \int \left( T_{\sigma }\square _{I}^{\sigma ,\mathbf{b
}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega \right\vert
\lesssim \sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }
\end{equation*}
\end{lemma}
\begin{proof}
Since $J$ and $I$ are separated by at least $\max \left\{ \ell \left(
J\right) ,\ell \left( I\right) \right\} $, we have the inequalit
\begin{equation*}
\mathrm{P}^{\alpha }\left( J,\left\vert \square _{I}^{\sigma ,\mathbf{b
}f\right\vert \sigma \right) \approx \int_{I}\frac{\ell \left( J\right) }
\left\vert y-c_{J}\right\vert ^{2-\alpha }}\left\vert \square _{I}^{\sigma
\mathbf{b}}f\left( y\right) \right\vert d\sigma \left( y\right) \lesssim
\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }\frac{\ell \left( J\right) \sqrt{\left\vert I\right\vert
_{\sigma }}}{d\left( I,J\right) ^{2-\alpha }},
\end{equation*
since $\int_{I}\left\vert \square _{I}^{\sigma ,\mathbf{b}}f\left( y\right)
\right\vert d\sigma \left( y\right) \leq \left\Vert \square _{I}^{\sigma
\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }\sqrt{\left\vert
I\right\vert _{\sigma }}$. Thus if $A\left( f,g\right) $ denotes the left
hand side of the conclusion of Lemma \ref{delta long}, we hav
\begin{eqnarray*}
A\left( f,g\right) &\lesssim &\sum_{I\in \mathcal{D}}\sum_{J\;:\;\ell \left(
J\right) \leq \ell \left( I\right) :\ d\left( I,J\right) \geq \ell \left(
I\right) }\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) } \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \frac{\ell \left( J\right) }{d\left(
I,J\right) ^{2-\alpha }}\sqrt{\left\vert I\right\vert _{\sigma }}\sqrt
\left\vert J\right\vert _{\omega }} \\
&\equiv &\sum_{\left( I,J\right) \in \mathcal{P}}\left\Vert \square
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }A\left( I,J\right) ; \\
\text{with }A\left( I,J\right) &\equiv &\frac{\ell \left( J\right) }{d\left(
I,J\right) ^{2-\alpha }}\sqrt{\left\vert I\right\vert _{\sigma }}\sqrt
\left\vert J\right\vert _{\omega }}; \\
\text{ and }\mathcal{P} &\equiv &\left\{ \left( I,J\right) \in \mathcal{D
\times \mathcal{G}:\ell \left( J\right) \leq \ell \left( I\right) \text{ and
}d\left( I,J\right) \geq \ell \left( I\right) \right\} .
\end{eqnarray*
Now let $\mathcal{D}_{N}\equiv \left\{ K\in \mathcal{D}:\ell \left( K\right)
=2^{N}\right\} $ for each $N\in \mathbb{Z}$. For $N\in \mathbb{Z}$ and $s\in
\mathbb{Z}_{+}$, we further decompose $A\left( f,g\right) $ by pigeonholing
the sidelengths of $I$ and $J$ by $2^{N}$ and $2^{N-s}$ respectively:
\begin{eqnarray*}
A\left( f,g\right) &=&\sum_{s=0}^{\infty }\sum_{N\in \mathbb{Z
}A_{N}^{s}\left( f,g\right) ; \\
A_{N}^{s}\left( f,g\right) &\equiv &\sum_{\left( I,J\right) \in \mathcal{P
_{N}^{s}}\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }A\left( I,J\right) \\
\text{where }\mathcal{P}_{N}^{s} &\equiv &\left\{ \left( I,J\right) \in
\mathcal{D}_{N}\times \mathcal{G}_{N-s}:d\left( I,J\right) \geq \ell \left(
I\right) \right\} .
\end{eqnarray*}
Now let $\mathsf{P}_{M}^{\sigma }=\dsum\limits_{K\in \mathcal{D}_{M}}\square
_{K}^{\sigma ,\mathbf{b}}$ denote the dual martingale pseudoprojection onto
\limfunc{Span}\left\{ \square _{K}^{\sigma ,\mathbf{b}}\right\} _{K\in
\mathcal{D}_{M}}$. Since the intervals $K$ in $\mathcal{D}_{M}$ are pairwise
disjoint, the pseudoprojections $\square _{K}^{\sigma ,\mathbf{b}}$ are
mutually orthogonal, which means that $\left\Vert \mathsf{P}_{M}^{\sigma
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}=\dsum\limits_{K\in \mathcal
D}_{M}}\left\Vert \square _{K}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{2}$. We claim tha
\begin{equation}
\left\vert A_{N}^{s}\left( f,g\right) \right\vert \leq C2^{-s}\sqrt
A_{2}^{\alpha }}\left\Vert \mathsf{P}_{N}^{\sigma }f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \mathsf{P
_{N-s}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar },\ \
\ \ \ \text{for }s\geq 0\text{ and }N\in \mathbb{Z}. \label{AsN}
\end{equation
With this proved, we can then obtai
\begin{eqnarray*}
A\left( f,g\right) &=&\sum_{s=0}^{\infty }\sum_{N\in \mathbb{Z
}A_{N}^{s}\left( f,g\right) \leq C\sqrt{A_{2}^{\alpha }}\sum_{s=0}^{\infty
}2^{-s}\sum_{N\in \mathbb{Z}}\left\Vert \mathsf{P}_{N}^{\sigma }f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \mathsf{P
_{N-s}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &C\sqrt{A_{2}^{\alpha }}\sum_{s=0}^{\infty }2^{-s}\left( \sum_{N\in
\mathbb{Z}}\left\Vert \mathsf{P}_{N}^{\sigma }f\right\Vert _{L^{2}\left(
\sigma \right) }^{\bigstar 2}\right) ^{\frac{1}{2}}\left( \sum_{N\in \mathbb
Z}}\left\Vert \mathsf{P}_{N-s}^{\omega }g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar 2}\right) ^{\frac{1}{2}} \\
&\leq &C\sqrt{A_{2}^{\alpha }}\sum_{s=0}^{\infty }2^{-s}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }=C\sqrt{A_{2}^{\alpha }}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) },
\end{eqnarray*
where in the last line we have used the lower frame inequality for $\square
_{I}^{\sigma ,\mathbf{b}}$ in Appendix A.
To prove (\ref{AsN}), we pigeonhole the distance between $I$ and $J$
\begin{eqnarray*}
A_{N}^{s}\left( f,g\right) &=&\dsum\limits_{\ell =0}^{\infty }A_{N,\ell
}^{s}\left( f,g\right) ; \\
A_{N,\ell }^{s}\left( f,g\right) &\equiv &\sum_{\left( I,J\right) \in
\mathcal{P}_{N,\ell }^{s}}\left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert \square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }A\left(
I,J\right) \\
\text{where }\mathcal{P}_{N,\ell }^{s} &\equiv &\left\{ \left( I,J\right)
\in \mathcal{D}_{N}\times \mathcal{G}_{N-s}:d\left( I,J\right) \approx
2^{N+\ell }\right\} .
\end{eqnarray*
If we define $\mathcal{H}\left( A_{N,\ell }^{s}\right) $ to be the bilinear
form on $\ell ^{2}\times \ell ^{2}$ with matrix $\left[ A\left( I,J\right)
\right] _{\left( I,J\right) \in \mathcal{P}_{N,\ell }^{s}}$, then it remains
to show that the norm $\left\Vert \mathcal{H}\left( A_{N,\ell }^{s}\right)
\right\Vert _{\ell ^{2}\rightarrow \ell ^{2}}$ of $\mathcal{H}\left(
A_{N,\ell }^{s}\right) $ on the sequence space $\ell ^{2}$ is bounded by
C2^{-s-\ell }\sqrt{A_{2}^{\alpha }}$. In turn, this is equivalent to showing
that the norm $\left\Vert \mathcal{H}\left( B_{N,\ell }^{s}\right)
\right\Vert _{\ell ^{2}\rightarrow \ell ^{2}}$ of the bilinear form
\mathcal{H}\left( B_{N,\ell }^{s}\right) \equiv \mathcal{H}\left( A_{N,\ell
}^{s}\right) ^{\limfunc{tr}}\mathcal{H}\left( A_{N,\ell }^{s}\right) $ on
the sequence space $\ell ^{2}$ is bounded by $C^{2}2^{-2s-2\ell }\mathfrak{A
_{2}^{\alpha }$. Here $\mathcal{H}\left( B_{N,\ell }^{s}\right) $ is the
quadratic form with matrix kernel $\left[ B_{N,\ell }^{s}\left( J,J^{\prime
}\right) \right] _{J,J^{\prime }\in \mathcal{D}_{N-s}}$ having entries
\begin{equation*}
B_{N,\ell }^{s}\left( J,J^{\prime }\right) \equiv \sum_{I\in \mathcal{D
_{N}:\ d\left( I,J\right) \approx d\left( I,J^{\prime }\right) \approx
2^{N+\ell }}A\left( I,J\right) A\left( I,J^{\prime }\right) ,\ \ \ \ \ \text
for }J,J^{\prime }\in \mathcal{G}_{N-s}.
\end{equation*}
We are reduced to showing the bilinear form inequality
\begin{equation*}
\left\Vert \mathcal{H}\left( B_{N,\ell }^{s}\right) \right\Vert _{\ell
^{2}\rightarrow \ell ^{2}}\leq C2^{-2s-2\ell }A_{2}^{\alpha }\ \ \ \ \text
for }s\geq 0\text{, }\ell \geq 0\text{ and }N\in \mathbb{Z}.
\end{equation*
We begin by computing $B_{N,\ell }^{s}\left( J,J^{\prime }\right) $
\begin{eqnarray*}
B_{N,\ell }^{s}\left( J,J^{\prime }\right) &=&\sum_{\substack{ I\in \mathcal
D}_{N} \\ d\left( I,J\right) \approx d\left( I,J^{\prime }\right) \approx
2^{N+\ell }}}\frac{\ell \left( J\right) }{d\left( I,J\right) ^{2-\alpha }
\sqrt{\left\vert I\right\vert _{\sigma }}\sqrt{\left\vert J\right\vert
_{\omega }}\frac{\ell \left( J^{\prime }\right) }{d\left( I,J^{\prime
}\right) ^{2-\alpha }}\sqrt{\left\vert I\right\vert _{\sigma }}\sqrt
\left\vert J^{\prime }\right\vert _{\omega }} \\
&=&\left\{ \sum_{\substack{ I\in \mathcal{D}_{N} \\ d\left( I,J\right)
\approx d\left( I,J^{\prime }\right) \approx 2^{N+\ell }}}\left\vert
I\right\vert _{\sigma }\frac{1}{d\left( I,J\right) ^{2-\alpha }d\left(
I,J^{\prime }\right) ^{2-\alpha }}\right\} \ell \left( J\right) \ell \left(
J^{\prime }\right) \sqrt{\left\vert J\right\vert _{\omega }}\sqrt{\left\vert
J^{\prime }\right\vert _{\omega }}.
\end{eqnarray*
Now we show tha
\begin{equation}
\left\Vert \mathcal{H}\left( B_{N,\ell }^{s}\right) \right\Vert _{\ell
^{2}\rightarrow \ell ^{2}}=\left\Vert B_{N,\ell }^{s}\right\Vert _{\ell
^{2}\rightarrow \ell ^{2}}\lesssim 2^{-2s-2\ell }A_{2}^{\alpha }\ ,
\label{Schur s}
\end{equation
by applying the proof of Schur's lemma. Fix $\ell \geq 0$ and $s\geq 0$.
Choose the Schur function $\beta \left( K\right) =\frac{1}{\sqrt{\left\vert
K\right\vert _{\omega }}}$. Fix $J\in \mathcal{D}_{N-s}$. We now group those
$I\in \mathcal{D}_{N}$ with $d\left( I,J\right) \approx 2^{N+\ell }$ into
finitely many groups $G_{1},...G_{C}$ for which the union of the $I$ in each
group is contained in an interval of side length roughly $\frac{1}{100
2^{N+\ell }$ , and we set $I_{k}^{\ast }\equiv \dbigcup\limits_{I\in G_{k}}I$
for $1\leq k\leq C$. We then hav
\begin{eqnarray*}
&&\sum_{J^{\prime }\in \mathcal{G}_{N-s}}\frac{\beta \left( J\right) }{\beta
\left( J^{\prime }\right) }B_{N,\ell }^{s}\left( J,J^{\prime }\right) \\
&=&\sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left( J^{\prime
},J\right) \leq \frac{1}{100}2^{N+\ell +2}}}\frac{\beta \left( J\right) }
\beta \left( J^{\prime }\right) }B_{N,\ell }^{s}\left( J,J^{\prime }\right)
+\sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left( J^{\prime
},J\right) >\frac{1}{100}2^{N+\ell +2}}}\frac{\beta \left( J\right) }{\beta
\left( J^{\prime }\right) }B_{N,\ell }^{s}\left( J,J^{\prime }\right) \\
&=&A+B,
\end{eqnarray*
wher
\begin{eqnarray*}
&&A\lesssim \sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left(
J,J^{\prime }\right) \leq \frac{1}{100}2^{N+\ell +2}}}\left\{ \sum
_{\substack{ I\in \mathcal{D}_{N} \\ d\left( I,J\right) \approx 2^{N+\ell }
}}\left\vert I\right\vert _{\sigma }\right\} \ \frac{2^{2\left( N-s\right)
}{2^{2\left( \ell +N\right) \left( 2-\alpha \right) }}\left\vert J^{\prime
}\right\vert _{\omega } \\
&=&\sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left( J,J^{\prime
}\right) \leq \frac{1}{100}2^{N+\ell +2}}}\left\{ \sum_{k=1}^{C}\left\vert
I_{k}^{\ast }\right\vert _{\sigma }\right\} \ \frac{2^{2\left( N-s\right) }}
2^{2\left( \ell +N\right) \left( 2-\alpha \right) }}\left\vert J^{\prime
}\right\vert _{\omega }=\frac{2^{2\left( N-s\right) }}{2^{2\left( \ell
+N\right) \left( 2-\alpha \right) }}\sum_{k=1}^{C}\sum_{\substack{ J^{\prime
}\in \mathcal{G}_{N-s} \\ d\left( J,J^{\prime }\right) \leq \frac{1}{100
2^{N+\ell +2}}}\left\vert I_{k}^{\ast }\right\vert _{\sigma }\ \left\vert
J^{\prime }\right\vert _{\omega } \\
&\lesssim &2^{-2s-2\ell }\sum_{k=1}^{C}\frac{\left\vert I_{k}^{\ast
}\right\vert _{\sigma }}{2^{\left( \ell +N\right) \left( 1-\alpha \right) }
\frac{\left\vert \frac{1}{100}2^{s+\ell +4}J\right\vert _{\omega }}
2^{\left( \ell +N\right) \left( 1-\alpha \right) }}\lesssim 2^{-2s-2\ell
}A_{2}^{\alpha },
\end{eqnarray*
since the intervals $I_{k}^{\ast }$ and $\frac{1}{100}2^{s+\ell +4}J$ are
well separated.
Defin
\begin{equation*}
E_{k}^{\limfunc{left}}\equiv \dbigcup\limits_{\substack{ J^{\prime }\in
\mathcal{G}_{N-s}:\ d\left( I_{k}^{\ast },J^{\prime }\right) \approx
2^{N+\ell } \\ d\left( J,J^{\prime }\right) >\frac{1}{100}2^{N+\ell +2
\text{ and }J^{\prime }\text{ is to the left of }I_{k}^{\ast }}}J^{\prime }\
\text{\ and }E_{k}^{\limfunc{right}}\equiv \dbigcup\limits_{\substack{
J^{\prime }\in \mathcal{G}_{N-s}:\ d\left( I_{k}^{\ast },J^{\prime }\right)
\approx 2^{N+\ell } \\ d\left( J,J^{\prime }\right) >\frac{1}{100}2^{N+\ell
+2}\text{ and }J^{\prime }\text{ is\ to the right of }I_{k}^{\ast }}
J^{\prime },
\end{equation*
and let $Q_{k}^{\limfunc{left}}$ (respectively $Q_{k}^{\limfunc{right}}$) be
the smallest interval containing $E_{k}^{\limfunc{left}}$ (respectively
E_{k}^{\limfunc{right}}$). Then we hav
\begin{eqnarray*}
B &\lesssim &\sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left(
J,J^{\prime }\right) >\frac{1}{100}2^{N+\ell +2}}}\left\{ \sum_{\substack{
I\in \mathcal{D}_{N} \\ d\left( I,J^{\prime }\right) \approx d\left(
I,J\right) \approx 2^{N+\ell }}}\left\vert I\right\vert _{\sigma }\right\} \
\frac{2^{2\left( N-s\right) }}{2^{2\left( \ell +N\right) \left( 2-\alpha
\right) }}\left\vert J^{\prime }\right\vert _{\omega } \\
&\lesssim &\sum_{\substack{ J^{\prime }\in \mathcal{G}_{N-s} \\ d\left(
J,J^{\prime }\right) >\frac{1}{100}2^{N+\ell +2}}}\left\{ \sum_{k:\ d\left(
I_{k}^{\ast },J^{\prime }\right) \approx 2^{N+\ell }}\left\vert I_{k}^{\ast
}\right\vert _{\sigma }\right\} \ \frac{2^{2\left( N-s\right) }}{2^{2\left(
\ell +N\right) \left( 2-\alpha \right) }}\left\vert J^{\prime }\right\vert
_{\omega } \\
&\lesssim &\frac{2^{2\left( N-s\right) }}{2^{2\left( \ell +N\right) \left(
2-\alpha \right) }}\sum_{k=1}^{C}\left\vert I_{k}^{\ast }\right\vert
_{\sigma }\left\vert E_{k}^{\limfunc{left}}\cup E_{k}^{\limfunc{right
}\right\vert _{\omega } \\
&\lesssim &2^{-2s-2\ell }\sum_{k=1}^{C}\frac{\left\vert I_{k}^{\ast
}\right\vert _{\sigma }}{2^{\left( \ell +N\right) \left( 1-\alpha \right) }
\frac{\left\vert Q_{k}^{\limfunc{left}}\right\vert _{\omega }+\left\vert
Q_{k}^{\limfunc{right}}\right\vert _{\omega }}{2^{\left( \ell +N\right)
\left( 1-\alpha \right) }}\lesssim 2^{-2s-2\ell }\mathfrak{A}_{2}^{\alpha },
\end{eqnarray*
since the interval $I_{k}^{\ast }$ is well separated from each of the
intervals $Q_{k}^{\limfunc{left}}$ and $Q_{k}^{\limfunc{right}}$.
Thus we can now apply Schur's argument with $\sum_{J}\left( a_{J}\right)
^{2}=\sum_{J^{\prime }}\left( b_{J^{\prime }}\right) ^{2}=1$ to obtai
\begin{eqnarray*}
&&\sum_{J,J^{\prime }\in \mathcal{G}_{N-s}}a_{J}b_{J^{\prime }}B_{N,\ell
}^{s}\left( J,J^{\prime }\right) \\
&=&\sum_{J,J^{\prime }\in \mathcal{G}_{N-s}}a_{J}\beta \left( J\right)
b_{J^{\prime }}\beta \left( J^{\prime }\right) \frac{B_{N,\ell }^{s}\left(
J,J^{\prime }\right) }{\beta \left( J\right) \beta \left( J^{\prime }\right)
} \\
&\leq &\sum_{J}\left( a_{J}\beta \left( J\right) \right) ^{2}\sum_{J^{\prime
}}\frac{B_{N,\ell }^{s}\left( J,J^{\prime }\right) }{\beta \left( J\right)
\beta \left( J^{\prime }\right) }+\sum_{J^{\prime }}\left( b_{J^{\prime
}}\beta \left( J^{\prime }\right) \right) ^{2}\sum_{J}\frac{B_{N,\ell
}^{s}\left( J,J^{\prime }\right) }{\beta \left( J\right) \beta \left(
J^{\prime }\right) } \\
&=&\sum_{J}\left( a_{J}\right) ^{2}\left\{ \sum_{J^{\prime }}\frac{\beta
\left( J\right) }{\beta \left( J^{\prime }\right) }B_{N,\ell }^{s}\left(
J,J^{\prime }\right) \right\} +\sum_{J^{\prime }}\left( b_{J^{\prime
}}\right) ^{2}\left\{ \sum_{J}\frac{\beta \left( J^{\prime }\right) }{\beta
\left( J\right) }B_{N,\ell }^{s}\left( J,J^{\prime }\right) \right\} \\
&\lesssim &2^{-2s-2\ell }A_{2}^{\alpha }\left( \sum_{J}\left( a_{J}\right)
^{2}+\sum_{J^{\prime }}\left( b_{J^{\prime }}\right) ^{2}\right)
=2^{1-2s-2\ell }A_{2}^{\alpha }.
\end{eqnarray*
This completes the proof of (\ref{Schur s}). We can now sum in $\ell $ to
get (\ref{AsN}) and we are done. This completes our proof of the long range
estimat
\begin{equation*}
\mathcal{A}\left( f,g\right) \lesssim \sqrt{A_{2}^{\alpha }}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }\ .
\end{equation*}
\end{proof}
\subsection{Short range form}
The form $\Theta _{1}^{\limfunc{short}}\left( f,g\right) $ is handled by the
following lemma.
\begin{lemma}
\label{delta short}We hav
\begin{equation*}
\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ \ell \left(
J\right) \leq \ell \left( I\right) \\ \ell \left( I\right) \geq d\left(
J,I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right)
^{1-\varepsilon }}}\left\vert \int \left( T_{\sigma }\square _{I}^{\sigma
\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega
\right\vert \lesssim \sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }
\end{equation*}
\end{lemma}
\begin{proof}
The pairs $\left( I,J\right) $ that occur in the sum above satisfy $J\subset
4I\setminus I$ and so we consider
\begin{equation*}
\mathcal{P}\equiv \left\{ \left( I,J\right) \in \mathcal{D}\times \mathcal{G
:\ell \left( J\right) \leq \ell \left( I\right) ,\ \ell \left( I\right) \geq
d\left( J,I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon },\text{ }J\subset 4I\setminus I\right\} .
\end{equation*
For $\left( I,J\right) \in \mathcal{P}$, the `pivotal' estimate from the
Energy Lemma \ref{ener} give
\begin{equation*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma
\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }\right\vert \lesssim \left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }\mathrm{P}^{\alpha
}\left( J,\left\vert \square _{I}^{\sigma ,\mathbf{b}}f\right\vert \sigma
\right) \sqrt{\left\vert J\right\vert _{\omega }}\,.
\end{equation*
Now we pigeonhole the lengths of $I$ and $J$ and the distance between them
by definin
\begin{equation*}
\mathcal{P}_{N,d}^{s}\equiv \left\{ \left( I,J\right) \in \mathcal{P}:\ell
\left( I\right) =2^{N},\ \ell \left( J\right) =2^{N-s},\ 2^{d-1}\leq d\left(
I,J\right) \leq 2^{d},\text{ }J\subset 4I\setminus I\right\} .
\end{equation*
Note that the closest an interval $J$ can come to $I$ is determined by:
\begin{eqnarray*}
&&2^{d}\geq 2\ell \left( I\right) ^{1-\varepsilon }\ell \left( J\right)
^{\varepsilon }=2^{1+N\left( 1-\varepsilon \right) +\left( N-s\right)
\varepsilon }=2^{1+N-\varepsilon s}; \\
&&\text{which implies }N-\varepsilon s+1\leq d\leq N.
\end{eqnarray*
Thus we hav
\begin{eqnarray*}
&&\dsum\limits_{\left( I,J\right) \in \mathcal{P}}\left\vert \left\langle
T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}\right\vert \lesssim \dsum\limits_{\left( I,J\right) \in \mathcal{P
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }\mathrm{P}^{\alpha }\left( J,\left\vert
\square _{I}^{\sigma ,\mathbf{b}}f\right\vert \sigma \right) \sqrt
\left\vert J\right\vert _{\omega }} \\
&&\ \ \ \ \ =\dsum\limits_{s=0}^{\infty }\ \sum_{N\in \mathbb{Z}}\
\sum_{d=N-\varepsilon s+1}^{N}\ \sum_{\left( I,J\right) \in \mathcal{P
_{N,d}^{s}}\ \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }\mathrm{P}^{\alpha }\left(
J,\left\vert \square _{I}^{\sigma ,\mathbf{b}}f\right\vert \sigma \right)
\sqrt{\left\vert J\right\vert _{\omega }}.
\end{eqnarray*
Now we us
\begin{eqnarray*}
\mathrm{P}^{\alpha }\left( J,\left\vert \square _{I}^{\sigma ,\mathbf{b
}f\right\vert \sigma \right) &=&\int_{I}\frac{\ell \left( J\right) }{\left(
\ell \left( J\right) +\left\vert y-c_{J}\right\vert \right) ^{2-\alpha }
\left\vert \square _{I}^{\sigma ,\mathbf{b}}f\left( y\right) \right\vert
d\sigma \left( y\right) \\
&\lesssim &\frac{2^{N-s}}{2^{d\left( 2-\alpha \right) }}\left\Vert \square
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }\sqrt
\left\vert I\right\vert _{\sigma }}
\end{eqnarray*
and apply Cauchy-Schwarz in $J$ and use $J\subset 4I\setminus I$ to ge
\begin{eqnarray*}
&&\dsum\limits_{\left( I,J\right) \in \mathcal{P}}\left\vert \left\langle
T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}\right\vert \\
&\lesssim &\dsum\limits_{s=0}^{\infty }\ \sum_{N\in \mathbb{Z}}\
\sum_{d=N-\varepsilon s+1}^{N}\ \sum_{I\in \mathcal{D}_{N}}\frac
2^{N-s}2^{N\left( 1-\alpha \right) }}{2^{d\left( 2-\alpha \right) }
\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }\frac{\sqrt{\left\vert I\right\vert _{\sigma }}\sqrt
\left\vert 4I\setminus I\right\vert _{\omega }}}{2^{N\left( 1-\alpha \right)
}} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \sqrt
\sum_{\substack{ J\in \mathcal{G}_{N-s} \\ J\subset 4I\setminus I\text{ and
}d\left( I,J\right) \approx 2^{d}}}\left\Vert \square _{J}^{\omega ,\mathbf{
}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{2}} \\
&\lesssim &\dsum\limits_{s=0}^{\infty }\ \sum_{N\in \mathbb{Z}}\frac
2^{N-s}2^{N\left( 1-\alpha \right) }}{2^{\left( N-\varepsilon s\right)
\left( 2-\alpha \right) }}\sqrt{A_{2}^{\alpha }}\sum_{I\in \mathcal{D
_{N}}\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }\sqrt{\sum_{\substack{ J\in \mathcal{G}_{N-s} \\ J\subset
4I\setminus I}}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{2}} \\
&\lesssim &\dsum\limits_{s=0}^{\infty }2^{-s\left[ 1-\varepsilon \left(
2-\alpha \right) \right] }\sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\lesssim \sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) },
\end{eqnarray*
where in the third line above we have summed the geometric series in $d$,
and in the last line $\frac{2^{N-s}2^{N\left( 1-\alpha \right) }}{2^{\left(
N-\varepsilon s\right) \left( 2-\alpha \right) }}=2^{-s\left[ 1-\varepsilon
\left( 2-\alpha \right) \right] }$ followed by Cauchy-Schwarz in $I$ and $N
, using that we have bounded overlap in the quadruples of $I$ for $I\in
\mathcal{D}_{N}$, and finally using the lower frame inequality for $\square
_{I}^{\sigma ,\mathbf{b}}$ from Appendix A. More precisely, if we define
f_{k}\equiv \Psi _{\mathcal{D}_{k}}^{\sigma ,\mathbf{b}}f=\sum_{I\in
\mathcal{D}_{k}}\square _{I}^{\sigma ,\mathbf{b}}f$ and $g_{k}\equiv \Psi _
\mathcal{G}_{k}}^{\sigma ,\mathbf{b}^{\ast }}g=\sum_{J\in \mathcal{G
_{k}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}g$, then we have the
quasi-orthogonality inequality
\begin{eqnarray*}
\sum_{N\in \mathbb{Z}}\left\Vert f_{N}\right\Vert _{L^{2}\left( \sigma
\right) }\left\Vert g_{N-s}\right\Vert _{L^{2}\left( \omega \right) } &\leq
&\left( \sum_{N\in \mathbb{Z}}\left\Vert f_{N}\right\Vert _{L^{2}\left(
\sigma \right) }^{2}\right) ^{\frac{1}{2}}\left( \sum_{N\in \mathbb{Z
}\left\Vert g_{N-s}\right\Vert _{L^{2}\left( \omega \right) }^{2}\right) ^
\frac{1}{2}} \\
&\lesssim &\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }.
\end{eqnarray*
We have assumed that
\begin{equation}
0<\varepsilon <\frac{1}{2-\alpha } \label{short requirement}
\end{equation
in the calculations above in order that the sum in $s$ converges, and this
completes the proof of Lemma \ref{delta short}.
\end{proof}
\section{Nearby form\label{Sec nearby}}
We dominate the nearby form $\Theta _{3}\left( f,g\right) $ b
\begin{equation*}
\left\vert \Theta _{3}\left( f,g\right) \right\vert \leq \sum_{I\in \mathcal
D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r}}\ell \left( I\right)
<\ell \left( J\right) \leq \ell \left( I\right) \\ d\left( J,I\right) \leq
2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}
\left\vert \int \left( T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b
}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega \right\vert ,
\end{equation*
and prove the following lemma that controls the expectation, over two
independent grids, of the nearby form $\Theta _{3}\left( f,g\right) $. It
should be noted that weak goodness plays no role in treating the nearby form.
\begin{lemma}
\label{nearby form}Suppose $T^{\alpha }$ is a standard fractional singular
integral with $0\leq \alpha <1$. Let $\theta \in \left( 0,1\right) $ be
sufficiently small depending only on $0\leq \alpha <1$. Then there is a
constant $C_{\theta }$ such that for $f\in L^{2}\left( \sigma \right) $ and
g\in L^{2}\left( \omega \right) $, and dual martingale differences $\square
_{I}^{\sigma ,\mathbf{b}}$ and $\square _{J}^{\omega ,\mathbf{b}^{\ast }}$
with $\infty $-strongly accretive families of test functions $\mathbf{b}$
and $\mathbf{b}^{\ast }$, we hav
\begin{eqnarray}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \left\langle T_{\sigma }^{\alpha
}\left( \square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert \label{delta near}
\\
&\lesssim &\left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak
N}_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ . \notag
\end{eqnarray}
\end{lemma}
\textbf{Proof:} As usual, we continue to write the independent grids for $f$
and $g$ as $\mathcal{D}$ and $\mathcal{G}$ respectively. Write the dual
martingale averages $\square _{I}^{\sigma ,\mathbf{b}}f$ and $\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g$ as linear combinations
\begin{eqnarray*}
\square _{I}^{\sigma ,\mathbf{b}}f &=&b_{I}\ \sum_{I^{\prime }\in \mathfrak{
}_{\limfunc{natural}}\left( I\right) }\mathbf{1}_{I^{\prime }}\ E_{I^{\prime
}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\mathbf{b}}f\right)
+\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right)
}b_{I^{\prime }}\ \mathbf{1}_{I^{\prime }}\widehat{\mathbb{F}}_{I^{\prime
}}^{\sigma ,b_{I^{\prime }}}f-b_{I}\ \sum_{I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( I\right) }\mathbf{1}_{I^{\prime }}\widehat{\mathbb{F
}_{I}^{\sigma ,b_{I}}f, \\
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g &=&b_{J}^{\ast }\ \sum_{J^{\prime
}\in \mathfrak{C}_{\limfunc{natural}}\left( J\right) }\mathbf{1}_{J^{\prime
}}\ E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\mathbf{
}^{\ast }}g\right) +\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken
}\left( J\right) }b_{J^{\prime }}^{\ast }\ \mathbf{1}_{J^{\prime }}\widehat
\mathbb{F}}_{J^{\prime }}^{\omega ,b_{J^{\prime }}^{\ast }}g-b_{J}^{\ast }\
\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right) }\mathbf
1}_{J^{\prime }}\widehat{\mathbb{F}}_{J}^{\omega ,b_{J}^{\ast }}g,
\end{eqnarray*
of the appropriate function $b$ times the indicators of their children,
denoted $I^{\prime }$ and $J^{\prime }$ respectively. We will regroup the
terms as needed below.
\begin{notation}
On the natural child $I^{\prime }$, the expression $\widehat{\square
_{I}^{\sigma ,\mathbf{b}}f=\frac{1}{b_{I}}\square _{I}^{\sigma ,\mathbf{b}}f$
simply denotes the dual martingale average with $b_{I}$ removed, so that we
need not assume $\left\vert b_{I}\right\vert $ is bounded below in order to
make sense of $\frac{1}{b_{I}}\square _{I}^{\sigma ,\mathbf{b}}f$. Similar
comments apply to the expressions $\widehat{\mathbb{F}}_{I^{\prime
}}^{\sigma ,b_{I^{\prime }}}f=\frac{1}{b_{I^{\prime }}}\mathbb{F}_{I^{\prime
}}^{\sigma ,b_{I^{\prime }}}f$ and $\widehat{\mathbb{F}}_{I}^{\sigma
,b_{I}}f=\frac{1}{b_{I}}\mathbb{F}_{I}^{\sigma ,b_{I}}f$. On the other hand,
we are assuming from Proposition \ref{lower bound} that the $PLBP$ (\ref{plb
) holds, which shows that $\frac{1}{b_{I^{\prime }}}$ is actually a bounded
function. This latter fact will be used shortly in (\ref{box hat bound})
below.
\end{notation}
Recall that the length of $J$ is at most the length of $I$, i.e. $\ell
\left( J\right) \leq \ell \left( I\right) $. If $J$ and $I$ are \emph
separated}, by which we mean here that $J\cap I=\emptyset $, then by (\re
{disj supp}) we have the satisfactory estimat
\begin{equation*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma
\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }\right\vert \lesssim \sqrt{\mathcal{A}_{2}^{\alpha ,\ast }
\mathcal{A}_{2}^{\alpha }}\left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert \square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }.
\end{equation*
Suppose now that $J\cap I\neq \emptyset $. Using (\ref{flat broken}) we hav
\begin{eqnarray}
\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b
}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
} &=&\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\flat
\mathbf{b}}f\right) ,\square _{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }+\left\langle T_{\sigma }^{\alpha }\left( \square
_{I,\limfunc{broken}}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square _{J
\limfunc{broken}}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
} \label{case d small} \\
&&+\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\flat
\mathbf{b}}f\right) ,\square _{J,\limfunc{broken}}^{\omega ,\flat ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }+\left\langle T_{\sigma }^{\alpha }\left(
\square _{I,\limfunc{broken}}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ . \notag
\end{eqnarray
The estimation of the latter three inner products, i.e. those in which a
broken operator $\square _{I,\limfunc{broken}}^{\sigma ,\flat ,\mathbf{b}}$
or $\square _{J,\limfunc{broken}}^{\omega ,\flat ,\mathbf{b}^{\ast }}$
arises, is easy. Indeed, recall that
\begin{eqnarray*}
\square _{I,\limfunc{broken}}^{\sigma ,\flat ,\mathbf{b}}f
&=&\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right)
\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{b}}f=\sum_{I^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( I\right) }\left( E_{I^{\prime
}}^{\sigma }\widehat{\mathbb{F}}_{I^{\prime }}^{\sigma ,\mathbf{b}}f\right)
b_{I^{\prime }}\ , \\
\square _{J,\limfunc{broken}}^{\omega ,\flat ,\mathbf{b}}g
&=&\sum_{J^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( J\right)
\mathbb{F}_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g=\sum_{J^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left( E_{J^{\prime
}}^{\omega }\widehat{\mathbb{F}}_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}g\right) b_{J^{\prime }}^{\ast }\ ,
\end{eqnarray*
so that if at least one broken difference appears in the inner product, as
is the case for the latter three inner products in (\ref{case d small}),
then testing and Cauchy-Schwarz are all that is needed. For example, the
fourth term satisfies
\begin{eqnarray*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( \square _{I,\limfunc
broken}}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert &=&\left\vert
\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) }\left(
E_{I^{\prime }}^{\sigma }\widehat{\mathbb{F}}_{I^{\prime }}^{\sigma ,\mathbf
b}}f\right) \left\langle T_{\sigma }^{\alpha }b_{I^{\prime }},\square
_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert
\\
&\lesssim &\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
I\right) }\left\vert E_{I^{\prime }}^{\sigma }\widehat{\mathbb{F}
_{I^{\prime }}^{\sigma ,\mathbf{b}}f\right\vert \mathfrak{T}_{T^{\alpha }}^
\mathbf{b}}\sqrt{\left\vert I^{\prime }\right\vert _{\sigma }}\left\Vert
\square _{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left(
\omega \right) } \\
&\lesssim &\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\left\Vert \nabla
_{I}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right) }\sum_{I^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) }\left( \left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }+\left\Vert \square _{J,\limfunc{broken}}^{\omega ,\flat ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }\right) \\
&\lesssim &\mathcal{NTV}_{\alpha }\left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{\bigstar }\ ,
\end{eqnarray*
and the third term can be written as $\left\langle \square _{I}^{\sigma
,\flat ,\mathbf{b}}f,T_{\omega }^{\alpha ,\ast }\left( \square _{J,\limfunc
broken}}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right) \right\rangle _{\sigma
} $ and handled similarly.
Thus it remains to consider the first inner product $\left\langle T_{\sigma
}^{\alpha }\left( \square _{I}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }$ on the
right hand side of (\ref{case d small}), which we call the problematic term,
and write it a
\begin{eqnarray}
P\left( I,J\right) &\equiv &\left\langle T_{\sigma }^{\alpha }\left( \square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right\rangle _{\omega } \label{def P(I,J)} \\
&=&\sum_{I^{\prime }\in \mathfrak{C}\left( I\right) \text{ and }J^{\prime
}\in \mathfrak{C}\left( J\right) }\left\langle T_{\sigma }^{\alpha }\left(
\mathbf{1}_{I^{\prime }}\square _{I}^{\sigma ,\flat ,\mathbf{b}}f\right)
\mathbf{1}_{J^{\prime }}\square _{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega } \notag \\
&=&\sum_{I^{\prime }\in \mathfrak{C}\left( I\right) \text{ and }J^{\prime
}\in \mathfrak{C}\left( J\right) }E_{I^{\prime }}^{\sigma }\left( \widehat
\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \left\langle T_{\sigma
}^{\alpha }\left( \mathbf{1}_{I^{\prime }}b_{I}\right) ,\mathbf{1
_{J^{\prime }}b_{J}^{\ast }\right\rangle _{\omega }\ E_{J^{\prime }}^{\omega
}\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right) .
\notag
\end{eqnarray}
It now remains to show tha
\begin{equation}
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert P\left( I,J\right) \right\vert
\lesssim \left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak{N
_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }.
\label{must show final}
\end{equation
We will repeatedly use the inequality $\left\Vert \widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}\lesssim \left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }$ which, upon noting (\ref{flat
broken}), follows from the $PLBP$ (\ref{plb}),
\begin{eqnarray}
\left\Vert \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) } &\lesssim &\left\Vert b_{I}\widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}=\left\Vert \square _{I}^{\sigma ,\flat ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) } \label{box hat bound} \\
&\leq &\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }+\left\Vert \square _{I,\limfunc{broken
}^{\sigma ,\flat ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}\lesssim \left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }. \notag
\end{eqnarray}
Suppose now that $I\in \mathcal{C}_{A}$ for $A\in \mathcal{A}$, and that
J\in \mathcal{C}_{B}$ for $B\in \mathcal{B}$. Then the inner product in the
third line of (\ref{def P(I,J)}) become
\begin{equation*}
\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{I^{\prime
}}\right) ,b_{J}^{\ast }\mathbf{1}_{J^{\prime }}\right\rangle _{\omega
}=\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{I^{\prime
}}\right) ,b_{B}^{\ast }\mathbf{1}_{J^{\prime }}\right\rangle _{\omega }\ ,
\end{equation*
and we will write this inner product in either form, depending on context.
We also introduce the following notation
\begin{equation*}
P_{\left( I,J\right) }\left( E,F\right) \equiv \left\langle T_{\sigma
}^{\alpha }\left( b_{I}\mathbf{1}_{E}\right) ,b_{J}^{\ast }\mathbf{1
_{F}\right\rangle _{\omega },\ \ \ \ \ \text{for any sets }E\text{ and }F,
\end{equation*
so tha
\begin{equation*}
P\left( I,J\right) =\sum_{I^{\prime }\in \mathfrak{C}\left( I\right) \text{
and }J^{\prime }\in \mathfrak{C}\left( J\right) }E_{I^{\prime }}^{\sigma
}\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \
P_{\left( I,J\right) }\left( I^{\prime },J^{\prime }\right) \ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) .
\end{equation*
The first thing we do is reduce matters to showing inequality (\ref{must
show final}) in the case of \emph{equal intervals}, by which we mean that
P_{\left( I,J\right) }\left( I^{\prime },J^{\prime }\right) $ is replaced by
$P_{\left( I,J\right) }\left( I^{\prime }\cap J^{\prime },I^{\prime }\cap
J^{\prime }\right) $ in the terms $P\left( I,J\right) $ appearing in (\re
{must show final}). To see this let $K\equiv I^{\prime }\cap J^{\prime }$,
writ
\begin{eqnarray*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{I^{\prime }}\right) ,b_{J}^{\ast }\mathbf{1}_{J^{\prime }}\right\rangle
_{\omega }\right\vert &=&\left\vert \left\langle T_{\sigma }^{\alpha }\left(
b_{I}\mathbf{1}_{I^{\prime }\setminus J^{\prime }}\right) ,b_{J}^{\ast
\mathbf{1}_{J^{\prime }}\right\rangle _{\omega }+\left\langle T_{\sigma
}^{\alpha }\left( b_{I}\mathbf{1}_{I^{\prime }\cap J^{\prime }}\right)
,b_{J}^{\ast }\mathbf{1}_{J^{\prime }\setminus I^{\prime }}\right\rangle
_{\omega }+\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{I^{\prime }\cap J^{\prime }}\right) ,b_{J}^{\ast }\mathbf{1}_{J^{\prime
}\cap I^{\prime }}\right\rangle _{\omega }\right\vert \\
&\leq &\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{I^{\prime }\setminus J^{\prime }}\right) ,b_{J}^{\ast }\mathbf{1
_{J^{\prime }}\right\rangle _{\omega }\right\vert +\left\vert \left\langle
T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{K}\right) ,b_{J}^{\ast }\mathbf
1}_{J^{\prime }\setminus I^{\prime }}\right\rangle _{\omega }\right\vert
+\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{K}\right) ,b_{J}^{\ast }\mathbf{1}_{K}\right\rangle \right\vert ,
\end{eqnarray*
and use (\ref{disj supp}) to obtai
\begin{equation*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{I^{\prime }\setminus J^{\prime }}\right) ,b_{J}^{\ast }\mathbf{1
_{J^{\prime }}\right\rangle _{\omega }\right\vert +\left\vert \left\langle
T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{K}\right) ,b_{J}^{\ast }\mathbf
1}_{J^{\prime }\setminus I^{\prime }}\right\rangle _{\omega }\right\vert
\lesssim \sqrt{\mathfrak{A}_{2}^{\alpha }}\sqrt{\left\vert I^{\prime
}\right\vert _{\sigma }\left\vert J^{\prime }\right\vert _{\omega }}.
\end{equation*
It thus remains to consider only the term $P_{\left( I,J\right) }\left(
K,K\right) =\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{K}\right) ,b_{J}^{\ast }\mathbf{1}_{K}\right\rangle _{\omega }$ where
K=I^{\prime }\cap J^{\prime }\neq \emptyset $, and to show tha
\begin{eqnarray}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ K=I^{\prime }\cap J^{\prime }\neq \emptyset }}E_{I^{\prime
}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f\right) \ P_{\left( I,J\right) }\left( K,K\right) \ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \label{ineq KK} \\
&\lesssim &\left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak
N}_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }, \notag
\end{eqnarray}
\subsection{Random surgery}
However, we wish to further reduce matters to the case where $K\in \mathcal{
}$ and contained in $I^{\prime }\cap J^{\prime }$, and for this we use
random surgery and (\ref{disj supp}). First, we note that $I^{\prime }$
cannot be strictly contained in $J^{\prime }$ since $\ell \left( J\right)
\leq \ell \left( I\right) $, and in the case that $J^{\prime }\subset
I^{\prime }$, then $K=I^{\prime }\cap J^{\prime }=J^{\prime }\in \mathcal{G}$
and there is nothing more to do in this case. So we may assume that
J^{\prime }$ intersects both $I^{\prime }$ and its complement $\left(
I^{\prime }\right) ^{c}$.
Our first step is to reduce matters to showing inequality (\ref{ineq KK}) in
the case where $\ell \left( K\right) \geq \lambda \ell \left( I^{\prime
}\right) $ for a small positive number $\lambda \ll 2^{-\mathbf{r}}$. This
small constant $\lambda $, as well as the constant $\eta _{0}$ introduced in
probability estimates below, will be chosen sufficiently small at the end of
the proof to result in the term $\sqrt{\theta }\mathfrak{N}_{T^{\alpha }}$
appearing on the right hand side of (\ref{ineq KK}). This is accomplished by
writing (recall that $K=I^{\prime }\cap J^{\prime }$ for the moment
\begin{eqnarray*}
&&\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ K\neq \emptyset }}E_{I^{\prime }}^{\sigma }\left( \widehat
\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ P_{\left( I,J\right)
}\left( K,K\right) \ E_{J^{\prime }}^{\omega }\left( \widehat{\square
_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert \\
&\leq &\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{
\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{
I^{\prime }\in \mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in
\mathfrak{C}\left( J\right) \\ \ell \left( K\right) \geq \lambda \ell
\left( I^{\prime }\right) }}E_{I^{\prime }}^{\sigma }\left( \widehat{\square
}_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ P_{\left( I,J\right) }\left(
K,K\right) \ E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert \\
&&+\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ 0<\ell \left( K\right) <\lambda \ell \left( I^{\prime }\right)
}}E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \ P_{\left( I,J\right) }\left( K,K\right) \ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \\
&\equiv &A+B.
\end{eqnarray*
Term $B$ is handled using the norm constant $\mathfrak{N}_{T^{\alpha }}$ and
probability, together with the estimat
\begin{eqnarray*}
&&\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in \mathfrak{C
\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left( J\right) \\
0<\ell \left( K\right) <\lambda \ell \left( I^{\prime }\right) }}\left\vert
\sqrt{\left\vert K\right\vert _{\omega }}\ E_{J^{\prime }}^{\omega }\left(
\widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right)
\right\vert ^{2} \\
&=&\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in \mathfrak{C
\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left( J\right) \\
0<\ell \left( K\right) <\lambda \ell \left( I^{\prime }\right) }}\left\Vert
\widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2} \\
&\lesssim &\sum_{J\in \mathcal{G}}\left\Vert \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}\lesssim \sum_{J\in \mathcal{G}}\left( \left\Vert \square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{2}+\left\Vert \widehat{\square }_{J,\limfunc{broken}}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right)
\lesssim \left\Vert g\right\Vert _{L^{2}\left( \omega \right) }^{2}\ ,
\end{eqnarray*
to obtai
\begin{eqnarray*}
\boldsymbol{E}_{\Omega }^{\mathcal{G}}B &\lesssim &\boldsymbol{E}_{\Omega }^
\mathcal{G}}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{
\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ 0<\ell \left( K\right) <\lambda \ell \left( I^{\prime }\right)
}}\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \ \mathfrak{N}_{T^{\alpha }}\sqrt{\left\vert
K\right\vert _{\sigma }\left\vert K\right\vert _{\omega }}\ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \\
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{G}}\mathfrak{N}_{T^{\alpha
}}\left( \sum_{I\in \mathcal{D}}\sum_{I^{\prime }\in \mathfrak{C}\left(
I\right) }\left( \sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r}}\ell
\left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\ d\left(
J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right)
^{1-\varepsilon }}}\sum_{\substack{ J^{\prime }\in \mathfrak{C}\left(
J\right) \\ 0<\ell \left( I^{\prime }\cap J^{\prime }\right) <\lambda \ell
\left( I^{\prime }\right) }}\left\vert I^{\prime }\cap J^{\prime
}\right\vert _{\sigma }\right) \left\vert E_{I^{\prime }}^{\sigma }\left(
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert
^{2}\right) ^{\frac{1}{2}}\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) } \\
&\leq &\mathfrak{N}_{T^{\alpha }}\left( \sum_{I\in \mathcal{D
}\sum_{I^{\prime }\in \mathfrak{C}\left( I\right) }\lambda \left\vert
I^{\prime }\right\vert _{\sigma }\left\vert E_{I^{\prime }}^{\sigma }\left(
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert
^{2}\right) ^{\frac{1}{2}}\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\lesssim \mathfrak{N}_{T^{\alpha }}\sqrt{\lambda }\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }
\end{eqnarray*
since the probability that, given an interval $I^{\prime }$, a grid
\mathcal{G}$ contains an interval $J^{\prime }$ with $2^{-\mathbf{r}}\ell
\left( I^{\prime }\right) <\ell \left( J^{\prime }\right) \leq \ell \left(
I^{\prime }\right) $ and $0<\ell \left( I^{\prime }\cap J^{\prime }\right)
<\lambda \ell \left( I^{\prime }\right) $, is at most $C_{\mathbf{r}}\lambda
$ for some large constant $C_{\mathbf{r}}$ depending on the goodness
parameter $\mathbf{r}$ (we could also have appealed to the more general halo
estimate (\ref{hand'})). This term of course contributes to the conclusion
of the lemma provided $\lambda >0$ is chosen sufficiently small. Thus we are
left to control term $A$ in which $\ell \left( K\right) \geq \lambda \ell
\left( I^{\prime }\right) $ as required.
Now choose $\eta _{0}\in \left( 0,1\right) \cap \left\{ 2^{-m}\right\}
_{m\in \mathbb{N}}$, say $\eta _{0}=2^{-m}$ for $m$ sufficiently large. We
will assume from now on that $\eta _{0}<\frac{1}{2}\lambda $, where $\lambda
$ is the small constant above. For any interval $L$ and $0<\eta \leq \frac{
}{2}$, define
\begin{equation}
\partial _{\eta }L\equiv \left( 1+\eta \right) L\setminus \left( 1-\eta
\right) L \label{def halo}
\end{equation
to be the `$\eta $-halo' around the boundary, i.e. endpoints, of $L$. One
should also recall that all intervals are assumed to be closed on the left
and open on the right. For $\eta _{1}\in \left( 0,\eta _{0}\right] $ we writ
\begin{eqnarray*}
I^{\prime }\cap J^{\prime } &=&\left\{ \left( I^{\prime }\setminus \partial
_{\eta _{1}}I^{\prime }\right) \cap J^{\prime }\right\} \cup \left\{ \left[
I^{\prime }\cap J^{\prime }\right] \setminus \left[ \left( I^{\prime
}\setminus \partial _{\eta _{1}}I^{\prime }\right) \cap J^{\prime }\right]
\right\} \\
&\equiv &M\overset{\cdot }{\cup }L,
\end{eqnarray*
where $\overset{\cdot }{\cup }$ denotes a disjoint union. At this point we
note that there is precisely one endpoint of $I^{\prime \prime }\equiv
I^{\prime }\setminus \partial _{\eta _{1}}I^{\prime }$ that lies in
J^{\prime }$ since $\ell \left( I^{\prime }\cap J^{\prime }\right) \geq
\lambda \ell \left( I^{\prime }\right) $ and $\eta _{1}\leq \eta _{0}<\frac{
}{2}\lambda $.
Moreover, and this is the key part of the argument, we can choose $\frac{1}{
}\eta _{0}\leq \eta _{1}\leq \eta _{0}$ so that the interval $M=I^{\prime
\prime }\cap J^{\prime }$ is a union of a number $B=B\left( I^{\prime
},J^{\prime }\right) =B\left( I^{\prime },J^{\prime },\eta _{1}\left(
I^{\prime },J^{\prime }\right) \right) $ of intervals $K_{i}\in \mathcal{G}$
each having side length $\ell \left( K_{i}\right) =2^{-m-1}\ell \left(
J^{\prime }\right) \equiv \frac{1}{2}\eta _{0}\ell \left( J^{\prime }\right)
$ and where $B\leq C\frac{1}{\eta _{0}}$. Indeed, we take $\frac{1}{2}\eta
_{0}\leq \eta _{1}\leq \eta _{0}=2^{-m}$, so that the endpoint of the
interval $I^{\prime \prime }\equiv I^{\prime }\setminus \partial _{\eta
_{1}}I^{\prime }$ that lies in $J^{\prime }$ coincides with an endpoint of
some $K\in \mathcal{G}$ with $\ell \left( K\right) =2^{-m-1}\ell \left(
J^{\prime }\right) $. This can be arranged by varying $\eta _{1}$ between
\frac{1}{2}\eta _{0}$ and $\eta _{0}$ until the endpoint in question lies
among the dyadic numbers that form the endpoints of intervals in the grid
\mathcal{G}$ with side length $2^{-m-1}\ell \left( J^{\prime }\right) $. The
choice of intervals $\left\{ K_{i}\right\} _{i=1}^{B}$ having common side
length $2^{-m-1}\ell \left( J^{\prime }\right) $ is then uniquely
determined, and it is easy to see that
\begin{equation*}
B\leq C\frac{1}{\eta _{0}}.
\end{equation*
Thus the choice of $\eta _{1}=\eta _{1}\left( I^{\prime },J^{\prime }\right)
>0$ is always at most $\eta _{0}$, and at least $\frac{1}{2}\eta _{0}$, but
changes according to the relative position of $I^{\prime }$ with respect to
J^{\prime }$ in order that $M$ is a union of intervals $K_{i}\in \mathcal{G}$
of side length at least $\frac{1}{2}\eta _{0}\ell \left( J^{\prime }\right)
, and specified in the manner described above. Defin
\begin{equation}
\mathcal{K}\left( I^{\prime },J^{\prime }\right) \equiv \left\{
K_{i}\right\} _{i=1}^{B\left( I^{\prime },J^{\prime }\right) }=\left\{
K_{i}\right\} _{i=1}^{B} \label{def K(I',J')}
\end{equation
to be this collection of consecutive adjacent intervals $K_{i}$ uniquely
defined here in terms of $I^{\prime }$, $J^{\prime }$ and $\eta _{1}=\eta
_{1}\left( I^{\prime },J^{\prime }\right) $.
Having chosen the parameter $\eta _{1}$ as above, we now momentarily ignore
the decomposition of $M=\dbigcup\limits_{i=1}^{B}K_{i}$ into subintervals,
and return to the representation $K=I^{\prime }\cap J^{\prime }=M\overset
\cdot }{\cup }L$ determined by our choice of $\eta _{1}$. We hav
\begin{eqnarray}
\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{K}\right)
,b_{J}^{\ast }\mathbf{1}_{K}\right\rangle _{\omega } &=&\left\langle
T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{M}\right) ,b_{J}^{\ast }\mathbf
1}_{L}\right\rangle _{\omega }+\left\langle T_{\sigma }^{\alpha }\left( b_{I
\mathbf{1}_{L}\right) ,b_{J}^{\ast }\mathbf{1}_{M}\right\rangle _{\omega }
\label{four terms} \\
&&+\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{L}\right)
,b_{J}^{\ast }\mathbf{1}_{L}\right\rangle _{\omega }+\left\langle T_{\sigma
}^{\alpha }\left( b_{I}\mathbf{1}_{M}\right) ,b_{J}^{\ast }\mathbf{1
_{M}\right\rangle _{\omega }\ . \notag
\end{eqnarray
Now we apply (\ref{disj supp}) to the first two terms in (\ref{four terms})
to obtain tha
\begin{eqnarray*}
&&\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{M}\right) ,b_{J}^{\ast }\mathbf{1}_{L}\right\rangle _{\omega }\right\vert
+\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{L}\right) ,b_{J}^{\ast }\mathbf{1}_{M}\right\rangle _{\omega }\right\vert
\\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}\left\{ \sqrt{\int_{M}\left\vert
b_{I}\right\vert ^{2}d\sigma }\sqrt{\int_{L}\left\vert b_{J}^{\ast
}\right\vert ^{2}d\omega }+\sqrt{\int_{L}\left\vert b_{I}\right\vert
^{2}d\sigma }\sqrt{\int_{M}\left\vert b_{J}^{\ast }\right\vert ^{2}d\omega
\right\} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lesssim \sqrt{\mathfrak{A
_{2}^{\alpha }}\sqrt{\left\vert I^{\prime }\right\vert _{\sigma }\left\vert
J^{\prime }\right\vert _{\omega }},
\end{eqnarray*
which when plugged appropriately into the left hand side of (\ref{must show
final}) is dominated by the right hand side of (\ref{must show final})
\begin{eqnarray*}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in \mathfrak{C
\left( I\right) \\ J^{\prime }\in \mathfrak{C}\left( J\right) }} \\
&&\ \ \ \ \ \ \ \ \ \ \times \left\vert E_{I^{\prime }}^{\sigma }\left(
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \left(
\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{M}\right) ,b_{J}^{\ast }\mathbf{1}_{L}\right\rangle _{\omega }\right\vert
+\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{L}\right) ,b_{J}^{\ast }\mathbf{1}_{M}\right\rangle _{\omega }\right\vert
\right) \ E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert \\
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^
\mathcal{G}}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{
\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \\ J^{\prime }\in \mathfrak{C}\left( J\right) }
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert \sqrt{\mathfrak{A}_{2}^{\alpha }
\sqrt{\left\vert I^{\prime }\right\vert _{\sigma }\left\vert J^{\prime
}\right\vert _{\omega }}\left\vert E_{J^{\prime }}^{\omega }\left( \widehat
\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\ ,
\end{eqnarray*
where in the last line we have used $\sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \\ J^{\prime }\in \mathfrak{C}\left( J\right) }
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert ^{2}\left\vert I^{\prime
}\right\vert _{\sigma }=\left\Vert \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}$ and (\ref{box
hat bound}) and the frame inequalities in Appendix A. Then we apply
Cauchy-Schwarz to the sums in the third term in (\ref{four terms}) using
L=\partial _{\eta _{1}}I^{\prime }\cap J^{\prime }$ to ge
\begin{eqnarray*}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\sum_{\substack{ I^{\prime }\in \mathfrak{C
\left( I\right) \\ J^{\prime }\in \mathfrak{C}\left( J\right) }}\left\vert
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \ \left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{
}_{L}\right) ,b_{J}^{\ast }\mathbf{1}_{L}\right\rangle _{\omega }\
E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right) \right\vert \\
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{G}}\mathfrak{N}_{T^{\alpha
}}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\boldsymbol{E
_{\Omega }^{\mathcal{D}}\sqrt{\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in
\mathcal{G}:\ 2^{-\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq
\ell \left( I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right)
^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }}}\sum_{\substack{
I^{\prime }\in \mathfrak{C}\left( I\right) \\ J^{\prime }\in \mathfrak{C
\left( J\right) }}\left( \int_{\partial _{\eta _{1}}I^{\prime }\cap
J^{\prime }}\left\vert b_{J}^{\ast }\right\vert ^{2}d\omega \right)
\left\vert E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert ^{2}},
\end{eqnarray*
using (\ref{box hat bound}) and the frame inequalities in Appendix A again.
Then using Cauchy-Schwarz on the expectation $\boldsymbol{E}_{\Omega }^
\mathcal{D}}$, this is dominated by
\begin{eqnarray*}
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{G}}\mathfrak{N}_{T^{\alpha
}}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\sqrt{\sum_{J\in
\mathcal{G}}\sum_{J^{\prime }\in \mathfrak{C}\left( J\right) }\left(
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\sum_{\substack{ I\in \mathcal{D}:\
2^{-\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon } \\ I^{\prime }\in \mathfrak{C
\left( I\right) }}\left\vert \partial _{\eta _{1}}I^{\prime }\cap J^{\prime
}\right\vert _{\omega }\right) \left\vert E_{J^{\prime }}^{\omega }\left(
\widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right)
\right\vert ^{2}} \\
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{G}}\mathfrak{N}_{T^{\alpha
}}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\sqrt{\sum_{J\in
\mathcal{G}}\sum_{J^{\prime }\in \mathfrak{C}\left( J\right) }C2^{\mathbf{r
}\left( \boldsymbol{E}_{\Omega }^{\mathcal{D}}\sum_{I^{\prime }\in \mathcal{
}:\ell \left( J^{\prime }\right) \leq \ell \left( I^{\prime }\right) \leq 2^
\mathbf{r}}\ell \left( J^{\prime }\right) }\left\vert \partial _{\eta
_{0}}I^{\prime }\cap J^{\prime }\right\vert _{\omega }\right) \left\vert
E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right) \right\vert ^{2}} \\
&\lesssim &\sqrt{\eta _{0}}\mathfrak{N}_{T^{\alpha }}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\ ,
\end{eqnarray*
where in the last line we have used $\eta _{1}\leq \eta _{0}$, and then
\begin{equation*}
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\sum_{I^{\prime }\in \mathcal{D}:\ell
\left( J^{\prime }\right) \leq \ell \left( I^{\prime }\right) \leq 2^
\mathbf{r}}\ell \left( J^{\prime }\right) }\left\vert \partial _{\eta
_{0}}I^{\prime }\cap J^{\prime }\right\vert _{\omega }\lesssim \eta
_{0}\left\vert J^{\prime }\right\vert _{\omega }
\end{equation*
from (\ref{hand'}) since $\eta _{0}\leq \frac{1}{2}\lambda \ll 2^{-\mathbf{r
}$, and finally quasiorthogonality and (\ref{box hat bound}) yet again, to
obtain
\begin{eqnarray*}
\sum_{J\in \mathcal{G}}\sum_{J^{\prime }\in \mathfrak{C}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert ^{2} &\lesssim &\sum_{J\in \mathcal{G}}\left\Vert
\widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( J\right) }^{2}\lesssim \sum_{J\in \mathcal{G}}\left\Vert
\square _{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left(
J\right) }^{2} \\
&\lesssim &\sum_{J\in \mathcal{G}}\left( \left\Vert \square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( J\right) }^{2}+\left\Vert
\nabla _{J}^{\omega }g\right\Vert _{L^{2}\left( J\right) }^{2}\right)
\lesssim \left\Vert g\right\Vert _{L^{2}\left( \omega \right) }^{2}\ .
\end{eqnarray*}
This leaves us to estimate the fourth term in (\ref{four terms}), i.e. the
inner product $\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1
_{M}\right) ,b_{J}^{\ast }\mathbf{1}_{M}\right\rangle $. It is at this point
that we will use the decomposition $M=\overset{\cdot }{\dbigcup }_{1\leq
i\leq B}K_{i}$ constructed above. We hav
\begin{equation*}
\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{M}\right)
,b_{J}^{\ast }\mathbf{1}_{M}\right\rangle _{\omega
}=\dsum\limits_{i,i^{\prime }=1}^{B}\left\langle T_{\sigma }^{\alpha }\left(
b_{I}\mathbf{1}_{K_{i}}\right) ,b_{J}^{\ast }\mathbf{1}_{K_{i^{\prime
}}}\right\rangle _{\omega }=\dsum\limits_{i=1}^{B}\left\langle T_{\sigma
}^{\alpha }\left( b_{I}\mathbf{1}_{K_{i}}\right) ,b_{J}^{\ast }\mathbf{1
_{K_{i}}\right\rangle _{\omega }+\dsum\limits_{i\neq i^{\prime
}}\left\langle T_{\sigma }^{\alpha }\left( b_{I}\mathbf{1}_{K_{i}}\right)
,b_{J}^{\ast }\dsum\limits_{i^{\prime }:\ i^{\prime }\neq i}\mathbf{1
_{K_{i^{\prime }}}\right\rangle _{\omega },
\end{equation*
and finally, we can use (\ref{disj supp}) once more on the second sum above
to reduce matters, modulo a constant multiple of $\frac{1}{\eta _{0}}$, to
the case of estimating the inner products $\left\langle T_{\sigma }^{\alpha
}\left( b_{I}\mathbf{1}_{K}\right) ,b_{J}^{\ast }\mathbf{1}_{K}\right\rangle
$ for the intervals $K=K_{i}\in \mathcal{G}$, $1\leq i\leq B\leq C\frac{1}
\eta _{0}}$, which are contained in $I^{\prime }\cap J^{\prime }$. Thus it
remains to sho
\begin{eqnarray}
&& \label{after prob} \\
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) }
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \ P_{\left( I,J\right) }\left( K,K\right) \ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \notag \\
&\lesssim &\left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak
N}_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }, \notag
\end{eqnarray
where we recall that $\mathcal{K}\left( I^{\prime },J^{\prime }\right)
=\left\{ K_{i}\right\} _{i=1}^{B}$, and where $B=B\left( I^{\prime
},J^{\prime }\right) $ depends on the pair $\left( I^{\prime },J^{\prime
}\right) $ but is bounded by $C\frac{1}{\eta _{0}}$ independent of $\left(
I^{\prime },J^{\prime }\right) $ and the choice of $\eta _{1}$, and
\begin{equation*}
K_{i}\in \mathcal{G},\ K_{i}\subset I^{\prime }\cap J^{\prime },\ \ell
\left( K_{i}\right) =2^{-m-1}\ell \left( J^{\prime }\right) ,\ \ \ \ \ 1\leq
i\leq B\mathfrak{.}
\end{equation*}
There will be just one more use of random probability in dealing with the
nearby form, and that will occur at the end of the finite iteration in
Subsection \ref{Subsection iteration} below.
\subsection{Return of the original testing function}
We now consider the inner product $\left\langle T_{\sigma }^{\alpha }\left(
b_{A}\mathbf{1}_{K}\right) ,b_{B}^{\ast }\mathbf{1}_{K}\right\rangle
_{\omega }$ and estimate the case whe
\begin{equation*}
K\in \mathcal{G},\ K\subset I^{\prime }\cap J^{\prime },\ I^{\prime }\in
\mathfrak{C}\left( I\right) ,\ J^{\prime }\in \mathfrak{C}\left( J\right) ,\
I\in \mathcal{C}_{A}^{\mathcal{A}},\ J\in \mathcal{C}_{B}^{\mathcal{B}},\
\ell \left( K\right) =2^{-m-1}\ell \left( J^{\prime }\right) \mathfrak{.}
\end{equation*
Recall that for $\eta \in \left( 0,\frac{1}{2}\right] $ and any interval $K$
, we defined $\partial _{\eta }K\equiv \left( 1+\eta \right) K\setminus
\left( 1-\eta \right) K$ to be the `$\eta $-halo' around the boundary, i.e.
endpoints, of $K$. In what follows we will now take $\eta =\frac{1}{2}$ and
invoke \emph{deterministic} surgery with $\eta $-halos (which are $\frac{1}{
}$-halos), together with the energy condition and one last application of
random surgery, as follows. For subsets $E,F\subset A\cap B$ and intervals
K\subset A\cap B$ we defin
\begin{equation}
\left\{ E,F\right\} \equiv \left\langle T_{\sigma }^{\alpha }\left( b_{A
\mathbf{1}_{E}\right) ,b_{B}^{\ast }\mathbf{1}_{F}\right\rangle _{\omega }\ ,
\label{def E,F}
\end{equation
an
\begin{equation*}
K_{\limfunc{in}}\equiv K\setminus \partial _{\eta }K\text{ and }K_{\limfunc
out}}\equiv K\cap \partial _{\eta }K\ ,
\end{equation*
and we writ
\begin{equation}
\left\{ K,K\right\} =\left\{ A,K_{\limfunc{in}}\right\} -\left\{ A\setminus
K,K_{\limfunc{in}}\right\} +\left\{ K_{\limfunc{out}},K_{\limfunc{out
}\right\} +\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} . \label{K,K}
\end{equation}
Note that the first two terms on the right hand side of (\ref{K,K})
decompose the inner product $\left\{ K,K_{\limfunc{in}}\right\} $, which
`includes' the difficult symmetric inner product $\left\{ K_{\limfunc{in
},K_{\limfunc{in}}\right\} $. Thus the difficult symmetric inner product is
ultimately controlled by testing on the interval $A$ to handle$\ \left\{
A,K_{\limfunc{in}}\right\} $, and by using a trick that resurrects the
original testing functions $\left\{ b_{J}^{\ast ,\limfunc{orig}}\right\}
_{J\in \mathcal{G}}$, discarded in the corona constructions above, to handle
$\left\{ A\setminus K,K_{\limfunc{in}}\right\} $. More precisely, these
original testing functions $b_{J}^{\ast ,\limfunc{orig}}$ are the testing
functions obtained after reducing matters to the case of bounded testing
functions with the pointwise lower bound property $PLBP$ as in Conclusion
\ref{bounded PLBP} above.
The first term on the right side of (\ref{K,K}) satisfie
\begin{equation}
\left\vert \left\{ A,K_{\limfunc{in}}\right\} \right\vert =\left\vert
\int_{K_{\limfunc{in}}}\left( T_{\sigma }^{\alpha }b_{A}\right) b_{B}^{\ast
}d\omega \right\vert \leq \left\Vert \mathbf{1}_{K_{\limfunc{in}}}T_{\sigma
}^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega \right) }\left\Vert \mathbf
1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\Vert _{L^{2}\left( \omega \right)
}\leq \left\Vert b_{B}^{\ast }\right\Vert _{\infty }\left\Vert \mathbf{1
_{K_{\limfunc{in}}}T_{\sigma }^{\alpha }b_{A}\right\Vert _{L^{2}\left(
\omega \right) }\sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\ .
\label{AKin}
\end{equation
Before proceeding further it will prove convenient to introduce some
additional notation, namely we will write the energy estimate in the second
display of the Energy Lemma a
\begin{equation}
\left\vert \left\langle T^{\alpha }\nu ,\Psi _{J}\right\rangle _{\omega
}\right\vert \lesssim C_{\gamma }\ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q
^{\omega }\left( J,\upsilon \right) \ \left\Vert \Psi _{J}\right\Vert
_{L^{2}\left( \mu \right) }^{\bigstar },\ \ \ \ \ \text{if }\int \Psi
_{J}d\omega =0\text{ and }\gamma J\cap \func{Supp}\nu =\emptyset ,\gamma >1,
\label{star}
\end{equation
wher
\begin{equation}
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( J,\upsilon \right)
\equiv \frac{\mathrm{P}^{\alpha }\left( J,\nu \right) }{\left\vert
J\right\vert }\left\Vert \mathsf{Q}_{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }+\frac{\mathrm{P
_{1+\delta }^{\alpha }\left( J,\nu \right) }{\left\vert J\right\vert
\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }\ .
\label{def compact}
\end{equation
The use of the compact notation $\mathrm{P}_{\delta }^{\alpha }\mathsf{Q
^{\omega }\left( J,\upsilon \right) $ to denote the complicated expression
on the right hand side will considerably reduce the size of many subsequent
displays.
Let $K_{\limfunc{in}}^{\limfunc{left}}$ and $K_{\limfunc{in}}^{\limfunc{righ
}}$ denote the left and right children of $K_{\limfunc{in}}$, which until
now have been written as $\left\{ K_{\ell }^{\prime \prime }\right\} _{\ell
=1}^{2}$ in no particular order, and we will continue to use both of these
notations. We now claim that the second term on the right side of (\ref{K,K
) satisfies
\begin{eqnarray}
\left\vert \left\{ A\setminus K,K_{\limfunc{in}}\right\} \right\vert
&\lesssim &\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left(
K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma \right)
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\limfunc{in}}^
\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right) \right\} \sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }} \label{second term} \\
&&+\left( \sqrt{\int_{K_{\limfunc{in}}}\left\vert T_{\sigma }^{\alpha
}b_{A}\right\vert ^{2}d\omega }+\left( \mathfrak{T}_{T^{\alpha }}+\sqrt
\mathfrak{A}_{2}^{\alpha }}\right) \sqrt{\left\vert K_{\limfunc{in
}\right\vert _{\sigma }}\right) \sqrt{\left\vert K_{\limfunc{in}}\right\vert
_{\omega }} \notag \\
&&+\sum_{\ell =1}^{2}\left\vert \left\langle T_{\sigma }^{\alpha }b_{A
\mathbf{1}_{K_{\limfunc{out}}},b_{K_{\ell }^{\prime \prime }}^{\ast
\limfunc{orig}}\right\rangle _{\omega }\right\vert , \notag
\end{eqnarray
upon using a trick with the \textbf{original} testing functions $b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}$ for the two grandchildren
\left\{ K_{\ell }^{\prime \prime }\right\} _{\ell =1}^{2}$ of the interval
K $ that lie strictly inside $K$, and whose union is $\frac{1}{2}K$. Indeed,
to prove (\ref{second term}), we use the following identity whose proof is
immediate (and whose origin will be made clear in the discussion below)
\begin{eqnarray}
&&\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega
}-\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle T_{\sigma }^{\alpha
}\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}\right\rangle _{\omega } \label{big identity} \\
&=&\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega }
\notag \\
&&-\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ \left\langle T_{\sigma
}^{\alpha }b_{A},b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}\right\rangle _{\omega }-\left\langle b_{A}\mathbf{1}_{K_{\limfunc{in
}},T_{\omega }^{\alpha ,\ast }b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc
orig}}\right\rangle _{\sigma }-\left\langle T_{\sigma }^{\alpha }b_{A
\mathbf{1}_{K_{\limfunc{out}}},b_{K_{\ell }^{\prime \prime }}^{\ast
\limfunc{orig}}\right\rangle _{\omega }\right\} \notag \\
&=&\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega
}+\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle T_{\sigma }^{\alpha
}b_{A}\mathbf{1}_{K_{\limfunc{out}}},b_{K_{\ell }^{\prime \prime }}^{\ast
\limfunc{orig}}\right\rangle _{\omega } \notag \\
&&-\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ \left\langle T_{\sigma
}^{\alpha }b_{A},b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}\right\rangle _{\omega }-\left\langle b_{A}\mathbf{1}_{K_{\limfunc{in
}},T_{\omega }^{\alpha ,\ast }b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc
orig}}\right\rangle _{\sigma }\right\} . \notag
\end{eqnarray
In fact we have the following estimate, more precise than (\ref{second term
).
\begin{lemma}
\label{preiterate}We have
\begin{eqnarray*}
&&\left\vert \left\{ A\setminus K,K_{\limfunc{in}}\right\} +\sum_{\ell
=1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ K_{\limfunc{out}},K_
\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}\right\vert \\
&\lesssim &\left( \left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega
}\left( K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma
\right) +\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_
\limfunc{in}}^{\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right)
\right\} +\sqrt{\int_{K_{\limfunc{in}}}\left\vert T_{\sigma }^{\alpha
}b_{A}\right\vert ^{2}d\omega }+\left( \mathfrak{T}_{T^{\alpha ,\ast }}
\sqrt{\mathfrak{A}_{2}^{\alpha }}\right) \sqrt{\left\vert K_{\limfunc{in
}\right\vert _{\sigma }}\right) \sqrt{\left\vert K_{\limfunc{in}}\right\vert
_{\omega }},
\end{eqnarray*
where $b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}$ is the
original testing function for the grandchild $K_{\ell }^{\prime \prime }$ of
$K$ in Conclusion \ref{bounded PLBP} above, and wher
\begin{equation*}
\left\{ K_{\limfunc{out}},K_{\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig
}\equiv \left\langle T_{\sigma }^{\alpha }b_{A}\mathbf{1}_{K_{\limfunc{out
}},b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\rangle
_{\omega }\ \text{for }\ell \in \left\{ 1,2\right\} .
\end{equation*}
\end{lemma}
Before starting the proof of the lemma, we motivate the identity (\ref{big
identity}) with the following discussion. A simpler way to start the
analysis for for $\left\{ A\setminus K,K_{\limfunc{in}}\right\}
=\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega
} $ would be to use instead of (\ref{big identity}), the more obvious
decomposition
\begin{eqnarray}
&&\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega
}=\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}\left( b_{B}^{\ast }-\frac{1}
\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\int_{K_{\limfunc{in
}}b_{B}^{\ast }d\omega \right) \right\rangle _{\omega } \label{reach''} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +\left\langle T_{\sigma }^{\alpha
}\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in
}}\left( \frac{1}{\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\int_{K_
\limfunc{in}}}b_{B}^{\ast }d\omega \right) \right\rangle _{\omega }\ .
\notag
\end{eqnarray
For the first term in (\ref{reach''}), we would like to apply the Energy
Lemma to obtai
\begin{eqnarray}
&&\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}\left( b_{B}^{\ast }
\frac{1}{\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\int_{K_{\limfunc
in}}}b_{B}^{\ast }d\omega \right) \right\rangle _{\omega }\right\vert
\label{after mon''} \\
&\lesssim &\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_
\limfunc{in}},\mathbf{1}_{A\setminus K}\sigma \right) \left\Vert \mathbf{1
_{K_{\limfunc{in}}}\left( b_{B}^{\ast }-\frac{1}{\left\vert K_{\limfunc{in
}\right\vert _{\omega }}\int_{K_{\limfunc{in}}}b_{B}^{\ast }d\omega \right)
\right\Vert _{L^{2}\left( \omega \right) }\ , \notag
\end{eqnarray
using that the function $h_{K_{\limfunc{in}}}^{\ast }\equiv \mathbf{1}_{K_
\limfunc{in}}}\left( b_{B}^{\ast }-\frac{1}{\left\vert K_{\limfunc{in
}\right\vert _{\omega }}\int_{K_{\limfunc{in}}}b_{B}^{\ast }d\omega \right) $
has $\omega $-mean value zero and has support $K_{\limfunc{in}}$ that is
\emph{strictly separated} from the support of $\left\vert b_{A}\right\vert
\mathbf{1}_{A\setminus K}$. But a problem arises here since $K_{\limfunc{in
} $ is not in the dyadic grid $\mathcal{G}$, despite the fact that $K$
itself is. Indeed, the Energy Lemma requires the dual martingale support of
h_{K_{\limfunc{in}}}^{\ast }$ to be contained in $K_{\limfunc{in}}$, so that
we can take $\mathcal{H}$ in the Energy Lemma to be pseudoprojection onto
K_{\limfunc{in}}$. However, if $K^{\prime }$ is a child of $K$, then
\square _{K^{\prime }}^{\omega ,\mathbf{b}^{\ast }}h_{K_{\limfunc{in
}}^{\ast }$ could be nonzero, yet $K^{\prime }\not\subset K_{\limfunc{in}}$.
This is easily fixed in two steps as follows.
First, recall $\eta =\frac{1}{2}$ so that
\begin{equation*}
K_{\limfunc{in}}=\left( 1-\eta \right) K=\frac{1}{2}K=\dbigcup \left\{
K^{\prime \prime }\in \mathfrak{C}^{\left( 2\right) }\left( K\right)
:\partial K^{\prime \prime }\cap \partial K=\emptyset \right\}
\end{equation*
is the union of the $2$ grandchildren $K^{\prime \prime }$ of $K$ whose
boundaries are disjoint from the boundary of $K$. Then
\begin{equation*}
K_{\limfunc{out}}=\dbigcup \left\{ K^{\prime \prime }\in \mathfrak{C
^{\left( 2\right) }\left( K\right) :\partial K^{\prime \prime }\cap \partial
K\neq \emptyset \right\}
\end{equation*
is the union of the $2$ grandchildren $K^{\prime \prime }$ of $K$ whose
boundaries intersect the boundary of $K$. The only possible dyadic
subintervals $K^{\prime }$ of $K$ for which both $\square _{K^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}h_{K_{\limfunc{in}}}^{\ast }\neq 0$ and
K^{\prime }\not\subset K_{\limfunc{in}}$ are the children of $K$. Enumerate
by $\left\{ K_{\ell }^{\prime \prime }\right\} _{\ell =1}^{2}$ the
grandchildren of $K$ whose boundaries are disjoint from the boundary of $K$.
Then second, instead of decomposing $\mathbf{1}_{K_{\limfunc{in
}}b_{B}^{\ast }$ as $h_{K_{\limfunc{in}}}^{\ast }$ plus $\mathbf{1}_{K_
\limfunc{in}}}\frac{1}{\left\vert K_{\limfunc{in}}\right\vert _{\omega }
\int_{K_{\limfunc{in}}}b_{B}^{\ast }d\omega $, we decompose $\mathbf{1}_{K_
\limfunc{in}}}b_{B}^{\ast }$ a
\begin{equation*}
\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }=\sum_{\ell =1}^{2}\mathbf{1
_{K_{\ell }^{\prime \prime }}\left( b_{B}^{\ast }-\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega \right) +\sum_{\ell =1}^{2}\mathbf{1}_{K_{\ell
}^{\prime \prime }}\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega ,
\end{equation*
and then apply the Energy Lemma to the functio
\begin{equation*}
k_{K_{\limfunc{in}}}^{\ast }\equiv \sum_{\ell =1}^{2}\mathbf{1}_{K_{\ell
}^{\prime \prime }}\left( b_{B}^{\ast }-\frac{1}{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime
}}b_{B}^{\ast }d\omega \right) \equiv k_{K_{\limfunc{in}}}^{\ast ,1}+k_{K_
\limfunc{in}}}^{\ast ,2},
\end{equation*
which does indeed satisfy $\square _{K^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}k_{K_{\limfunc{in}}}^{\ast }=0$ unless $K^{\prime }$ is a dyadic
subinterval of $K$ that is contained in $K_{\limfunc{in}}$. (Furthermore, we
could even replace grandchildren by $m$-grandchildren in this argument in
order that $\square _{K^{\prime }}^{\omega ,\mathbf{b}^{\ast }}k_{K_
\limfunc{in}}}^{\ast }=0$ unless $K^{\prime }$ is a dyadic $m$-grandchild of
$K$ that is contained in $K_{\limfunc{in}}$, but we will not need this.) If
we now use $k_{K_{\limfunc{in}}}^{\ast }$ instead of $h_{K_{\limfunc{in
}}^{\ast }$ in (\ref{reach''}) and (\ref{after mon''}), we obtai
\begin{eqnarray}
\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega
} &=&\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,k_{K_{\limfunc{in}}}^{\ast }\right\rangle _{\omega }
\label{reach'''} \\
&&+\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\sum_{\ell =1}^{2}\mathbf{1}_{K_{\ell }^{\prime \prime }}\left(
\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega }
\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega \right) \right\rangle
_{\omega }\ , \notag
\end{eqnarray
an
\begin{eqnarray}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus K}\right) ,k_{K_{\limfunc{in}}}^{\ast }\right\rangle _{\omega
}\right\vert &\leq &\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A
\mathbf{1}_{A\setminus K}\right) ,k_{K_{\limfunc{in}}}^{\ast
,1}\right\rangle _{\omega }\right\vert +\left\vert \left\langle T_{\sigma
}^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,k_{K_{\limfunc{in
}}^{\ast ,2}\right\rangle _{\omega }\right\vert \label{after mon'''} \\
&\leq &C_{\eta }\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega
}\left( K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma
\right) +\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_
\limfunc{in}}^{\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right)
\right\} \left\Vert k_{K_{\limfunc{in}}}^{\ast }\right\Vert _{L^{2}\left(
\omega \right) }\ , \notag
\end{eqnarray
where the constant $C_{\eta }$ depends on the constant $C_{\gamma }$ in the
statement of the Monotonicity Lemma with $\gamma =\frac{1}{1-\eta }$ since
\frac{1}{1-\eta }K_{\limfunc{in}}\cap \left( A\setminus K\right) =\emptyset
, and where we have written $\left\{ K_{\ell }^{\prime \prime }\right\}
_{\ell =1}^{2}=\left\{ K_{\limfunc{in}}^{\limfunc{left}},K_{\limfunc{in}}^
\limfunc{right}}\right\} $ with $K_{\limfunc{in}}^{\limfunc{left}}$ and $K_
\limfunc{in}}^{\limfunc{right}}$ denoting the left hand child and right hand
child of $K_{\limfunc{in}}$ respectively.
\begin{conclusion}
Thus we see that $\mathsf{P}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}$ and
$\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}$ in the Energy Lemma
can be taken to be pseudoprojection onto $K_{\limfunc{in}}$, i.e. $\mathsf{P
_{K_{\limfunc{in}}}^{\omega ,\mathbf{b}^{\ast }}=\sum_{J\in \mathcal{G}:\
J\subset K_{\limfunc{in}}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}$ and
\mathsf{Q}_{K_{\limfunc{in}}}^{\omega ,\mathbf{b}^{\ast }}=\sum_{J\in
\mathcal{G}:\ J\subset K_{\limfunc{in}}}\bigtriangleup _{J}^{\omega ,\mathbf
b}^{\ast }}$, and we will see below that the intervals $K_{\limfunc{in}}$
that arise in subsequent arguments will be pairwise disjoint. Furthermore,
the energy condition will be used to control these full pseudoprojections
\mathsf{P}_{K_{\limfunc{in}}}^{\omega ,\mathbf{b}^{\ast }}$ when taken over
pairwise disjoint decompositions of intervals by subintervals of the form
K_{\limfunc{in}}$.
\end{conclusion}
However, the second line of (\ref{reach'''}) remains problematic, and this
is our point of departure for beginning the proof of Lemma \ref{preiterate},
in which we exploit the original testing functions $b_{K_{\ell }^{\prime
\prime }}^{\ast ,\limfunc{orig}}$ in identity (\ref{big identity}) in order
to handle the second line of (\ref{reach'''}).
\begin{proof}[Proof of Lemma \protect\ref{preiterate}]
We begin by rewriting identity (\ref{big identity}) in the for
\begin{eqnarray*}
\boldsymbol{A} &=&\boldsymbol{B}+\boldsymbol{C}\text{ where} \\
\boldsymbol{A} &\equiv &\left\langle T_{\sigma }^{\alpha }\left( b_{A
\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast
}\right\rangle _{\omega }+\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle
T_{\sigma }^{\alpha }b_{A}\mathbf{1}_{K_{\limfunc{out}}},b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\rangle _{\omega }\ , \\
\boldsymbol{B} &\equiv &\left\langle T_{\sigma }^{\alpha }\left( b_{A
\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast
}\right\rangle _{\omega }-\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle
T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus K}\right)
,b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\rangle _{\omega
}\ , \\
\boldsymbol{C} &\equiv &\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{
\left\langle T_{\sigma }^{\alpha }b_{A},b_{K_{\ell }^{\prime \prime }}^{\ast
,\limfunc{orig}}\right\rangle _{\omega }-\left\langle b_{A}\mathbf{1}_{K_
\limfunc{in}}},T_{\omega }^{\alpha ,\ast }b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}\right\rangle _{\sigma }\right\} \ .
\end{eqnarray*
From the discussion above, we recall the identity (\ref{reach'''}) and the
estimate (\ref{after mon'''}). We also have the analogous identity and
estimate with $b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}$ in
place of $\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast }$
\begin{eqnarray}
\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}\right\rangle _{\omega } &=&\left\langle T_{\sigma }^{\alpha }\left( b_{A
\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\ell }^{\prime \prime
}}\left( b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}-\frac{1}
\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell
}^{\prime \prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}d\omega \right) \right\rangle _{\omega } \label{reach''''} \\
&&+\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,\mathbf{1}_{K_{\ell }^{\prime \prime }}\left( \frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega
\right) \right\rangle _{\omega }\ , \notag
\end{eqnarray
an
\begin{eqnarray}
&&\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus K}\right) ,\mathbf{1}_{K_{\ell }^{\prime \prime }}\left(
b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}-\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega
\right) \right\rangle _{\omega }\right\vert \label{after mon''''} \\
&\lesssim &\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\ell
}^{\prime \prime },\mathbf{1}_{A\setminus K}\sigma \right) \left\Vert
\mathbf{1}_{K_{\ell }^{\prime \prime }}\left( b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}-\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega \right) \right\Vert
_{L^{2}\left( \omega \right) }\ , \notag
\end{eqnarray
for $1\leq \ell \leq 2$, where the implied constants depend on $L^{\infty }$
norms of testing functions and the constant in the Energy Lemma. We will
typically suppress dependence on $\eta =\frac{1}{2}$ from now on since there
are no other values of $\eta $ used below. Thus we have, using that $K_{\ell
}^{\prime \prime }$ is close to $K_{\limfunc{in}}$ in both scale and
position,
\begin{eqnarray}
&&\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast
}\right\rangle _{\omega }-\sum_{\ell =1}^{2}\left( \frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega \right) \left\langle T_{\sigma }^{\alpha
}\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\ell }^{\prime
\prime }}\right\rangle _{\omega }\right\vert \label{scale and position} \\
&\lesssim &\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left(
K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma \right)
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\limfunc{in}}^
\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right) \right\} \left\Vert
\mathbf{1}_{K_{\limfunc{in}}}\left( b_{B}^{\ast }-\frac{1}{\left\vert K_
\limfunc{in}}\right\vert _{\omega }}\int_{K_{\limfunc{in}}}b_{B}^{\ast
}d\omega \right) \right\Vert _{L^{2}\left( \omega \right) } \notag \\
&\lesssim &\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left(
K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma \right)
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\limfunc{in}}^
\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right) \right\} \sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\ . \notag
\end{eqnarray
Then we obtain, upon applying (\ref{star}) with the function $\Psi
_{J}^{\ell }$ equal to
\begin{equation*}
\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega }{\frac{1}
\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell
}^{\prime \prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}d\omega }\right) b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}-\left( \frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega
}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega \right) \mathbf{1
_{K_{\ell }^{\prime \prime }}
\end{equation*
for $\ell =1,2$, and also using (\ref{scale and position}), tha
\begin{eqnarray*}
&&\left\vert \boldsymbol{B}\right\vert =\left\vert \left\langle T_{\sigma
}^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_
\limfunc{in}}}b_{B}^{\ast }\right\rangle _{\omega }-\sum_{\ell =1}^{2}\left(
\frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega }
\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega
\right) \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}\right\rangle _{\omega }\right\vert \\
&=&\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus K}\right) ,\mathbf{1}_{K_{\limfunc{in}}}b_{B}^{\ast
}\right\rangle _{\omega }-\sum_{\ell =1}^{2}\left( \frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega \right) \left\langle T_{\sigma }^{\alpha
}\left( b_{A}\mathbf{1}_{A\setminus K}\right) ,\mathbf{1}_{K_{\ell }^{\prime
\prime }}\right\rangle _{\omega }\right\vert \\
&&+O\left\{ \sum_{\ell =1}^{2}\left( \frac{\mathrm{P}^{\alpha }\left(
K_{\ell }^{\prime \prime },\mathbf{1}_{A\setminus K}\sigma \right) }
\left\vert K_{\ell }^{\prime \prime }\right\vert }\left\Vert \mathsf{Q
_{K_{\ell }^{\prime \prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit }+\frac{\mathrm{P}_{1+\delta
}^{\alpha }\left( K_{\ell }^{\prime \prime },\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K_{\ell }^{\prime \prime }\right\vert }\left\Vert
x-m_{K_{\ell }^{\prime \prime }}\right\Vert _{L^{2}\left( \mathbf{1
_{K_{\ell }^{\prime \prime }}\omega \right) }\right) \sqrt{\left\vert K_
\limfunc{in}}\right\vert _{\omega }}\right\} \\
&\lesssim &\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left(
K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma \right)
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\limfunc{in}}^
\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right) \right\} \sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }}\ ,
\end{eqnarray*
since by the triangle inequality
\begin{eqnarray*}
&&\left\Vert \Psi _{J}^{\ell }\right\Vert _{L^{2}\left( \omega \right)
}=\left\Vert \left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}-\left( \frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega \right) \mathbf{1}_{K_{\ell }^{\prime \prime }}\right\Vert
_{L^{2}\left( \omega \right) } \\
&\lesssim &\left\vert \frac{\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right\vert \sqrt{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}+\sqrt{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\lesssim \sqrt{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\leq \sqrt{\left\vert K_{\limfunc{in
}\right\vert _{\omega }},\ \ \ \ \ 1\leq \ell \leq 2.
\end{eqnarray*}
Finally, turning our attention to term $\boldsymbol{C}$, the reason for
using the identity (\ref{big identity}) now becomes clear - namely the terms
$\left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus
K}\right) ,b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig
}\right\rangle _{\omega }$, for which multiples are subtracted on the left
side above, involves the original testing functions $b_{K_{\ell }^{\prime
\prime }}^{\ast ,\limfunc{orig}}$ for which we have the\ full testing
condition. Thus we have
\begin{eqnarray*}
&&\left\vert \sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime
}}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle
T_{\sigma }^{\alpha }b_{A},b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc
orig}}\right\rangle _{\omega }\right\vert \\
&\lesssim &\sum_{\ell =1}^{2}\sqrt{\int_{K_{\ell }^{\prime \prime
}}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert ^{2}d\omega }\sqrt
\int_{K_{\ell }^{\prime \prime }}\left\vert b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}\right\vert ^{2}d\omega }\lesssim \sqrt{\int_{K_
\limfunc{in}}}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert ^{2}d\omega
\sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\omega }},
\end{eqnarray*
and similarl
\begin{eqnarray*}
&&\left\vert \sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime
}}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\langle b_{A
\mathbf{1}_{K_{\limfunc{in}}},T_{\omega }^{\alpha ,\ast }b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\rangle _{\sigma
}\right\vert \lesssim \sqrt{\int_{K_{\limfunc{in}}}\left\vert
b_{A}\right\vert ^{2}d\sigma }\sqrt{\int_{K_{\limfunc{in}}}\left\vert
T_{\omega }^{\alpha ,\ast }\sum_{\ell =1}^{2}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}\right\vert ^{2}d\sigma } \\
&\lesssim &\left( \mathfrak{T}_{T^{\alpha ,\ast }}+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \sqrt{\int_{K_{\limfunc{in}}}\left\vert
b_{A}\right\vert ^{2}d\sigma }\sum_{\ell =1}^{2}\sqrt{\int \left\vert
b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\vert ^{2}d\omega
}\lesssim \left( \mathfrak{T}_{T^{\alpha ,\ast }}+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \sqrt{\int_{K_{\limfunc{in}}}\left\vert
b_{A}\right\vert ^{2}d\sigma }\sqrt{\left\vert K_{\limfunc{in}}\right\vert
_{\omega }},
\end{eqnarray*
which together prov
\begin{eqnarray*}
\left\vert \boldsymbol{C}\right\vert &=&\left\vert \sum_{\ell =1}^{2}\left(
\frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert _{\omega }
\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{K_{\ell }^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega
\right) \left\{ \left\langle T_{\sigma }^{\alpha }b_{A},b_{K_{\ell }^{\prime
\prime }}^{\ast ,\limfunc{orig}}\right\rangle _{\omega }-\left\langle b_{A
\mathbf{1}_{K_{\limfunc{in}}},T_{\omega }^{\alpha ,\ast }b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}\right\rangle _{\sigma }\right\}
\right\vert \\
&\lesssim &\left( \sqrt{\int_{K_{\limfunc{in}}}\left\vert T_{\sigma
}^{\alpha }b_{A}\right\vert ^{2}d\omega }+\left( \mathfrak{T}_{T^{\alpha
,\ast }}+\sqrt{\mathfrak{A}_{2}^{\alpha }}\right) \sqrt{\left\vert K_
\limfunc{in}}\right\vert _{\sigma }}\right) \sqrt{\left\vert K_{\limfunc{in
}\right\vert _{\omega }}\ .
\end{eqnarray*
This completes the proof of Lemma \ref{preiterate}, and hence also that of
\ref{second term}) since $\left\vert \frac{\frac{1}{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime
}}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right\vert \lesssim 1$.
\end{proof}
The term $\left\{ K_{\limfunc{out}},K_{\limfunc{in}}^{\ell }\right\} ^
\limfunc{orig}}$ will be handled below by relatively crude estimates. If we
writ
\begin{eqnarray*}
&&\left\{ K,K\right\} =\left\{ A,K_{\limfunc{in}}\right\} -\left\{
A\setminus K,K_{\limfunc{in}}\right\} +\left\{ K_{\limfunc{out}},K_{\limfunc
out}}\right\} +\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} \\
&=&\left\{ A,K_{\limfunc{in}}\right\} -\left( \left\{ A\setminus K,K_
\limfunc{in}}\right\} +\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert
K_{\ell }^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime
\prime }}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ K_
\limfunc{out}},K_{\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}\right) \\
&&+\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime
\prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ K_{\limfunc{out}},K_
\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}+\left\{ K_{\limfunc{out}},K_
\limfunc{out}}\right\} +\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} ,
\end{eqnarray*
then, using (\ref{AKin}) and Lemma \ref{preiterate}, we see that we have
reduced control of $\left\{ K,K\right\} $ to control o
\begin{equation*}
\sum_{\ell =1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ K_{\limfunc{out}},K_
\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}+\left\{ K_{\limfunc{out}},K_
\limfunc{out}}\right\} +\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} .
\end{equation*}
Altogether then, using the above estimates, we have prove
\begin{eqnarray}
&& \label{KK bound} \\
&&\left\vert \left\{ K,K\right\} -\left\{ K_{\limfunc{out}},K_{\limfunc{out
}\right\} -\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} -\sum_{\ell
=1}^{2}\left( \frac{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{B}^{\ast
}d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime }\right\vert
_{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell }^{\prime \prime
}}^{\ast ,\limfunc{orig}}d\omega }\right) \left\{ K_{\limfunc{out}},K_
\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}\right\vert \notag \\
&\lesssim &\left\Vert \mathbf{1}_{K_{\limfunc{in}}}T_{\sigma }^{\alpha
}b_{A}\right\Vert _{L^{2}\left( \omega \right) }\sqrt{\left\vert K_{\limfunc
in}}\right\vert _{\omega }}+C\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q
^{\omega }\left( K_{\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus
K}\sigma \right) +\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left(
K_{\limfunc{in}}^{\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right)
\right\} \sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\omega }} \notag \\
&\lesssim &C\left( \mathfrak{T}_{T^{\alpha ,\ast }}+\mathfrak{A}_{2}^{\alpha
}\right) \sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\sigma }}\sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }}. \notag
\end{eqnarray
We emphasize that this bound did not use any special information regarding
K $ being in the coronas $\mathcal{C}_{A},\mathcal{C}_{B}$ or not, and thus
holds for any interval $K$ contained in $A\cap B$. For clarity of notation
we defin
\begin{eqnarray}
\Phi ^{A,B}\left( K_{\limfunc{in}}\right) &\equiv &\left\Vert \mathbf{1}_{K_
\limfunc{in}}}T_{\sigma }^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega
\right) }\sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\omega }}
\label{def PHI} \\
&&+C\left\{ \mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_
\limfunc{in}}^{\limfunc{left}},\mathbf{1}_{A\setminus K}\sigma \right)
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( K_{\limfunc{in}}^
\limfunc{right}},\mathbf{1}_{A\setminus K}\sigma \right) \right\} \sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }} \notag \\
&&+C\left( \mathfrak{T}_{T^{\alpha ,\ast }}+\mathfrak{A}_{2}^{\alpha
}\right) \sqrt{\left\vert K_{\limfunc{in}}\right\vert _{\sigma }}\sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\omega }}, \notag
\end{eqnarray
where $\Phi ^{A,B}$ should not be confused with the notation $\Phi ^{\alpha
} $ introduced for the Monotonicity Lemma, and we also define the constant
\begin{equation*}
A_{K_{\limfunc{in}}^{\ell }}\equiv \frac{\frac{1}{\left\vert K_{\ell
}^{\prime \prime }\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime
}}b_{B}^{\ast }d\omega }{\frac{1}{\left\vert K_{\ell }^{\prime \prime
}\right\vert _{\omega }}\int_{K_{\ell }^{\prime \prime }}b_{K_{\ell
}^{\prime \prime }}^{\ast ,\limfunc{orig}}d\omega },\ \ \ \ \ \ell \in
\left\{ 1,2\right\} ,
\end{equation*
so that we can rewrite (\ref{KK bound}) a
\begin{equation}
\left\vert \left\{ K,K\right\} -\left\{ K_{\limfunc{out}},K_{\limfunc{out
}\right\} -\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} -\sum_{\ell
=1}^{2}A_{K_{\limfunc{in}}^{\ell }}\left\{ K_{\limfunc{out}},K_{\limfunc{in
}^{\ell }\right\} ^{\limfunc{orig}}\right\vert \lesssim \Phi ^{A,B}\left( K_
\limfunc{in}}\right) ,\ \ \ \ \ K\in \mathcal{G},K\subset A\cap B\ .
\label{KK bound rewrite}
\end{equation
We can further simplify notation by defining
\begin{equation}
\left\{ K_{\limfunc{out}},K_{\limfunc{in}}\right\} ^{\limfunc{orig}}\equiv
\sum_{\ell =1}^{2}A_{K_{\limfunc{in}}^{\ell }}\left\{ K_{\limfunc{out}},K_
\limfunc{in}}^{\ell }\right\} ^{\limfunc{orig}}, \label{def orig}
\end{equation
which we will often use below. At this point, as we will see below, the only
problematic inner product subtracted from $\left\{ K,K\right\} $ on the left
hand side of (\ref{KK bound rewrite}) is $\left\{ K_{\limfunc{out}},K_
\limfunc{out}}\right\} $, and we will handle this by iterating (\ref{KK
bound rewrite}) finitely many times and then appealing to a final
probability argument starting in (\ref{future prob}) below.
\subsection{A finite iteration and final random surgery\label{Subsection
iteration}}
For $K$ an interval, we write $K_{\limfunc{out}}=K_{\limfunc{left}}\cup K_
\limfunc{right}}$ where $K_{\limfunc{left}}$ and $K_{\limfunc{right}}$ are
the two small subintervals on the left and right hand sides of $K$
respectively, and then we have
\begin{equation*}
\left\{ K_{\limfunc{out}},K_{\limfunc{out}}\right\} =\left\{ K_{\limfunc{lef
}},K_{\limfunc{left}}\right\} +\left\{ K_{\limfunc{right}},K_{\limfunc{right
}\right\} +\left\{ K_{\limfunc{left}},K_{\limfunc{right}}\right\} +\left\{
K_{\limfunc{right}},K_{\limfunc{left}}\right\} ,
\end{equation*
so that (\ref{KK bound rewrite}) can be written using (\ref{def orig}) a
\begin{eqnarray}
\left\{ K,K\right\} &=&\left\{ K_{\limfunc{out}},K_{\limfunc{out}}\right\}
+\left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} +\left\{ K_{\limfunc{out
},K_{\limfunc{in}}\right\} ^{\limfunc{orig}}+O\left[ \Phi ^{A,B}\left( K_
\limfunc{in}}\right) \right] \label{KK bound again} \\
&=&\left\{ K_{\limfunc{left}},K_{\limfunc{left}}\right\} +\left\{ K_
\limfunc{right}},K_{\limfunc{right}}\right\} +\left\{ K_{\limfunc{in}},K_
\limfunc{out}}\right\} +\left\{ K_{\limfunc{out}},K_{\limfunc{in}}\right\} ^
\limfunc{orig}} \notag \\
&&+\left\{ K_{\limfunc{left}},K_{\limfunc{right}}\right\} +\left\{ K_
\limfunc{right}},K_{\limfunc{left}}\right\} +O\left[ \Phi ^{A,B}\left( K_
\limfunc{in}}\right) \right] . \notag
\end{eqnarray}
At this point we observe that the terms $\left\{ K_{\limfunc{in}},K_
\limfunc{out}}\right\} $, $\left\{ K_{\limfunc{out}},K_{\limfunc{in
}\right\} ^{\limfunc{orig}}$, $\left\{ K_{\limfunc{left}},K_{\limfunc{right
}\right\} $, and $\left\{ K_{\limfunc{right}},K_{\limfunc{left}}\right\} $
can all be handled somewhat crudely by separation (\ref{disj supp})
\begin{eqnarray*}
&&\left\vert \left\{ K_{\limfunc{in}},K_{\limfunc{out}}\right\} \right\vert
=\left\vert \int_{K_{\limfunc{out}}}\left[ T_{\sigma }^{\alpha }\left(
\mathbf{1}_{K_{\limfunc{in}}}b_{A}\right) \right] b_{B}^{\ast }d\omega
\right\vert \\
&\lesssim &\sqrt{\mathcal{A}_{2}^{\alpha }}\left( \int_{K_{\limfunc{in
}}\left\vert b_{A}\right\vert ^{2}d\sigma \right) ^{\frac{1}{2}}\left(
\int_{K_{\limfunc{out}}}\left\vert b_{B}^{\ast }\right\vert ^{2}d\omega
\right) ^{\frac{1}{2}}\lesssim \sqrt{\mathcal{A}_{2}^{\alpha }}\sqrt
\left\vert K_{\limfunc{in}}\right\vert _{\sigma }}\sqrt{\left\vert K_
\limfunc{out}}\right\vert _{\omega }}\ ,
\end{eqnarray*
and similarly
\begin{eqnarray*}
&&\left\vert \left\{ K_{\limfunc{out}},K_{\limfunc{in}}\right\} ^{\limfunc
orig}}\right\vert \lesssim \sqrt{\mathcal{A}_{2}^{\alpha }}\sqrt{\left\vert
K_{\limfunc{out}}\right\vert _{\sigma }}\sqrt{\left\vert K_{\limfunc{in
}\right\vert _{\omega }}\ , \\
&&\left\vert \left\{ K_{\limfunc{left}},K_{\limfunc{right}}\right\}
\right\vert +\left\vert \left\{ K_{\limfunc{right}},K_{\limfunc{left
}\right\} \right\vert \lesssim \sqrt{\mathcal{A}_{2}^{\alpha }}\sqrt
\left\vert K_{\limfunc{out}}\right\vert _{\sigma }}\sqrt{\left\vert K_
\limfunc{out}}\right\vert _{\omega }}\ .
\end{eqnarray*
Thus we hav
\begin{equation*}
\left\{ K,K\right\} =\left\{ K_{\limfunc{left}},K_{\limfunc{left}}\right\}
+\left\{ K_{\limfunc{right}},K_{\limfunc{right}}\right\} +O\left[ \Phi
^{A,B}\left( K_{\limfunc{in}}\right) \right] +\mathcal{A}_{2}^{\alpha }\sqrt
\left\vert K\right\vert _{\sigma }}\sqrt{\left\vert K\right\vert _{\omega }}.
\end{equation*
Upon application of a single iteration we obtai
\begin{eqnarray*}
\left\{ K,K\right\} &=&\left\{ K_{\limfunc{left}\limfunc{left}},K_{\limfunc
left}\limfunc{left}}\right\} +\left\{ K_{\limfunc{left}\limfunc{right}},K_
\limfunc{left}\limfunc{right}}\right\} +\left\{ K_{\limfunc{right}\limfunc
left}},K_{\limfunc{right}\limfunc{left}}\right\} +\left\{ K_{\limfunc{right
\limfunc{right}},K_{\limfunc{right}\limfunc{right}}\right\} \\
&&+O\left[ \Phi ^{A,B}\left( K_{\limfunc{in}}\right) +\Phi ^{A,B}\left( K_
\limfunc{left}\limfunc{in}}\right) +\Phi ^{A,B}\left( K_{\limfunc{right
\limfunc{in}}\right) \right] \\
&&+\mathcal{A}_{2}^{\alpha }\left( \sqrt{\left\vert K\right\vert _{\sigma }
\sqrt{\left\vert K\right\vert _{\omega }}+\sqrt{\left\vert K_{\limfunc{left
}\right\vert _{\sigma }}\sqrt{\left\vert K_{\limfunc{left}}\right\vert
_{\omega }}+\sqrt{\left\vert K_{\limfunc{right}}\right\vert _{\sigma }}\sqrt
\left\vert K_{\limfunc{right}}\right\vert _{\omega }}\right) ,
\end{eqnarray*
and then iterating finitely many more times gives for $n\in \mathbb{N}$
\begin{eqnarray}
\left\{ K,K\right\} &=&\sum_{M\in \mathcal{M}_{n}}\left\{ M,M\right\}
+O\left( \sum_{M\in \mathcal{M}_{n}^{\ast }}\left[ \Phi ^{A,B}\left( M_
\limfunc{in}}\right) \right] \right) +\sqrt{\mathcal{A}_{2}^{\alpha }
\sum_{M\in \mathcal{M}_{n}^{\ast }}\sqrt{\left\vert M\right\vert _{\sigma }
\sqrt{\left\vert M\right\vert _{\omega }} \label{K iterated} \\
&\equiv &A\left( K\right) +B\left( K\right) +C\left( K\right) =A_{\left(
I^{\prime },J^{\prime }\right) }\left( K\right) +B_{\left( I^{\prime
},J^{\prime }\right) }\left( K\right) +C_{\left( I^{\prime },J^{\prime
}\right) }\left( K\right) , \notag
\end{eqnarray
where the collections of intervals $\mathcal{M}_{n}=\mathcal{M}_{n}\left(
K\right) $ and $\mathcal{M}_{n}^{\ast }=\mathcal{M}_{n}^{\ast }\left(
K\right) $ are defined recursively by
\begin{eqnarray*}
\mathcal{M}_{0} &\equiv &\left\{ K\right\} , \\
\mathcal{M}_{k+1} &\equiv &\dbigcup \left\{ M_{\limfunc{left}},M_{\limfunc
right}}:M\in \mathcal{M}_{k}\right\} ,\ \ \ \ \ k\geq 0, \\
\mathcal{M}_{n}^{\ast } &\equiv &\dbigcup_{k=0}^{n}\mathcal{M}_{k}\ .
\end{eqnarray*
We will include the subscript $\left( I^{\prime },J^{\prime }\right) $ in
the notation when we want to indicate the pair $\left( I^{\prime },J^{\prime
}\right) $ for which $K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) $
as defined in (\ref{def K(I',J')}) above. Now the term $C\left( K\right) $
can be estimated by the crude estimate
\begin{equation}
C\left( K\right) =\sqrt{\mathcal{A}_{2}^{\alpha }}\sum_{M\in \mathcal{M
_{n}^{\ast }}\sqrt{\left\vert M\right\vert _{\sigma }}\sqrt{\left\vert
M\right\vert _{\omega }}\leq C_{n}\sqrt{\mathcal{A}_{2}^{\alpha }}\sqrt
\left\vert K\right\vert _{\sigma }}\sqrt{\left\vert K\right\vert _{\omega }
\ , \label{C est}
\end{equation
where $n$ is chosen below depending on $\eta _{0}$. For the first term
A\left( K\right) $, we will apply the norm inequality and use probability,
namel
\begin{eqnarray*}
\left\vert A\left( K\right) \right\vert &\leq &\sqrt{C_{\mathbf{b}}C_
\mathbf{b}^{\ast }}}\mathfrak{N}_{T^{\alpha }}\sum_{M\in \mathcal{M}_{n}
\sqrt{\left\vert M\right\vert _{\sigma }}\sqrt{\left\vert M\right\vert
_{\omega }} \\
&\leq &\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }}}\mathfrak{N}_{T^{\alpha }
\sqrt{\sum_{M\in \mathcal{M}_{n}}\left\vert M\right\vert _{\sigma }}\sqrt
\sum_{M\in \mathcal{M}_{n}}\left\vert M\right\vert _{\omega }}\leq \sqrt{C_
\mathbf{b}}C_{\mathbf{b}^{\ast }}}\mathfrak{N}_{T^{\alpha }}\sqrt{\sum_{M\in
\mathcal{M}_{n}}\left\vert M\right\vert _{\sigma }}\sqrt{\left\vert
K\right\vert _{\omega }},
\end{eqnarray*
where $\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }}}$ is an upper bound for
the testing functions involved, followed by
\begin{equation*}
\boldsymbol{E}_{\Omega }^{\mathcal{G}}\left( \sum_{M\in \mathcal{M
_{n}}\left\vert M\right\vert _{\sigma }\right) \leq \varepsilon \left\vert
I^{\prime }\right\vert _{\sigma }\ ,
\end{equation*
for a sufficiently small $\varepsilon >0$, where \emph{roughly speaking}, we
use the fact that the intervals $M\in \mathcal{M}_{n}$ depend on the grid
\mathcal{G}$ and form a relatively small proportion of $I^{\prime }$, which
captures only a small amount of the total mass $\left\vert I^{\prime
}\right\vert _{\sigma }$ as the grid is translated relative to the grid
\mathcal{D}$ that contains $I^{\prime }$. To be a bit more precise, recall
that the intervals $K$ are uniquely constructed as consecutive adjacent
intervals of equal length in the grid $\mathcal{G}$ that start out at the
endpoint of $J^{\prime }$ that lies inside $I^{\prime }$, and then progress
toward the constructed endpoint of $I^{\prime \prime }=I^{\prime }\setminus
\partial _{\eta _{1}}I^{\prime }$ lying in the interior of $J^{\prime }$.
Thus translates of the grid $\mathcal{G}$ result in translating the
intervals $K$ across the interval $I^{\prime }$ a distance comparable to at
least the length of $K$, and where $I^{\prime }$ is fixed in the grid
\mathcal{D}$. As a consequence the intervals $M\in \mathcal{M}_{n}\left(
K\right) $ are also translated across the fixed interval $I^{\prime }$ a
distance comparable to at least $\ell \left( K\right) $, and the standard
halo estimate (\ref{hand'}) applies since the sum of the lengths of the
intervals $M\in \mathcal{M}_{n}$ is a small proportion of the length of
J^{\prime }$, whose length is at most that of $I^{\prime }$.
Here are the specific details. Recall that the intervals $K$ are taken from
the set of consecutive intervals $\left\{ K_{i}\right\} _{i=1}^{B}$ that lie
in $I^{\prime }\cap J^{\prime }$, that the intervals $M\in \mathcal{M
_{n}\left( K_{i}\right) $ have length $\frac{1}{4^{n}}\ell \left(
K_{i}\right) $, and that there are $2^{n}$ such intervals in $\mathcal{M
_{n}\left( K_{i}\right) $ for each $i$. Thus we have
\begin{eqnarray}
\sum_{M\in \mathcal{M}_{n}\left( K\right) }\left\vert M\right\vert
&=&\sum_{M\in \mathcal{M}_{n}\left( K\right) }\frac{1}{4^{n}}\left\vert
K\right\vert =2^{n}\frac{1}{4^{n}}\left\vert K\right\vert =\frac{1}{2^{n}
\left\vert K\right\vert \label{future prob} \\
&\Longrightarrow &\boldsymbol{E}_{\Omega }^{\mathcal{G}}\left(
\sum_{i=1}^{B}\sum_{M\in \mathcal{M}_{n}\left( K_{i}\right) }\left\vert
M\right\vert _{\sigma }\right) \leq C\frac{1}{\eta _{0}}\frac{1}{2^{n}
\left\vert I^{\prime }\right\vert _{\sigma }\leq \eta _{0}\left\vert
I^{\prime }\right\vert _{\sigma }\ , \notag
\end{eqnarray
where we have used that the variable $B$ is at most $\frac{1}{\eta _{0}}$,
and where the final inequality holds if $n$ is chosen large enough that
\frac{1}{2^{n}}\leq \eta _{0}^{2}$. Then we have by Cauchy-Schwarz applied
first to $\sum_{i=1}^{B}\sum_{M\in \mathcal{M}_{n}\left( K_{i}\right) }$ and
then to $\boldsymbol{E}_{\Omega }^{\mathcal{G}}$,
\begin{eqnarray}
&&\boldsymbol{E}_{\Omega }^{\mathcal{G}}\left( \sum_{i=1}^{B}\left\vert
A\left( K_{i}\right) \right\vert \right) \leq \boldsymbol{E}_{\Omega }^
\mathcal{G}}\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }}}\mathfrak{N
_{T^{\alpha }}\sqrt{\sum_{i=1}^{B}\sum_{M\in \mathcal{M}_{n}\left(
K_{i}\right) }\left\vert M\right\vert _{\sigma }}\sqrt{\left\vert J^{\prime
}\right\vert _{\omega }} \label{A est} \\
&\leq &\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }}}\mathfrak{N}_{T^{\alpha }
\sqrt{\boldsymbol{E}_{\Omega }^{\mathcal{G}}\sum_{i=1}^{B}\sum_{M\in
\mathcal{M}_{n}\left( K_{i}\right) }\left\vert M\right\vert _{\sigma }}\sqrt
\left\vert J^{\prime }\right\vert _{\omega }} \notag \\
&\leq &\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }}}\mathfrak{N}_{T^{\alpha }
\sqrt{\eta _{0}\left\vert I^{\prime }\right\vert _{\sigma }}\sqrt{\left\vert
J^{\prime }\right\vert _{\omega }}=\sqrt{C_{\mathbf{b}}C_{\mathbf{b}^{\ast }
}\sqrt{\eta _{0}}\mathfrak{N}_{T^{\alpha }}\sqrt{\left\vert I^{\prime
}\right\vert _{\sigma }}\sqrt{\left\vert J^{\prime }\right\vert _{\omega }},
\notag
\end{eqnarray
as required.
Now we turn to summing up the remaining terms $B\left( K\right) =C\sum_{M\in
\mathcal{M}_{n}^{\ast }}\Phi ^{A,B}\left( M_{\limfunc{in}}\right) $ above.
We begin by claiming that in the case when the interval $I^{\prime }$ is a
\emph{natural} child of $I$, i.e. $I^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( I\right) $ so that $I^{\prime }\in \mathcal{C}_{A}^{\mathcal
A}}$, we have
\begin{eqnarray}
&&\left( \sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\Vert
\mathbf{1}_{M_{\limfunc{in}}}T_{\sigma }^{\alpha }b_{A}\right\Vert
_{L^{2}\left( \omega \right) }^{2}\right) ^{\frac{1}{2}}\left( \sum_{M\in
\mathcal{M}_{n}^{\ast }\left( K\right) }\left\vert M_{\limfunc{in
}\right\vert _{\omega }\right) ^{\frac{1}{2}} \label{begin claim} \\
&&+\left( \sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\{ \mathrm
P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc
left}},\mathbf{1}_{A}\sigma \right) ^{2}+\mathrm{P}_{\delta }^{\alpha
\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc{right}},\mathbf{1
_{A}\sigma \right) ^{2}\right\} \right) ^{\frac{1}{2}}\left( \sum_{M\in
\mathcal{M}_{n}^{\ast }\left( K\right) }\left\vert M_{\limfunc{in
}\right\vert _{\omega }\right) ^{\frac{1}{2}} \notag \\
&\lesssim &\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\mathcal{E
_{2}^{\alpha }+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }+\mathcal{A}_{2}^{\alpha
}}\right) \sqrt{\left\vert I^{\prime }\right\vert _{\sigma }\left\vert
J^{\prime }\right\vert _{\omega }}\lesssim \mathcal{NTV}_{\alpha }\sqrt
\left\vert I^{\prime }\right\vert _{\sigma }\left\vert J^{\prime
}\right\vert _{\omega }}, \notag
\end{eqnarray
where the last line is a consequence of the crucial fact that the intervals
\left\{ M_{\limfunc{in}}\right\} _{M\in \mathcal{M}_{n}^{\ast }\left(
K\right) }$ form a pairwise disjoint subdecomposition of $K\subset I^{\prime
}\cap J^{\prime }$ (for any $n\geq 1$). Indeed, we then have the following
inequalities
\begin{enumerate}
\item the first sum on the left hand side satisfie
\begin{equation*}
\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\Vert \mathbf{1}_{M_
\limfunc{in}}}T_{\sigma }^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega
\right) }^{2}=\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\int_{M_
\limfunc{in}}}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert ^{2}d\omega
\leq \int_{I^{\prime }}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert
^{2}d\omega \lesssim \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\right)
^{2}\left\vert I^{\prime }\right\vert _{\sigma }\ ,
\end{equation*
by the weak testing condition for $I^{\prime }$ in the corona $\mathcal{C
_{A}$,
\item an
\begin{equation*}
\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\vert M_{\limfunc{in
}\right\vert _{\omega }\leq \left\vert K\right\vert _{\omega }\leq
\left\vert J^{\prime }\right\vert _{\omega }\ ,
\end{equation*}
\item and, using the definition of $\mathrm{P}_{\delta }^{\alpha }\mathsf{Q
^{\omega }\left( J,\upsilon \right) $ in (\ref{def compact}),
\begin{eqnarray*}
&&\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\{ \mathrm{P
_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc
left}},\mathbf{1}_{A}\sigma \right) ^{2}+\mathrm{P}_{\delta }^{\alpha
\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc{right}},\mathbf{1
_{A}\sigma \right) ^{2}\right\} \\
&\lesssim &\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\{ \left(
\frac{\mathrm{P}^{\alpha }\left( M_{\limfunc{in}}^{\limfunc{left}}
\boldsymbol{1}_{A}\sigma \right) }{\left\vert M_{\limfunc{in}}^{\limfunc{lef
}}\right\vert }\right) ^{2}\left\Vert x-m_{M_{\limfunc{in}}^{\limfunc{left
}}\right\Vert _{L^{2}\left( \mathbf{1}_{M_{\limfunc{in}}^{\limfunc{left
}}\omega \right) }^{2}+\left( \frac{\mathrm{P}^{\alpha }\left( M_{\limfunc{i
}}^{\limfunc{right}},\boldsymbol{1}_{A}\sigma \right) }{\left\vert M_
\limfunc{in}}^{\limfunc{right}}\right\vert }\right) ^{2}\left\Vert x-m_{M_
\limfunc{in}}^{\limfunc{right}}}\right\Vert _{L^{2}\left( \mathbf{1}_{M_
\limfunc{in}}^{\limfunc{right}}}\omega \right) }^{2}\right\} \\
&\lesssim &\mathcal{NTV}_{\alpha }\left\vert I^{\prime }\right\vert _{\sigma
}\ ,
\end{eqnarray*
upon using the stopping energy condition for $I^{\prime }$ in the corona
\mathcal{C}_{A}$, i.e. the failure of (\ref{def stop 3}), in the corona
\mathcal{C}_{A}$ with the subdecomposition $I^{\prime }\supset \overset
\cdot }{\dbigcup }_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left( M_
\limfunc{in}}^{\limfunc{left}}\overset{\cdot }{\dbigcup }M_{\limfunc{in}}^
\limfunc{right}}\right) $.
\end{enumerate}
This completes the proof of (\ref{begin claim}). Our next claim is the
inequalit
\begin{eqnarray}
&&\left( \sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\int_{M_
\limfunc{in}}}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert ^{2}d\omega
+\left( \mathfrak{T}_{T^{\alpha ,\ast }}+\mathfrak{A}_{2}^{\alpha }\right)
^{2}\left\vert M_{\limfunc{in}}\right\vert _{\sigma }\right) ^{\frac{1}{2
}\left( \sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\left\vert M_
\limfunc{in}}\right\vert _{\omega }\right) ^{\frac{1}{2}} \label{next claim}
\\
&\lesssim &\left( \int_{I^{\prime }}\left\vert T_{\sigma }^{\alpha
}b_{A}\right\vert ^{2}d\omega +\left( \mathfrak{T}_{T^{\alpha ,\ast }}
\mathfrak{A}_{2}^{\alpha }\right) ^{2}\left\vert I^{\prime }\right\vert
_{\sigma }\right) ^{\frac{1}{2}}\left( \left\vert J^{\prime }\right\vert
_{\omega }\right) ^{\frac{1}{2}}\lesssim \mathcal{NTV}_{\alpha }\sqrt
\left\vert I^{\prime }\right\vert _{\sigma }\left\vert J^{\prime
}\right\vert _{\omega }}, \notag
\end{eqnarray
and combining this with (\ref{begin claim}), together with the definition of
$\Phi ^{A,B}$ in (\ref{def PHI}), give
\begin{equation*}
\sum_{M\in \mathcal{M}_{n}^{\ast }\left( K\right) }\Phi ^{A,B}\left( M_
\limfunc{in}}\right) \lesssim L{\small eft}H{\small and}S{\small ide}\left(
\ref{begin claim}\right) +L{\small eft}H{\small and}S{\small ide}\left( \re
{next claim}\right) \lesssim \mathcal{NTV}_{\alpha }\sqrt{\left\vert
I^{\prime }\right\vert _{\sigma }\left\vert J^{\prime }\right\vert _{\omega
}.
\end{equation*}
In order to deal with this sum in the case when the child $I^{\prime }$ is
broken, we must take the estimate one step further and sum over those broken
intervals $I^{\prime }$ whose parents belong to the corona $\mathcal{C}_{A}
, i.e. $\left\{ I^{\prime }\in \mathcal{D}:I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( I\right) \text{ for some }I\in \mathcal{C
_{A}\right\} $. Of course this collection is precisely the set of $\mathcal{
}$ -children of $A$, i.e
\begin{equation}
\left\{ I^{\prime }\in \mathcal{D}:I^{\prime }\in \mathfrak{C}_{\limfunc
broken}}\left( I\right) \text{ for some }I\in \mathcal{C}_{A}\right\}
\mathfrak{C}_{\mathcal{A}}\left( A\right) . \label{precisely}
\end{equation}
To help motivate this, we first recall that we denote the term $B\left(
K\right) $ by $B_{\left( I^{\prime },J^{\prime }\right) }\left( K\right) $
when we wish to indicate the pair $\left( I^{\prime },J^{\prime }\right) $
to which $K$ is associated, i.e. $K\in \mathcal{K}\left( I^{\prime
},J^{\prime }\right) $ as in (\ref{def K(I',J')}). Of course the intervals
M\in \mathcal{M}\left( K\right) $ also depend on the pair of intervals
I^{\prime }$ and $J^{\prime }$, but we will suppress notation to this
effect, and the reader should keep this in mind. In particular then, if we
now sum over \emph{natural} children $I^{\prime }$ of $I$ $\in \mathcal{C
_{A}$ and the associated children $J^{\prime }$ of $J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{nearby}}\left( I\right) $, wher
\begin{equation*}
\mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left( I\right) \equiv \left\{
J\in \mathcal{G}:2^{-\mathbf{r}}\ell \left( I\right) <\ell \left( J\right)
\leq \ell \left( I\right) \text{ and }d\left( J,I\right) \leq 2\ell \left(
J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon }\right\} ,\ \
\ \ \ \text{for }I\in \mathcal{C}_{A}\ ,
\end{equation*
we obtain the following corona estimate, using the collection $\mathcal{K
\left( I^{\prime },J^{\prime }\right) $ that is defined in (\ref{def
K(I',J')}) above with $B\leq C\frac{1}{\eta _{0}}$,
\begin{eqnarray}
&&\sum_{I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{nearby}}\left( I\right) }\sum_{\substack{ I^{\prime }\in \mathfrak{
}_{\limfunc{natural}}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C
\left( J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) }
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert \ \left\vert B_{\left( I^{\prime
},J^{\prime }\right) }\left( K\right) \right\vert \ \left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \label{prelim corona natural} \\
&\lesssim &\frac{1}{\eta _{0}}\mathcal{NTV}_{\alpha }\sum_{I\in \mathcal{C
_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left(
I\right) }\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{natural}}\left(
I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left( J\right) }\left\vert
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \right\vert \ \sqrt{\left\vert I^{\prime }\right\vert
_{\sigma }\left\vert J^{\prime }\right\vert _{\omega }}\ \left\vert
E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat
\mathbf{b}^{\ast }}g\right) \right\vert \notag \\
&\lesssim &\frac{1}{\eta _{0}}\mathcal{NTV}_{\alpha }\left( \sum_{I\in
\mathcal{C}_{A}}\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{natural}}\left(
I\right) }\left\vert I^{\prime }\right\vert _{\sigma }\left\vert
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \right\vert ^{2}\right) ^{\frac{1}{2}}\left( \sum_{I\in
\mathcal{C}_{A}}\sum_{J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby
}\left( I\right) }\sum_{J^{\prime }\in \mathfrak{C}\left( J\right)
}\left\vert J^{\prime }\right\vert _{\omega }\ \left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert ^{2}\right) ^{\frac{1}{2}} \notag \\
&\lesssim &\frac{1}{\eta _{0}}\mathcal{NTV}_{\alpha }\left\Vert \mathsf{P}_
\mathcal{C}_{A}}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right)
}^{\bigstar }\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
nearby}}}^{\omega }g\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar }\ ,
\notag
\end{eqnarray
where $\mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}=\dbigcup\limits_{I\in
\mathcal{C}_{A}}\mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left(
I\right) $, and the final line uses (\ref{box hat bound}) to obtain
\begin{equation*}
\sum_{I\in \mathcal{C}_{A}}\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( I\right) }\left\vert I^{\prime }\right\vert _{\sigma
}\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert ^{2}=\sum_{I\in \mathcal{C
_{A}}\left\Vert \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}\lesssim \sum_{I\in \mathcal
C}_{A}}\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{2}\leq \left\Vert \mathsf{P}_{\mathcal{C
_{A}}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar 2},
\end{equation*
and similarly for the sum in $J$ and $J^{\prime }$, once we note that given
J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}$, there are only
boundedly many $I\in \mathcal{C}_{A}$ for which $J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{nearby}}\left( I\right) $.
To obtain the same corona estimate when summing over broken $I^{\prime }$,
we will exploit the fact that the intervals $A^{\prime }\in \mathfrak{C}_
\mathcal{A}}\left( A\right) $ are pairwise disjoint. But first we note that
when $I^{\prime }$ is a broken child, neither weak testing nor stopping
energy is available. But if we sum over such broken $I^{\prime }$, and use
\ref{precisely}) to see that the broken children are pairwise disjoint, we
obtain the following estimate where for convenience we writ
\begin{equation*}
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^
\limfunc{left}/\limfunc{right}},\mathbf{1}_{A}\sigma \right) ^{2}\equiv
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^
\limfunc{left}},\mathbf{1}_{A}\sigma \right) ^{2}+\mathrm{P}_{\delta
}^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc{right}}
\mathbf{1}_{A}\sigma \right) ^{2},
\end{equation*
and we use the notation $\mathcal{M}_{n}^{\ast }\left( I^{\prime },J^{\prime
}\right) \equiv \dbigcup\limits_{K\in \mathcal{K}\left( I^{\prime
},J^{\prime }\right) }\mathcal{M}_{n}^{\ast }\left( K\right) $:
\begin{eqnarray*}
&&\sum_{I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{nearby}}\left( I\right) }\sum_{\substack{ I^{\prime }\in \mathfrak{
}_{\limfunc{broken}}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C
\left( J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) }
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert \ \left\vert B_{\left( I^{\prime
},J^{\prime }\right) }\left( K\right) \right\vert \ \left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \\
&\lesssim &\mathcal{NTV}_{\alpha }\sum_{I\in \mathcal{C}_{A}\text{ and }J\in
\mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left( I\right)
}\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) \text{
and }J^{\prime }\in \mathfrak{C}\left( J\right) }\left\vert E_{I^{\prime
}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f\right) \right\vert \\
&&\times \sqrt{\sum_{M\in \mathcal{M}_{n}^{\ast }\left( I^{\prime
},J^{\prime }\right) }\left\Vert \mathbf{1}_{M_{\limfunc{in}}}T_{\sigma
}^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega \right) }^{2}+\sum_{M\in
\mathcal{M}_{n}^{\ast }\left( I^{\prime },J^{\prime }\right) }\mathrm{P
_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc
left}/\limfunc{right}},\mathbf{1}_{A}\sigma \right) ^{2}+\sum_{M\in \mathcal
M}_{n}^{\ast }\left( I^{\prime },J^{\prime }\right) }\left\vert M_{\limfunc
in}}\right\vert _{\sigma }} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \sqrt
\left\vert J^{\prime }\right\vert _{\omega }}\ \left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert ,
\end{eqnarray*
which give
\begin{eqnarray}
&& \label{prelim corona broken} \\
&&\sum_{I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{nearby}}\left( I\right) }\sum_{\substack{ I^{\prime }\in \mathfrak{
}_{\limfunc{broken}}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C
\left( J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) }
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert \ \left\vert B_{\left( I^{\prime
},J^{\prime }\right) }\left( K\right) \right\vert \ \left\vert E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \notag \\
&\lesssim &\mathcal{NTV}_{\alpha }\left( \sum_{\substack{ I\in \mathcal{C
_{A} \\ I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) }
\sum _{\substack{ J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left(
I\right) \\ J^{\prime }\in \mathfrak{C}\left( J\right) }}\sum_{M\in
\mathcal{M}_{n}^{\ast }\left( I^{\prime },J^{\prime }\right) }\left\{
\left\Vert \mathbf{1}_{M_{\limfunc{in}}}T_{\sigma }^{\alpha
}b_{A}\right\Vert _{L^{2}\left( \omega \right) }^{2}+\mathrm{P}_{\delta
}^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc{left}
\limfunc{right}},\mathbf{1}_{A}\sigma \right) ^{2}+\left\vert M_{\limfunc{in
}\right\vert _{\sigma }\right\} \right) ^{\frac{1}{2}} \notag \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\times \left( \frac{1}{\left\vert A\right\vert _{\sigma }}\int_{A}\left\vert
f\right\vert d\sigma \right) \left( \sum_{\substack{ J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{nearby}} \\ J^{\prime }\in \mathfrak{C}\left( J\right)
}}\sum_{\substack{ I\in \mathcal{C}_{A}:\ J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{nearby}}\left( I\right) \\ I^{\prime }\in \mathfrak{C}_{\limfunc
broken}}\left( I\right) }}\left\vert J^{\prime }\right\vert _{\omega
}\left\vert E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert ^{2}\right) ^{\frac{1}{2}}
\notag \\
&\lesssim &\mathcal{NTV}_{\alpha }\sqrt{\left\vert A\right\vert _{\sigma
}\left( \frac{1}{\left\vert A\right\vert _{\sigma }}\int_{A}\left\vert
f\right\vert d\sigma \right) ^{2}}\left\Vert \mathsf{P}_{\mathcal{C}_{A}^
\mathcal{G},\limfunc{nearby}}}^{\omega }g\right\Vert _{L^{2}\left( \sigma
\right) }^{\bigstar }\ , \notag
\end{eqnarray
becaus
\begin{equation*}
\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert =\left\vert \frac{1}
\int_{I}b_{I}d\sigma }\int_{I}fd\sigma \right\vert \lesssim \frac{1}
\left\vert I\right\vert _{\sigma }}\int_{I}\left\vert f\right\vert d\sigma
\lesssim \frac{1}{\left\vert A\right\vert _{\sigma }}\int_{A}\left\vert
f\right\vert d\sigma
\end{equation*
if $I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) $ and
I\in \mathcal{C}_{A}$, and because
\begin{eqnarray}
&& \label{last inequ} \\
&&\sum_{\substack{ I\in \mathcal{C}_{A} \\ I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( I\right) }}\sum_{\substack{ J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{nearby}}\left( I\right) \\ J^{\prime }\in \mathfrak{C
\left( J\right) }}\sum_{M\in \mathcal{M}_{n}^{\ast }\left( I^{\prime
},J^{\prime }\right) }\left\{ \left\Vert \mathbf{1}_{M_{\limfunc{in
}}T_{\sigma }^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega \right) }^{2}
\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^
\limfunc{left}},\mathbf{1}_{A}\sigma \right) ^{2}+\mathrm{P}_{\delta
}^{\alpha }\mathsf{Q}^{\omega }\left( M_{\limfunc{in}}^{\limfunc{right}}
\mathbf{1}_{A}\sigma \right) ^{2}+\left\vert M_{\limfunc{in}}\right\vert
_{\sigma }\right\} \notag \\
&\leq &\int_{A}\left\vert T_{\sigma }^{\alpha }b_{A}\right\vert ^{2}d\omega
\mathfrak{E}_{2}^{\alpha }\left\vert A\right\vert _{\sigma }+\left\vert
A\right\vert _{\sigma }\leq \left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}
\mathfrak{E}_{2}^{\alpha }+1\right) ^{2}\left\vert A\right\vert _{\sigma }\ .
\notag
\end{eqnarray}
Indeed, in this last inequality (\ref{last inequ}), we have used first the
testing condition, which applies since the collection of $\mathcal{G}
-dyadic intervals
\begin{equation*}
\mathcal{R}\equiv \dbigcup\limits_{I\in \mathcal{C}_{A}}\dbigcup\limits_{I^
\prime }\in \mathfrak{C}_{\limfunc{broken}}}\dbigcup\limits_{J\in \mathcal{C
_{A}^{\mathcal{G},\limfunc{nearby}}\left( I\right)
}\dbigcup\limits_{J^{\prime }\in \mathfrak{C}\left( J\right)
}\dbigcup\limits_{M\in \mathcal{M}_{n}^{\ast }\left( I^{\prime },J^{\prime
}\right) }\left\{ M_{\limfunc{in}}^{\limfunc{left}},M_{\limfunc{in}}^
\limfunc{right}}\right\}
\end{equation*
has bounded overlap counting repetitions of intervals. Indeed, given an
interval $L\in \mathcal{G}$, there are only a bounded, say $B$, number of
pairs $\left( I^{\prime },J^{\prime }\right) $, comparable in both scale and
position, for which $\mathcal{M}_{n}^{\ast }\left( I^{\prime },J^{\prime
}\right) $ contains an interval $M$ with $M_{\limfunc{in}}^{\limfunc{left}}$
or $M_{\limfunc{in}}^{\limfunc{right}}$ equal to $L$. Thus any tower of such
intervals $M_{\limfunc{in}}^{\limfunc{left}}$ or $M_{\limfunc{in}}^{\limfunc
right}}$, that contains a fixed point $x\in \mathbb{R}$, has at most $B$
intervals counting repetitions.
Next we used the energy condition in (\ref{last inequ}), which applies since
if $\mathcal{R}$, considered now without repetitions, has bounded overlap $B
, then $\mathcal{R}$ can be decomposed as $B$ pairwise disjoint families
\left\{ \mathcal{R}_{i}\right\} _{i=1}^{B}$. Indeed, since all of the
intervals lie in the dyadic grid $\mathcal{G}$ and are contained in a fixed
interval $A$, the family $\mathcal{R}_{1}$ of maximal intervals in $\mathcal
R}$ are pairwise disjoint, and after removing them, the remaining collection
of intervals $\mathcal{R}\setminus \mathcal{R}_{1}$ has bounded overlap $B-1
. Let $\mathcal{R}_{2}$ be the family of maximal dyadic intervals in
\mathcal{R}\setminus \mathcal{R}_{1}$ and continue until all the intervals
are exhausted after removing $\mathcal{R}_{R}$.
The inequality (\ref{prelim corona broken}) is a suitable estimate sinc
\begin{equation*}
\sum_{A\in \mathcal{A}}\sqrt{\left\vert A\right\vert _{\sigma }\left( \frac{
}{\left\vert A\right\vert _{\sigma }}\int_{A}\left\vert f\right\vert d\sigma
\right) ^{2}}\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
nearby}}}^{\omega }g\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar
}\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \sigma \right) }
\end{equation*
by quasiorthogonality and the frame inequalities in Appendix A, (\ref{Car
embed}) and (\ref{low frame}), together with the bounded overlap of the
`nearby' coronas $\left\{ \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby
}\right\} _{A\in \mathcal{A}}$.
Recall that after an initial application of random surgery, we reduced the
proof of Lemma \ref{nearby form} to establishing inequality (\ref{after prob
), in which $P_{\left( I,J\right) }\left( K,K\right) =\left\{ K,K\right\} $
in the notation used in (\ref{K iterated}). Now putting all of the above
estimates (\ref{C est}), (\ref{A est}), (\ref{prelim corona natural}) and
\ref{prelim corona broken}) together with (\ref{K iterated}) establishes
probabilistic control of the sum of all the inner products $\left\{
K,K\right\} $ taken over appropriate intervals $K$, yielding (\ref{after
prob}) as required if we choose $\lambda $ and $\eta _{0}$ sufficiently
small
\begin{eqnarray*}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{-\mathbf{r
}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left( I\right) \\
d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }}}\left\vert \sum_{\substack{ I^{\prime }\in
\mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in \mathfrak{C}\left(
J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime }\right) }
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) \ P_{\left( I,J\right) }\left( K,K\right) \ E_{J^{\prime
}}^{\omega }\left( \widehat{\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast
}}g\right) \right\vert \\
&\leq &\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^
\mathcal{G}}\sum_{A\in \mathcal{A}}\sum_{I\in \mathcal{C}_{A}\text{ and
J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{nearby}}\left( I\right) }\sum
_{\substack{ I^{\prime }\in \mathfrak{C}\left( I\right) \text{ and
J^{\prime }\in \mathfrak{C}\left( J\right) \\ K\in \mathcal{K}\left(
I^{\prime },J^{\prime }\right) }}\left\vert E_{I^{\prime }}^{\sigma }\left(
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert \
\left\vert A_{\left( I^{\prime },J^{\prime }\right) }\left( K\right)
\right\vert \ E_{J^{\prime }}^{\omega }\left( \widehat{\square }_{J}^{\omega
,\flat ,\mathbf{b}^{\ast }}g\right) \\
&&+\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{
}}\sum_{A\in \mathcal{A}}\sum_{I\in \mathcal{C}_{A}\text{ and }J\in \mathcal
C}_{A}^{\mathcal{G},\limfunc{nearby}}\left( I\right) }\sum_{\substack{
I^{\prime }\in \mathfrak{C}\left( I\right) \text{ and }J^{\prime }\in
\mathfrak{C}\left( J\right) \\ K\in \mathcal{K}\left( I^{\prime },J^{\prime
}\right) }}\left\vert E_{I^{\prime }}^{\sigma }\left( \widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert \ \left( \left\vert
B_{\left( I^{\prime },J^{\prime }\right) }\left( K\right) \right\vert
+\left\vert C_{\left( I^{\prime },J^{\prime }\right) }\left( K\right)
\right\vert \right) \ \left\vert E_{J^{\prime }}^{\omega }\left( \widehat
\square }_{J}^{\omega ,\flat ,\mathbf{b}^{\ast }}g\right) \right\vert \\
&\lesssim &\left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak
N}_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{eqnarray*
This completes the proof of Lemma \ref{nearby form}.
\section{Main below form\label{Sec Main below}}
Now we turn to controlling the main below form (\ref{def Theta 2 good}
\footnote
While it remains the case with $Tb$ arguments that this form is the most
challenging, the nearby form also poses great difficulties with $Tb$
arguments, especially in contrast with $T1$ arguments, where the nearby form
was handled almost trivially.}
\begin{equation*}
\Theta _{2}^{\limfunc{good}}\left( f,g\right) =\sum_{I\in \mathcal{D
}\sum_{J^{\maltese }\subsetneqq I:\ \ell \left( J\right) \leq 2^{-\mathbf{r
}\ell \left( I\right) }\int \left( T_{\sigma }^{\alpha }\square _{I}^{\sigma
,\mathbf{b}}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega .
\end{equation*
where we recall that $\mathbf{r}$ is the goodness parameter fixed in (\re
{choice of r}) given $0<\varepsilon <\frac{1}{2}$. The corresponding main
below form in \cite{SaShUr6} was denoted $\mathsf{B}_{\Subset _{\mathbf{r
}}\left( f,g\right) =\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}\left(
f,g\right) $. However, there are significant differences between the forms
\Theta _{2}^{\limfunc{good}}\left( f,g\right) $ and $\mathsf{B}_{\Subset _
\mathbf{r}}}\left( f,g\right) $. In \cite{SaShUr6}, the Haar martingale
averages $\bigtriangleup _{I}^{\sigma }$ and $\bigtriangleup _{J}^{\omega }$
in $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) $ are orthogonal
projections, and the intervals $I$ and $J$ that appear in $\mathsf{B
_{\Subset _{\mathbf{r}}}\left( f,g\right) $ are all good in the traditional
sense. Here, on the other hand, the dual martingale averages $\square
_{I}^{\sigma ,\mathbf{b}}$ and $\square _{J}^{\omega ,\mathbf{b}^{\ast }}$
in $\Theta _{2}^{\limfunc{good}}\left( f,g\right) $ are no longer orthogonal
projections, and while the intervals $J$ paired with $I$ remain $\varepsilon
$-good inside $I$ and beyond, the lack of orthogonal projections is
compensated by the fact that the collections of intervals $I$ associated
with any fixed $J$ in $\Theta _{2}^{\limfunc{good}}\left( f,g\right) $ are
tree-connected. Nevertheless, in order to efficiently import the methods for
controlling $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) $ from \cit
{SaShUr6}, we will relabel the main below form $\Theta _{2}^{\limfunc{good
}\left( f,g\right) $ as $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right)
=\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}\left( f,g\right) $ from now
on, keeping in mind the aforementioned differences.
To control $\Theta _{2}^{\limfunc{good}}\left( f,g\right) =\mathsf{B
_{\Subset _{\mathbf{r}}}\left( f,g\right) $ we first perform the \emph
canonical corona splitting} of $\mathsf{B}_{\Subset _{\mathbf{r}}}\left(
f,g\right) $ into a diagonal form $\mathsf{T}_{\limfunc{diagonal}}\left(
f,g\right) $ and a far below form $\mathsf{T}_{\limfunc{far}\limfunc{below
}\left( f,g\right) $ as in \cite{SaShUr6}. This \emph{canonical splitting}
of the form $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) $ involves
the corona pseudoprojections $\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{D
}}^{\sigma ,\mathbf{b}}$ acting on $f$ and the \emph{shifted} corona
pseudoprojections $\mathsf{P}_{\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}$ acting on $g$, where $B$ is a stopping
interval in $\mathcal{A}$. The stopping intervals $\mathcal{B}$ constructed
relative to $g\in L^{2}\left( \omega \right) $ play no role in the analysis
here, except to guarantee that the frame and weak Riesz inequalities hold
for $g$ and the pseudprojections $\left\{ \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\} _{J\in \mathcal{G}}$ and Carleson averaging operators
\left\{ \nabla _{J,\mathcal{G}}^{\omega }g\right\} _{J\in \mathcal{G}}$.
Here the shifted corona $\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}$ is
defined for $B\in \mathcal{A}$ - and \textbf{not} for $B\in \mathcal{B}$ -
to include those intervals $J\in \mathcal{G}$ such that $J^{\maltese }\in
\mathcal{C}_{B}^{\mathcal{D}}$.
\begin{definition}
\label{shifted corona}For $B\in \mathcal{A}$ we define the shifted $\mathcal
G}$-corona b
\begin{equation*}
\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}=\left\{ J\in \mathcal{G
:J^{\maltese }\in \mathcal{C}_{B}^{\mathcal{D}}\right\} .
\end{equation*}
\end{definition}
The Carleson averaging operator $\nabla _{J,\mathcal{G}}^{\omega }$ is taken
over the `broken' children of $J$ which depend on the grid $\mathcal{G}$
(see (\ref{Carleson avg op}) in Appendix A). We will use repeatedly the fact
that the shifted coronas $\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}$ are
pairwise disjoint in $B$
\begin{equation}
\sum_{B\in \mathcal{A}}\mathbf{1}_{\mathcal{C}_{B}^{\mathcal{G},\limfunc
shift}}}\left( J\right) \leq \mathbf{1},\ \ \ \ \ J\in \mathcal{G}.
\label{tau overlap}
\end{equation
It is convenient at this point to introduce the following shorthand notation
\begin{equation}
\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C}_{A}^
\mathcal{D}}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C}_{B}^
\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }^{\Subset _{\mathbf{r},\varepsilon }}\equiv \sum_{\substack{ I\in
\mathcal{C}_{A}^{\mathcal{D}}\text{ and }J\in \mathcal{C}_{B}^{\mathcal{G}
\limfunc{shift}}:\ J^{\maltese }\subsetneqq I \\ \ell \left( J\right) \leq
2^{-\mathbf{r}}\ell \left( I\right) }}\left\langle T_{\sigma }^{\alpha
}\left( \square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ . \label{def shorthand}
\end{equation}
\begin{description}
\item[Caution] One musn't assume, from the notation on the left hand side
above, that the function $T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C
_{A}^{\mathcal{D}}}^{\sigma ,\mathbf{b}}f\right) $ is simply integrated
against the function $\mathsf{P}_{\mathcal{C}_{B}^{\mathcal{G},\limfunc{shif
}}}^{\omega ,\mathbf{b}^{\ast }}g$. Indeed, the sum on the right hand side
is taken over pairs $\left( I,J\right) $ such that $J^{\maltese }\in
\mathcal{C}_{B}^{\mathcal{D}}$ and $J^{\maltese }\subsetneqq I$ and $\ell
\left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) $.
\end{description}
Here is the relevant portion of the brief schematic diagram (\ref{schematic
) of the decompositions, with bounds in $\fbox{}$, used in the next
subsections
\begin{equation*}
\fbox{
\begin{array}{ccccccc}
\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) & & & & & & \\
\downarrow & & & & & & \\
\mathsf{T}_{\limfunc{diagonal}}\left( f,g\right) & + & \mathsf{T}_{\limfunc
far}\limfunc{below}}\left( f,g\right) & + & \mathsf{T}_{\limfunc{far
\limfunc{above}}\left( f,g\right) & + & \mathsf{T}_{\limfunc{disjoint
}\left( f,g\right) \\
\downarrow & & \fbox{$\mathcal{NTV}_{\alpha }$} & & \fbox{$\emptyset $} &
& \fbox{$\emptyset $} \\
\mathsf{B}_{\Subset _{\mathbf{r}}}^{A}\left( f,g\right) & & & & & & \\
\downarrow & & & & & & \\
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc
paraproduct}}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc{neighbour
}^{A}\left( f,g\right) & + & \mathsf{B}_{\limfunc{broken}}^{A}\left(
f,g\right) \\
\fbox{$\mathcal{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}+\sqrt{A_{2}^{\alpha
\limfunc{punct}}}$} & & \fbox{$\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}$} &
& \fbox{$\sqrt{A_{2}^{\alpha }}$} & & \fbox{$\mathfrak{T}_{T^{\alpha }}^
\mathbf{b}}$
\end{array
$}
\end{equation*}
\subsection{The canonical splitting and local below forms}
We begin with an informal description of decompositions and estimates. The
canonical splitting is determined by the coronas $\mathcal{C}_{A}^{\mathcal{
}}$ for $A\in \mathcal{A}$ - note that the stopping times $\mathcal{B}$ play
no explicit role in the canonical splitting of the below form, other than to
guarantee weak Riesz inequalities for $\square _{J}^{\omega ,\mathbf{b
^{\ast }}$ and $\nabla _{J,\mathcal{G}}^{\omega }$
\begin{eqnarray}
&&\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}\left( f,g\right)
\label{parallel corona decomp'} \\
&=&\sum_{A,B\in \mathcal{A}}\left\langle T_{\sigma }^{\alpha }\left( \mathsf
P}_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C
_{B}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }^{\Subset _{\mathbf{r},\varepsilon }} \notag \\
&=&\sum_{A\in \mathcal{A}}\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P
_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C
_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }^{\Subset _{\mathbf{r},\varepsilon }}+\sum
_{\substack{ A,B\in \mathcal{A} \\ B\subsetneqq A}}\left\langle T_{\sigma
}^{\alpha }\left( \mathsf{P}_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}f\right)
\mathsf{P}_{\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{
}^{\ast }}g\right\rangle _{\omega }^{\Subset _{\mathbf{r},\varepsilon }}
\notag \\
&&+\sum_{\substack{ A,B\in \mathcal{A} \\ B\supsetneqq A}}\left\langle
T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b
}f\right) ,\mathsf{P}_{\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }^{\Subset _{\mathbf{
},\varepsilon }}+\sum_{\substack{ A,B\in \mathcal{A} \\ A\cap B=\emptyset }
\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C
_{A}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C}_{B}^{\mathcal{G}
\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}^{\Subset _{\mathbf{r},\varepsilon }} \notag \\
&\equiv &\mathsf{T}_{\limfunc{diagonal}}\left( f,g\right) +\mathsf{T}_
\limfunc{far}\limfunc{below}}\left( f,g\right) +\mathsf{T}_{\limfunc{far
\limfunc{above}}\left( f,g\right) +\mathsf{T}_{\limfunc{disjoint}}\left(
f,g\right) . \notag
\end{eqnarray
Now the final two terms $\mathsf{T}_{\limfunc{far}\limfunc{above}}\left(
f,g\right) $ and $\mathsf{T}_{\limfunc{disjoint}}\left( f,g\right) $ each
vanish since there are no pairs $\left( I,J\right) \in \mathcal{C}_{A}^
\mathcal{D}}\times \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}$ with both
\textbf{i}) $J^{\maltese }\subsetneqq I$ and (\textbf{ii}) either
B\supsetneqq A$ or $B\cap A=\emptyset $. The far below form $\mathsf{T}_
\limfunc{far}\limfunc{below}}\left( f,g\right) $ requires functional energy,
which we discuss in a moment.
Next we follow this splitting by a further decomposition of the diagonal
form into local below forms $\mathsf{B}_{\Subset _{\mathbf{r}}}^{A}\left(
f,g\right) $ given by the individual corona pieces
\begin{equation}
\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left( f,g\right) \equiv
\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C
_{A}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}^{\Subset _{\mathbf{r},\varepsilon }}\ , \label{def local}
\end{equation
and prove the following estimate where $\mathcal{NTV}_{\alpha }$ is defined
in (\ref{def NTV})
\begin{equation}
\left\vert \mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left(
f,g\right) \right\vert \lesssim \mathcal{NTV}_{\alpha }\ \left( \alpha _
\mathcal{A}}\left( A\right) \sqrt{\left\vert A\right\vert _{\sigma }
+\left\Vert \mathsf{P}_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\right) \ \left\Vert \mathsf{P}_
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\ .
\label{local est}
\end{equation
This reduces matters to the local forms since we then have from
Cauchy-Schwarz tha
\begin{eqnarray*}
&&\sum_{A\in \mathcal{A}}\left\vert \mathsf{B}_{\Subset _{\mathbf{r
,\varepsilon }}^{A}\left( f,g\right) \right\vert \lesssim \mathcal{NTV
_{\alpha }\ \left( \sum_{A\in \mathcal{A}}\alpha _{\mathcal{A}}\left(
A\right) ^{2}\left\vert A\right\vert _{\sigma }+\left\Vert \mathsf{P}_
\mathcal{C}_{A}^{\mathcal{D}}}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar 2}\right) ^{\frac{1}{2}}\left(
\sum_{A\in \mathcal{A}}\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left(
\omega \right) }^{\bigstar 2}\right) ^{\frac{1}{2}} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lesssim
\mathcal{NTV}_{\alpha }\ \left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ ,
\end{eqnarray*
by the lower frame inequalities $\sum_{A\in \mathcal{A}}\left\Vert \mathsf{P
_{\mathcal{C}_{A}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }^{\bigstar 2}\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }^{2}$ and $\sum_{A\in \mathcal{A}}\left\Vert \mathsf{P}_{\mathcal{C
_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2}\lesssim \left\Vert g\right\Vert
_{L^{2}\left( \omega \right) }^{2}$, using quasi-orthogonality $\sum_{A\in
\mathcal{A}}\alpha _{\mathcal{A}}\left( f\right) ^{2}\left\vert A\right\vert
_{\sigma }\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}^{2} $ in the stopping intervals $\mathcal{A}$, and the pairwise
disjointedness of the shifted coronas $\mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}$:
\begin{equation*}
\sum_{A\in \mathcal{A}}\mathbf{1}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}}\leq \mathbf{1}_{\mathcal{D}}.
\end{equation*
From now on we will often write $\mathcal{C}_{A}$ in place of $\mathcal{C
_{A}^{\mathcal{D}}$ when no confusion is possible.
Finally, the local forms $\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon
}}^{A}\left( f,g\right) $ are decomposed into stopping forms $\mathsf{B}_
\limfunc{stop}}^{A}\left( f,g\right) $, paraproduct forms $\mathsf{B}_
\limfunc{paraproduct}}^{A}\left( f,g\right) $, neighbour forms $\mathsf{B}_
\limfunc{neighbour}}^{A}\left( f,g\right) $ forms and broken forms $\mathsf{
}_{\limfunc{broken}}^{A}\left( f,g\right) $. The paraproduct and neighbour
terms are handled as in \cite{SaShUr6}, which in turn follows the treatment
originating in \cite{NTV3}, and the broken form is handled with Carleson
measure methods, leaving only the stopping form $\mathsf{B}_{\limfunc{stop
}^{A}\left( f,g\right) $ to be bounded, which we treat in the next section
below, Section \ref{Sec stop}, by refining the bottom/up stopping time in
the argument of M. Lacey in \cite{Lac} with an additional top/down
`indented' corona construction to handle weak goodness.
However, in order to complete the required bounds of the above forms into
which the below form $\mathsf{B}_{\Subset _{\mathbf{r}}}\left( f,g\right) $
was decomposed, we need functional energy for the far below form. Recall
that the vector-valued function $\mathbf{b}$ in the accretive coronas
`breaks' only at a collection of intervals satisfying a Carleson condition.
\begin{definition}
\label{def Whitney}Define the \emph{Whitney} subintervals $\mathcal{W}\left(
F\right) $ of an interval $F\in \mathcal{D}$ to consist of the \emph{maximal}
dyadic $\mathcal{D}$-subintervals of a $\mathcal{D}$-interval $F$ that have
their triples contained in $F$.
\end{definition}
See (\ref{def pseudo rest}) in Appendix B below for more detail on this and
the remaining terms in (\ref{e.funcEnergy n}) below.
\begin{definition}
\label{functional energy n}Let $\mathfrak{F}_{\alpha }=\mathfrak{F}_{\alpha
}\left( \mathcal{D},\mathcal{G}\right) =\mathfrak{F}_{\alpha }^{\mathbf{b
^{\ast }}\left( \mathcal{D},\mathcal{G}\right) $ be the smallest constant in
the `\textbf{f}unctional energy' inequality below, holding for all $h\in
L^{2}\left( \sigma \right) $ and all $\sigma $-Carleson collections
\mathcal{F}\subset \mathcal{D}$ with Carleson norm $C_{\mathcal{F}}$ bounded
by a fixed constant $C$:
\begin{equation}
\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) }\left( \frac
\mathrm{P}^{\alpha }\left( M,h\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}};M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\leq \left( \mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast
}}\left( \mathcal{D},\mathcal{G}\right) \right) ^{2}\left\Vert h\right\Vert
_{L^{2}\left( \sigma \right) }^{2}\,, \label{e.funcEnergy n}
\end{equation}
\end{definition}
The main ingredient used in reducing control of the below form $\mathsf{B
_{\Subset _{\mathbf{r}}}\left( f,g\right) $ to control of the functional
energy $\mathfrak{F}_{\alpha }=\mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast
}}\left( \mathcal{D},\mathcal{G}\right) $ constant and the stopping form
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) $, is the Intertwining
Proposition from \cite{SaShUr7} and/or \cite{SaShUr6}. The control of the
functional energy condition by the energy and Muckenhoupt conditions must
also be adapted in light of the $\infty $-strongly accretive function
\mathbf{b}$ that only `breaks' at a collection of intervals satisfying a
Carleson condition, but this poses no real difficulties. The fact that the
usual Haar bases are orthonormal is here replaced by the weaker condition
that the corresponding broken dual martingale `bases' are merely weak frames
satisfying certain\ weak lower and weak upper Riesz inequalities, but again
this poses no real difference in the arguments. Finally, the fact that
goodness for $J$ has been replaced with weak goodness, namely $J^{\maltese
}\subsetneqq I$ whenever the pair $\left( I,J\right) $ occurs in a sum,
forces the use of a Whitney decomposition $\mathcal{W}$ of intervals instead
of the deeply embedded decomposition $\mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep}}$ used in \cite{SaShUr7}.
We then use the paraproduct / neighbour / broken / stopping splitting
mentioned above to reduce boundedness of $\mathsf{B}_{\Subset _{\mathbf{r
,\varepsilon }}^{A}\left( f,g\right) $ to boundedness of the associated
stopping form
\begin{equation}
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) \equiv \sum_{\substack{
I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}} \\ J^{\maltese }\subset I\text{, }J^{\maltese }\neq I\text{ and
\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }
E_{I_{J}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f\right) \left\langle T_{\sigma }^{\alpha }b_{A}\mathbf{1}_{A\setminus
I_{J}},\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ ,
\label{bounded stopping form}
\end{equation
where $f$ is supported in the interval $A$ and its expectations $\mathbb{E
_{I}^{\sigma }\left\vert f\right\vert $ are bounded by $\alpha _{\mathcal{A
}\left( A\right) $ for $I\in \mathcal{C}_{A}$, the dual martingale support
of $f$ is contained in the corona $\mathcal{C}_{A}^{\sigma }$, and the dual
martingale support of $g$\ is contained in $\mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}$, and where $I_{J}$ is the $\mathcal{D}$-child of $I$ that
contains $J$.
\subsection{Diagonal and far below forms}
Now we turn to estimating the \emph{diagonal term} $\mathsf{T}_{\limfunc
diagonal}}\left( f,g\right) $ and the \emph{far below} term $\mathsf{T}_
\limfunc{far}\limfunc{below}}\left( f,g\right) $, where in \cite{SaShUr7}
and/or \cite{SaShUr6}, the far below terms were bounded using the
Intertwining Proposition and the control of functional energy condition by
the energy and Muckenhoupt conditions, but of course under the restriction
there that the intervals $J$ were good. Here we writ
\begin{eqnarray}
&&\mathsf{T}_{\limfunc{far}\limfunc{below}}\left( f,g\right) =\sum
_{\substack{ A,B\in \mathcal{A} \\ B\subsetneqq A}}\sum_{\substack{ I\in
\mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift
} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq 2^{
\mathbf{r}}\ell \left( I\right) }}\left\langle T_{\sigma }^{\alpha }\left(
\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega } \label{write} \\
&=&\sum_{B\in \mathcal{A}}\sum_{I\in \mathcal{D}:\ B\subsetneqq
I}\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b
}f\right) ,\sum_{J\in \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}}\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }-\sum_{B\in
\mathcal{A}}\sum_{I\in \mathcal{D}:\ B\subsetneqq I}\left\langle T_{\sigma
}^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b}}f\right) ,\sum_{\substack{
J\in \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}} \\ \ell \left( J\right)
>2^{-\mathbf{r}}\ell \left( I\right) }}\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega } \notag \\
&=&\sum_{B\in \mathcal{A}}\sum_{I\in \mathcal{D}:\ B\subsetneqq
I}\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b
}f\right) ,g_{B}\right\rangle _{\omega }-\sum_{B\in \mathcal{A}}\sum_{I\in
\mathcal{D}:\ B\subsetneqq I}\left\langle T_{\sigma }^{\alpha }\left(
\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\sum_{\substack{ J\in \mathcal{C
_{B}^{\mathcal{G},\limfunc{shift}} \\ \ell \left( J\right) >2^{-\mathbf{r
}\ell \left( I\right) }}\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }; \notag \\
&&\text{where }g_{B}=\sum_{J\in \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift
}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}g=\mathsf{P}_{\mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\ , \notag
\end{eqnarray
since if $I\in \mathcal{C}_{A}$ and $J\in \mathcal{C}_{B}^{\mathcal{G}
\limfunc{shift}}$, with $J^{\maltese }\subsetneqq I$ and $B\subsetneqq A$,
then we must have $B\subsetneqq I$. First, we note that expectation of the
second sum on the right hand side of (\ref{write}) is controlled by (\re
{delta near}) in Lemma \ref{nearby form}, i.e
\begin{eqnarray*}
&&\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\left\vert \sum_{B\in \mathcal{A}}\sum_{I\in \mathcal{D}:\ B\subsetneqq
I}\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b
}f\right) ,\sum_{\substack{ J\in \mathcal{C}_{B}^{\mathcal{G},\limfunc{shift
} \\ \ell \left( J\right) >2^{-\mathbf{r}}\ell \left( I\right) }}\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert \\
&\lesssim &\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^
\mathcal{G}}\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in \mathcal{G}:\ 2^{
\mathbf{r}}\ell \left( I\right) <\ell \left( J\right) \leq \ell \left(
I\right) \\ d\left( J,I\right) \leq 2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }}}\left\vert \left\langle T_{\sigma
}^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert \\
&\lesssim &\left( C_{\theta }\mathcal{NTV}_{\alpha }+\sqrt{\theta }\mathfrak
N}_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{eqnarray*
Second, we note that the Intertwining Proposition \ref{strongly adapted},
which controls sums of the for
\begin{equation*}
\sum_{F\in \mathcal{F}}\ \sum_{I:\ I\supsetneqq F}\ \left\langle T_{\sigma
}^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f,\mathsf{P}_{\mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega },
\end{equation*
can be applied to the first sum on the right hand side of (\ref{write}) to
show that it is bounded by $\left( \mathcal{NTV}_{\alpha }+\mathfrak{F
_{\alpha }^{\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G}\right) \right)
\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }$, where the goodness parameter
\varepsilon >0$ is chosen sufficiently small. Then Proposition \ref{func
ener control}\ can be applied to show that $\mathfrak{F}_{\alpha }^{\mathbf{
}^{\ast }}\left( \mathcal{D},\mathcal{G}\right) \lesssim \mathfrak{A
_{2}^{\alpha }+\mathfrak{E}_{2}^{\alpha }$, which completes the proof tha
\begin{equation}
\left\vert \mathsf{T}_{\limfunc{far}\limfunc{below}}\left( f,g\right)
\right\vert \lesssim \mathcal{NTV}_{\alpha }\ \left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\ . \label{far below bound}
\end{equation}
\subsection{Intertwining Proposition}
First we adapt the relevant definitions from \cite{SaShUr6}.
\begin{definition}
\label{sigma carleson n}A collection $\mathcal{F}$ of dyadic intervals is
\sigma $\emph{-Carleson} i
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\subset S}\left\vert F\right\vert _{\sigma }\leq
C_{\mathcal{F}}\left\vert S\right\vert _{\sigma },\ \ \ \ \ S\in \mathcal{F}.
\end{equation*
The constant $C_{\mathcal{F}}$ is referred to as the Carleson norm of
\mathcal{F}$.
\end{definition}
\begin{definition}
\label{def shift}Let $\mathcal{F}$ be a collection of dyadic intervals in a
grid $\mathcal{D}$. Then for $F\in \mathcal{F}$, we define the shifted
corona $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$ in analogy with
Definition \ref{shifted corona} b
\begin{equation*}
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}\equiv \left\{ J\in \mathcal{G
:J^{\maltese }\in \mathcal{C}_{F}\right\} ,
\end{equation*
where $J^{\maltese }$ is defined in Definition \ref{def sharp cross}.
\end{definition}
Note that the collections $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$
are pairwise disjoint in $F$. Let $\mathfrak{C}_{\mathcal{F}}\left( F\right)
$ denote the set of $\mathcal{F}$-children of $F$. Given any collection
\mathcal{H}\subset \mathcal{G}$ of intervals, a family $\mathbf{b}^{\ast }$
of dual testing functions, and an arbitrary interval $K\in \mathcal{P}$, we
define the corresponding dual pseudoprojection $\mathsf{P}_{\mathcal{H
}^{\omega ,\mathbf{b}^{\ast }}$ and its localization $\mathsf{P}_{\mathcal{H
;K}^{\omega ,\mathbf{b}^{\ast }}$ to $K$ b
\begin{equation}
\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}=\sum_{H\in \mathcal{H
}\bigtriangleup _{H}^{\omega ,\mathbf{b}^{\ast }}\text{ and }\mathsf{Q}_
\mathcal{H};K}^{\omega ,\mathbf{b}^{\ast }}=\sum_{H\in \mathcal{H}:\
H\subset K}\bigtriangleup _{H}^{\omega ,\mathbf{b}^{\ast }}\ .
\label{def localization}
\end{equation
Recall from Definition \ref{functional energy n} that $\mathfrak{F}_{\alpha
}=\mathfrak{F}_{\alpha }\left( \mathcal{D},\mathcal{G}\right) =\mathfrak{F
_{\alpha }^{\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G}\right) $ is
the best constant in (\ref{e.funcEnergy n}), i.e.
\begin{equation*}
\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) }\left( \frac
\mathrm{P}^{\alpha }\left( M,h\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}};M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\leq \mathfrak{F}_{\alpha }\lVert h\rVert
_{L^{2}\left( \sigma \right) }\,.
\end{equation*}
\begin{remark}
\label{explaining funct ener}If in (\ref{e.funcEnergy n}), we take $h
\mathbf{1}_{I}$ and $\mathcal{F}$ to be the trivial Carleson collection
\left\{ I_{r}\right\} _{r=1}^{\infty }$ where the intervals $I_{r}$ are
pairwise disjoint in $I$, then we essentially obtain the Whitney energy
condition in Definition \ref{energy condition}, but with $\mathsf{Q}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};M}^{\omega ,\mathbf{b}^{\ast
}} $ in place of $\mathsf{Q}_{M}^{\limfunc{weak}\limfunc{good},\omega }$.
However, the pseudoprojection $\mathsf{Q}_{M}^{\limfunc{weak}\limfunc{good
,\omega }$ is larger than $\mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}};J}^{\omega ,\mathbf{b}^{\ast }}$, and so we just miss obtaining the
Whitney energy condition as a consequence of the functional energy
condition. Nevertheless, this near miss with $h=\mathbf{1}_{I}$ explains the
terminology `functional' energy.
\end{remark}
We will need an `indicator' version of the estimate proved above for the
disjoint for
\begin{equation*}
\Theta _{1}\left( f,g\right) =\sum_{I\in \mathcal{D}}\sum_{\substack{ J\in
\mathcal{G}:\ \ell \left( J\right) \leq \ell \left( I\right) \\ d\left(
J,I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right)
^{1-\varepsilon }}}\int \left( T_{\sigma }\square _{I}^{\sigma ,\mathbf{b
}f\right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega .
\end{equation*}
\begin{lemma}
\label{standard indicator}Fix dyadic grids $\mathcal{D}$ and $\mathcal{G}$.
Suppose $T^{\alpha }$ is a standard fractional singular integral with $0\leq
\alpha <1$, that $f\in L^{2}\left( \sigma \right) $ and $g\in L^{2}\left(
\omega \right) $, that $\mathcal{F}\subset \mathcal{D}$ and $\mathcal{H
\subset \mathcal{G}$ are $\sigma $-Carleson and $\omega $-Carleson
collections respectively, i.e.
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}:\ F^{\prime }\subset F}\left\vert F^{\prime
}\right\vert _{\sigma }\lesssim \left\vert F\right\vert _{\sigma },\ \ \ \ \
F\in \mathcal{F},\text{ and }\sum_{H^{\prime }\in \mathcal{H}:\ H^{\prime
}\subset H}\left\vert H^{\prime }\right\vert _{\omega }\lesssim \left\vert
H\right\vert _{\omega },\ \ \ \ \ H\in \mathcal{H},
\end{equation*
and that there are numerical sequences $\left\{ \alpha _{\mathcal{F}}\left(
F\right) \right\} _{F\in \mathcal{F}}$ and $\left\{ \beta _{\mathcal{H
}\left( H\right) \right\} _{H\in \mathcal{H}}$ such tha
\begin{equation}
\sum_{F\in \mathcal{F}}\alpha _{\mathcal{F}}\left( F\right) ^{2}\left\vert
F\right\vert _{\sigma }\leq \left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }^{2}\text{ and }\sum_{H\in \mathcal{H}}\beta _{\mathcal{H}}\left(
H\right) ^{2}\left\vert H\right\vert _{\sigma }\leq \left\Vert g\right\Vert
_{L^{2}\left( \sigma \right) }^{2}\ . \label{qo}
\end{equation
The
\begin{eqnarray}
&&\sum_{F\in \mathcal{F}}\sum_{\substack{ J\in \mathcal{G}:\ \ell \left(
J\right) \leq \ell \left( F\right) \\ d\left( J,F\right) >2\ell \left(
J\right) ^{\varepsilon }\ell \left( F\right) ^{1-\varepsilon }}}\left\vert
\int \left( T_{\sigma }^{\alpha }\mathbf{1}_{F}\alpha _{\mathcal{F}}\left(
F\right) \right) \square _{J}^{\omega ,\mathbf{b}^{\ast }}gd\omega
\right\vert \label{indicator far} \\
&&+\sum_{G\in \mathcal{G}}\sum_{\substack{ I\in \mathcal{D}:\ \ell \left(
I\right) \leq \ell \left( G\right) \\ d\left( I,G\right) >2\ell \left(
I\right) ^{\varepsilon }\ell \left( G\right) ^{1-\varepsilon }}}\left\vert
\int \left( T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f\right)
\mathbf{1}_{G}\beta _{\mathcal{G}}\left( G\right) d\omega \right\vert \notag
\\
&\lesssim &\sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left(
\sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }.
\notag
\end{eqnarray}
\end{lemma}
The proof of this lemma is similar to those of Lemmas \ref{delta long} and
\ref{delta short} in Section \ref{Sec disj form}\ above, using the square
function inequalities for $\square _{I}^{\sigma ,\mathbf{b}}$, $\nabla _{I
\mathcal{F}}^{\sigma }$ and $\square _{J}^{\omega ,\mathbf{b}^{\ast }}$,
\nabla _{J,\mathcal{G}}^{\omega }$ in Appendix A, as well as the
quasiorthogonal inequalities assumed in (\ref{qo}), which substitute for the
square function inequalities when dealing with indicators $\mathbf{1
_{F}\alpha _{\mathcal{F}}\left( F\right) $ instead instead of dual
martingale differences $\square _{I}^{\sigma ,\mathbf{b}}f$. We note that
there is no explicit restriction of the type $\ell \left( J\right) \leq 2^{
\mathbf{\rho }}\ell \left( I\right) $ in any of Lemmas \ref{delta long}, \re
{delta short}, or \ref{standard indicator}.
There is one more simple lemma that we will use in the proof of the
Intertwining Proposition, namely that for small $\varepsilon >0$, an
interval $J$ is $\varepsilon $-$\limfunc{good}$ inside an interval $I$ only
if $J$ is many scales smaller in size than $I$. Recall from Definition \re
{good arb} that if an interval $J$ is $\varepsilon $-$\limfunc{good}$ inside
an interval $I$, then
\begin{equation*}
J\subset I\text{ and }d\left( J,\limfunc{skel}I\right) \geq d\left( J
\limfunc{body}I\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left(
I\right) ^{1-\varepsilon }.
\end{equation*}
\begin{lemma}
\label{good scale}If $J$ is $\varepsilon $-$\limfunc{good}$ inside $I$, then
$\ell \left( J\right) <2^{-\frac{3}{\varepsilon }}\ell \left( I\right) $.
\end{lemma}
\begin{proof}
We hav
\begin{equation*}
\frac{1}{4}\ell \left( I\right) \geq d\left( J,\limfunc{skel}I\right) >2\ell
\left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon
}=2\left( \frac{\ell \left( J\right) }{\ell \left( I\right) }\right)
^{\varepsilon }\ell \left( I\right) ,
\end{equation*
which gives $\frac{1}{8}>\left( \frac{\ell \left( J\right) }{\ell \left(
I\right) }\right) ^{\varepsilon }$, i.e. $\frac{\ell \left( J\right) }{\ell
\left( I\right) }<\left( \frac{1}{8}\right) ^{\frac{1}{\varepsilon }}=2^{
\frac{3}{\varepsilon }}$.
\end{proof}
\begin{proposition}[The Intertwining Proposition]
\label{strongly adapted}Let $\mathcal{D}$ and $\mathcal{G}$ be grids, and
suppose that $\mathbf{b}$ and $\mathbf{b}^{\ast }$ are $\infty $-strongly
\sigma $-accretive families of intervals in $\mathcal{D}$ and $\mathcal{G}$
respectively. Suppose that $\mathcal{F}\subset \mathcal{D}$ is $\sigma
-Carleson and that the $\mathcal{F}$-coronas
\begin{equation*}
\mathcal{C}_{F}\equiv \left\{ I\in \mathcal{D}:I\subset F\text{ but
I\not\subset F^{\prime }\text{ for }F^{\prime }\in \mathfrak{C}_{\mathcal{F
}\left( F\right) \right\}
\end{equation*
satisf
\begin{equation*}
E_{I}^{\sigma }\left\vert f\right\vert \lesssim E_{F}^{\sigma }\left\vert
f\right\vert \text{ and }b_{I}=\mathbf{1}_{I}b_{F},\ \ \ \ \ \text{for all
I\in \mathcal{C}_{F}\mathfrak{\ ,\ }F\in \mathcal{F}.
\end{equation*
Then with the shifted corona in Definition \ref{def shift}, i.e. $\mathcal{C
_{F}^{\mathcal{G},\limfunc{shift}}=\left\{ J\in \mathcal{G}:J^{\maltese }\in
\mathcal{C}_{F}\right\} $ with $J^{\maltese }$ as in Definition \ref{def
sharp cross} that depends on $\varepsilon >0$, we hav
\begin{equation*}
\left\vert \sum_{F\in \mathcal{F}}\ \sum_{I:\ I\supsetneqq F}\ \left\langle
T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f,\mathsf{P}_{\mathcal
C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\right\vert \lesssim \left( \mathfrak{F}_{\alpha
}^{\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G}\right) +\mathcal{NTV
_{\alpha }\right) \ \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) },
\end{equation*
where the implied constant depends on the $\sigma $-Carleson norm $C_
\mathcal{F}}$ of the family $\mathcal{F}$.
\end{proposition}
\begin{proof}[Proof of Proposition \protect\ref{strongly adapted}]
We write the sum on the left hand side of the display above a
\begin{eqnarray*}
&&\sum_{F\in \mathcal{F}}\ \sum_{I:\ I_{\infty }\supset I\supsetneqq F}\
\left\langle T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f,\mathsf
P}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle
_{\omega }=\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\left(
\sum_{I:\ I_{\infty }\supset I\supsetneqq F}\square _{I}^{\sigma ,\mathbf{b
}f\right) ,\mathsf{P}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}}^{\omega }g\right\rangle _{\omega }=\sum_{F\in \mathcal{F}}\ \left\langle
T_{\sigma }^{\alpha }\left( f_{F}^{\ast }\right) ,g_{F}\right\rangle
_{\omega }; \\
&&\text{where }f_{F}^{\ast }\equiv \sum_{I:\ I_{\infty }\supset I\supsetneqq
F}\square _{I}^{\sigma ,\mathbf{b}}f\text{ and }g_{F}\equiv \mathsf{P}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g,
\end{eqnarray*
where $I_{\infty }$ is the starting interval for corona constructions in
\mathcal{D}$ as in (\ref{top control}) above. Note that $g_{F}$ is supported
in $F$. By the telescoping identity for $\square _{I}^{\sigma ,\mathbf{b}}$,
the function $f_{F}^{\ast }$ satisfie
\begin{equation*}
\mathbf{1}_{F}f_{F}^{\ast }=\sum_{I:\ I_{\infty }\supset I\supsetneqq
F}\square _{I}^{\sigma ,\mathbf{b}}f=\mathbb{F}_{F}^{\sigma ,\mathbf{b}}f
\mathbf{1}_{F}\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f=b_{F}\frac
E_{F}^{\sigma }f}{E_{F}^{\sigma }b_{F}}-\mathbf{1}_{F}b_{I_{\infty }}\frac
E_{I_{\infty }}^{\sigma }f}{E_{I_{\infty }}^{\sigma }b_{I_{\infty }}}\ .
\end{equation*
However, we cannot apply the testing condition to the function $\mathbf{1
_{F}b_{I_{\infty }}$, and since $E_{I_{\infty }}^{\sigma }f$ does not vanish
in general, we will instead add and subtract the term $\mathbb{F}_{I_{\infty
}}^{\sigma ,\mathbf{b}}f$ to get
\begin{eqnarray*}
\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\left(
f_{F}^{\ast }\right) ,g_{F}\right\rangle _{\omega } &=&\sum_{F\in \mathcal{F
}\ \left\langle T_{\sigma }^{\alpha }\left( \sum_{I:\ I_{\infty }\supset
I\supsetneqq F}\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle
_{\omega } \\
&=&\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\left( \mathbb
F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f+\sum_{I:\ I_{\infty }\supset
I\supsetneqq F}\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle
_{\omega }-\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\left(
\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{
}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle _{\omega }\ ,
\end{eqnarray*
where the second sum on the right hand side of the identity satisfie
\begin{eqnarray*}
\left\vert \sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\left(
\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{
}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle _{\omega
}\right\vert &=&\left\vert \left\langle T_{\sigma }^{\alpha }\left( \mathbb{
}_{I_{\infty }}^{\sigma ,\mathbf{b}}f\right) ,\sum_{F\in \mathcal{F}}\mathsf
P}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega }g\right\rangle
_{\omega }\right\vert \\
&\leq &\left\Vert T_{\sigma }^{\alpha }\left( \mathbb{F}_{I_{\infty
}}^{\sigma ,\mathbf{b}}f\right) \right\Vert _{L^{2}\left( \omega \right)
}\left\Vert \sum_{F\in \mathcal{F}}\mathsf{P}_{\mathcal{C}_{F;\mathbf{r}}^
\mathcal{G},\limfunc{shift}}}^{\omega }g\right\Vert _{L^{2}\left( \omega
\right) } \\
&\lesssim &\mathfrak{FT}_{T_{\alpha }}^{\mathbf{b}}\left\vert E_{I_{\infty
}}^{\sigma }f\right\vert \left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\lesssim \left( \mathfrak{T}_{T_{\alpha }}^{\mathbf{b}}+\mathfrak{A
_{2}^{\alpha }\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }
\end{eqnarray*
by (\ref{full proved}) above, and the Riesz inequalities in Appendix A. The
advantage now is that wit
\begin{equation*}
f_{F}\equiv \mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f+f_{F}^{\ast }
\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f+\sum_{I:\ I_{\infty }\supset
I\supsetneqq F}\square _{I}^{\sigma ,\mathbf{b}}f
\end{equation*
then in the first term on the right hand side of the identity, the
telescoping identity give
\begin{equation*}
\mathbf{1}_{F}f_{F}=\mathbf{1}_{F}\left( \mathbb{F}_{I_{\infty }}^{\sigma
\mathbf{b}}f+\sum_{I:\ I_{\infty }\supset I\supsetneqq F}\square
_{I}^{\sigma ,\mathbf{b}}f\right) =\mathbb{F}_{F}^{\sigma ,\mathbf{b}}f=b_{F
\frac{E_{F}^{\sigma }f}{E_{F}^{\sigma }b_{F}},
\end{equation*
which shows that $f_{F}$ is a controlled constant times $b_{F}$ on $F$.
The intervals $I$ occurring in this sum are linearly and consecutively
ordered by inclusion, along with the intervals $F^{\prime }\in \mathcal{F}$
that contain $F$. More precisely we can writ
\begin{equation*}
F\equiv F_{0}\subsetneqq F_{1}\subsetneqq F_{2}\subsetneqq ...\subsetneqq
F_{n}\subsetneqq F_{n+1}\subsetneqq ...F_{N}=I_{\infty }
\end{equation*
where $F_{m}=\pi _{\mathcal{F}}^{m}F$ for all $m\geq 1$. We can also writ
\begin{equation*}
F=F_{0}\equiv I_{0}\subsetneqq I_{1}\subsetneqq I_{2}\subsetneqq
...\subsetneqq I_{k}\subsetneqq I_{k+1}\subsetneqq ...\subsetneqq
I_{K}=F_{N}=I_{\infty }
\end{equation*
where $I_{k}=\pi _{\mathcal{D}}^{k}F$ for all $k\geq 1$. There is a (unique)
subsequence $\left\{ k_{m}\right\} _{m=1}^{N}$ such tha
\begin{equation*}
F_{m}=I_{k_{m}},\ \ \ \ \ 1\leq m\leq N.
\end{equation*}
Then we hav
\begin{eqnarray*}
f_{F}\left( x\right) &\equiv &\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b
}f\left( x\right) +\sum_{\ell =1}^{K}\square _{I_{\ell }}^{\sigma ,\mathbf{b
}f\left( x\right) , \\
g_{F} &\equiv &\sum_{J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}g.
\end{eqnarray*
Assume now that $k_{m}\leq k<k_{m+1}$. We denote the sibling of $I$ by
\theta \left( I\right) $, i.e. $\left\{ \theta \left( I\right) \right\}
\mathfrak{C}_{\mathcal{D}}\left( \pi _{\mathcal{D}}I\right) \setminus
\left\{ I\right\} $. There are two cases to consider here
\begin{equation*}
\theta \left( I_{k}\right) \notin \mathcal{F}\text{ and }\theta \left(
I_{k}\right) \in \mathcal{F}.
\end{equation*
We first note that in either case, using a telescoping sum, we compute that
for
\begin{equation*}
x\in \theta \left( I_{k}\right) =I_{k+1}\setminus I_{k}\subset
F_{m+1}\setminus F_{m},
\end{equation*
we have the formula
\begin{eqnarray*}
f_{F}\left( x\right) &=&\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f\left(
x\right) +\sum_{\ell \geq k+1}\square _{I_{\ell }}^{\sigma ,\mathbf{b
}f\left( x\right) \\
&=&\mathbb{F}_{\theta \left( I_{k}\right) }^{\sigma ,\mathbf{b}}f\left(
x\right) -\mathbb{F}_{I_{k+1}}^{\sigma ,\mathbf{b}}f\left( x\right)
+\sum_{\ell =k+1}^{K-1}\left( \mathbb{F}_{I_{\ell }}^{\sigma ,\mathbf{b
}f\left( x\right) -\mathbb{F}_{I_{\ell +1}}^{\sigma ,\mathbf{b}}f\left(
x\right) \right) +\mathbb{F}_{I_{\infty }}^{\sigma ,\mathbf{b}}f\left(
x\right) =\mathbb{F}_{\theta \left( I_{k}\right) }^{\sigma ,\mathbf{b
}f\left( x\right) \ .
\end{eqnarray*
Now fix $x\in \theta \left( I_{k}\right) $. If $\theta \left( I_{k}\right)
\notin \mathcal{F}$, then $\theta \left( I_{k}\right) \in \mathcal{C
_{F_{m+1}}^{\sigma }$, and we have
\begin{equation}
\left\vert f_{F}\left( x\right) \right\vert =\left\vert \mathbb{F}_{\theta
\left( I_{k}\right) }^{\sigma ,\mathbf{b}}f\left( x\right) \right\vert
\lesssim \left\vert b_{\theta \left( I_{k}\right) }\left( x\right)
\right\vert \ \frac{E_{\theta \left( I_{k}\right) }^{\sigma }\left\vert
f\right\vert }{\left\vert E_{\theta \left( I_{k}\right) }^{\sigma }b_{\theta
\left( I_{k}\right) }\right\vert }\lesssim E_{F_{m+1}}^{\sigma }\left\vert
f\right\vert \ , \label{bound for f_F}
\end{equation
since the testing functions $b_{\theta \left( I_{k}\right) }$ are bounded
and accretive, and $E_{\theta \left( I_{k}\right) }^{\sigma }\left\vert
f\right\vert \lesssim E_{F_{m+1}}^{\sigma }\left\vert f\right\vert $ by
hypothesis. On the other hand, if $\theta \left( I_{k}\right) \in \mathcal{F}
$, then $I_{k+1}\in \mathcal{C}_{F_{m+1}}^{\sigma }$ and we hav
\begin{equation*}
\left\vert f_{F}\left( x\right) \right\vert =\left\vert \mathbb{F}_{\theta
\left( I_{k}\right) }^{\sigma ,\mathbf{b}}f\left( x\right) \right\vert
\lesssim E_{\theta \left( I_{k}\right) }^{\sigma }\left\vert f\right\vert \ .
\end{equation*
Note that $F^{c}=\overset{\cdot }{\dbigcup }_{k\geq 0}\theta \left(
I_{k}\right) $. Now we writ
\begin{eqnarray*}
f_{F} &=&\varphi _{F}+\psi _{F}, \\
\varphi _{F} &\equiv &\sum_{k:\ \theta \left( I_{k}\right) \in \mathcal{F}
\mathbb{F}_{\theta \left( I_{k}\right) }^{\sigma ,\mathbf{b}}f\text{ and
\psi _{F}=f_{F}-\varphi _{F}\ ; \\
\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha
}f_{F},g_{F}\right\rangle _{\omega } &=&\sum_{F\in \mathcal{F}}\
\left\langle T_{\sigma }^{\alpha }\varphi _{F},g_{F}\right\rangle _{\omega
}+\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\psi
_{F},g_{F}\right\rangle _{\omega }\ ,
\end{eqnarray*
and note that $\varphi _{F}=0$ on $F$, and $\psi _{F}=b_{F}\frac
E_{F}^{\sigma }f}{E_{F}^{\sigma }b_{F}}$ on $F$. We can apply the first line
in (\ref{indicator far}) using $\theta \left( I_{k}\right) \in \mathcal{F}$
to the first sum above since $J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}}$ implies $J\subset J^{\maltese }\subset F\subset \theta \left(
I_{k}\right) ^{c}$, which implies that $d\left( J,\theta \left( I_{k}\right)
\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left( \theta \left(
I_{k}\right) \right) ^{1-\varepsilon }$. Thus we obtain after substituting
F^{\prime }$ for $\theta \left( I_{k}\right) $ below,
\begin{eqnarray*}
\left\vert \sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha
}\varphi _{F},g_{F}\right\rangle _{\omega }\right\vert &=&\left\vert
\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shif
}}}\left\langle T_{\sigma }^{\alpha }\left( \sum_{k:\ \theta \left(
I_{k}\right) \in \mathcal{F}}\mathbb{F}_{\theta \left( I_{k}\right)
}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\right\vert \\
&\leq &\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{C}_{F}^{\mathcal{G}
\limfunc{shift}}}\sum_{k:\ \theta \left( I_{k}\right) \in \mathcal{F
}\left\vert \left\langle T_{\sigma }^{\alpha }\left( \mathbb{F}_{\theta
\left( I_{k}\right) }^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert \\
&\leq &\sum_{F^{\prime }\in \mathcal{F}}\sum_{\substack{ J\in \mathcal{G}:\
\ell \left( J\right) \leq \ell \left( F^{\prime }\right) \\ d\left(
J,F^{\prime }\right) >2\ell \left( J\right) ^{\varepsilon }\ell \left(
F^{\prime }\right) ^{1-\varepsilon }}}\left\vert \left\langle T_{\sigma
}^{\alpha }\left( \mathbb{F}_{F^{\prime }}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}\right\vert \\
&\lesssim &\sqrt{A_{2}^{\alpha }}\left\Vert f\right\Vert _{L^{2}\left(
\sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{eqnarray*}
Turning to the second sum, we note that for $k_{m}\leq k<k_{m+1}$ and $x\in
\theta \left( I_{k}\right) $ with $\theta \left( I_{k}\right) \notin
\mathcal{F}$, we hav
\begin{equation*}
\left\vert \psi _{F}\left( x\right) \right\vert \lesssim \left\vert
b_{\theta \left( I_{k}\right) }\right\vert \ E_{\theta \left( I_{k}\right)
}^{\sigma }\left\vert f\right\vert \ \mathbf{1}_{F_{m+1}\setminus
F_{m}}\left( x\right) \lesssim \alpha _{\mathcal{F}}\left( F_{m+1}\right) \
\mathbf{1}_{F_{m+1}\setminus F_{m}}\left( x\right) \ ,
\end{equation*
and hence the following inequality for $x\notin F$,
\begin{equation}
\left\vert \psi _{F}\left( x\right) \right\vert \lesssim \sum_{F^{\prime
}\in \mathcal{F}:\ F\subset F^{\prime }}\alpha _{\mathcal{F}}\left( \pi _
\mathcal{F}}F^{\prime }\right) \ \mathbf{1}_{\pi _{\mathcal{F}}F^{\prime
}\setminus F^{\prime }}\left( x\right) =\Phi \left( x\right) \ \mathbf{1
_{F^{c}}\left( x\right) \ , \label{Psi_F bound}
\end{equation
wher
\begin{equation*}
\Phi \equiv \sum_{F^{\prime \prime }\in \mathcal{F}}\alpha _{\mathcal{F
}\left( F^{\prime \prime }\right) \ \mathbf{1}_{F^{\prime \prime }\setminus
\cup \mathfrak{C}_{\mathcal{F}}\left( F^{\prime \prime }\right) }=\sum_{F\in
\mathcal{F}}\alpha _{\mathcal{F}}\left( F\right) \mathbf{1}_{F\setminus \cup
\mathfrak{C}_{\mathcal{F}}\left( F\right) }\ .
\end{equation*}
Now we writ
\begin{equation*}
\sum_{F\in \mathcal{F}}\ \left\langle T_{\sigma }^{\alpha }\psi
_{F},g_{F}\right\rangle _{\omega }=\sum_{F\in \mathcal{F}}\ \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{F}\psi _{F}\right)
,g_{F}\right\rangle _{\omega }+\sum_{F\in \mathcal{F}}\ \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{F^{c}}\psi _{F}\right)
,g_{F}\right\rangle _{\omega }\equiv I+II,
\end{equation*
where $I$ and $II$ are defined at the end of the display. Then by interval
testing,
\begin{equation*}
\left\vert \left\langle T_{\sigma }^{\alpha }\left( b_{F}\mathbf{1
_{F}\right) ,g_{F}\right\rangle _{\omega }\right\vert =\left\vert
\left\langle \mathbf{1}_{F}T_{\sigma }^{\alpha }\left( b_{F}\mathbf{1
_{F}\right) ,g_{F}\right\rangle _{\omega }\right\vert \lesssim \mathfrak{T
_{T^{\alpha }}\sqrt{\left\vert F\right\vert _{\sigma }}\left\Vert
g_{F}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\ ,
\end{equation*
and so quasi-orthogonality, together with the fact that on $F$, $\psi
_{F}=b_{F}\frac{E_{F}^{\sigma }f}{E_{F}^{\sigma }b_{F}}$ is a constant $c
\frac{E_{F}^{\sigma }f}{E_{F}^{\sigma }b_{F}}$ times $b_{F}$, where
\left\vert c\right\vert $ is bounded by $\alpha _{\mathcal{F}}\left(
F\right) $, give
\begin{eqnarray*}
&&\left\vert I\right\vert =\left\vert \sum_{F\in \mathcal{F}}\ \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{F}cb_{F}\right) ,g_{F}\right\rangle
_{\omega }\right\vert \lesssim \sum_{F\in \mathcal{F}}\ \alpha _{\mathcal{F
}\left( F\right) \ \left\vert \left\langle T_{\sigma }^{\alpha
}b_{F},g_{F}\right\rangle _{\omega }\right\vert \\
&\lesssim &\sum_{F\in \mathcal{F}}\ \alpha _{\mathcal{F}}\left( F\right)
\mathcal{NTV}_{\alpha }\sqrt{\left\vert F\right\vert _{\sigma }}\left\Vert
g_{F}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\lesssim \mathcal
NTV}_{\alpha }\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left[
\sum_{F\in \mathcal{F}}\left\Vert g_{F}\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar 2}\right] ^{\frac{1}{2}}.
\end{eqnarray*}
Now $\mathbf{1}_{F^{c}}\psi _{F}$ is supported outside $F$, and each $J$ in
the dual martingale support $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$
of $g_{F}=\mathsf{P}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}}^{\omega
}g$ is in particular $\limfunc{good}$ in the interval $F$, and as a
consequence, each such interval $J$ as above is contained in some interval
M $ for $M\in \mathcal{W}\left( F\right) $. This containment will be used in
the analysis of the term $II_{G}$ below.
In addition, each $J$ in the dual martingale support $\mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}}$ of $g_{F}=\mathsf{P}_{\mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}}}^{\omega }g$ is $\left( \left[ \frac{3}
\varepsilon }\right] ,\varepsilon \right) $-deeply embedded in $F$, i.e.
J\Subset _{\left[ \frac{3}{\varepsilon }\right] ,\varepsilon }F$, by Lemma
\ref{good scale} and the definition of $\mathcal{C}_{F}^{\mathcal{G}
\limfunc{shift}}$ in Definition \ref{def shift}. As a consequence, each such
interval $J$ as above is contained in some interval $M$ for $M\in \mathcal{M
_{\left( \left[ \frac{3}{\varepsilon }\right] ,\varepsilon \right) -\limfunc
deep},\mathcal{D}}\left( F\right) $. This containment will be used in the
analysis of the term $II_{B}$ below.
\begin{notation}
Define $\mathbf{\rho }\equiv \left[ \frac{3}{\varepsilon }\right] $, so that
for every $J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$, there is
M\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{G}}\left( F\right) $ such that $J\subset M$.
\end{notation}
The collections $\mathcal{W}\left( F\right) $ and $\mathcal{M}_{\left(
\mathbf{\rho },\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left(
F\right) $ used here, and in the display below, are defined in (\ref{def
M_r-deep}) in Appendix B. Finally, since the intervals $M\in \mathcal{W
\left( F\right) $, as well as the intervals $M\in \mathcal{M}_{\left( \left[
\frac{3}{\varepsilon }\right] ,\varepsilon \right) -\limfunc{deep},\mathcal{
}}\left( F\right) $, satisfy $3M\subset F$, we can apply (\ref{estimate}) in
the Monotonicity Lemma \ref{mono} using (\ref{Psi_F bound}) with $\mu
\mathbf{1}_{F^{c}}\psi _{F}$ and $J^{\prime }$ in place of $J$ there, to
obtai
\begin{eqnarray*}
\left\vert II\right\vert &=&\left\vert \sum_{F\in \mathcal{F}}\left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{F^{c}}\psi _{F}\right)
,g_{F}\right\rangle _{\omega }\right\vert =\left\vert \sum_{F\in \mathcal{F
}\sum_{J^{\prime }\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}}\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{F^{c}}\psi
_{F}\right) ,\square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\right\vert \\
&\lesssim &\sum_{F\in \mathcal{F}}\sum_{J^{\prime }\in \mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}}}\frac{\mathrm{P}^{\alpha }\left( J^{\prime }
\mathbf{1}_{F^{c}}\Phi \sigma \right) }{\left\vert J^{\prime }\right\vert
\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&&+\sum_{F\in \mathcal{F}}\sum_{J^{\prime }\in \mathcal{C}_{F}^{\mathcal{G}
\limfunc{shift}}}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime }
\mathbf{1}_{F^{c}}\Phi \sigma \right) }{\left\vert J^{\prime }\right\vert
\left\Vert x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \omega \right)
}\left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\lesssim &\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right)
\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{F^{c}}\Phi \sigma \right) }
\left\vert M\right\vert }\left\Vert \mathsf{Q}_{\mathcal{C}_{F;M}^{\mathcal{
},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit }\left\Vert g_{F;M}\right\Vert _{L^{2}\left(
\omega \right) }^{\bigstar } \\
&&+\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right)
}\sum_{J^{\prime }\in \mathcal{C}_{F;J}^{\mathcal{G},\limfunc{shift}}}\frac
\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime },\mathbf{1}_{F^{c}}\Phi
\sigma \right) }{\left\vert J^{\prime }\right\vert }\left\Vert
x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \mathbf{1}_{J^{\prime }}\omega
\right) }\left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\equiv &II_{G}+II_{B}\ .
\end{eqnarray*
where $g_{F;M}$ denotes the pseudoprojection $g_{F;M}=\sum_{J^{\prime }\in
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}:\ J^{\prime }\subset M}\square
_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g$.
\smallskip
\textbf{Note}: We could also bound $II_{G}$ by using the decomposition
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{G}}\left( F\right) $ of $F$ into certain maximal $\mathcal{G}
-intervals, but the `smaller' choice $\mathcal{W}\left( F\right) $ of
\mathcal{D}$-intervals is needed for $II_{G}$ in order to bound it by the
corresponding functional energy constant $\mathfrak{F}_{\alpha }^{\mathbf{b
^{\ast }}$, which can then be controlled by the energy and Muckenhoupt
constants in Appendix B.
\smallskip
Then from Cauchy-Schwarz, the functional energy condition, and
\begin{equation*}
\left\Vert \Phi \right\Vert _{L^{2}\left( \sigma \right) }^{2}\leq
\sum_{F\in \mathcal{F}}\alpha _{\mathcal{F}}\left( F\right) ^{2}\left\vert
F\right\vert _{\sigma }\lesssim \left\Vert f\right\Vert _{L^{2}\left( \sigma
\right) }^{2}\ ,
\end{equation*
we obtai
\begin{eqnarray*}
\left\vert II_{G}\right\vert &\leq &\left( \sum_{F\in \mathcal{F}}\sum_{M\in
\mathcal{W}\left( F\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M
\mathbf{1}_{F^{c}}\Phi \sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{\mathcal{C}_{F;M}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\right) ^{\frac{1}{2}} \\
&&\times \left( \sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right)
}\left\Vert g_{F;M}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar
2}\right) ^{\frac{1}{2}} \\
&\lesssim &\mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast }}\left\Vert \Phi
\right\Vert _{L^{2}\left( \sigma \right) }\left[ \sum_{F\in \mathcal{F
}\left\Vert g_{F}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2
\right] ^{\frac{1}{2}}\lesssim \mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast
}}\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) },
\end{eqnarray*
by the pairwise disjointedness of the coronas $\mathcal{C}_{F;M}^{\mathcal{G
,\limfunc{shift}}$ jointly in $F$ and $M$, which in turn follows from the
pairwise disjointedness (\ref{tau overlap}) of the shifted coronas $\mathcal
C}_{F}^{\mathcal{G},\limfunc{shift}}$ in $F$, together with the pairwise
disjointedness of the cubes $M$. Thus we obtain the pairwise disjointedness
of both of the pseudoprojections $\mathsf{P}_{\mathcal{C}_{F;M}^{\mathcal{G}
\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}$ and $\mathsf{Q}_{\mathcal{C
_{F;M}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}$ jointly
in $F$ and $M$.
In term $II_{B}$ the quantities $\left\Vert x-m_{J^{\prime }}\right\Vert
_{L^{2}\left( \mathbf{1}_{J^{\prime }}\omega \right) }^{2}$ are no longer
additive except when the intervals $J^{\prime }$ are pairwise disjoint. As a
result we will use (\ref{Haar trick}) in the form
\begin{eqnarray}
\sum_{J^{\prime }\subset J}\left( \frac{\mathrm{P}_{1+\delta }^{\alpha
}\left( J^{\prime },\nu \right) }{\left\vert J^{\prime }\right\vert }\right)
^{2}\left\Vert x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \mathbf{1
_{J^{\prime }}\omega \right) }^{2} &\lesssim &\frac{1}{\gamma ^{2\delta
^{\prime }}}\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left(
J,\nu \right) }{\left\vert J\right\vert }\right) ^{2}\sum_{J^{\prime \prime
}\subset J}\left\Vert \bigtriangleup _{J^{\prime \prime }}^{\omega
}x\right\Vert _{L^{2}\left( \omega \right) }^{2} \label{Haar trick'} \\
&\lesssim &\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left(
J,\nu \right) }{\left\vert J\right\vert }\right) ^{2}\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2}\ ,
\notag
\end{eqnarray
and exploit the decay in the Poisson integral $\mathrm{P}_{1+\delta ^{\prime
}}^{\alpha }$ along with weak goodness of the intervals $J$. As a
consequence we will be able to bound $II_{B}$ \emph{directly} by the strong
energy condition (\ref{strong b* energy}), without having to invoke the more
difficult functional energy condition. For the decay we compute that for
J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{G}}\left( F\right)
\begin{eqnarray*}
\frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left( J,\mathbf{1
_{F^{c}}\Phi \sigma \right) }{\left\vert J\right\vert } &\approx
&\int_{F^{c}}\frac{\left\vert J\right\vert ^{\delta ^{\prime }}}{\left\vert
y-c_{J}\right\vert ^{2+\delta ^{\prime }-\alpha }}\Phi \left( y\right)
d\sigma \left( y\right) \\
&\leq &\sum_{t=0}^{\infty }\int_{\pi _{\mathcal{F}}^{t+1}F\setminus \pi _
\mathcal{F}}^{t}F}\left( \frac{\left\vert J\right\vert }{\limfunc{dist
\left( c_{J},\left( \pi _{\mathcal{F}}^{t}F\right) ^{c}\right) }\right)
^{\delta ^{\prime }}\frac{1}{\left\vert y-c_{J}\right\vert ^{2-\alpha }}\Phi
\left( y\right) d\sigma \left( y\right) \\
&\lesssim &\sum_{t=0}^{\infty }\left( \frac{\left\vert J\right\vert }
\limfunc{dist}\left( c_{J},\left( \pi _{\mathcal{F}}^{t}F\right) ^{c}\right)
}\right) ^{\delta ^{\prime }}\frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1
_{\pi _{\mathcal{F}}^{t+1}F\setminus \pi _{\mathcal{F}}^{t}F}\Phi \sigma
\right) }{\left\vert J\right\vert },
\end{eqnarray*
and then use the weak goodness inequalit
\begin{equation*}
\limfunc{dist}\left( c_{J},\left( \pi _{\mathcal{F}}^{t}F\right) ^{c}\right)
\geq 2\ell \left( \pi _{\mathcal{F}}^{t}F\right) ^{1-\varepsilon }\ell
\left( J\right) ^{\varepsilon }\geq 2\cdot 2^{t\left( 1-\varepsilon \right)
}\ell \left( F\right) ^{1-\varepsilon }\ell \left( J\right) ^{\varepsilon
}\geq 2^{t\left( 1-\varepsilon \right) +1}\ell \left( J\right) ,
\end{equation*
to conclude tha
\begin{eqnarray}
\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left( J,\mathbf{1
_{F^{c}}\Phi \sigma \right) }{\left\vert J\right\vert }\right) ^{2}
&\lesssim &\left( \sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left(
1-\varepsilon \right) }\frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\pi _
\mathcal{F}}^{t+1}F\setminus \pi _{\mathcal{F}}^{t}F}\Phi \sigma \right) }
\left\vert J\right\vert }\right) ^{2} \label{decay in t} \\
&\lesssim &\sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left( 1-\varepsilon
\right) }\left( \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\pi _
\mathcal{F}}^{t+1}F\setminus \pi _{\mathcal{F}}^{t}F}\Phi \sigma \right) }
\left\vert J\right\vert }\right) ^{2}. \notag
\end{eqnarray
Now we first apply Cauchy-Schwarz and (\ref{Haar trick'}) to obtai
\begin{eqnarray*}
II_{B} &=&\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho
},\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right)
}\sum_{J^{\prime }\in \mathcal{C}_{F;J}^{\mathcal{G},\limfunc{shift}}}\frac
\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime },\mathbf{1}_{F^{c}}\Phi
\sigma \right) }{\left\vert J^{\prime }\right\vert }\left\Vert
x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \mathbf{1}_{J^{\prime }}\omega
\right) }\left\Vert \square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\left( \sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M}_{\left( \mathbf
\rho },\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right)
}\sum_{J^{\prime }\in \mathcal{C}_{F;J}^{\mathcal{G},\limfunc{shift}}}\left(
\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J^{\prime },\mathbf{1
_{F^{c}}\Phi \sigma \right) }{\left\vert J^{\prime }\right\vert }\right)
^{2}\left\Vert x-m_{J^{\prime }}\right\Vert _{L^{2}\left( \mathbf{1
_{J^{\prime }}\omega \right) }^{2}\right) ^{\frac{1}{2}}\left[
\sum_{F}\left\Vert g_{F}\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar
2}\right] ^{\frac{1}{2}} \\
&\leq &\left( \sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M}_{\left( \mathbf
\rho },\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right)
}\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha }\left( J,\mathbf{1
_{F^{c}}\Phi \sigma \right) }{\left\vert J\right\vert }\right)
^{2}\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}^{2}\right) ^{\frac{1}{2}}\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) } \\
&\equiv &\sqrt{II_{\limfunc{energy}}}\left\Vert g\right\Vert _{L^{2}\left(
\omega \right) },
\end{eqnarray*
and it remains to estimate $II_{\limfunc{energy}}$. From (\ref{decay in t})
and the strong energy condition (\ref{strong b* energy}), we hav
\begin{eqnarray*}
&&II_{\limfunc{energy}}=\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M
_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep},\mathcal{G
}\left( F\right) }\left( \frac{\mathrm{P}_{1+\delta ^{\prime }}^{\alpha
}\left( J,\mathbf{1}_{F^{c}}\Phi \sigma \right) }{\left\vert J\right\vert
\right) ^{2}\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega
\right) }^{2} \\
&\leq &\sum_{F\in \mathcal{F}}\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right)
}\sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left( 1-\varepsilon \right)
}\left( \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\pi _{\mathcal{F
}^{t+1}F\setminus \pi _{\mathcal{F}}^{t}F}\Phi \sigma \right) }{\left\vert
J\right\vert }\right) ^{2}\left\Vert x-m_{J}\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }^{2} \\
&=&\sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left( 1-\varepsilon \right)
}\sum_{G\in \mathcal{F}}\sum_{F\in \mathfrak{C}_{\mathcal{F}}^{\left(
t+1\right) }\left( G\right) }\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( F\right) }\left(
\frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{G\setminus \pi _{\mathcal{F
}^{t}F}\Phi \sigma \right) }{\left\vert J\right\vert }\right) ^{2}\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2} \\
&\lesssim &\sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left( 1-\varepsilon
\right) }\sum_{G\in \mathcal{F}}\alpha _{\mathcal{F}}\left( G\right)
^{2}\sum_{F\in \mathfrak{C}_{\mathcal{F}}^{\left( t+1\right) }\left(
G\right) }\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right)
\limfunc{deep}}\left( F\right) }\left( \frac{\mathrm{P}^{\alpha }\left( J
\mathbf{1}_{G\setminus \pi _{\mathcal{F}}^{t}F}\sigma \right) }{\left\vert
J\right\vert }\right) ^{2}\left\Vert x-m_{J}\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }^{2} \\
&\lesssim &\sum_{t=0}^{\infty }2^{-t\delta ^{\prime }\left( 1-\varepsilon
\right) }\sum_{G\in \mathcal{F}}\alpha _{\mathcal{F}}\left( G\right)
^{2}\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\left\vert G\right\vert
_{\sigma }\lesssim \left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\left\Vert
f\right\Vert _{L^{2}\left( \sigma \right) }^{2}.
\end{eqnarray*}
This completes the proof of the Intertwining Proposition \ref{strongly
adapted}.
\end{proof}
The task of controlling functional energy is taken up in Appendix B below.
\subsection{Paraproduct, neighbour and broken forms}
In this subsection we reduce boundedness of the local below form $\mathsf{B
_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left( f,g\right) $ defined in (\re
{def local}) to boundedness of the associated stopping for
\begin{equation}
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) \equiv \sum_{\substack{
I\in \mathcal{C}_{A}^{\mathcal{D}}\text{ and }J\in \mathcal{C}_{A}^{\mathcal
G},\limfunc{shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }}\left( E_{I_{J}}^{\sigma
}\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{A\setminus I_{J}}b_{A}\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ ,
\label{def stop}
\end{equation
where the modified difference $\widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}$ must be carefully chosen (see (\ref{flat box}) and (\ref{factor
b_A}) in Appendix A) in order to control the corresponding paraproduct form
below. Indeed, below we will decompose
\begin{equation*}
\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left( f,g\right)
\mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right) -\mathsf{B}_
\limfunc{stop}}^{A}\left( f,g\right) +\mathsf{B}_{\limfunc{neighbour
}^{A}\left( f,g\right) +\mathsf{B}_{\limfunc{broken}}^{A}\left( f,g\right) ,
\end{equation*
and then prove in (\ref{est para}), (\ref{est neigh}) and (\ref{broken
vanish}) the estimat
\begin{eqnarray*}
\left\vert \mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left(
f,g\right) +\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) \right\vert
&\leq &\left\vert \mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right)
\right\vert +\left\vert \mathsf{B}_{\limfunc{neighbour}}^{A}\left(
f,g\right) \right\vert +\left\vert \mathsf{B}_{\limfunc{broken}}^{A}\left(
f,g\right) \right\vert \\
&\lesssim &\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\ \alpha _{\mathcal{A
}\left( A\right) \ \sqrt{\left\vert A\right\vert _{\sigma }}\ \left\Vert
\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{
}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&&+\sqrt{\mathfrak{A}_{2}^{\alpha }}\left( \left\Vert \mathsf{P}_{\mathcal{C
_{A}}^{\sigma }f\right\Vert _{L^{2}(\sigma )}^{\bigstar }+\sqrt
\sum_{A^{\prime }\in \mathfrak{C}_{A}\left( A\right) }\left\vert A^{\prime
}\right\vert _{\sigma }\alpha _{\mathcal{A}}\left( A^{\prime }\right) ^{2}
\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}(\omega )}^{\bigstar }\ ,
\end{eqnarray*
which can of course then be summed in $A\in \mathcal{A}$ to conclude tha
\begin{eqnarray*}
&&\sum_{A\in \mathcal{A}}\left\vert \mathsf{B}_{\Subset _{\mathbf{r
,\varepsilon }}^{A}\left( f,g\right) +\mathsf{B}_{\limfunc{stop}}^{A}\left(
f,g\right) \right\vert \\
&\lesssim &\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \sqrt{\sum_{A\in \mathcal{A}}\left\{ \alpha _
\mathcal{A}}\left( A\right) ^{2}\left\vert A\right\vert _{\sigma
}+\left\Vert \mathsf{P}_{\mathcal{C}_{A}}^{\sigma }f\right\Vert
_{L^{2}(\sigma )}^{\bigstar 2}+\sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) }\alpha _{\mathcal{A}}\left( A^{\prime }\right)
^{2}\left\vert A^{\prime }\right\vert _{\sigma }\right\} }\sqrt{\sum_{A\in
\mathcal{A}}\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar 2}} \\
&\lesssim &\left( \mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) }\ .
\end{eqnarray*
The stopping form is the subject of the section following this one.
Note from (\ref{factor b_A}) and (\ref{telescoping})\ in Appendix A, that
the modified dual martingale differences $\square _{I}^{\sigma ,\flat
\mathbf{b}}$ and $\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}$,
\begin{equation*}
\square _{I}^{\sigma ,\flat ,\mathbf{b}}f\equiv \square _{I}^{\sigma
\mathbf{b}}f-\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{b}}f=b_{A}\sum_{I^
\prime }\in \mathfrak{C}\left( I\right) }\mathbf{1}_{I^{\prime
}}E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) =b_{A}\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f,
\end{equation*
satisfy the following telescoping property for all $K\in \left( \mathcal{C
_{A}\setminus \left\{ A\right\} \right) \cup \left( \bigcup_{A^{\prime }\in
\mathfrak{C}_{\mathcal{A}}\left( A\right) }A^{\prime }\right) $ and $L\in
\mathcal{C}_{A}$ with $K\subset L$
\begin{equation*}
\sum_{I:\ \pi K\subset I\subset L}E_{I_{K}}^{\sigma }\left( \widehat{\square
}_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) =\left\{
\begin{array}{ccc}
-E_{L}^{\sigma }\widehat{\mathbb{F}}_{L}^{\sigma ,\mathbf{b}}f & \text{ if }
& K\in \mathfrak{C}_{\mathcal{A}}\left( A\right) \\
E_{K}^{\sigma }\widehat{\mathbb{F}}_{K}^{\sigma ,\mathbf{b}}f-E_{L}^{\sigma
\widehat{\mathbb{F}}_{L}^{\sigma ,\mathbf{b}}f & \text{ if } & K\in \mathcal
C}_{A
\end{array
\right. .
\end{equation*
Fix $I\in \mathcal{C}_{A}$ for the moment. We will us
\begin{eqnarray*}
\mathbf{1}_{I} &=&\mathbf{1}_{I_{J}}+\mathbf{1}_{\theta \left( I_{J}\right)
}\ , \\
\mathbf{1}_{I_{J}} &=&\mathbf{1}_{A}-\mathbf{1}_{A\setminus I_{J}}\ ,
\end{eqnarray*
where $\theta \left( I_{J}\right) \in \mathfrak{C}_{\mathcal{D}}\left(
I\right) \setminus \left\{ I_{J}\right\} $ is the $\mathcal{D}$-child of $I$
other than the child $I_{J}$ that contains $J$. We begin with the splittin
\begin{eqnarray*}
&&\left\langle T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b
}f,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}=\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{I_{J}}\square
_{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }+\left\langle T_{\sigma }^{\alpha }\left( \mathbf
1}_{\theta \left( I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega } \\
&=&\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{I_{J}}\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }+\left\langle T_{\sigma }^{\alpha }\left(
\mathbf{1}_{I_{J}}\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }+\left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{\theta \left( I_{J}\right) }\square
_{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega } \\
&\equiv &I+II+III\ .
\end{eqnarray*
From (\ref{factor b_A}) we hav
\begin{eqnarray*}
I &=&\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{I_{J}}\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }=\left\langle T_{\sigma }^{\alpha }\left[
b_{A}\left( \mathbf{1}_{I_{J}}\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf
b}}f\right) \right] ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega } \\
&=&E_{I_{J}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{
}}f\right) \left\langle T_{\sigma }^{\alpha }\left( \mathbf{1
_{I_{J}}b_{A}\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega } \\
&=&E_{I_{J}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{
}}f\right) \left\langle T_{\sigma }^{\alpha }b_{A},\square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\rangle _{\omega }-E_{I_{J}}^{\sigma }\left(
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{A\setminus I_{J}}b_{A}\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ .
\end{eqnarray*
Since the function $\mathbb{F}_{I_{J}}^{\sigma ,\mathbf{b}}f$ is a constant
multiple of $b_{I_{J}}$ on $I_{J}$, we can define $\widehat{\mathbb{F}
_{I_{J}}^{\sigma ,\mathbf{b}}f\equiv \frac{1}{b_{I_{J}}}\mathbb{F
_{I_{J}}^{\sigma ,\mathbf{b}}f$ (or simply use the $PLBP$ we are assuming)
and the
\begin{equation*}
II=\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{I_{J}}\sum_{I^
\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) }\mathbb{F
_{I^{\prime }}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }=\mathbf{1}_{\mathfrak{C}_{\mathcal{A
}\left( A\right) }\left( I_{J}\right) \ E_{I_{J}}^{\sigma }\left( \widehat
\mathbb{F}}_{I_{J}}^{\sigma ,\mathbf{b}}f\right) \ \left\langle T_{\sigma
}^{\alpha }b_{I_{J}},\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }\ ,
\end{equation*
where the presence of the indicator function $\mathbf{1}_{\mathfrak{C}_
\mathcal{A}}\left( A\right) }\left( I_{J}\right) $ simply means that term
II $ vanishes unless $I_{J}$ is an $\mathcal{A}$-child of $A$. We now write
these terms a
\begin{eqnarray*}
\left\langle T_{\sigma }^{\alpha }\square _{I}^{\sigma ,\mathbf{b}}f,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }
&=&E_{I_{J}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{
}}f\right) \left\langle T_{\sigma }^{\alpha }b_{A},\square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\rangle _{\omega } \\
&&-E_{I_{J}}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{
}}f\right) \left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{A\setminus
I_{J}}b_{A}\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega } \\
&&+\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{\theta \left(
I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega } \\
&&+\mathbf{1}_{\left\{ I_{J}\in \mathfrak{C}_{\mathcal{A}}\left( A\right)
\right\} }\ E_{I_{J}}^{\sigma }\left( \widehat{\mathbb{F}}_{I_{J}}^{\sigma
\mathbf{b}}f\right) \ \left\langle T_{\sigma }^{\alpha }b_{I_{J}},\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ ,
\end{eqnarray*
where the four lines are respectively a paraproduct, stopping, neighbour and
broken term.
The corresponding NTV splitting of $\mathsf{B}_{\Subset _{\mathbf{r
,\varepsilon }}^{A}\left( f,g\right) $ using (\ref{def local}) and (\ref{def
shorthand}) become
\begin{eqnarray*}
\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left( f,g\right)
&=&\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C
_{A}}^{\sigma }f\right) ,\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}}^{\omega }g\right\rangle _{\omega }^{\Subset _{\mathbf{r
,\varepsilon }}=\sum_{\substack{ I\in \mathcal{C}_{A}\text{ and }J\in
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}} \\ J^{\maltese }\subsetneqq
\text{ and }\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }
\left\langle T_{\sigma }^{\alpha }\left( \square _{I}^{\sigma ,\mathbf{b
}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }
\\
&=&\mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right) -\mathsf{B}_
\limfunc{stop}}^{A}\left( f,g\right) +\mathsf{B}_{\limfunc{neighbour
}^{A}\left( f,g\right) +\mathsf{B}_{\limfunc{broken}}^{A}\left( f,g\right) ,
\end{eqnarray*
wher
\begin{eqnarray*}
\mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right) &\equiv &\sum
_{\substack{ I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{
},\limfunc{shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }}E_{I_{J}}^{\sigma
}\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right)
\left\langle T_{\sigma }^{\alpha }b_{A},\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\rangle _{\omega }\ , \\
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) &\equiv &\sum_{\substack{
I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq
2^{-\mathbf{r}}\ell \left( I\right) }}E_{I_{J}}^{\sigma }\left( \widehat
\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle T_{\sigma
}^{\alpha }\left( \mathbf{1}_{A\setminus I_{J}}b_{A}\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ , \\
\mathsf{B}_{\limfunc{neighbour}}^{A}\left( f,g\right) &\equiv &\sum
_{\substack{ I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{
},\limfunc{shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }}\left\langle T_{\sigma
}^{\alpha }\left( \mathbf{1}_{\theta \left( I_{J}\right) }\square
_{I}^{\sigma ,\mathbf{b}}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\ ,
\end{eqnarray*
correspond to the three original NTV forms associated with $1$-testing, and
where
\begin{equation}
\mathsf{B}_{\limfunc{broken}}^{A}\left( f,g\right) \equiv \sum_{\substack{
I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq
2^{-\mathbf{r}}\ell \left( I\right) }}\mathbf{1}_{\left\{ I_{J}\in \mathfrak
C}_{\mathcal{A}}\left( A\right) \right\} }\ E_{I_{J}}^{\sigma }\left(
\widehat{\mathbb{F}}_{I_{J}}^{\sigma ,\mathbf{b}}f\right) \ \left\langle
T_{\sigma }^{\alpha }b_{I_{J}},\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }=0\ , \label{broken vanish}
\end{equation
since $J^{\maltese }\subsetneqq I$ and $I_{J}\in \mathfrak{C}_{\mathcal{A
}\left( A\right) $ imply that $J^{\maltese }\notin \mathcal{C}_{A}^{\mathcal
G},\limfunc{shift}}$, contradicting $J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}$.
\begin{remark}
The inquisitive reader will note that the pairs $\left( I,J\right) $ arising
in the above sum with $J^{\maltese }\subsetneqq I$ replaced by $J^{\maltese
}=I$ are handled in the probabilistic estimate (\ref{HM bad}) for the bad
form $\Theta _{2}^{\limfunc{bad}\natural }$ defined in (\ref{Theta_2^bad
sharp}).
\end{remark}
\subsubsection{The paraproduct form}
The paraproduct form $\mathsf{B}_{\limfunc{paraproduct}}^{A}\left(
f,g\right) $ is easily controlled by the testing condition for $T^{\alpha }$
together with weak Riesz inequalities for dual martingale differences.
Indeed, recalling the telescoping identity (\ref{telescoping}), and that the
collection $\left\{ I\in \mathcal{C}_{A}\text{:\ }\ell \left( J\right) \leq
2^{-\mathbf{r}}\ell \left( I\right) \right\} $ is tree connected for all
J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}$, we hav
\begin{eqnarray*}
\mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right) &=&\sum_{\substack{
I\in \mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq
2^{-\mathbf{r}}\ell \left( I\right) }}E_{I_{J}}^{\sigma }\left( \widehat
\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle T_{\sigma
}^{\alpha }b_{A},\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega } \\
&=&\sum_{J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}\left\langle
T_{\sigma }^{\alpha }b_{A},\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\left\{ \sum_{I\in \mathcal{C}_{A}\text{:\
J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq 2^{-\mathbf{
}}\ell \left( I\right) }E_{I_{J}}^{\sigma }\left( \widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\} \\
&=&\sum_{J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}\left\langle
T_{\sigma }^{\alpha }b_{A},\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }\left\{ \mathbf{1}_{\left\{ J:I^{\natural }\left(
J\right) _{J}\in \mathcal{C}_{A}\right\} }E_{I^{\natural }\left( J\right)
_{J}}^{\sigma }\widehat{\mathbb{F}}_{I^{\natural }\left( J\right)
_{J}}^{\sigma ,\mathbf{b}}f-E_{A}^{\sigma }\widehat{\mathbb{F}}_{A}^{\sigma
\mathbf{b}}f\right\} \\
&=&\left\langle T_{\sigma }^{\alpha }b_{A},\sum_{J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{shift}}}\left\{ \mathbf{1}_{\left\{ J:I^{\natural
}\left( J\right) _{J}\in \mathcal{C}_{A}\right\} }E_{I^{\natural }\left(
J\right) _{J}}^{\sigma }\widehat{\mathbb{F}}_{I^{\natural }\left( J\right)
_{J}}^{\sigma ,\mathbf{b}}f-E_{A}^{\sigma }\widehat{\mathbb{F}}_{A}^{\sigma
\mathbf{b}}f\right\} \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }\ ,
\end{eqnarray*
where $I^{\natural }\left( J\right) $ denotes the smallest interval $I\in
\mathcal{C}_{A}$ such that $J^{\maltese }\subsetneqq I$ and $\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) $, and of course
I^{\natural }\left( J\right) _{J}$ denotes its child containing $J$. Note
that by construction of the modified difference operator $\square
_{I}^{\sigma ,\flat ,\mathbf{b}}$, the only time the average $\widehat
\mathbb{F}}_{I^{\natural }\left( J\right) _{J}}^{\sigma }f$ appears in the
above sum is when $I^{\natural }\left( J\right) _{J}\in \mathcal{C}_{A}$,
since the case $I^{\natural }\left( J\right) _{J}\in \mathcal{A}$ has been
removed to the broken term. This is reflected above with the inclusion of
the indicator $\mathbf{1}_{\left\{ J:I^{\natural }\left( J\right) _{J}\in
\mathcal{C}_{A}\right\} }$. It follows that we have the bound $\left\vert
\mathbf{1}_{\left\{ J:I^{\natural }\left( J\right) _{J}\in \mathcal{C
_{A}\right\} }E_{I^{\natural }\left( J\right) _{J}}^{\sigma }\widehat
\mathbb{F}}_{I^{\natural }\left( J\right) _{J}}^{\sigma ,\mathbf{b
}f\right\vert +\left\vert E_{A}^{\sigma }\widehat{\mathbb{F}}_{A}^{\sigma
\mathbf{b}}f\right\vert \lesssim E_{A}^{\sigma }\left\vert f\right\vert \leq
\alpha _{\mathcal{A}}\left( A\right) $.
Thus from Cauchy-Schwarz, the upper weak Riesz inequalities Proposition \re
{half Riesz} for the pseudoprojections $\square _{J}^{\omega ,\mathbf{b
^{\ast }}g$ and the bound on the coefficients $\lambda _{J}\equiv \left(
\mathbf{1}_{\left\{ J:I^{\natural }\left( J\right) _{J}\in \mathcal{C
_{A}\right\} }E_{I^{\natural }\left( J\right) _{J}}^{\sigma }\widehat
\mathbb{F}}_{I^{\natural }\left( J\right) _{J}}^{\sigma ,\mathbf{b
}f-E_{A}^{\sigma }\widehat{\mathbb{F}}_{A}^{\sigma ,\mathbf{b}}f\right) $
given by $\left\vert \lambda _{J}\right\vert \lesssim \alpha _{\mathcal{A
}\left( A\right) $, we hav
\begin{eqnarray}
&& \label{est para} \\
\left\vert \mathsf{B}_{\limfunc{paraproduct}}^{A}\left( f,g\right)
\right\vert &=&\left\vert \left\langle T_{\sigma }^{\alpha }b_{A},\sum_{J\in
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}\left\{ \left( \mathbf{1
_{\left\{ J:I^{\natural }\left( J\right) _{J}\in \mathcal{C}_{A}\right\}
}E_{I^{\natural }\left( J\right) _{J}}^{\sigma }\widehat{\mathbb{F}
_{I^{\natural }\left( J\right) _{J}}^{\sigma ,\mathbf{b}}f-E_{A}^{\sigma
\widehat{\mathbb{F}}_{A}^{\sigma ,\mathbf{b}}f\right) \right\} \square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert \notag
\\
&\leq &\left\Vert \mathbf{1}_{A}T_{\sigma }^{\alpha }b_{A}\right\Vert
_{L^{2}\left( \omega \right) }\left\Vert \sum_{J\in \mathcal{C}_{A}^
\mathcal{G},\limfunc{shift}}}\lambda _{J}\square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) } \notag \\
&\lesssim &\alpha _{\mathcal{A}}\left( A\right) \ \left\Vert \mathbf{1
_{A}T_{\sigma }^{\alpha }b_{A}\right\Vert _{L^{2}\left( \omega \right) }\
\left\Vert \sum_{J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{\bigstar } \notag \\
&\leq &\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}}\ \alpha _{\mathcal{A}}\left(
A\right) \ \sqrt{\left\vert A\right\vert _{\sigma }}\ \left\Vert \mathsf{P}_
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }. \notag
\end{eqnarray}
\subsubsection{The neighbour form}
Next, the neighbour form $\mathsf{B}_{\limfunc{neighbour}}^{A}\left(
f,g\right) $ is easily controlled by the $\mathfrak{A}_{2}^{\alpha }$
condition using the pivotal estimate in Energy Lemma \ref{ener} and the fact
that the intervals $J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}$ are
good in $I$ and beyond when the pair $\left( I,J\right) $ occurs in the sum.
In particular, the information encoded in the stopping tree $\mathcal{A}$
plays no role here, apart from appearing in the corona projections on the
right hand side of (\ref{est neigh}) below. We hav
\begin{equation}
\mathsf{B}_{\limfunc{neighbour}}^{A}\left( f,g\right) =\sum_{\substack{ I\in
\mathcal{C}_{A}\text{ and }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift
} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left( J\right) \leq 2^{
\mathbf{r}}\ell \left( I\right) }}\left\langle T_{\sigma }^{\alpha }\left(
\mathbf{1}_{\theta \left( I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b
}f\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
}, \label{def neighbour}
\end{equation
where we keep in mind that the pairs $\left( I,J\right) \in \mathcal{D
\times \mathcal{G}$ that arise in the sum for $\mathsf{B}_{\limfunc{neighbou
}}^{A}\left( f,g\right) $ satisfy the property that $J^{\maltese
}\subsetneqq I$, so that $J$ is good with respect to all intervals $K$ of
size at least that of $J^{\maltese }$, which includes $I$. Recall that
I_{J} $ is the child of $I$ that contains $J$, and that $\theta \left(
I_{J}\right) $ denotes its sibling in $I$, i.e. $\theta \left( I_{J}\right)
\in \mathfrak{C}_{\mathcal{D}}\left( I\right) \setminus \left\{
I_{J}\right\} $. Fix $\left( I,J\right) $ momentarily, and an integer $s\geq
\mathbf{r}$. Using $\square _{I}^{\sigma ,\mathbf{b}}=\square _{I}^{\sigma
,\flat ,\mathbf{b}}+\square _{I,\limfunc{broken}}^{\sigma ,\flat ,\mathbf{b
} $ and the fact that $\square _{I}^{\sigma ,\flat ,\mathbf{b}}f$ is a
constant multiple of $b_{\theta \left( I_{J}\right) }$ on the interval
\theta \left( I_{J}\right) $, we have the estimates
\begin{eqnarray*}
\left\vert \mathbf{1}_{\theta \left( I_{J}\right) }\square _{I}^{\sigma
,\flat ,\mathbf{b}}f\right\vert &=&\left\vert \left( E_{\theta \left(
I_{J}\right) }^{\sigma }\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b
}f\right) b_{\theta \left( I_{J}\right) }\right\vert \leq C_{\mathbf{b
}\left\vert E_{\theta \left( I_{J}\right) }^{\sigma }\widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\vert , \\
\left\vert \mathbf{1}_{\theta \left( I_{J}\right) }\square _{I,\limfunc
broken}}^{\sigma ,\flat ,\mathbf{b}}f\right\vert &\leq &\mathbf{1}_
\mathfrak{C}_{A}\left( A\right) }\left( \theta \left( I_{J}\right) \right) \
E_{\theta \left( I_{J}\right) }^{\sigma }\left\vert f\right\vert ,
\end{eqnarray*
and henc
\begin{equation}
\mathbf{1}_{\theta \left( I_{J}\right) }\left\vert \square _{I}^{\sigma
\mathbf{b}}f\right\vert \leq C\mathbf{1}_{\theta \left( I_{J}\right) }\left(
\left\vert E_{\theta \left( I_{J}\right) }^{\sigma }\widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\vert +\mathbf{1}_{\mathfrak{C
_{A}\left( A\right) }\left( \theta \left( I_{J}\right) \right) \ E_{\theta
\left( I_{J}\right) }^{\sigma }\left\vert f\right\vert \right) ,
\label{box bound}
\end{equation
which will be used below after an application of the Energy Lemma. We can
writ
\begin{equation*}
\mathsf{B}_{\limfunc{neighbour}}^{A}\left( f,g\right) =\sum_{\substack{ I\in
\mathcal{C}_{A}\text{ and }J\in \mathcal{G}_{\left( \kappa \left(
I_{J},J\right) ,\varepsilon \right) -\limfunc{good}}^{\mathcal{D}}\cap
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}\text{ and }J^{\maltese
}\subsetneqq I \\ d\left( J,\theta \left( I_{J}\right) \right) >2\ell
\left( J\right) ^{\varepsilon }\ell \left( \theta \left( I_{J}\right)
\right) ^{1-\varepsilon }\text{ and }\ell \left( J\right) \leq 2^{-\mathbf{r
}\ell \left( I\right) }}\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1
_{\theta \left( I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }
\end{equation*
where we have included the conditions $J\in \mathcal{G}_{\left( \kappa
\left( I_{J},J\right) ,\varepsilon \right) -\limfunc{good}}^{\mathcal{D}}$
and $d\left( J,\theta \left( I_{J}\right) \right) >2\ell \left( J\right)
^{\varepsilon }\ell \left( \theta \left( I_{J}\right) \right)
^{1-\varepsilon }$ in the summation since they are already implied the
remaining four conditions, and will be used in estimates below.
We will also use the following fractional analogue of the Poisson inequality
in \cite{Vol}.
\begin{lemma}
\label{Poisson inequality}Suppose $0\leq \alpha <1$ and $J\subset I\subset K$
and that $d\left( J,\partial I\right) >2\ell \left( J\right) ^{\varepsilon
}\ell \left( I\right) ^{1-\varepsilon }$ for some $0<\varepsilon <\frac{1}
2-\alpha }$. Then for a positive Borel measure $\mu $ we hav
\begin{equation}
\mathrm{P}^{\alpha }(J,\mu \mathbf{1}_{K\setminus I})\lesssim \left( \frac
\ell \left( J\right) }{\ell \left( I\right) }\right) ^{1-\varepsilon \left(
2-\alpha \right) }\mathrm{P}^{\alpha }(I,\mu \mathbf{1}_{K\setminus I}).
\label{e.Jsimeq}
\end{equation}
\end{lemma}
\begin{proof}
We hav
\begin{equation*}
\mathrm{P}^{\alpha }\left( J,\mu \mathbf{1}_{K\setminus I}\right) \approx
\sum_{k=0}^{\infty }2^{-k}\frac{1}{\left\vert 2^{k}J\right\vert ^{1-\alpha }
\int_{\left( 2^{k}J\right) \cap \left( K\setminus I\right) }d\mu ,
\end{equation*
and $\left( 2^{k}J\right) \cap \left( K\setminus I\right) \neq \emptyset $
require
\begin{equation*}
d\left( J,K\setminus I\right) \leq c2^{k}\ell \left( J\right) ,
\end{equation*
for some dimensional constant $c>0$. Let $k_{0}$ be the smallest such $k$.
By our distance assumption we must then hav
\begin{equation*}
2\ell \left( J\right) ^{\varepsilon }\ell \left( I\right) ^{1-\varepsilon
}\leq d\left( J,\partial I\right) \leq c2^{k_{0}}\ell \left( J\right) ,
\end{equation*
o
\begin{equation*}
2^{-k_{0}+1}\leq c\left( \frac{\ell \left( J\right) }{\ell \left( I\right)
\right) ^{1-\varepsilon }.
\end{equation*
Now let $k_{1}$ be defined by $2^{k_{1}}\equiv \frac{\ell \left( I\right) }
\ell \left( J\right) }$. Then assuming $k_{1}>k_{0}$ (the case $k_{1}\leq
k_{0}$ is similar) we hav
\begin{eqnarray*}
\mathrm{P}^{\alpha }\left( J,\mu \mathbf{1}_{K\setminus I}\right) &\approx
&\left\{ \sum_{k=k_{0}}^{k_{1}}+\sum_{k=k_{1}}^{\infty }\right\} 2^{-k}\frac
1}{\left\vert 2^{k}J\right\vert ^{1-\alpha }}\int_{\left( 2^{k}J\right) \cap
\left( K\setminus I\right) }d\mu \\
&\lesssim &2^{-k_{0}}\frac{\left\vert I\right\vert ^{1-\alpha }}{\left\vert
2^{k_{0}}J\right\vert ^{1-\alpha }}\left( \frac{1}{\left\vert I\right\vert
^{1-\alpha }}\int_{\left( 2^{k_{1}}J\right) \cap \left( K\setminus I\right)
}d\mu \right) +2^{-k_{1}}\mathrm{P}^{\alpha }\left( I,\mu \mathbf{1
_{K\setminus I}\right) \\
&\lesssim &\left( \frac{\ell \left( J\right) }{\ell \left( I\right) }\right)
^{\left( 1-\varepsilon \right) \left( 2-\alpha \right) }\left( \frac{\ell
\left( I\right) }{\ell \left( J\right) }\right) ^{1-\alpha }\mathrm{P
^{\alpha }\left( I,\mu \mathbf{1}_{K\setminus I}\right) +\frac{\ell \left(
J\right) }{\ell \left( I\right) }\mathrm{P}^{\alpha }\left( I,\mu \mathbf{1
_{K\setminus I}\right) ,
\end{eqnarray*
which is the inequality (\ref{e.Jsimeq}).
\end{proof}
Now fix $I_{0}=I_{J},I_{\theta }=\theta \left( I_{J}\right) \in \mathfrak{C
_{\mathcal{D}}\left( I\right) $ and assume that $J\Subset _{\mathbf{r
,\varepsilon }I_{0}$. Let $\frac{\ell \left( J\right) }{\ell \left(
I_{0}\right) }=2^{-s}$ in the pivotal estimate from Energy Lemma \ref{ener}
with $J\subset I_{0}\subset I$ to obtain
\begin{align*}
& \left\vert \langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{\theta \left(
I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\rangle _{\omega }\right\vert \\
& \lesssim \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }\sqrt{\left\vert J\right\vert _{\omega }
\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\theta \left( I_{J}\right)
}\left\vert \square _{I}^{\sigma ,\mathbf{b}}f\right\vert \sigma \right) \\
& \lesssim \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }\sqrt{\left\vert J\right\vert _{\omega }}\cdot
2^{-\left( 1-\varepsilon \left( 2-\alpha \right) \right) s}\mathrm{P
^{\alpha }\left( I_{0},\mathbf{1}_{\theta \left( I_{J}\right) }\left\vert
\square _{I}^{\sigma ,\mathbf{b}}f\right\vert \sigma \right) \\
& \lesssim \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }\sqrt{\left\vert J\right\vert _{\omega }}\cdot
2^{-\left( 1-\varepsilon \left( 2-\alpha \right) \right) s}\mathrm{P
^{\alpha }\left( I_{0},\mathbf{1}_{\theta \left( I_{J}\right) }\left(
\left\vert E_{\theta \left( I_{J}\right) }^{\sigma }\widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\vert +\mathbf{1}_{\mathfrak{C
_{A}\left( A\right) }\left( \theta \left( I_{J}\right) \right) \ E_{\theta
\left( I_{J}\right) }^{\sigma }\left\vert f\right\vert \right) \sigma
\right) .
\end{align*
Here we are using (\ref{e.Jsimeq}) in the third line, which applies since
J\subset I_{0}$, and we have used (\ref{box bound}) in the fourth line. It
will be convenient to use the shorthand notatio
\begin{equation*}
\mathbf{E}_{\theta \left( I_{J}\right) }^{\sigma }f\equiv \left\vert
E_{\theta \left( I_{J}\right) }^{\sigma }\widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right\vert +\mathbf{1}_{\mathfrak{C}_{A}\left( A\right)
}\left( \theta \left( I_{J}\right) \right) \ E_{\theta \left( I_{J}\right)
}^{\sigma }\left\vert f\right\vert
\end{equation*
where the intervals $I$ and $I_{J}$ on the right hand side are determined
uniquely by the interval $\theta \left( I_{J}\right) $.
In the sum below, we keep the side lengths of the intervals $J$ fixed at
2^{-s}$ times that of $I_{0}$, and of course take $J\subset I_{0}$. We also
keep the underlying assumptions that $J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}$ and that $J\in \mathcal{G}_{\left( \kappa \left(
I_{J},J\right) ,\varepsilon \right) -\limfunc{good}}^{\mathcal{D}}$ in mind
without necessarily pointing to them in the notation. Matters will shortly
be reduced to estimating the following term:
\begin{align*}
A(I,I_{0},I_{\theta },s)& \equiv \sum_{J\;:\;2^{s+1}\ell \left( J\right)
=\ell \left( I\right) :J\subset I_{0}}\left\vert \langle T_{\sigma }^{\alpha
}\left( \mathbf{1}_{I_{\theta }}\square _{I}^{\sigma ,\mathbf{b}}f\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\rangle _{\omega }\right\vert \\
& \leq 2^{-\left( 1-\varepsilon \left( 2-\alpha \right) \right) s}\left(
\mathbf{E}_{\theta \left( I_{J}\right) }^{\sigma }f\right) \ \mathrm{P
^{\alpha }(I_{0},\mathbf{1}_{\theta \left( I_{J}\right) }\sigma
)\sum_{J\;:\;2^{s+1}\ell \left( J\right) =\ell \left( I\right) :\ J\subset
I_{0}}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }\sqrt{\left\vert J\right\vert _{\omega }} \\
& \leq 2^{-\left( 1-\varepsilon \left( 2-\alpha \right) \right) s}\left(
\mathbf{E}_{\theta \left( I_{J}\right) }^{\sigma }f\right) \ \mathrm{P
^{\alpha }(I_{0},\mathbf{1}_{\theta \left( I_{J}\right) }\sigma )\sqrt
\left\vert I_{0}\right\vert _{\omega }}\Lambda (I,I_{0},I_{\theta },s), \\
& \text{where }\Lambda (I,I_{0},I_{\theta },s)^{2}\equiv \sum_{J\in \mathcal
C}_{A}^{\mathcal{G},\limfunc{shift}}:\;2^{s+1}\ell \left( J\right) =\ell
\left( I\right) :\ J\subset I_{0}}\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{2}\,.
\end{align*
The last line follows upon using the Cauchy-Schwarz inequality and the fact
that $J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}$. We also note that
since $2^{s+1}\ell \left( J\right) =\ell \left( I\right) $,
\begin{eqnarray}
\sum_{I_{0}\in \mathfrak{C}_{\mathcal{D}}\left( I\right) }\Lambda
(I,I_{0},I_{\theta },s)^{2} &\equiv &\sum_{J\in \mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}:\;2^{s+1}\ell \left( J\right) =\ell \left( I\right) :\
J\subset I}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{2}\ ; \label{g} \\
\sum_{I\in \mathcal{C}_{A}}\sum_{I_{0}\in \mathfrak{C}_{\mathcal{D}}\left(
I\right) }\Lambda (I,I_{0},I_{\theta },s)^{2} &\leq &\left\Vert \mathsf{P}_
\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}(\omega )}^{\bigstar 2}\ . \notag
\end{eqnarray}
Using (\ref{box hat bound}) we obtain
\begin{equation}
\left\vert E_{I_{\theta }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) \right\vert \leq \sqrt{E_{I_{\theta }}^{\sigma
}\left\vert \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right\vert
^{2}}\lesssim \left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\ \left\vert I_{\theta
}\right\vert _{\sigma }^{-\frac{1}{2}}, \label{e.haarAvg}
\end{equation
and henc
\begin{equation*}
\mathbf{E}_{\theta \left( I_{J}\right) }^{\sigma }f\equiv \left\vert
E_{\theta \left( I_{J}\right) }^{\sigma }\widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right\vert +\mathbf{1}_{\mathfrak{C}_{A}\left( A\right)
}\left( \theta \left( I_{J}\right) \right) \ E_{\theta \left( I_{J}\right)
}^{\sigma }\left\vert f\right\vert \lesssim \left( \left\Vert \square
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{\bigstar }\ +\mathbf{1}_{\mathfrak{C}_{A}\left( A\right) }\left( \theta
\left( I_{J}\right) \right) \ \left\vert I_{\theta }\right\vert _{\sigma }^
\frac{1}{2}}E_{\theta \left( I_{J}\right) }^{\sigma }\left\vert f\right\vert
\right) \left\vert I_{\theta }\right\vert _{\sigma }^{-\frac{1}{2}},
\end{equation*
and we can thus estimate $A(I,I_{0},I_{\theta },s)$ as follows:
\begin{eqnarray*}
&&A(I,I_{0},I_{\theta },s) \\
&\lesssim &2^{-\left( 1-\varepsilon \left( 2-\alpha \right) \right) s}\left(
\left\Vert \square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }^{\bigstar }+\mathbf{1}_{\mathfrak{C}_{A}\left( A\right)
}\left( I_{\theta }\right) \ \left\vert I_{\theta }\right\vert _{\sigma }^
\frac{1}{2}}E_{I_{\theta }}^{\sigma }\left\vert f\right\vert \right) \Lambda
(I,I_{0},I_{\theta },s)\cdot \left\vert I_{\theta }\right\vert _{\sigma }^{
\frac{1}{2}}\mathrm{P}^{\alpha }(I_{0},\mathbf{1}_{\theta \left(
I_{J}\right) }\sigma )\sqrt{\left\vert I_{0}\right\vert _{\omega }} \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}2^{-\left( 1-\varepsilon \left(
2-\alpha \right) \right) s}\left( \left\Vert \square _{I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar }+\mathbf{1}_
\mathfrak{C}_{A}\left( A\right) }\left( I_{\theta }\right) \ \left\vert
I_{\theta }\right\vert _{\sigma }^{\frac{1}{2}}E_{I_{\theta }}^{\sigma
}\left\vert f\right\vert \right) \Lambda (I,I_{0},I_{\theta },s)\,,
\end{eqnarray*
since $\mathrm{P}^{\alpha }(I_{0},\mathbf{1}_{\theta \left( I_{J}\right)
}\sigma )\lesssim \frac{\left\vert I_{\theta }\right\vert _{\sigma }}
\left\vert I_{\theta }\right\vert ^{1-\alpha }}$ shows that
\begin{equation*}
\left\vert I_{\theta }\right\vert _{\sigma }^{-\frac{1}{2}}\mathrm{P
^{\alpha }(I_{0},\mathbf{1}_{\theta \left( I_{J}\right) }\sigma )\ \sqrt
\left\vert I_{0}\right\vert _{\omega }}\lesssim \frac{\sqrt{\left\vert
I_{\theta }\right\vert _{\sigma }}\sqrt{\left\vert I_{0}\right\vert _{\omega
}}}{\left\vert I\right\vert ^{1-\alpha }}\lesssim \sqrt{\mathfrak{A
_{2}^{\alpha }}.
\end{equation*}
An application of Cauchy-Schwarz to the sum over $I$ using (\ref{g}) then
shows that
\begin{eqnarray*}
&&\sum_{I\in \mathcal{C}_{A}}\sum_{\substack{ I_{0},I_{\theta }\in \mathfrak
C}_{\mathcal{D}}\left( I\right) \\ I_{0}\neq I_{\theta }}
A(I,I_{0},I_{\theta },s) \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}2^{-\left( 1-\varepsilon \left(
2-\alpha \right) \right) s}\sqrt{\sum_{I\in \mathcal{C}_{A}}\left\Vert
\square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{\bigstar 2}+\sum_{I_{\theta }\in \mathfrak{C}_{A}\left( A\right)
}\left\vert I_{\theta }\right\vert _{\sigma }\left( E_{I_{\theta }}^{\sigma
}\left\vert f\right\vert \right) ^{2}}\sqrt{\sum_{I\in \mathcal{C
_{A}}\left( \sum_{\substack{ I_{0},I_{\theta }\in \mathfrak{C}_{\mathcal{D
}\left( I\right) \\ I_{0}\neq I_{\theta }}}\Lambda (I,I_{0},I_{\theta
},s)\right) ^{2}} \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}2^{-\left( 1-\varepsilon \left(
2-\alpha \right) \right) s}\sqrt{\left\Vert \mathsf{P}_{\mathcal{C
_{A}}^{\sigma }f\right\Vert _{L^{2}(\sigma )}^{\bigstar 2}+\sum_{A^{\prime
}\in \mathfrak{C}_{A}\left( A\right) }\left\vert A^{\prime }\right\vert
_{\sigma }\left( E_{A^{\prime }}^{\sigma }\left\vert f\right\vert \right)
^{2}}\sqrt{\sum_{I\in \mathcal{C}_{A}}\left( \sum_{\substack{ I_{0}\in
\mathfrak{C}_{\mathcal{D}}\left( I\right) \\ I_{0}\neq I_{\theta }}}\Lambda
(I,I_{0},I_{\theta },s)\right) ^{2}} \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}2_{{}}^{-\left( 1-\varepsilon
\left( 2-\alpha \right) \right) s}\left( \lVert \mathsf{P}_{\mathcal{C
_{A}}^{\sigma }f\rVert _{L^{2}(\sigma )}^{\bigstar }+\sqrt{\sum_{A^{\prime
}\in \mathfrak{C}_{A}\left( A\right) }\left\vert A^{\prime }\right\vert
_{\sigma }\left( E_{A^{\prime }}^{\sigma }\left\vert f\right\vert \right)
^{2}}\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}(\omega
)}^{\bigstar }\,.
\end{eqnarray*
This estimate is summable in $s\geq \mathbf{r}$ since $\varepsilon <\frac{1}
2-\alpha }$, and so the proof of
\begin{eqnarray}
\left\vert \mathsf{B}_{\limfunc{neighbour}}^{A}\left( f,g\right) \right\vert
&=&\left\vert \sum_{\substack{ I\in \mathcal{C}_{A}\text{ and }J\in \mathcal
C}_{A}^{\mathcal{G},\limfunc{shift}} \\ J^{\maltese }\subsetneqq I\text{
and }\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }
\left\langle T_{\sigma }^{\alpha }\left( \mathbf{1}_{\theta \left(
I_{J}\right) }\square _{I}^{\sigma ,\mathbf{b}}f\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert
\label{est neigh} \\
&\leq &\sum_{I\in \mathcal{C}_{A}}\sum_{\substack{ I_{0},I_{\theta }\in
\mathfrak{C}_{\mathcal{D}}\left( I\right) \\ I_{0}\neq I_{\theta }}}\sum_{s
\mathbf{r}}^{\infty }A(I,I_{0},I_{\theta },s) \notag \\
&\lesssim &\sqrt{\mathfrak{A}_{2}^{\alpha }}\left( \left\Vert \mathsf{P}_
\mathcal{C}_{A}}^{\sigma }f\right\Vert _{L^{2}(\sigma )}^{\bigstar }+\sqrt
\sum_{A^{\prime }\in \mathfrak{C}_{A}\left( A\right) }\left\vert A^{\prime
}\right\vert _{\sigma }\alpha _{\mathcal{A}}\left( A^{\prime }\right) ^{2}
\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}(\omega )}^{\bigstar }
\notag
\end{eqnarray
is complete since $E_{A^{\prime }}^{\sigma }\left\vert f\right\vert \lesssim
\alpha _{\mathcal{A}}\left( A^{\prime }\right) $.
\section{Stopping form\label{Sec stop}}
In this section, we modify our adaptation in \cite{SaShUr7}, \cite{SaShUr9}
and \cite{SaShUr10}\footnote
And correct an error in \cite{SaShUr7} related to the restricted norms of
stopping forms for admissible collections.} of the argument of M. Lacey in
\cite{Lac} to apply in the setting of a local $Tb$ theorem for an $\alpha
-fractional Calder\'{o}n-Zygmund operator $T^{\alpha }$ in $\mathbb{R}$
using the Monotonicity Lemma \ref{mono}, the energy condition, and the weak
goodness of Hyt\"{o}nen and Martikainen \cite{HyMa}. Following Lacey in \cit
{Lac}, we construct $\mathcal{L}\,$-coronas from the `bottom up' with
stopping times involving the energies $\left\Vert \bigtriangleup
_{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{2}$, but then overlay this with an additional top/down `indented' corona
construction in order to accommodate the weaker goodness of Hyt\"{o}nen and
Martikainen. We directly control the pairs $\left( I,J\right) $ in the
stopping form `essentially' according to the $\mathcal{L}\,$-coronas to
which $I$ and $J^{\maltese }$ are associated, by absorbing the case when
both $I$ and $J^{\maltese }$ belong to the same $\mathcal{L}\,$-corona, and
by using the Straddling and Subtraddling Lemmas to control the case when $I$
and $J^{\maltese }$ lie in different coronas, with a geometric gain coming
from the separation of the coronas in the `indented' construction overlaying
Lacey's bottom/up construction (we actually use the grandchild $J^{\flat }$
of $J^{\maltese }$ that contains $J$ to distinguish aborption cases from
straddling cases). We also use a Corona-straddling Lemma to control certain
extremal pairs $\left( I,J\right) $ that straddle two $\mathcal{A}$-coronas.
As in \cite{Lac}, an Orthogonality Lemma proves useful in all cases.
Finally, since we are using two independent dyadic grids, we must enlarge
the $\limfunc{skeleton}$ of an interval to include an infinite sequence of
points we call the $\limfunc{body}$ of the interval.
Apart from these changes, the remaining modifications are more obvious, such
as
\begin{itemize}
\item the use of the weak goodness of Hyt\"{o}nen and Martikainen \cite{HyMa}
for pairs $\left( I,J\right) $ arising in the stopping form, rather than
goodness for all intervals $I$ and $J$ that was available in \cite{Lac},
\cite{SaShUr7}, \cite{SaShUr9} and \cite{SaShUr10}. For the most part
definitions such as admissible collections are modified to require
J^{\maltese }\subsetneqq I$. In paricular, Lacey's size functional is
enlarged to include more intervals $K\in \mathcal{D}$ that are not good;
\item the pseudoprojections $\square _{I}^{\sigma ,\mathbf{b}},\square
_{J}^{\omega ,\mathbf{b}^{\ast }}$ and Carleson averaging operators $\nabla
_{I}^{\sigma },\nabla _{J}^{\omega }$ are used in place of the orthogonal
Haar projections, and the frame and weak Riesz inequalities compensate for
the lack of orthogonality.
\end{itemize}
Fix grids $\mathcal{D}$ and $\mathcal{G}$. In Section 6 we reduced matters
to proving (\ref{local est}), i.e
\begin{equation}
\left\vert \mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) \right\vert
\lesssim \mathcal{NTV}_{\alpha }\left( \left\Vert \mathsf{P}_{\mathcal{C
_{A}^{\mathcal{D}}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }^{\bigstar }+\alpha _{\mathcal{A}}\left( A\right) \sqrt{\left\vert
A\right\vert _{\sigma }}\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}^
\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }\ , \label{B stop form 3}
\end{equation
where we recall that $\mathcal{NTV}_{\alpha }$ is defined in (\ref{def NTV
), and the nonstandard `norms' are given in Notation \ref{nonstandard norm}
by
\begin{eqnarray*}
\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{D}}}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar 2} &\equiv
&\sum_{I\in \mathcal{C}_{A}^{\mathcal{D}}}\left( \left\Vert \square
_{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{2}+\left\Vert \nabla _{I}^{\sigma }f\right\Vert _{L^{2}\left( \sigma
\right) }^{2}\right) , \\
\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift
}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{\bigstar 2} &\equiv &\sum_{J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc
shift}}}\left( \left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{2}+\left\Vert \nabla
_{J}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) }^{2}\right) ,
\end{eqnarray*
and that the stopping form is given in (\ref{def stop}) b
\begin{equation*}
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) \equiv \sum_{\substack{
I\in \mathcal{C}_{A}^{\mathcal{D}}\text{ and }J\in \mathcal{C}_{A}^{\mathcal
G},\limfunc{shift}} \\ J^{\maltese }\subsetneqq I\text{ and }\ell \left(
J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }}\left( E_{I_{J}}^{\sigma
}\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle
T_{\sigma }^{\alpha }\left( \mathbf{1}_{A\setminus I_{J}}b_{A}\right)
,\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ .
\end{equation*}
It is important to note that $J^{\maltese }\subsetneqq I$ implies
J^{\maltese }\subset I_{J}$, and it follows that we \emph{cannot} have
I_{J}\in \mathfrak{C}_{\mathcal{A}}\left( A\right) $, i.e. we cannot have
that the child of $I$ containing $J$ is a stopping interval in $\mathcal{A}
, since this would then contradict the assumption that $J\in \mathcal{C
_{A}^{\mathcal{G},\limfunc{shift}}$. Furthermore, the pair $\left(
I,J\right) =\left( J^{\maltese },J\right) $ does not arise in the sum simply
because of the requirement $J^{\maltese }\subsetneqq I$. For convenience in
notation, and without loss of generality, we now reindex the stopping form
with this in mind by replacing the pairs $\left( I,J\right) $ in the sum
above with new pairs $\left( I^{\prime },J^{\prime }\right) \equiv \left(
I_{J},J\right) $ (recall that the child of $I$ that contains $J$ is denoted
I_{J}$). The result is tha
\begin{equation*}
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) =\sum_{\substack{ I^{\prime
}\in \mathcal{C}_{A}^{\mathcal{D},\limfunc{restrict}}\text{ and }J^{\prime
}\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}} \\ J^{\maltese }\subset
I^{\prime }\text{ and }\ell \left( J^{\prime }\right) \leq 2^{1-\mathbf{r
}\ell \left( I^{\prime }\right) }}\left( E_{I^{\prime }}^{\sigma }\widehat
\square }_{\pi I^{\prime }}^{\sigma ,\flat ,\mathbf{b}}f\right) \left\langle
T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus I^{\prime }}\right)
,\square _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega
},
\end{equation*
where
\begin{equation}
\mathcal{C}_{A}^{\mathcal{D},\limfunc{restrict}}\equiv \mathcal{C
_{A}\setminus \left\{ A\right\} \label{def restrict}
\end{equation
is the $A$-corona with its top interval $A$ removed. Now we simply drop the
primes from the dummy variables $I^{\prime }$ and $J^{\prime }$ and relabel
1-\mathbf{r}$ as $-\mathbf{r}$ to obtain
\begin{equation}
\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) =\sum_{\substack{ I\in
\mathcal{C}_{A}^{\mathcal{D},\limfunc{restrict}}\text{ and }J\in \mathcal{C
_{A}^{\mathcal{G},\limfunc{shift}} \\ J^{\maltese }\subset I\text{ and
\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) }}\left(
E_{I}^{\sigma }\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right) \left\langle T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1
_{A\setminus I}\right) ,\square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\rangle _{\omega }. \label{dummy}
\end{equation}
\begin{definition}
Suppose that $A\in \mathcal{A}$ and that $\mathcal{P}\subset \mathcal{C
_{A}^{\mathcal{D},\limfunc{restrict}}\times \mathcal{C}_{A}^{\mathcal{G}
\limfunc{shift}}$. We say that the collection of pairs $\mathcal{P}$ is $A
\emph{-admissible} if
\end{definition}
\begin{itemize}
\item (good and $\left( \mathbf{r},\varepsilon \right) $-deeply embedded)
\ell \left( J\right) \leq 2^{-\mathbf{r}}\ell \left( I\right) $ and
J^{\maltese }\subset I\varsubsetneqq A$ for every $\left( I,J\right) \in
\mathcal{P},$
\item (tree-connected in the first component) if $I_{1}\subset I_{2}$ and
both $\left( I_{1},J\right) \in \mathcal{P}$ and $\left( I_{2},J\right) \in
\mathcal{P}$, then $\left( I,J\right) \in \mathcal{P}$ for every $I$ in the
geodesic $\left[ I_{1},I_{2}\right] =\left\{ I\in \mathcal{D}:I_{1}\subset
I\subset I_{2}\right\} $.
\end{itemize}
From now on we often write $\mathcal{C}_{A}$ and $\mathcal{C}_{A}^{\limfunc
restrict}}$ in place of $\mathcal{C}_{A}^{\mathcal{D}}$ and $\mathcal{C
_{A}^{\mathcal{D},\limfunc{restrict}}$ respectively, i.e. we drop the
superscript $\mathcal{D}$, when there is no possiblility of confusion. The
basic example of an admissible collection of pairs is obtained from the
pairs of intervals summed in the stopping form $\mathsf{B}_{\limfunc{stop
}^{A}\left( f,g\right) $ in (\ref{dummy}) which occurs in the inequality
\ref{B stop form 3}) above;
\begin{equation}
\mathcal{P}^{A}\equiv \left\{ \left( I,J\right) :I\in \mathcal{C}_{A}^
\limfunc{restrict}}\text{, }J\in \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift
}\text{, }J^{\maltese }\subset I\text{ and\ }\ell \left( J\right) \leq 2^{
\mathbf{r}}\ell \left( I\right) \right\} . \label{initial P}
\end{equation}
\begin{definition}
\label{def stop P}Suppose that $A\in \mathcal{A}$ and that $\mathcal{P}$ is
an $A$\emph{-admissible} collection of pairs. Define the associated \emph
stopping} form $\mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}$ b
\begin{equation*}
\mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}\left( f,g\right) \equiv
\sum_{\left( I,J\right) \in \mathcal{P}}\left( E_{I}^{\sigma }\widehat
\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \left\langle
T_{\sigma }^{\alpha }\left( b_{A}\mathbf{1}_{A\setminus I}\right) ,\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\ .
\end{equation*
where $\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}$ is the
modified dual martingale difference defined in (\ref{flat box}) and (\re
{flat box hat}).
\end{definition}
Recall the strong energy condition constant $\mathcal{E}_{2}^{\alpha }$
defined b
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\equiv \sup_{I=\dot{\cup}I_{r}
\frac{1}{\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\left( \frac
\mathrm{P}^{\alpha }\left( I_{r},\mathbf{1}_{I}\sigma \right) }{\left\vert
I_{r}\right\vert }\right) ^{2}\left\Vert x-m_{I_{r}}^{\omega }\right\Vert
_{L^{2}\left( \mathbf{1}_{I_{r}}\omega \right) }^{2}\ .
\end{equation*}
\begin{proposition}
\label{stopping bound}Suppose that $A\in \mathcal{A}$ and that $\mathcal{P}$
is an $A$\emph{-admissible} collection of pairs. Then the stopping form
\mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}$ satisfies the boun
\begin{equation}
\left\vert \mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}\left( f,g\right)
\right\vert \lesssim \left( \mathcal{E}_{2}^{\alpha }+\sqrt{\mathfrak{A
_{2}^{\alpha }}\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}}^{\sigma
\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert
\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{
}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\ .
\label{B stop form}
\end{equation}
\end{proposition}
The above proposition proves (\ref{B stop form 3}) - even without the term
\alpha _{\mathcal{A}}\left( A\right) \sqrt{\left\vert A\right\vert _{\sigma
}$ on the right - with the choice $\mathcal{P}=\mathcal{P}^{A}$. To prove
Proposition \ref{stopping bound}, we begin by letting
\begin{eqnarray*}
\Pi _{1}\mathcal{P} &\equiv &\left\{ I\in \mathcal{C}_{A}^{\mathcal{D}
\limfunc{restrict}}:\left( I,J\right) \in \mathcal{P}\text{ for some }J\in
\mathcal{C}_{A}^{\mathcal{G},\func{shift}}\right\} , \\
\Pi _{2}\mathcal{P} &\equiv &\left\{ J\in \mathcal{C}_{A}^{\mathcal{G},\func
shift}}:\left( I,J\right) \in \mathcal{P}\text{ for some }I\in \mathcal{C
_{A}^{\mathcal{D},\limfunc{restrict}}\right\} ,
\end{eqnarray*
consist of the first and second components respectively of the pairs in
\mathcal{P}$, and writing
\begin{eqnarray}
\mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}\left( f,g\right) &=&\sum_{J\in
\Pi _{2}\mathcal{P}}\left\langle T_{\sigma }^{\alpha }\varphi _{J}^{\mathcal
P}},\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega };
\label{def phi P} \\
\text{where }\varphi _{J}^{\mathcal{P}} &\equiv &\sum_{I\in \mathcal{C}_{A}^
\limfunc{restrict}}:\ \left( I,J\right) \in \mathcal{P}}b_{A}\left(
E_{I}^{\sigma }\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right) \ \mathbf{1}_{A\setminus I}\ , \notag
\end{eqnarray
where $E_{I}^{\sigma }h\equiv \frac{1}{\left\vert I\right\vert _{\sigma }
\int_{I}hd\sigma $ denotes the $\sigma $-average of $h$ on\thinspace $I$,
and where we note that the function $\widehat{\square }_{\pi I}^{\sigma
,\flat ,\mathbf{b}}f$ is constant on $I$, so that $E_{I}^{\sigma }\widehat
\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f$ simply picks out the value
of $\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f$ on $I$. By the
tree-connected property of $\mathcal{P}$, and the telescoping property (\re
{telescoping}), together with the bound $\alpha _{\mathcal{A}}\left(
A\right) $ on the averages of $f$ in the corona $\mathcal{C}_{A}$, we hav
\begin{equation}
\left\vert \varphi _{J}^{\mathcal{P}}\right\vert \lesssim \alpha _{\mathcal{
}}\left( A\right) \mathbf{1}_{A\setminus I_{\mathcal{P}}\left( J\right) },
\label{phi bound}
\end{equation
where $I_{\mathcal{P}}\left( J\right) \equiv \dbigcap \left\{ I:\left(
I,J\right) \in \mathcal{P}\right\} $ is the smallest interval $I$ for which
\left( I,J\right) \in \mathcal{P}$. It is important to note that $J$ is good
with respect to $I_{\mathcal{P}}\left( J\right) $ and beyond by the infusion
of weak goodness above.
Another important property of these functions is the sublinearity
\begin{equation}
\left\vert \varphi _{J}^{\mathcal{P}}\right\vert \leq \left\vert \varphi
_{J}^{\mathcal{P}_{1}}\right\vert +\left\vert \varphi _{J}^{\mathcal{P
_{2}}\right\vert ,\ \ \ \ \ \mathcal{P}=\mathcal{P}_{1}\dot{\cup}\mathcal{P
_{2}\ , \label{phi sublinear}
\end{equation
which is an immediate consequence of
\begin{equation*}
\varphi _{J}^{\mathcal{P}_{1}\dot{\cup}\mathcal{P}_{2}}=\sum_{I\in \mathcal{
}_{A}^{\limfunc{restrict}}:\ \left( I,J\right) \in \mathcal{P}_{1}\dot{\cup
\mathcal{P}_{2}}\left\{ ...\right\} =\sum_{I\in \mathcal{C}_{A}^{\limfunc
restrict}}:\ \left( I,J\right) \in \mathcal{P}_{1}}\left\{ ...\right\}
+\sum_{I\in \mathcal{C}_{A}^{\limfunc{restrict}}:\ \left( I,J\right) \in
\mathcal{P}_{2}}\left\{ ...\right\} =\varphi _{J}^{\mathcal{P}_{1}}+\varphi
_{J}^{\mathcal{P}_{2}}.
\end{equation*
Now apply the Monotonicity Lemma \ref{mono} to the inner product
\left\langle T_{\sigma }^{\alpha }\varphi _{J},\square _{J}^{\omega ,\mathbf
b}^{\ast }}g\right\rangle _{\omega }$ (which applies since $J$ is good in
I_{\mathcal{P}}\left( J\right) $) to obtai
\begin{eqnarray*}
\left\vert \left\langle T_{\sigma }^{\alpha }\varphi _{J},\square
_{J}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle _{\omega }\right\vert
&\lesssim &\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^
\mathcal{P}}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{P}}\left(
J\right) }\sigma \right) }{\left\vert J\right\vert }\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit }\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&&+\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\left\vert \varphi _{J}^
\mathcal{P}}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{P}}\left(
J\right) }\sigma \right) }{\left\vert J\right\vert }\left\Vert
x-m_{J}^{\omega }\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }.
\end{eqnarray*
Thus we hav
\begin{eqnarray}
\left\vert \mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}\left( f,g\right)
\right\vert &\leq &\sum_{J\in \Pi _{2}\mathcal{P}}\frac{\mathrm{P}^{\alpha
}\left( J,\left\vert \varphi _{J}^{\mathcal{P}}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma \right) }{\left\vert
J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \label{def mod B} \\
&&+\sum_{J\in \Pi _{2}\mathcal{P}}\frac{\mathrm{P}_{1+\delta }^{\alpha
}\left( J,\left\vert \varphi _{J}^{\mathcal{P}}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma \right) }{\left\vert
J\right\vert }\left\Vert x-m_{J}^{\omega }\right\Vert _{L^{2}\left( \mathbf{
}_{J}\omega \right) }\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \notag \\
&\equiv &\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup
^{\omega }}^{A,\mathcal{P}}\left( f,g\right) +\left\vert \mathsf{B
\right\vert _{\limfunc{stop},1+\delta ,\mathsf{P}^{\omega }}^{A,\mathcal{P
}\left( f,g\right) , \notag
\end{eqnarray
where we have dominated the stopping form by two sublinear stopping forms
that involve the Poisson integrals of order $1$ and $1+\delta $
respectively, and where the smaller Poisson integral $\mathrm{P}_{1+\delta
}^{\alpha }$ is multiplied by the larger quantity $\left\Vert
x-m_{J}^{\omega }\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }$.
It remains to show the following two inequalities where we abbreviate
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}$ to $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}$ and $\left\vert \mathsf{B
\right\vert _{\limfunc{stop},1+\delta ,\mathsf{P}^{\omega }}^{A,\mathcal{P}}$
to $\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal
P}}$
\begin{equation}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) \lesssim \left( \mathcal{E}_{2}^{\alpha
}+\sqrt{\mathfrak{A}_{2}^{\alpha }}\right) \left\Vert \mathsf{P}_{\pi \left(
\Pi _{1}\mathcal{P}\right) }^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }^{\bigstar }\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P
}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right)
}^{\bigstar }, \label{First inequality}
\end{equation
for $f\in L^{2}\left( \sigma \right) $ satisfying $E_{I}^{\sigma }\left\vert
f\right\vert \lesssim \alpha _{\mathcal{A}}\left( A\right) $ for all $I\in
\mathcal{C}_{A}$, and where $\pi \left( \Pi _{1}\mathcal{P}\right) \equiv
\left\{ \pi _{\mathcal{D}}I:I\in \Pi _{1}\mathcal{P}\right\} $; an
\begin{equation}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal{P
}\left( f,g\right) \lesssim \left( \mathcal{E}_{2}^{\alpha }+\sqrt
A_{2}^{\alpha }}\right) \left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{D
}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
\mathsf{P}_{\mathcal{C}_{A}^{\mathcal{G},\func{shift}}}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) },
\label{Second inequality}
\end{equation
where we only need the case $\mathcal{P}=\mathcal{P}^{A}$ in this latter
inequality as there is no absorption involved in treating this second
sublinear form. We consider first the easier inequality (\ref{Second
inequality}) that does not require absorption.
\subsection{The bound for the second sublinear inequality}
Here we turn to proving (\ref{Second inequality}), i.e
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal{P
}\left( f,g\right) &=&\sum_{J\in \Pi _{2}\mathcal{P}}\frac{\mathrm{P
_{1+\delta }^{\alpha }\left( J,\left\vert \varphi _{J}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma \right) }{\left\vert
J\right\vert }\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1
_{J}\omega \right) }\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\lesssim &\left( \mathcal{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}\right)
\left\Vert \mathsf{P}_{\mathcal{C}_{A}^{\mathcal{D}}}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert \mathsf{P}_{\mathcal{
}_{A}^{\mathcal{G},\func{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) },
\end{eqnarray*
where since
\begin{equation*}
\left\vert \varphi _{J}\right\vert =\left\vert \sum_{I\in \mathcal{C
_{A}^{\prime }:\ \left( I,J\right) \in \mathcal{P}}\left( E_{I}^{\sigma
}\square _{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \mathbf{1
_{A\setminus I}\right\vert \leq \sum_{I\in \mathcal{C}_{A}^{\prime }:\
\left( I,J\right) \in \mathcal{P}}\left\vert E_{I}^{\sigma }\square _{\pi
I}^{\sigma ,\flat ,\mathbf{b}}\left( f\right) \ \mathbf{1}_{A\setminus
I}\right\vert ,
\end{equation*
the sublinear form $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,1+\delta }^{A,\mathcal{P}}$ can be decomposed by pigeonholing the ratio of
side lengths of $J$ and $I$
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal{P
}\left( f,g\right) &=&\sum_{J\in \Pi _{2}\mathcal{P}}\frac{\mathrm{P
_{1+\delta }^{\alpha }\left( J,\left\vert \varphi _{J}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma \right) }{\left\vert
J\right\vert ^{\frac{1}{n}}}\left\Vert x-m_{J}\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\sum_{\left( I,J\right) \in \mathcal{P}}\frac{\mathrm{P}_{1+\delta
}^{\alpha }\left( J,\left\vert E_{I}^{\sigma }\left( \square _{\pi
I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert \mathbf{1}_{A\setminus
I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}{n}}}\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \\
&\equiv &\sum_{s=0}^{\infty }\left\vert \mathsf{B}\right\vert _{\limfunc{sto
},1+\delta }^{A,\mathcal{P};s}\left( f,g\right) ; \\
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal{P
;s}\left( f,g\right) &\equiv &\sum_{\substack{ \left( I,J\right) \in
\mathcal{P} \\ \ell \left( J\right) =2^{-s}\ell \left( I\right) }}\frac
\mathrm{P}_{1+\delta }^{\alpha }\left( J,\left\vert \left( E_{I}^{\sigma
}\square _{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert \mathbf{1
_{A\setminus I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}{n}}
\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }.
\end{eqnarray*
We will now adapt the argument for the stopping term starting on page 42 of
\cite{LaSaUr2}, where the geometric gain from the assumed Energy Hypothesis
there will be replaced by a geometric gain from the smaller Poisson integral
$\mathrm{P}_{1+\delta }^{\alpha }$ used here.
We exploit the additional decay in the Poisson integral $\mathrm{P
_{1+\delta }^{\alpha }$ as follows. Suppose that $J$ is good in $I$ with
\ell \left( J\right) =2^{-s}\ell \left( I\right) $. We then obtain from (\re
{Poisson decay}) above tha
\begin{equation}
\left( \frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{A\setminus
I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}{n}}}\right) \lesssim
2^{-s\delta \left( 1-\varepsilon \right) }\frac{\mathrm{P}^{\alpha }\left( J
\mathbf{1}_{A\setminus I}\sigma \right) }{\left\vert J\right\vert ^{\frac{1}
n}}}. \label{Poisson decay'}
\end{equation
We next claim that for $s\geq 0$ an integer
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A,\mathcal{P
;s}\left( f,g\right) &=&\sum_{\substack{ \left( I,J\right) \in \mathcal{P}
\\ \ell \left( J\right) =2^{-s}\ell \left( I\right) }}\frac{\mathrm{P
_{1+\delta }^{\alpha }\left( J,\left\vert \left( E_{I}^{\sigma }\square
_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \right\vert \mathbf{1
_{A\setminus I}\sigma \right) }{\left\vert J\right\vert }\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \\
&\lesssim &2^{-s\delta \left( 1-\varepsilon \right) }\ \left( \mathcal{E
_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}\right) \ \left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\ \left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\,,
\end{eqnarray*
from which (\ref{Second inequality}) follows upon summing in $s\geq 0$. Now
using bot
\begin{eqnarray*}
\left\vert E_{I}^{\sigma }\square _{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right\vert &=&\frac{1}{\left\vert I\right\vert _{\sigma }
\int_{I}\left\vert \square _{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right\vert
d\sigma \leq \left\Vert \square _{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }\frac{1}{\sqrt{\left\vert
I\right\vert _{\sigma }}}, \\
\sum_{I\in \mathcal{D}}\left\Vert \square _{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2} &\lesssim &\sum_{I\in
\mathcal{D}}\left( \left\Vert \square _{\pi I}^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{2}+\left\Vert \nabla _{\pi
I}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right) }^{2}\right) \approx
\left\Vert f\right\Vert _{L^{2}(\sigma )}^{2}\ ,
\end{eqnarray*
where the second line uses frame inequalities in Proposition \ref{dual frame}
and displays (\ref{low frame}) and (\ref{corr upper}) from Appendix A below,
we apply Cauchy-Schwarz in the $I$ variable above to see that
\begin{eqnarray*}
&&\left[ \left\vert \mathsf{B}\right\vert _{\limfunc{stop},1+\delta }^{A
\mathcal{P};s}\left( f,g\right) \right] ^{2}\lesssim \left\Vert f\right\Vert
_{L^{2}(\sigma )}^{2} \\
&&\times \left[ \sum_{I\in \mathcal{C}_{A}^{\limfunc{restrict}}}\left( \frac
1}{\sqrt{\left\vert I\right\vert _{\sigma }}}\sum_{\substack{ J:\ \left(
I,J\right) \in \mathcal{P} \\ \ell \left( J\right) =2^{-s}\ell \left(
I\right) }}\frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1
_{A\setminus I}\sigma \right) }{\left\vert J\right\vert }\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar }\right) ^{2}\right] ^{\frac{1}{2}}.
\end{eqnarray*
We can then estimate the sum inside the square brackets b
\begin{equation*}
\sum_{I\in \mathcal{C}_{A}^{\prime }}\left\{ \sum_{\substack{ J:\ \left(
I,J\right) \in \mathcal{P} \\ \ell \left( J\right) =2^{-s}\ell \left(
I\right) }}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2}\right\} \sum_{\substack{ J:\
\left( I,J\right) \in \mathcal{P} \\ \ell \left( J\right) =2^{-s}\ell
\left( I\right) }}\frac{1}{\left\vert I\right\vert _{\sigma }}\left( \frac
\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{A\setminus I}\sigma
\right) }{\left\vert J\right\vert }\right) ^{2}\left\Vert x-m_{J}\right\Vert
_{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2}\lesssim \left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }^{2}A\left( s\right) ^{2},
\end{equation*
wher
\begin{equation*}
A\left( s\right) ^{2}\equiv \sup_{I\in \mathcal{C}_{A}^{\prime }}\sum
_{\substack{ J:\ \left( I,J\right) \in \mathcal{P} \\ \ell \left( J\right)
=2^{-s}\ell \left( I\right) }}\frac{1}{\left\vert I\right\vert _{\sigma }
\left( \frac{\mathrm{P}_{1+\delta }^{\alpha }\left( J,\mathbf{1}_{A\setminus
I}\sigma \right) }{\left\vert J\right\vert }\right) ^{2}\left\Vert
x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right) }^{2}\,.
\end{equation*
Finally then we turn to the analysis of the supremum in last display. From
the Poisson decay (\ref{Poisson decay'}) we have
\begin{eqnarray*}
A\left( s\right) ^{2} &\lesssim &\sup_{I\in \mathcal{C}_{A}^{\prime }}\frac{
}{\left\vert I\right\vert _{\sigma }}2^{-2s\delta \left( 1-\varepsilon
\right) }\sum_{\substack{ J:\ \left( I,J\right) \in \mathcal{P} \\ \ell
\left( J\right) =2^{-s}\ell \left( I\right) }}\left( \frac{\mathrm{P
^{\alpha }\left( J,\mathbf{1}_{A\setminus I}\sigma \right) }{\left\vert
J\right\vert }\right) ^{2}\left\Vert x-m_{J}\right\Vert _{L^{2}\left(
\mathbf{1}_{J}\omega \right) }^{2} \\
&\lesssim &2^{-2s\delta \left( 1-\varepsilon \right) }\left[ \left( \mathcal
E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha }\right] \,,
\end{eqnarray*
where the last inequality is the stopping energy inquality (\ref{def
stopping energy 3}) in the corona $\mathcal{C}_{A}$. Indeed, from Definition
\ref{def energy corona 3}, as $(I,J)\in \mathcal{P}$, we have that $I$ is
\emph{not} a stopping interval in $\mathcal{A}$, and hence that (\ref{def
stop 3}) \emph{fails} to hold, delivering the estimate above since the terms
$\left\Vert x-m_{J}\right\Vert _{L^{2}\left( \mathbf{1}_{J}\omega \right)
}^{2}$ are additive, as the $J^{\prime }s$ are pigeonholed by $\ell \left(
J\right) =2^{-s}\ell \left( I\right) $ and hence pairwise disjoint.
\subsection{The bound for the first sublinear inequality}
Recall the definition of the sublinear form $\left\vert \mathsf{B
\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P
}\left( f,g\right) $ in display (\ref{def mod B}), and the inequality (\re
{First inequality}) that we wish to prove.
\begin{definition}
\label{Norm hat}Denote by $\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}$ the best constant in
\begin{equation}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) \leq \widehat{\mathfrak{N}}_{\limfunc
stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P}}\left\Vert \mathsf{P}_{\pi
\left( \Pi _{1}\mathcal{P}\right) }^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \mathsf{P}_{\Pi _{2
\mathcal{P}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar }, \label{best hat}
\end{equation
where $f\in L^{2}\left( \sigma \right) $ satisfies $E_{I}^{\sigma
}\left\vert f\right\vert \leq \alpha _{\mathcal{A}}\left( A\right) $ for all
$I\in \mathcal{C}_{A}$, and $g\in L^{2}\left( \omega \right) $, and $\pi
\left( \Pi _{1}\mathcal{P}\right) =\left\{ \pi I:I\in \Pi _{1}\mathcal{P
\right\} $. We refer to $\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}$ as the \emph{restricted} norm
relative to the collection $\mathcal{P}$.
\end{definition}
Inequality (\ref{First inequality}) will follow once we have shown that
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}}\lesssim \mathcal{E}_{2}^{\alpha }+\sqrt{\mathfrak{A
_{2}^{\alpha }}$. To this end, the following general result on mutually
orthogonal admissible collections, given as (3.5) in \cite{Lac}, will prove
very useful. Given a set $\left\{ \mathcal{Q}_{m}\right\} _{m=0}^{\infty }$
of admissible collections for $A$, we say that the collections $\mathcal{Q
_{m}$ are \emph{mutually orthogonal}, if each collection $\mathcal{Q}_{m}$
satisfie
\begin{equation*}
\mathcal{Q}_{m}\subset \dbigcup\limits_{j=0}^{\infty }\left\{ \mathcal{A
_{m,j}\times \mathcal{B}_{m,j}\right\} \ ,
\end{equation*
where the sets $\left\{ \mathcal{A}_{m,j}\right\} _{m,j}$ and $\left\{
\mathcal{B}_{m,j}\right\} _{m,j}$ are each pairwise disjoint in their
respective dyadic grids $\mathcal{D}$ and $\mathcal{G}$:
\begin{equation*}
\sum_{m,j=0}^{\infty }\mathbf{1}_{\mathcal{A}_{m,j}}\leq \mathbf{1}_
\mathcal{D}}\text{ and }\sum_{m,j=0}^{\infty }\mathbf{1}_{\mathcal{B
_{m,j}}\leq \mathbf{1}_{\mathcal{G}}.
\end{equation*}
\begin{lemma}
\label{mut orth}Suppose that $\left\{ \mathcal{Q}_{m}\right\} _{m=0}^{\infty
}$ is a set of admissible collections for $A$ that are \emph{mutually
orthogonal}. Then $\mathcal{Q}\equiv \dbigcup\limits_{m=0}^{\infty }\mathcal
Q}_{m}$ is admissible, and the sublinear stopping form $\left\vert \mathsf{B
\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q
}\left( f,g\right) $ has its localized norm $\widehat{\mathfrak{N}}_
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}$ controlled by the
\emph{supremum} of the localized norms $\widehat{\mathfrak{N}}_{\limfunc{sto
},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}$:
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{Q}}\leq \sup_{m\geq 0}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}.
\end{equation*}
\end{lemma}
\begin{proof}
If $J\in \Pi _{2}\mathcal{Q}_{m}$, then $\varphi _{J}^{\mathcal{Q}}=\varphi
_{J}^{\mathcal{Q}_{m}}$ and $I_{\mathcal{Q}}\left( J\right) =I_{\mathcal{Q
_{m}}\left( J\right) $, since the collection $\left\{ \mathcal{Q
_{m}\right\} _{m=0}^{\infty }$ is mutually orthogonal. Thus we have
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &=&\sum_{J\in \Pi _{2}\mathcal{Q}}\frac
\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&=&\sum_{m\geq 0}\sum_{J\in \Pi _{2}\mathcal{Q}_{m}}\frac{\mathrm{P}^{\alpha
}\left( J,\left\vert \varphi _{J}^{\mathcal{Q}_{m}}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{Q}_{m}}\left( J\right) }\sigma \right) }{\left\vert
J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar }=\sum_{m\geq 0}\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}\left(
f,g\right) ,
\end{eqnarray*
and we can continue with the definition of $\widehat{\mathfrak{N}}_{\limfunc
stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}$ and Cauchy-Schwarz to
obtai
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &\leq &\sum_{m\geq 0}\widehat{\mathfrak{
}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}\left\Vert
\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}_{m}\right) }^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \mathsf{P
_{\Pi _{2}\mathcal{Q}_{m}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\left( \sup_{m\geq 0}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}\right) \sqrt{\sum_{m\geq
0}\left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}_{m}\right) }^{\sigma
,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar 2}}\sqrt
\sum_{m\geq 0}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{Q}_{m}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}}
\\
&\leq &\left( \sup_{m\geq 0}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{m}}\right) \sqrt{\left\Vert
\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar 2}}\sqrt{\left\Vert
\mathsf{P}_{\Pi _{2}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar 2}}.
\end{eqnarray*}
\end{proof}
Now we turn to proving inequality (\ref{First inequality}) for the sublinear
form $\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup
^{\omega }}^{A,\mathcal{P}}\left( f,g\right) $, i.e
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) &\equiv &\sum_{J\in \Pi _{2}\mathcal{P}
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}\right\vert
\mathbf{1}_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma \right) }
\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \\
&\lesssim &\left( \mathcal{E}_{2}^{\alpha }+\sqrt{\mathfrak{A}_{2}^{\alpha }
\right) \left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{P}\right)
}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar
}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }; \\
\text{where }\varphi _{J} &\equiv &\sum_{I\in \mathcal{C}_{A}^{\limfunc
restrict}}:\ \left( I,J\right) \in \mathcal{P}}\left( E_{I}^{\sigma
\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) b_{A}\
\mathbf{1}_{A\setminus I}\ \text{\ is supported in }A\setminus I_{\mathcal{P
}\left( J\right) ,
\end{eqnarray*
and $I_{\mathcal{P}}\left( J\right) $ denotes the smallest interval $I\in
\mathcal{D}$ for which $\left( I,J\right) \in \mathcal{P}$. We recall the
stopping energy from (\ref{def stopping energy 3})
\begin{equation*}
\mathbf{X}_{\alpha }\left( \mathcal{C}_{A}\right) ^{2}\equiv \sup_{I\in
\mathcal{C}_{A}}\frac{1}{\left\vert I\right\vert _{\sigma }}\sup_{I\supset
\overset{\cdot }{\cup }J_{r}}\sum_{r=1}^{\infty }\left( \frac{\mathrm{P
^{\alpha }\left( J_{r},\mathbf{1}_{A}\sigma \right) }{\left\vert
J_{r}\right\vert }\right) ^{2}\left\Vert x-m_{J_{r}}\right\Vert
_{L^{2}\left( \mathbf{1}_{J_{r}}\omega \right) }^{2}\ ,
\end{equation*
where the intervals $J_{r}\in \mathcal{G}$ are pairwise disjoint in $I$.
What now follows is an adaptation to our sublinear form $\left\vert \mathsf{
}\right\vert _{\limfunc{stop},\square ^{\omega }}^{A,\mathcal{P}}$ of the
arguments of M. Lacey in \cite{Lac}, together with an additional `indented'
corona construction. We have the following Poisson inequality for intervals
B\subset A\subset I$
\begin{eqnarray}
\frac{\mathrm{P}^{\alpha }\left( A,\mathbf{1}_{I\setminus A}\sigma \right) }
\left\vert A\right\vert } &\approx &\int_{I\setminus A}\frac{1}{\left(
\left\vert y-c_{A}\right\vert \right) ^{2-\alpha }}d\sigma \left( y\right)
\label{BAI} \\
&\lesssim &\int_{I\setminus A}\frac{1}{\left( \left\vert y-c_{B}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) \approx \frac{\mathrm{P
^{\alpha }\left( B,\mathbf{1}_{I\setminus A}\sigma \right) }{\left\vert
B\right\vert }. \notag
\end{eqnarray}
Fix $A\in \mathcal{A}$. Following \cite{Lac} we will use a `decoupled'
modification of the stopping energy $\mathbf{X}_{\alpha }\left( \mathcal{C
_{A}\right) $ to define a `size functional' of an $A$-admissible collection
\mathcal{P}$. So suppose that $\mathcal{P}$ is an $A$-admissible collection
of pairs of intervals, and recall that $\Pi _{1}\mathcal{P}$ and $\Pi _{2
\mathcal{P}$ denote the intervals in the first and second components of the
pairs in $\mathcal{P}$ respectively.
\begin{definition}
\label{rest K}For an $A$-admissible collection of pairs of intervals
\mathcal{P}$, and an interval $K\in \Pi _{1}\mathcal{P}$, define the
projection of $\mathcal{P}$ `relative to $K$' by
\begin{equation*}
\Pi _{2}^{K}\mathcal{P}\equiv \left\{ J\in \Pi _{2}\mathcal{P}:\ J^{\maltese
}\subset K\right\} ,
\end{equation*
where we have suppressed dependence on $A$.
\end{definition}
\begin{definition}
\label{Pi below}We will use as the `size testing collection' of intervals
for $\mathcal{P}$ the collection
\begin{equation*}
\Pi _{1}^{\limfunc{below}}\mathcal{P}\equiv \left\{ K\in \mathcal{D
:K\subset I\text{ for some }I\in \Pi _{1}\mathcal{P}\right\} ,
\end{equation*
which consists of all intervals contained in an interval from $\Pi _{1
\mathcal{P}$.
\end{definition}
Continuing to follow Lacey \cite{Lac}, we define a `size functional' of
\mathcal{P}$, actually two of them, as follows. Recall from Notation \re
{nonstandard norm} that for a pseudoprojection $\mathsf{Q}_{\mathcal{H
}^{\omega }$ on $x$ we have
\begin{equation}
\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}=\sum_{J\in \mathcal{H
}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}=\sum_{J\in \mathcal{H}}\left(
\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\inf_{z\in \mathbb{R}}\sum_{J^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( J\right) }\left\vert J^{\prime
}\right\vert _{\omega }\left( E_{J^{\prime }}^{\omega }\left\vert
x-z\right\vert \right) ^{2}\right) . \label{h}
\end{equation}
\begin{definition}
\label{def ext size}If $\mathcal{P}$ is $A$-admissible, define an \emph
initial} size condition $\mathcal{S}_{\limfunc{init}\limfunc{size}}^{\alpha
,A}\left( \mathcal{P}\right) $ b
\begin{equation}
\mathcal{S}_{\limfunc{init}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}\equiv \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}}\frac{1}
\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K
\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert K\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{\Pi _{2}^{K}\mathcal{P}}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}.
\label{def P stop energy 3}
\end{equation}
\end{definition}
The following key fact is essential.
\textbf{Key Fact \#1:
\begin{equation}
K\subset A\text{ and }K\notin \mathcal{C}_{A}\Longrightarrow \Pi _{2}^{K
\mathcal{P}=\emptyset \ . \label{later use}
\end{equation
To see this, suppose that $K\subset A$ and $K\notin \mathcal{C}_{A}$. Then
K\subset A^{\prime }$ for some $A^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) $, and so if there is $J^{\prime }\in \Pi _{2}^{K}\mathcal{
}$, then $\left( J^{\prime }\right) ^{\maltese }\subset K\subset A^{\prime }
, which implies that $\left( J^{\prime }\right) ^{\maltese }\notin \mathcal{
}_{A}^{\mathcal{G},\limfunc{shift}}$, which contradicts $\Pi _{2}^{K
\mathcal{P}\subset \mathcal{C}_{A}^{\mathcal{G},\limfunc{shift}}$. We now
observe from (\ref{later use}) that we may also write the initial size
functional a
\begin{equation}
\mathcal{S}_{\limfunc{init}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}\equiv \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
C_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert _{\sigma }
\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{\Pi
_{2}^{K}\mathcal{P}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2}. \label{rewrite size}
\end{equation}
However, we will also need to control certain pairs $\left( I,J\right) \in
\mathcal{P}$ using testing intervals $K$ which are strictly smaller than
J^{\maltese }$, namely those $K\in \mathcal{C}_{A}$ such that $K\subset
J^{\maltese }\subset \pi _{\mathcal{D}}^{\left( 2\right) }K$. For this, we
need a second key fact regarding the intervals $J^{\maltese }$, that will
also play a crucial role in controlling pairs in the indented corona below,
and which is that $J$ is always contained in one of the \emph{inner} two
grandchildren of $J^{\maltese }$.
\textbf{Key Fact \#2:
\begin{eqnarray}
&&\text{\textbf{either} }3J\subset J_{-/+}^{\maltese }\text{ \textbf{or}
3J\subset J_{+/-}^{\maltese }\ , \label{indentation} \\
&&\text{where }\mathfrak{C}_{J^{\maltese }}^{\left( 1\right) }=\left\{
J_{-}^{\maltese },J_{+}^{\maltese }\right\} \text{ and }\mathfrak{C
_{J^{\maltese }}^{\left( 2\right) }=\left\{ J_{-/-}^{\maltese
},J_{-/+}^{\maltese },J_{+/-}^{\maltese },J_{+/+}^{\maltese }\right\} ,
\notag \\
&&\text{and the children and grandchildren are listed left to right.} \notag
\end{eqnarray
To see this, suppose without loss of generality that the child
J_{J}^{\maltese }$ of $J^{\maltese }$ that contains $J$ is the left child
J_{-}^{\maltese }$ (which exists because $J$ is good in $J^{\maltese }$).
Then observe that $J$ is by definition $\varepsilon -\limfunc{bad}$ in
J_{-}^{\maltese }=J_{J}^{\maltese }$, i.e. $\limfunc{dist}\left( J,\limfunc
body}J_{-}^{\maltese }\right) \leq 2\left\vert J\right\vert ^{\varepsilon
}\left\vert J_{-}^{\maltese }\right\vert ^{1-\varepsilon }$, and so cannot
lie in the leftmost grandchild $J_{-/-}^{\maltese }$. Indeed, if $J\subset
J_{-/-}^{\maltese }$, the
\begin{equation*}
\limfunc{dist}\left( J,\limfunc{body}J^{\maltese }\right) =\limfunc{dist
\left( J,\limfunc{body}J_{-}^{\maltese }\right) \leq 2\left\vert
J\right\vert ^{\varepsilon }\left\vert J_{-}^{\maltese }\right\vert
^{1-\varepsilon }=2^{\varepsilon }\left\vert J\right\vert ^{\varepsilon
}\left\vert J^{\maltese }\right\vert ^{1-\varepsilon }<2\left\vert
J\right\vert ^{\varepsilon }\left\vert J^{\maltese }\right\vert
^{1-\varepsilon },
\end{equation*
contradicting the fact that $J$ is $\varepsilon -\limfunc{good}$ in
J^{\maltese }$. Thus we must have $J\subset J_{-/+}^{\maltese }$ (where the
body of $J^{\maltese }$ does not intersect the interior of
J_{-/+}^{\maltese }$, thus permitting $J$ to be $\varepsilon -\limfunc{good}$
in $J^{\maltese }$). Finally, the fact that $J$ is $\varepsilon -\limfunc
good}$ in $J^{\maltese }$ implies that $3J\subset J_{-/+}^{\maltese }$. See
Figure \ref{gra}.
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the two inner grandchildren of $J^{\maltese }$, namely $J_{-/+}^{\maltese }$
or $J_{+/-}^{\maltese }$. The $\limfunc{body}$ of $J^{\maltese }$ consists
of the infinitely many red dots.}}{\Qlb{gra}}{grandchild.wmf}{\specia
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This second key fact is what underlies the construction of the indented
corona below, and motivates the next definition of augmented projection, in
which we allow intervals $K$ satisfying $J\subset K\subsetneqq J^{\maltese
}\subset \pi _{\mathcal{D}}^{\left( 2\right) }K$, as well as $K\in C_{A}$,
to be tested over in the augmented size condition below.
\begin{definition}
\label{augs}Suppose $\mathcal{P}$ is an $A$-admissible collection.
\begin{enumerate}
\item For $K\in \Pi _{1}\mathcal{P}$, define the \emph{augmented} projection
of $\mathcal{P}$ relative to $K$ b
\begin{equation*}
\Pi _{2}^{K,\limfunc{aug}}\mathcal{P}\equiv \left\{ J\in \Pi _{2}\mathcal{P
:J\subset K\text{ and }J^{\maltese }\subset \pi _{\mathcal{D}}^{\left(
2\right) }K\right\} =\left\{ J\in \Pi _{2}\mathcal{P}:J^{\flat }\subset
K\right\} .
\end{equation*}
\item Define the corresponding \emph{augmented} size functional $\mathcal{S
_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}\right) $ by
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}\equiv \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
C_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert _{\sigma }
\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{\Pi
_{2}^{K,\limfunc{aug}}\mathcal{P}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\ .
\end{equation*}
\end{enumerate}
\end{definition}
We note that the augmented projection $\Pi _{2}^{K,\limfunc{aug}}\mathcal{P}$
includes intervals $J$ for which $J\subset K\subsetneqq J^{\maltese }\subset
\pi _{\mathcal{D}}^{\left( 2\right) }K$, and hence $J$ need not be
\varepsilon -\limfunc{good}$ inside $K$. For $M\in \mathcal{D}$, denote by
M_{J}$ and $M_{\searrow J}$ the child and grandchild respectively of $M$
that contains $J$, provided they exist. Then by the second key fact (\re
{indentation}), and using that the endpoints of both $J_{-/+}^{\maltese }$
and $J_{+/-}^{\maltese }$ lie in the $\limfunc{body}$ of $J^{\maltese }$, we
have two consequences
\begin{equation*}
K\in \left\{ J_{J}^{\maltese },J_{\searrow J}^{\maltese }\right\} \text{ and
}3J\subset J_{\searrow J}^{\maltese }\subset 3J_{\searrow J}^{\maltese
}\subset J^{\maltese }\text{ for }J\in \Pi _{2}^{K,\limfunc{nar}}\mathcal{P},
\end{equation*
which will play an important role below.
The augmented size functional $\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) $ is a `decoupled' form of the
stopping energy $\mathbf{X}_{\alpha }\left( \mathcal{C}_{A}\right) $
restricted to $\mathcal{P}$, in which the intervals $J$ appearing in
\mathbf{X}_{\alpha }\left( \mathcal{C}_{A}\right) $ no longer appear in the
Poisson integral in $\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha
,A}\left( \mathcal{P}\right) $, and it plays a crucial role in Lacey's
argument in \cite{Lac}. We note two essential properties of this definition
of size functional:
\begin{enumerate}
\item \textbf{Monotonicity} of size: $\mathcal{S}_{\limfunc{aug}\limfunc{siz
}}^{\alpha ,A}\left( \mathcal{P}\right) \leq \mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q}\right) $ if $\mathcal{P
\subset \mathcal{Q}$,
\item \textbf{Control} by energy and Muckenhoupt conditions: $\mathcal{S}_
\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}\right) \lesssim
\mathcal{E}_{2}^{\alpha }+\sqrt{\mathfrak{A}_{2}^{\alpha }}$.
\end{enumerate}
The monotonicity property follows from $\Pi _{1}^{\limfunc{below}}\mathcal{P
\subset \Pi _{1}^{\limfunc{below}}\mathcal{Q}$ and $\Pi _{2}^{K}\mathcal{P
\subset \Pi _{2}^{K}\mathcal{Q}$. The control property is contained in the
next lemma, which uses the stopping energy control for the form $\mathsf{B}_
\limfunc{stop}}^{A}\left( f,g\right) $ associated with $A$.
\begin{lemma}
\label{energy control}If $\mathcal{P}^{A}$ is as in (\ref{initial P}) and
\mathcal{P}\subset \mathcal{P}^{A}$, then
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) \leq \mathbf{X}_{\alpha }\left( \mathcal{C}_{A}\right) \lesssim
\mathcal{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}+\sqrt{A_{2}^{\alpha
\limfunc{punct}}}\ .
\end{equation*}
\end{lemma}
\begin{proof}
We hav
\begin{eqnarray*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2} &=&\sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert
_{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus
K}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathsf{Q
_{\Pi _{2}^{K}\mathcal{P}\cup \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}}^{\omega
,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} \\
&\lesssim &\sup_{K\in \mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}
\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K
\mathbf{1}_{A}\sigma \right) }{\left\vert K\right\vert }\right)
^{2}\left\Vert x-m_{K}\right\Vert _{L^{2}\left( \mathbf{1}_{K}\omega \right)
}^{2}\leq \mathbf{X}_{\alpha }\left( \mathcal{C}_{A}\right) ^{2},
\end{eqnarray*
which is the first inequality in the statement of the lemma. The second
inequality follows from (\ref{def stopping bounds 3}).
\end{proof}
There is an important special circumstance, introduced by M. Lacey in \cit
{Lac}, in which we can bound our forms by the size functional, namely when
the pairs all straddle a subpartition of $A$, and we present this in the
next subsection. In order to handle the fact that the intervals in $\Pi
_{1}^{\limfunc{below}}\mathcal{P}\cap C_{A}^{\limfunc{restrict}}$ need no
longer enjoy any goodness, we will need to formulate a Substraddling Lemma
to deal with this situation as well. See \textbf{Remark on lack of usual
goodness} after (\ref{N_L}), where it is explained how this applies to the
proof of (\ref{rem}). Then in the following subsection, we use the bottom/up
stopping time construction of M. Lacey, together with an additional
`indented' top/down corona construction, to reduce control of the sublinear
stopping form $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}\left( f,g\right) $ in inequality
(\ref{First inequality}) to the three special cases addressed by the
Orthogonality Lemma, the Straddling Lemma and the Substraddling Lemma.
\subsection{Straddling, Substraddling and Corona-straddling Lemmas}
We begin with the Corona-straddling Lemma in which the straddling collection
is the set of $\mathcal{A}$-children of $A$, and applies to the `corona
straddling' subcollection of the initial admissible collection $\mathcal{P
^{A}$ - see (\ref{initial P}). Define the `corona straddling' collection
\mathcal{P}_{\func{cor}}^{A}$ b
\begin{equation}
\mathcal{P}_{\func{cor}}^{A}\equiv \dbigcup\limits_{A^{\prime }\in \mathfrak
C}_{\mathcal{A}}\left( A\right) }\left\{ \left( I,J\right) \in \mathcal{P
^{A}:J\subset A^{\prime }\varsubsetneqq J^{\maltese }\subset \pi _{\mathcal{
}}^{\left( 2\right) }A^{\prime }\right\} . \label{def cor}
\end{equation
Note that $\mathcal{P}_{\func{cor}}^{A}$ is an $A$-admissible collection
that consists of just those pairs $\left( I,J\right) $ for which
J^{\maltese }$ is either the $\mathcal{D}$-parent or the $\mathcal{D}
-grandparent of a stopping interval $A^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) $. The bound for the norm of the corresponding form is
controlled by the energy condition.
\begin{lemma}
\label{cor strad 1}We have the sublinear form boun
\begin{equation*}
\mathfrak{N}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P}_
\func{cor}}^{A}}\leq C\mathcal{E}_{2}^{\alpha }.
\end{equation*}
\end{lemma}
\begin{proof}
The key point here is our assumption that $J\subset A^{\prime
}\varsubsetneqq J^{\maltese }\subset \pi _{\mathcal{D}}^{\left( 2\right)
}A^{\prime }$ for $\left( I,J\right) \in \mathcal{P}_{\func{cor}}^{A}$,
which implies that in fact $3J\subset A^{\prime }$ since $J\cap \limfunc{bod
}\left( \pi _{\mathcal{D}}^{\left( 2\right) }A^{\prime }\right) =\emptyset $
because $J$ is $\varepsilon -\limfunc{good}$ in $\pi _{\mathcal{D}}^{\left(
2\right) }A^{\prime }$. We start wit
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}_{\func{cor}}^{A}}\left( f,g\right) &=&\sum_{J\in \Pi _{2
\mathcal{P}_{\func{cor}}^{A}}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert
\varphi _{J}^{\mathcal{P}_{\func{cor}}^{A}}\right\vert \mathbf{1
_{A\setminus I_{\mathcal{P}_{\func{cor}}^{A}}\left( J\right) }\sigma \right)
}{\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \\
&=&\sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right)
}\sum_{J\in \Pi _{2}\mathcal{P}_{\func{cor}}^{A}:\ 3J\subset A^{\prime }
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{P}_
\func{cor}}^{A}}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{P}_{\func{cor
}^{A}}\left( J\right) }\sigma \right) }{\left\vert J\right\vert }\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit }\left\Vert \square _{J}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }; \\
\text{where }\varphi _{J}^{\mathcal{P}_{\func{cor}}^{A}} &\equiv &\sum_{I\in
\Pi _{1}\mathcal{P}_{\func{cor}}^{A}:\mathcal{\ }\left( I,J\right) \in
\mathcal{P}_{\func{cor}}^{A}}b_{A}E_{I}^{\sigma }\left( \widehat{\square
_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \mathbf{1}_{A\setminus I}\ .
\end{eqnarray*
If $J\in \Pi _{2}\mathcal{P}_{\func{cor}}^{A}$ and $J\subset A^{\prime }\in
\mathfrak{C}_{\mathcal{A}}\left( A\right) $, then either (1) $A^{\prime
}=J_{-/+}^{\maltese }$ or $J_{+/-}^{\maltese }$ or (2) $A^{\prime
}=J_{-}^{\maltese }$ or $J_{+}^{\maltese }$, and we have
\begin{equation*}
\frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{A\setminus I_{\mathcal{P}_
\func{cor}}^{A}}\left( J\right) }\sigma \right) }{\left\vert J\right\vert
\approx \left\{
\begin{array}{ccc}
\frac{\mathrm{P}^{\alpha }\left( A^{\prime },\mathbf{1}_{A\setminus I_
\mathcal{P}_{\func{cor}}^{A}}}\sigma \right) }{\left\vert A^{\prime
}\right\vert }\leq \frac{\mathrm{P}^{\alpha }\left( A^{\prime },\mathbf{1
_{A}\sigma \right) }{\left\vert A^{\prime }\right\vert } & \text{ if } &
A^{\prime }=J_{-/+}^{\maltese }\text{ or }J_{+/-}^{\maltese } \\
\frac{\mathrm{P}^{\alpha }\left( A_{J}^{\prime },\mathbf{1}_{A\setminus I_
\mathcal{P}_{\func{cor}}^{A}}}\sigma \right) }{\left\vert A_{J}^{\prime
}\right\vert }\lesssim \frac{\mathrm{P}^{\alpha }\left( A^{\prime },\mathbf{
}_{A}\sigma \right) }{\left\vert A^{\prime }\right\vert } & \text{ if } &
A^{\prime }=J_{-}^{\maltese }\text{ or }J_{+}^{\maltese
\end{array
\right. .
\end{equation*
Since $\left\vert \varphi _{J}^{\mathcal{P}_{\func{cor}}^{A}}\right\vert
\lesssim \alpha _{\mathcal{A}}\left( A\right) \mathbf{1}_{A}$ by (\ref{phi
bound}), we can then bound $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}_{\func{cor}}^{A}}\left(
f,g\right) $ b
\begin{eqnarray*}
&&\alpha _{\mathcal{A}}\left( A\right) \sum_{A^{\prime }\in \mathfrak{C}_
\mathcal{A}}\left( A\right) }\left( \frac{\mathrm{P}^{\alpha }\left(
A^{\prime },\mathbf{1}_{A}\sigma \right) }{\left\vert A^{\prime }\right\vert
}\right) \left\Vert \mathsf{Q}_{\Pi _{2}\mathcal{P}_{\func{cor
}^{A};A^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit }\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P}_
\func{cor}}^{A};A^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\alpha _{\mathcal{A}}\left( A\right) \left( \sum_{A^{\prime }\in
\mathfrak{C}_{\mathcal{A}}\left( A\right) }\left( \frac{\mathrm{P}^{\alpha
}\left( A^{\prime },\mathbf{1}_{A}\sigma \right) }{\left\vert A^{\prime
}\right\vert }\right) ^{2}\left\Vert x-m_{A^{\prime }}^{\sigma }\right\Vert
_{L^{2}\left( \mathbf{1}_{A^{\prime }}\sigma \right) }^{\spadesuit 2}\right)
^{\frac{1}{2}} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \times \left( \sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) }\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P}_{\func{cor
}^{A};A^{\prime }}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left(
\omega \right) }^{\bigstar 2}\right) ^{\frac{1}{2}} \\
&\leq &\mathcal{E}_{2}^{\alpha }\alpha _{\mathcal{A}}\left( A\right) \sqrt
\left\vert A\right\vert _{\sigma }}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P
_{\func{cor}}^{A}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left(
\omega \right) }^{\bigstar }\leq \mathcal{E}_{2}^{\alpha }\alpha _{\mathcal{
}}\left( A\right) \sqrt{\left\vert A\right\vert _{\sigma }}\left\Vert
\mathsf{P}_{\mathcal{C}_{A}^{\func{shift}}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar },
\end{eqnarray*
where in the last line we have used the strong energy constant $\mathcal{E
_{2}^{\alpha }$ in (\ref{strong b* energy}).
\end{proof}
\begin{definition}
We say that an admissible collection of pairs $\mathcal{P}$ is \emph{reduced}
if it contains no pairs from $\mathcal{P}_{\func{cor}}^{A}$, i.e
\begin{equation*}
\mathcal{P}\cap \mathcal{P}_{\func{cor}}^{A}=\emptyset .
\end{equation*}
\end{definition}
\begin{definition}
\label{def aug}We define $J^{\flat }=J_{\searrow J}^{\maltese }$ to be the
inner grandchild of $J^{\maltese }$ that contains $J$.
\end{definition}
Recall that in terms of $J^{\flat }$ we rewrit
\begin{equation*}
\Pi _{2}^{K,\limfunc{aug}}\mathcal{P}=\left\{ J\in \Pi _{2}\mathcal{P
:J\subset K\text{ and }J^{\maltese }\subset \pi _{\mathcal{D}}^{\left(
2\right) }K\right\} =\left\{ J\in \Pi _{2}\mathcal{P}:J\subset K\text{ and
J^{\flat }\subset K\right\} .
\end{equation*}
\begin{definition}
\label{flat straddles}Given a \emph{reduced admissible} collection of pairs
\mathcal{Q}$ for $A$, and a subpartition $\mathcal{S}\subset \Pi _{1}^
\limfunc{below}}\mathcal{Q}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ of
pairwise disjoint intervals in $A$, we say that $\mathcal{Q}$ $\flat
\textbf{straddles} $\mathcal{S}$ if for every pair $\left( I,J\right) \in
\mathcal{Q}$ there is $S\in \mathcal{S}\cap \left[ J,I\right] $ with
J^{\flat }\subset S$. To avoid trivialities, we further assume that for
every $S\in \mathcal{S}$, there is at least one pair $\left( I,J\right) \in
\mathcal{Q}$ with $J^{\flat }\subset S\subset I$. Here $\left[ J,I\right] $
denotes the geodesic in the dyadic tree $\mathcal{D}$ that connects $J^
\mathcal{D}}$ to $I$, where $J^{\mathcal{D}}$ is the minimal interval in
\mathcal{D}$ that contains $J$.
\end{definition}
\begin{definition}
\label{def Whit}For any dyadic interval $S\in \mathcal{D}$, define the
Whitney collection $\mathcal{W}\left( S\right) $ to consist of the maximal
subintervals $K$ of $S$ whose triples $3K$ are contained in $S$. Then set
\mathcal{W}^{\ast }\left( S\right) \equiv \mathcal{W}\left( S\right) \cup
\left\{ S\right\} $.
\end{definition}
The following geometric proposition will prove useful in proving the $\flat
Straddling Lemma \ref{straddle 3 ref} below.
\begin{proposition}
\label{flatness}Suppose $\mathcal{Q}$ is reduced admissible and $\flat
straddles a subpartition $\mathcal{S}$ of $A$. Fix $S\in \mathcal{S}$.
Define
\begin{equation*}
\varphi _{J}^{\mathcal{Q}^{S}}\left[ h\right] \equiv \sum_{I\in \Pi _{1
\mathcal{Q}^{S}:\mathcal{\ }\left( I,J\right) \in \mathcal{Q
^{S}}b_{A}E_{I}^{\sigma }\left( \widehat{\square }_{\pi I}^{\sigma ,\flat
\mathbf{b}}h\right) \ \mathbf{1}_{A\setminus I}\ ,
\end{equation*
assume that $h\in L^{2}\left( \sigma \right) $ is supported in the interval
A$, and that there is an interval $H\in \mathcal{C}_{A}$ with $H\supset S$
such that
\begin{equation*}
E_{I}^{\sigma }\left\vert h\right\vert \leq CE_{H}^{\sigma }\left\vert
h\right\vert ,\ \ \ \ \ \text{for all }I\in \Pi _{1}^{\limfunc{below}
\mathcal{Q}\cap \mathcal{C}_{A}^{\limfunc{restrict}}\text{ with }I\supset S.
\end{equation*
The
\begin{eqnarray*}
&&\sum_{J\in \Pi _{2}\mathcal{Q}:\ J^{\flat }\subset S}\frac{\mathrm{P
^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q}}\left[ h\right]
\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\lesssim &\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha
}\left( S,\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert S\right\vert
\left\Vert \mathsf{Q}_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \mathsf{P}_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&&+\alpha _{\mathcal{H}}\left( H\right) \dsum\limits_{K\in \mathcal{W}\left(
S\right) }\frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\left\Vert \mathsf{Q}_{\Pi _{2}^{K
\limfunc{aug}}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert \mathsf{P}_{\Pi
_{2}^{K,\limfunc{aug}}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }\ .
\end{eqnarray*
The sum over Whitney intervals $K\in \mathcal{W}\left( S\right) $ is only
required to bound the sum of those terms on the left for which $J^{\flat
}\subset S^{\prime \prime }$ for some $S^{\prime \prime }\in \mathfrak{C}_
\mathcal{D}}^{\left( 2\right) }\left( S\right) $.
\end{proposition}
\begin{proof}
Suppose first that $J^{\flat }=S\in \mathcal{C}_{A}^{\limfunc{restrict}}$.
Then $3S=3J^{\flat }\subset J^{\maltese }\subset I_{\mathcal{Q}}\left(
J\right) $ and using (\ref{phi bound}) with $\alpha _{\mathcal{H}}\left(
H\right) $ in place of $\alpha _{\mathcal{A}}\left( A\right) $, we hav
\begin{eqnarray*}
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert } &\lesssim &\alpha _{\mathcal{H}}\left(
H\right) \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{A\setminus
J^{\maltese }}\sigma \right) }{\left\vert J\right\vert } \\
&\lesssim &\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha
}\left( S,\mathbf{1}_{A\setminus J^{\maltese }}\sigma \right) }{\left\vert
S\right\vert }\leq \alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P
^{\alpha }\left( S,\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert
S\right\vert }.
\end{eqnarray*
Suppose next that $J^{\flat }=S^{\prime }\in \mathfrak{C}_{\mathcal{D
}\left( S\right) $. Then $3S^{\prime }=3J^{\flat }\subset J^{\maltese
}\subset I_{\mathcal{Q}}\left( J\right) $ and (\ref{phi bound}) giv
\begin{eqnarray*}
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert } &\lesssim &\alpha _{\mathcal{H}}\left(
H\right) \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{A\setminus
J^{\maltese }}\sigma \right) }{\left\vert J\right\vert } \\
&\lesssim &\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha
}\left( S^{\prime },\mathbf{1}_{A\setminus J^{\maltese }}\sigma \right) }
\left\vert S^{\prime }\right\vert } \\
&\leq &\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha }\left(
S^{\prime },\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert S^{\prime
}\right\vert }\approx \alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P
^{\alpha }\left( S,\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert
S\right\vert }.
\end{eqnarray*
Thus in these two cases, by Cauchy-Schwarz, the left hand side of our
conclusion is bounded by a multiple o
\begin{eqnarray*}
&&\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha }\left( S
\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert S\right\vert }\left(
\sum_{J\in \Pi _{2}\mathcal{Q}:\ J^{\flat }\subset S}\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2}\right) ^{\frac{1}{2}}\left( \sum_{J\in \Pi
_{2}\mathcal{Q}:\ J^{\flat }\subset S}\left\Vert \square _{J}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar
2}\right) ^{\frac{1}{2}} \\
&=&\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha }\left( S
\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert S\right\vert
\left\Vert \mathsf{Q}_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \mathsf{P}_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\ .
\end{eqnarray*
Finally, suppose that $J^{\flat }\subset S^{\prime \prime }$ for some
S^{\prime \prime }\in \mathfrak{C}_{\mathcal{D}}^{\left( 2\right) }\left(
S\right) $. Then $J^{\maltese }\subset S$, and Key Fact \#2 in (\re
{indentation}) shows that $3J^{\flat }\subset J^{\maltese }$, so that
3J^{\flat }\subset J^{\maltese }\subset S\subset I_{\mathcal{Q}}\left(
J\right) $. Thus we have $J^{\flat }\subset K=K\left[ J\right] $ for some
K\in \mathcal{W}\left( S\right) $ and so by (\ref{phi bound}) again
\begin{eqnarray*}
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert } &\lesssim &\alpha _{\mathcal{H}}\left(
H\right) \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{A\setminus S}\sigma
\right) }{\left\vert J\right\vert } \\
&\lesssim &\alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha
}\left( K,\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert K\right\vert
\leq \alpha _{\mathcal{H}}\left( H\right) \frac{\mathrm{P}^{\alpha }\left( K
\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert K\right\vert }.
\end{eqnarray*
Now we apply Cauchy-Schwarz again, but noting that $J^{\flat }\subset K$
this time, to obtain that the left hand side of our conclusion is bounded by
a multiple o
\begin{eqnarray*}
&&\alpha _{\mathcal{H}}\left( H\right) \dsum\limits_{K\in \mathcal{W}\left(
S\right) }\frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\left( \sum_{J\in \Pi _{2}\mathcal{Q}:\
J^{\flat }\subset K}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\right) ^{\frac{
}{2}}\left( \sum_{J\in \Pi _{2}\mathcal{Q}:\ J^{\flat }\subset K}\left\Vert
\square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar 2}\right) ^{\frac{1}{2}} \\
&=&\alpha _{\mathcal{H}}\left( H\right) \dsum\limits_{K\in \mathcal{W}\left(
S\right) }\frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\left\Vert \mathsf{Q}_{\Pi _{2}^{K
\limfunc{aug}}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert \mathsf{P}_{\Pi
_{2}^{K,\limfunc{aug}}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }\ .
\end{eqnarray*
This completes the proof of Proposition \ref{flatness}.
\end{proof}
Recall the family of operators $\left\{ \square _{I}^{\sigma ,\pi ,\mathbf{b
}\right\} _{I\in \mathcal{C}_{A}^{\mathcal{A}}}$, where for $I\in \mathcal{C
_{A}^{\mathcal{A}}$, the dual martingale difference $\square _{I}^{\sigma
,\pi ,\mathbf{b}}$ is defined in (\ref{def pi box}) of Appendix A below, and
satisfie
\begin{equation*}
\square _{I}^{\sigma ,\pi ,\mathbf{b}}f=\left[ \sum_{I^{\prime }\in
\mathfrak{C}\left( I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,\pi ,\mathbf{
}}f\right] -\mathbb{F}_{I}^{\sigma ,\mathbf{b}}f=\sum_{I^{\prime }\in
\mathfrak{C}\left( I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,b_{A}}f
\mathbb{F}_{I}^{\sigma ,b_{A}}f\ .
\end{equation*
Since $\square _{I}^{\sigma ,\pi ,\mathbf{b}}$ is the transpose of
\triangle _{I}^{\sigma ,\pi ,\mathbf{b}}$ for $I\in \mathcal{C}_{A}^
\mathcal{A}}$, the first line of Lemma \ref{b proj} (where the superscript
\pi $ is suppressed for convenience) shows that $\left\{ \square
_{I}^{\sigma ,\pi ,\mathbf{b}}\right\} _{I\in \mathcal{C}_{A}^{\mathcal{A}}}$
is a family of projections, and the second line of Lemma \ref{b proj} shows
it is an orthogonal family, i.e.
\begin{equation*}
\square _{I}^{\sigma ,\pi ,\mathbf{b}}\square _{K}^{\sigma ,\pi ,\mathbf{b
}=\left\{
\begin{array}{ccc}
\square _{I}^{\sigma ,\pi ,\mathbf{b}} & \text{ if } & I=K \\
0 & \text{ if } & I\not=
\end{array
\right. ,\ \ \ \ \ I,K\in \mathcal{C}_{A}^{\mathcal{A}}.
\end{equation*
The orthogonal projections
\begin{eqnarray*}
\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi ,\mathbf{b
} &\equiv &\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\square
_{I}^{\sigma ,\pi ,\mathbf{b}}=\sum_{I\in \Pi _{1}\mathcal{Q}}\square _{\pi
I}^{\sigma ,\pi ,\mathbf{b}}, \\
\text{where }\pi \left( \Pi _{1}\mathcal{Q}\right) &\equiv &\left\{ \pi _
\mathcal{D}}I:I\in \Pi _{1}\mathcal{Q}\right\} \text{ and }\Pi _{1}\mathcal{
}\subset \mathcal{C}_{A}^{\mathcal{A},\limfunc{restrict}}\ ,
\end{eqnarray*
thus satisfy the equalitie
\begin{equation}
\square _{\pi I}^{\sigma ,\pi ,\mathbf{b}}f=\square _{\pi I}^{\sigma ,\pi
\mathbf{b}}\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi
\mathbf{b}}f\text{ and }\widehat{\square }_{\pi I}^{\sigma ,\pi ,\mathbf{b
}f=\widehat{\square }_{\pi I}^{\sigma ,\pi ,\mathbf{b}}\mathsf{P}_{\pi
\left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi ,\mathbf{b}}f,\ \ \ \ \
\text{for }I\in \Pi _{1}\mathcal{Q}\subset \mathcal{C}_{A}^{\mathcal{A}
\limfunc{restrict}}, \label{sat}
\end{equation
which will permit us to apply certain projection tricks used for Haar
projections in the proof of $T1$ theorems.
However, in our sublinear stopping form $\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}$, the dual
martingale projections in use in the functio
\begin{equation}
\varphi _{J}^{\mathcal{Q}^{S}}\equiv \sum_{I\in \Pi _{1}\mathcal{Q}^{S}
\mathcal{\ }\left( I,J\right) \in \mathcal{Q}^{S}}b_{A}E_{I}^{\sigma }\left(
\widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \ \mathbf{1
_{A\setminus I}\ , \label{in use}
\end{equation
given in (\ref{def phi P}) above, are the modified pseudoprojections
\left\{ \widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}\right\}
_{I\in \Pi _{1}\mathcal{Q}}$, where $\square _{\pi I}^{\sigma ,\flat
\mathbf{b}}$ differs from the orthogonal projection $\square _{\pi
I}^{\sigma ,\pi ,\mathbf{b}}$ for $I\in \Pi _{1}\mathcal{Q}$ b
\begin{equation*}
\square _{\pi I}^{\sigma ,\flat ,\mathbf{b}}f-\square _{\pi I}^{\sigma ,\pi
\mathbf{b}}f=\left\{ \left( \sum_{I^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( \pi I\right) }\mathbb{F}_{I^{\prime }}^{\sigma
,b_{A}}f\right) -\mathbb{F}_{\pi I}^{\sigma ,b_{A}}f\right\} -\left\{ \left(
\sum_{I^{\prime }\in \mathfrak{C}\left( \pi I\right) }\mathbb{F}_{I^{\prime
}}^{\sigma ,b_{A}}f\right) -\mathbb{F}_{\pi I}^{\sigma ,b_{A}}f\right\}
=-\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( \pi I\right)
\mathbb{F}_{I^{\prime }}^{\sigma ,b_{A}}f.
\end{equation*
But the "box support" $\func{Supp}_{\limfunc{box}}$ of this last expression
\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( \pi I\right)
\mathbb{F}_{I^{\prime }}^{\sigma ,b_{A}}f$ consists of the broken children
of $\pi I$, $\mathfrak{C}_{\limfunc{broken}}\left( \pi I\right) $, and is
contained in the set $\dbigcup\limits_{I\in \mathcal{C}_{A}^{\limfunc
restrict}}}\dbigcup\limits_{I^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left(
A\right) \cap \mathfrak{C}_{\mathcal{D}}\left( \pi I\right) }\left\{
I^{\prime }\right\} $, i.e.
\begin{eqnarray*}
\func{Supp}_{\limfunc{box}}\left( \sum_{I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( \pi I\right) }\mathbb{F}_{I^{\prime }}^{\sigma
,b_{A}}f\right) &\subset &\left\{ I^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) :I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( \pi
I\right) \text{ for some }I\in \mathcal{C}_{A}^{\limfunc{restrict}}\right\}
\\
&=&\dbigcup\limits_{I\in \mathcal{C}_{A}^{\limfunc{restrict
}}\dbigcup\limits_{I^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right)
\cap \mathfrak{C}_{\mathcal{D}}\left( \pi I\right) }\left\{ I^{\prime
}\right\} .
\end{eqnarray*
But $I\in \Pi _{1}\mathcal{Q}^{S}\subset \mathcal{C}_{A}^{\limfunc{restrict
} $ is a \emph{natural} child of $\pi I$, and s
\begin{equation*}
I\cap \func{Supp}_{\limfunc{box}}\left( \sum_{I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( \pi I\right) }\mathbb{F}_{I^{\prime }}^{\sigma
,b_{A}}f\right) =\emptyset .
\end{equation*
It now follows that we hav
\begin{equation}
E_{I}^{\sigma }\left( \widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b
}f\right) =E_{I}^{\sigma }\left( \widehat{\square }_{\pi I}^{\sigma ,\pi
\mathbf{b}}f\right) ,\ \ \ \ \ \text{for }I\in \mathcal{C}_{A}^{\limfunc
restrict}}. \label{fol}
\end{equation}
Returning to (\ref{in use}), we have from (\ref{sat}) and (\ref{fol}) the
identity
\begin{eqnarray}
\varphi _{J}^{\mathcal{Q}^{S}} &\equiv &\sum_{I\in \Pi _{1}\mathcal{Q}^{S}
\mathcal{\ }\left( I,J\right) \in \mathcal{Q}^{S}}b_{A}E_{I}^{\sigma }\left(
\widehat{\square }_{\pi I}^{\sigma ,\pi ,\mathbf{b}}f\right) \ \mathbf{1
_{A\setminus I} \label{iden} \\
&=&\sum_{I\in \Pi _{1}\mathcal{Q}^{S}:\mathcal{\ }\left( I,J\right) \in
\mathcal{Q}^{S}}b_{A}E_{I}^{\sigma }\left( \widehat{\square }_{\pi
I}^{\sigma ,\pi ,\mathbf{b}}\left( \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q
\right) }^{\sigma ,\pi ,\mathbf{b}}f\right) \right) \ \mathbf{1}_{A\setminus
I}\ , \notag
\end{eqnarray
which will play a critical role in proving the following $\flat $Straddling
and Substraddling lemmas. The $\flat $Straddling Lemma is an adaptation of
Lemmas 3.19 and 3.16 in \cite{Lac}.
\begin{lemma}
\label{straddle 3 ref}Let $\mathcal{Q}$ be a reduced admissible collection
of pairs for $A$, and suppose that $\mathcal{S}\subset \Pi _{1}^{\limfunc
below}}\mathcal{Q}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ is a
subpartition of $A$ such that $\mathcal{Q}$ $\flat $straddles $\mathcal{S}$.
Then we have the restricted sublinear norm boun
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{Q}}\leq C_{\mathbf{r}}\sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc
loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \leq C_{\mathbf{r
}\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q
\right) , \label{sub loc bound}
\end{equation
where $\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}$ is an $S
-localized size condition with an $S$-hole given b
\begin{equation}
\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q
\right) ^{2}\equiv \sup_{K\in \mathcal{W}^{\ast }\left( S\right) \cap
\mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert
_{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus
S}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\sum_{J\in \Pi
_{2}^{K,\limfunc{aug}}\mathcal{Q}}\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}. \label{localized size ref}
\end{equation}
\end{lemma}
\begin{proof}
For $S\in \mathcal{S}$ let $\mathcal{Q}^{S}\equiv \left\{ \left( I,J\right)
\in \mathcal{Q}:J^{\flat }\subset S\subset I\right\} $. We begin by using
that the reduced collection $\mathcal{Q}$ straddles $\mathcal{S}$, together
with the sublinearity property (\ref{phi sublinear}) of $\varphi _{J}^
\mathcal{Q}}$, and with $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( f,g\right) $ as in (\re
{def mod B}), to writ
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &=&\sum_{J\in \Pi _{2}\mathcal{Q}}\frac
\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\sum_{S\in \mathcal{S}}\sum_{J\in \Pi _{2}^{S,\limfunc{aug}}\mathcal{
}}\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{Q
^{S}}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{Q}}\left( J\right)
}\sigma \right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup
_{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit }\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }; \\
\text{where }\varphi _{J}^{\mathcal{Q}^{S}} &\equiv &\sum_{I\in \Pi _{1
\mathcal{Q}^{S}:\mathcal{\ }\left( I,J\right) \in \mathcal{Q
^{S}}b_{A}E_{I}^{\sigma }\left( \widehat{\square }_{\pi I}^{\sigma ,\flat
\mathbf{b}}f\right) \ \mathbf{1}_{A\setminus I}\ .
\end{eqnarray*}
At this point we invoke the identity (\ref{iden})
\begin{equation*}
\varphi _{J}^{\mathcal{Q}^{S}}=\sum_{I\in \Pi _{1}\mathcal{Q}^{S}:\mathcal{\
}\left( I,J\right) \in \mathcal{Q}^{S}}b_{A}E_{I}^{\sigma }\left( \widehat
\square }_{\pi I}^{\sigma ,\pi ,\mathbf{b}}\left( \mathsf{P}_{\pi \left( \Pi
_{1}\mathcal{Q}\right) }^{\sigma ,\pi ,\mathbf{b}}f\right) \right) \ \mathbf
1}_{A\setminus I}\ ,
\end{equation*
so tha
\begin{equation*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) =\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( h,g\right)
,\ \ \ \ \ \text{where }h\equiv \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q
\right) }^{\sigma ,\pi ,\mathbf{b}}f\ .
\end{equation*
We will treat the sublinear form $\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( h,g\right) $
with $h=\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi
\mathbf{b}}f$ using a small variation on the corresponding argument in Lacey
\cite{Lac}\footnote
There is a gap in the treatment of the Straddling Lemma 11.10 on page 166 of
\cite{SaShUr7}. The wrong restricted norm is used there, but can be fixed by
using the corresponding argument of Lacey in \cite{Lac}, equivalently
adapting the argument here. See Appendix C for a full discussion.}. Namely,
we will apply a Calder\'{o}n-Zygmund stopping time decomposition to the
function $h=\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi
,\mathbf{b}}f$ on the interval $A$ \ with `obstacle' $\mathcal{S}\cup
\mathfrak{C}_{A}$ $\left( A\right) $, to obtain stopping times $\mathcal{H}$
$\subset \mathcal{C}_{A}$ with the property that for all $H\in \mathcal{H
\setminus \left\{ A\right\} $ we have
\begin{eqnarray*}
&&H\in \mathcal{C}_{A}\text{ is not strictly contained in any interval from
\mathcal{S}, \\
&&E_{H}^{\sigma }\left\vert h\right\vert >\Gamma E_{\pi _{\mathcal{H
}H}^{\sigma }\left\vert h\right\vert , \\
&&E_{H^{\prime }}^{\sigma }\left\vert h\right\vert \leq \Gamma E_{\pi _
\mathcal{H}}H}^{\sigma }\left\vert h\right\vert \text{ for all }H\subsetneqq
H^{\prime }\subset \pi _{\mathcal{H}}H\text{ with }H^{\prime }\in \mathcal{C
_{A}.
\end{eqnarray*
More precisely, define generation $0$ of $\mathcal{H}$ to consist of the
single interval $A$. Having defined generation $n$, let generation $n+1$
consist of the union over all intervals $M$ in generation $n$ of the maximal
intervals $M^{\prime }$ in $\mathcal{C}_{A}$ that are contained in $M$ with
E_{M^{\prime }}^{\sigma }\left\vert h\right\vert >\Gamma E_{M}^{\sigma
}\left\vert h\right\vert $, but are \emph{not} strictly contained in any
interval $S$ from $\mathcal{S}$ or contained in any interval $A^{\prime }$
from $\mathfrak{C}_{A}$ $\left( A\right) $ - thus the construction stops at
the obstacle $\mathcal{S}\cup \mathfrak{C}_{A}$ $\left( A\right) $. Then
\mathcal{H}$ is the union of all generations $n\geq 0$.
Denote by
\begin{equation*}
\mathcal{C}_{H}^{\mathcal{H}}\equiv \left\{ H^{\prime }\in \mathcal{C
_{A}:H^{\prime }\subset H\text{ but }H^{\prime }\not\subset H^{\prime \prime
}\text{ for any }H^{\prime \prime }\in \mathfrak{C}_{\mathcal{H}}\left(
H\right) \right\}
\end{equation*
the usual $\mathcal{H}$-corona associated with the stopping interval $H$,
but restricted to $\mathcal{C}_{A}$, and let $\alpha _{\mathcal{H}}\left(
H\right) =E_{H}^{\sigma }\left\vert f\right\vert $ as is customary for a
Calder\'{o}n-Zygmund corona. Since these coronas $\mathcal{C}_{H}^{\mathcal{
}}$ are all contained in $\mathcal{C}_{A}$, we have the stopping energy from
the $\mathcal{A}$-corona $\mathcal{C}_{A}$ at our disposal, which as in \cit
{Lac}, is crucial for the argument. Furthermore, we denote b
\begin{equation}
\mathcal{Q}_{H}\equiv \left\{ \left( I,J\right) \in \mathcal{Q}:J\in
\mathcal{C}_{H}^{\mathcal{H},\flat \func{shift}}\right\} ,\ \ \ \ \ \text
with }\mathcal{C}_{H}^{\mathcal{H},\flat \func{shift}}\equiv \left\{ J\in
\Pi _{2}\mathcal{Q}:J^{\flat }\in \mathcal{C}_{H}^{\mathcal{H}}\right\} ,
\label{def Q H}
\end{equation
the restriction of the pairs $\left( I,J\right) $ in $\mathcal{Q}$ to those
for which $J$ lies in the flat shifted $\mathcal{H}$-corona $\mathcal{C
_{H}^{\mathcal{H},\flat \func{shift}}$. Since the $\mathcal{H}$-stopping
intervals satisfy a $\sigma $-Carleson condition for $\Gamma $ chosen large
enough, we have the quasiorthogonal inequality
\begin{equation}
\sum_{H\in \mathcal{H}}\alpha _{\mathcal{H}}\left( H\right) ^{2}\left\vert
H\right\vert _{\sigma }\lesssim \left\Vert h\right\Vert _{L^{2}\left( \sigma
\right) }^{2}, \label{qor}
\end{equation
which below we will see reduces matters to proving inequality (\ref{sub loc
bound}) for the family of reduced admissible collections $\left\{ \mathcal{Q
_{H}\right\} _{H\in \mathcal{H}}$ with constants independent of $H$
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{Q}_{H}}\leq C_{\mathbf{r}}\sup_{S\in \mathcal{S}}\mathcal{S}_
\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}_{H}\right) \leq
C_{\mathbf{r}}\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{Q}_{H}\right) ,\ \ \ \ \ H\in \mathcal{H}.
\end{equation*}
Given $S\in \mathcal{S}$, define $H_{S}\in \mathcal{H}$ to be the minimal
interval in $\mathcal{H}$ that contains $S$, and then define
\begin{equation*}
\mathcal{H}_{\mathcal{S}}\equiv \left\{ H_{S}\in \mathcal{H}:S\in \mathcal{S
\right\} .
\end{equation*
Note that a given $H\in \mathcal{H}_{\mathcal{S}}$ may have many intervals
S\in \mathcal{S}$ such that $H=H_{S}$, and we denote the collection of these
intervals by $\mathcal{S}_{H}\equiv \left\{ S\in \mathcal{S}:H_{S}=H\
\right\} $. We will organize the straddling intervals $\mathcal{S}$ a
\begin{equation*}
\mathcal{S}=\dbigcup\limits_{H\in \mathcal{H}_{\mathcal{S
}}\dbigcup\limits_{S\in \mathcal{S}_{H}}
\end{equation*
where each $S\in \mathcal{S}$ occurs exactly once in the union on the right
hand side, i.e. the collections $\left\{ \mathcal{S}_{H}\right\} _{H\in
\mathcal{H}_{\mathcal{S}}}$ are pairwise disjoint.
We now momentarily fix $H\in \mathcal{H}_{\mathcal{S}}$, and consider the
reduced admissible collection $\mathcal{Q}_{H}$, so that its projection onto
the second component $\Pi _{2}\mathcal{Q}_{H}$ of $\mathcal{Q}_{H}$\ is
\emph{contained} in the corona $\mathcal{C}_{H}^{\mathcal{H},\flat \func
shift}}$. Then the collection $\mathcal{Q}_{H}$ $\flat $straddles the set
\mathcal{S}_{H}=\left\{ S\in \mathcal{S}:H_{S}=H\ \right\} $. Moreover,
\mathcal{Q}_{H}=\dbigcup\limits_{S\in \mathcal{S}:\ S\subset H}\mathcal{Q
_{H}^{S}$ and $\Pi _{2}\mathcal{Q}_{H}^{S}=\Pi _{2}^{S,\limfunc{aug}
\mathcal{Q}_{H}$.
Recall that a Whitney interval $K$ was required in the right hand side of
the conclusion of Proposition \ref{flatness} only in the case that $J^{\flat
}\subset S^{\prime \prime }$ for some $S^{\prime \prime }\in \mathfrak{C}_
\mathcal{D}}^{\left( 2\right) }\left( S\right) $, which of course implies
3J^{\flat }\subset J^{\maltese }\subset S$. In this case we claim that $K\in
\mathcal{C}_{A}$. Indeed, suppose in order to derive a contradiction, that
K\not\in \mathcal{C}_{A}$. Then $J^{\maltese }\not\subset K$, and hence
3J^{\maltese }\not\subset S$. Since $J^{\maltese }\subset S$, it follows
that $J^{\maltese }$ shares an endpoint with $S$ (since if not, then
3J^{\maltese }\subset S$, a contradiction). Now Key Fact \#2 in (\re
{indentation}) implies that the inner grandchild containing $J$, either
J_{-/+}^{\maltese }$ or $J_{+/-}^{\maltese }$, is contained in $K$ where
K\not\in \mathcal{C}_{A}$. This then implies that the pair $\left(
I,J\right) $ belongs to the corona straddling subcollection $\mathcal{P}_
\func{cor}}^{A}$, contradicting the assumption that $\mathcal{Q}$ is reduced.
Thus we have $S\in \Pi _{1}^{\limfunc{below}}\mathcal{Q}\cap \mathcal{C
_{A}^{\limfunc{restrict}}$ and $K\in \mathcal{W}\left( S\right) \cap
\mathcal{C}_{A}^{\limfunc{restrict}}$ and we can use Proposition \re
{flatness} with $H=H_{S}$ to bound $\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( f,g\right) $
by first summing over $H\in \mathcal{H}_{\mathcal{S}}$ and then over $S\in
\mathcal{S}_{H}$. Indeed, $\mathcal{Q}_{H}$ $\flat $straddles $\mathcal{S
_{H}\equiv \left\{ S\in \mathcal{S}:H_{S}=H\ \right\} $, so that $\left\vert
\varphi _{J}^{\mathcal{Q}_{H}}\right\vert \lesssim \alpha _{\mathcal{H
}\left( H\right) \mathbf{1}_{A\setminus I_{\mathcal{Q}_{H}}\left( J\right) }$
by (\ref{phi bound}), and so the sum over $S\in \mathcal{S}_{H}$ of the
first term on the right side of the conclusion of Proposition \ref{flatness}
is bounded b
\begin{eqnarray*}
&&\alpha _{\mathcal{H}}\left( H\right) \sum_{S\in \mathcal{S}_{H}}\sqrt
\left\vert S\right\vert _{\sigma }}\frac{1}{\sqrt{\left\vert S\right\vert
_{\sigma }}}\left( \frac{\mathrm{P}^{\alpha }\left( S,\mathbf{1}_{A\setminus
S}\sigma \right) }{\left\vert S\right\vert }\right) \left\Vert \mathsf{Q
_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\mathsf{P}_{\Pi _{2}^{S,\limfunc{aug}}\mathcal{Q}_{H}}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\alpha _{\mathcal{H}}\left( H\right) \left\{ \sup_{S\in \mathcal{S
_{H}}\frac{1}{\sqrt{\left\vert S\right\vert _{\sigma }}}\left( \frac{\mathrm
P}^{\alpha }\left( S,\mathbf{1}_{A\setminus S}\sigma \right) }{\left\vert
S\right\vert }\right) \left\Vert \mathsf{Q}_{\Pi _{2}^{S,\limfunc{aug}
\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit }\right\} \sum_{S\in \mathcal{S}_{H}}\sqrt
\left\vert S\right\vert _{\sigma }}\left\Vert \mathsf{P}_{\Pi _{2}^{S
\limfunc{aug}}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\alpha _{\mathcal{H}}\left( H\right) \left\{ \sup_{S\in \mathcal{S
_{H}}\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{
}_{H}\right) \right\} \sqrt{\left\vert H\right\vert _{\sigma }}\left\Vert
\mathsf{P}_{\Pi _{2}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }\ ,
\end{eqnarray*
where $\Pi _{2}^{K,\limfunc{aug}}\mathcal{Q}_{H}$ is as in Definition \re
{def aug}, and the corresponding sum over $S\in \mathcal{S}_{H}$ of the
second term is bounded b
\begin{eqnarray*}
&&\alpha _{\mathcal{H}}\left( H\right) \sum_{S\in \mathcal{S}_{H}}\sum_{K\in
\mathcal{W}\left( S\right) \cap \mathcal{C}_{A}^{\limfunc{restrict}}}\sqrt
\left\vert K\right\vert _{\sigma }}\frac{1}{\sqrt{\left\vert K\right\vert
_{\sigma }}}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus
S}\sigma \right) }{\left\vert K\right\vert }\right) \left\Vert \mathsf{Q
_{\Pi _{2}^{K,\limfunc{aug}}\mathcal{Q}_{H}^{S}}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit }\left\Vert
\mathsf{P}_{\Pi _{2}^{K,\limfunc{aug}}\mathcal{Q}_{H}^{S}}^{\omega ,\mathbf{
}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\lesssim &\alpha _{\mathcal{H}}\left( H\right) \sup_{S\in \mathcal{S}_
\mathcal{H}}}\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left(
\mathcal{Q}_{H}\right) \left( \sum_{S\in \mathcal{S}}\sum_{K\in \mathcal{W
\left( S\right) }\left\vert K\right\vert _{\sigma }\right) ^{\frac{1}{2
}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar } \\
&\leq &\left\{ \sup_{S\in \mathcal{S}_{\mathcal{H}}}\mathcal{S}_{\limfunc{lo
}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}_{H}\right) \right\} \alpha
_{\mathcal{H}}\left( H\right) \sqrt{\left\vert H\right\vert _{\sigma }
\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{Q}_{\mathcal{H}}}^{\omega ,\mathbf{b
^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar }.
\end{eqnarray*}
Using the definition of $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( f,g\right) $ in (\ref{def
mod B}), we now sum the previous inequalities over the intervals $H\in
\mathcal{H}_{\mathcal{S}}$ to obtain the following string of inequalities
(explained in detail after the display
\begin{eqnarray}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &\leq &\left\{ \sup_{S\in \mathcal{S}
\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q
\right) \right\} \sum_{H\in \mathcal{H}_{\mathcal{S}}}\alpha _{\mathcal{H
}\left( H\right) \sqrt{\left\vert H\right\vert _{\sigma }}\left\Vert \mathsf
P}_{\Pi _{2}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar } \label{unfix} \\
&\leq &\left\{ \sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{loc}\limfunc{siz
}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \right\} \sqrt{\sum_{H\in \mathcal
H}_{\mathcal{S}}}\alpha _{\mathcal{H}}\left( H\right) ^{2}\left\vert
H\right\vert _{\sigma }}\sqrt{\sum_{H\in \mathcal{H}_{\mathcal{S
}}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{Q}_{H}}^{\omega ,\mathbf{b}^{\ast
}}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}} \notag \\
&\lesssim &\left\{ \sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{loc}\limfunc
size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \right\} \left\Vert
h\right\Vert _{L^{2}\left( \sigma \right) }\sqrt{\sum_{H\in \mathcal{H}_
\mathcal{S}}}\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{Q}_{H}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar 2}}
\notag \\
&\leq &\left\{ \sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{loc}\limfunc{siz
}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \right\} \left\Vert \mathsf{P
_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi ,\mathbf{b
}f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert \mathsf{P}_{\Pi _{2
\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar } \notag \\
&\lesssim &\left\{ \sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{loc}\limfunc
size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \right\} \left\Vert \mathsf{P
_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \sigma \right) }^{\bigstar }\left\Vert \mathsf{P}_{\Pi _{2
\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega
\right) }^{\bigstar }\ , \notag
\end{eqnarray
where in the first line we have used $\mathcal{Q}=\dbigcup\limits_{H\in
\mathcal{H}_{\mathcal{S}}}\mathcal{Q}_{H}$, which follows from the fact that
each $J^{\flat }$ is contained in a unique $S\in \mathcal{S}$; in the third
line we have used the quasiorthogonal inequality (\ref{qor}); in the fourth
line we have used that the sets $\Pi _{2}\mathcal{Q}_{H}\subset \mathcal{C
_{H}^{\mathcal{H},\flat \func{shift}}$ are pairwise disjoint in $H$ and have
union $\Pi _{2}\mathcal{Q}=$ $\overset{\cdot }{\dbigcup }_{H\in \mathcal{H}_
\mathcal{S}}}\Pi _{2}\mathcal{Q}_{H}$. In the final line, we have used first
the equality (\ref{box pi equals}), second the fact that the functions
\square _{I,\limfunc{broken}}^{\sigma ,\pi ,\mathbf{b}}f$ have pairwise
disjoint supports, third the upper weak Riesz inequality in Proposition \re
{half Riesz}, and fourth the estimate (\ref{F est}) - which relies on the
reverse H\"{o}lder property for children in Lemma \ref{prelim control of
corona} - to obtai
\begin{eqnarray}
\left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma ,\pi
\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{2} &=&\left\Vert
\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\square _{I}^{\sigma
\mathbf{b}}f-\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\square _{I
\limfunc{broken}}^{\sigma ,\pi ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }^{2} \label{needed for unfix} \\
&\lesssim &\left\Vert \sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right)
}\square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{2}+\left\Vert \sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\square
_{I,\limfunc{broken}}^{\sigma ,\pi ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }^{2} \notag \\
&\lesssim &\left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right)
}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{2}+\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\left\Vert \square
_{I,\limfunc{broken}}^{\sigma ,\pi ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\sigma \right) }^{2} \notag \\
&\lesssim &\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\left\Vert
\square _{I}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right)
}^{2}+\sum_{I\in \pi \left( \Pi _{1}\mathcal{Q}\right) }\left\Vert
\bigtriangledown _{I}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right)
}^{2}\lesssim \left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right)
}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma \right) }^{\bigstar
2}\ . \notag
\end{eqnarray}
\medskip
We now use the fact that the supremum in the definition of $\mathcal{S}_
\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) $ is
taken over $K\in \mathcal{W}^{\ast }\left( S\right) \cap \mathcal{C}_{A}^
\limfunc{restrict}}$ to conclude that
\begin{equation*}
\sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha
,A;S}\left( \mathcal{Q}\right) \leq \mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{Q}\right) ,
\end{equation*
and this completes the proof of Lemma \ref{straddle 3 ref}.
\end{proof}
In a similar fashion we can obtain the following Substraddling Lemma.
\begin{definition}
\label{def substraddles}Given a \emph{reduced admissible} collection of
pairs $\mathcal{Q}$ for $A$, and a $\mathcal{D}$-interval $L$ contained in
A $, we say that $\mathcal{Q}$ \textbf{substraddles} $L$ if for every pair
\left( I,J\right) \in \mathcal{Q}$ there is $K\in \mathcal{W}\left( L\right)
\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ with $J\subset K\subset 3K\subset
I\subset L$.
\end{definition}
\begin{lemma}
\label{substraddle ref}Let $L$ be a $\mathcal{D}$-interval contained in $A$,
and suppose that $\mathcal{Q}$ is an admissible collection of pairs that
substraddles $L$. Then we have the sublinear form boun
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{Q}}\leq C\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha
,A}\left( \mathcal{Q}\right) .
\end{equation*}
\end{lemma}
\begin{proof}
We will show that $\mathcal{Q}$ $\flat $straddles the subset $\mathcal{W
_{L} $ of Whitney intervals for $L$ given b
\begin{equation*}
\mathcal{W}^{\mathcal{Q}}\left( L\right) \equiv \left\{ K\in \mathcal{W
\left( L\right) \cap \mathcal{C}_{A}^{\limfunc{restrict}}:J\subset K\subset
3K\subset I\subset L\text{ for some }\left( I,J\right) \in \mathcal{Q
\right\} .
\end{equation*
It is clear that $\mathcal{W}^{\mathcal{Q}}\left( L\right) \subset \Pi _{1}^
\limfunc{below}}\mathcal{Q}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ is a
subpartition of $A$. It remains to show that for every pair $\left(
I,J\right) \in \mathcal{Q}$ there is $K\in \mathcal{W}^{\mathcal{Q}}\left(
L\right) \cap \left[ J,I\right] $ such that $J^{\flat }\subset K$. But our
hypothesis implies that there is $K\in \mathcal{W}^{\mathcal{Q}}\left(
L\right) $ with $J\subset K\subset 3K\subset I\subset L$. We now consider
two cases.
\textbf{Case 1}: If $\pi _{\mathcal{D}}^{\left( 3\right) }K\subset L$, then
by Key Fact \#2 in (\ref{indentation}), i.e. $3J$ is contained in an \emph
inner} grandchild of $J^{\maltese }$. But $K$ is contained in an \emph{outer}
grandchild of $\pi _{\mathcal{D}}^{\left( 3\right) }K$ since $\pi _{\mathcal
D}}^{\left( 1\right) }K$ shares an endpoint with $L$, and so then does $\pi
_{\mathcal{D}}^{\left( 3\right) }K$). We thus have $J^{\maltese }\subset \pi
_{\mathcal{D}}^{\left( 2\right) }K$, which implies that $J^{\flat }\subset K
.
\textbf{Case 2}: If $\pi _{\mathcal{D}}^{\left( 3\right) }K\varsupsetneqq L
, then $K\subset 3K\subset I\subset L$ implies that $I=L=\pi _{\mathcal{D
}^{\left( 2\right) }K$. Thus we have $J^{\maltese }\subset I=\pi _{\mathcal{
}}^{\left( 2\right) }K$, which again gives $J^{\flat }\subset K$.
Now that we know $\mathcal{Q}$ $\flat $straddles the subset $\mathcal{W}^
\mathcal{Q}}\left( L\right) $, we can apply Lemma \ref{straddle 3 ref} to
obtain the required bound $\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\leq C\mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q}\right) $.
\end{proof}
\subsection{The bottom/up stopping time argument of M. Lacey}
Before introducing Lacey's stopping times, we note that the
Corona-straddling Lemma \ref{cor strad 1} allows us to remove the `corona
straddling' collection $\mathcal{P}_{\func{cor}}^{A}$ of pairs of intervals
in (\ref{def cor}) from the collection $\mathcal{P}^{A}$ in (\ref{initial P
) used to define the stopping form $\mathsf{B}_{\limfunc{stop}}^{A}\left(
f,g\right) $. The collection $\mathcal{P}^{A}\setminus \mathcal{P}_{\func{co
}}^{A}$ is of course also $A$-admissible.
\begin{conclusion}
\label{assume}We assume for the remainder of the proof that all admissible
collections $\mathcal{P}$ are reduced, i.e.
\begin{equation}
\mathcal{P}^{A}\cap \mathcal{P}_{\func{cor}}^{A}=\emptyset ,\text{ as well
as }\mathcal{P}\cap \mathcal{P}_{\func{cor}}^{A}=\emptyset \text{ for all }
\text{-admissible }\mathcal{P}. \label{empty assumption}
\end{equation}
\end{conclusion}
We remind the reader again that we will generally use $\left\vert
J\right\vert $ in the Poisson integrals and estimates, but will usually use
\ell \left( J\right) $ when defining collections of intervals. For an
interval $K\in \mathcal{D}$, we defin
\begin{equation*}
\mathcal{G}\left[ K\right] \equiv \left\{ J\in \mathcal{G}:J\subset K\right\}
\end{equation*
to consist of all intervals $J$ in the other grid $\mathcal{G}$ that are
contained in $K$. For an $A$-admissible collection $\mathcal{P}$ of pairs,
define two atomic measures $\omega _{\mathcal{P}}$ and $\omega _{\flat
\mathcal{P}}$ in the upper half space $\mathbb{R}_{+}^{2}$ b
\begin{equation}
\omega _{\mathcal{P}}\equiv \sum_{J\in \Pi _{2}\mathcal{P}}\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2}\ \delta _{\left( c_{J^{\maltese }},\ell
\left( J^{\maltese }\right) \right) }\text{ and }\omega _{\flat \mathcal{P
}\equiv \sum_{J\in \Pi _{2}\mathcal{P}}\left\Vert \bigtriangleup
_{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\ \delta _{\left( c_{J^{\flat }},\ell \left( J^{\flat
}\right) \right) }, \label{def atomic}
\end{equation
where $J^{\flat }=J_{\searrow J}^{\maltese }$ is the inner grandchild of
J^{\maltese }$ that contains $J$, i.e. $J^{\flat }=\left\{
\begin{array}{ccc}
J_{+/-}^{\maltese } & \text{ if } & J\subset J_{+/-}^{\maltese } \\
J_{-/+}^{\maltese } & \text{ if } & J\subset J_{-/+}^{\maltese
\end{array
\right. $. Note that each interval $J\in \Pi _{2}\mathcal{P}$ has its
`energy' $\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}$ in the measure
$\omega _{\flat \mathcal{P}}$ assigned to exactly one of the two points
\left( c_{J_{-/+}^{\maltese }},\frac{1}{4}\ell \left( J^{\maltese }\right)
\right) $ and $\left( c_{J_{+/-}^{\maltese }},\frac{1}{4}\ell \left(
J^{\maltese }\right) \right) $ in the upper half plane $\mathbb{R}_{+}^{2}$
since $J$ is either contained in $J_{-/+}^{\maltese }$\ or in
J_{+/-}^{\maltese }$ by Key Fact \#2 in (\ref{indentation}). Note also that
the atomic measure $\omega _{\flat \mathcal{P}}$ differs from the measure
\mu $ in (\ref{def mu n}) in Appendix B below - which is used there to
control the functional energy condition - in that here we bundle together
all the $J^{\prime }s$ having a common $J^{\flat }$. This is in order to
rewrite the \emph{augmented} size functional in terms of the measure $\omega
_{\flat \mathcal{P}}$. We can get away with this here, as opposed to in
Appendix B, due to the `smaller and decoupled' nature of the augmented size
functional to which we will relate $\omega _{\flat \mathcal{P}}$.
Define the tent $\mathbf{T}\left( L\right) $ over an interval $L$ to be the
convex hull of the interval $L\times \left\{ 0\right\} $ and the point
\left( c_{L},\ell \left( L\right) \right) \in \mathbb{R}_{+}^{2}$. Then for
J\in \Pi _{2}\mathcal{P}$ we have $J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}
$ \emph{iff} $\left\{ J\subset K\text{ and }J^{\maltese }\subset \pi _
\mathcal{D}}^{\left( 2\right) }K\right\} $ \emph{iff} $J^{\flat
}=J_{\searrow J}^{\maltese }\subset K$ \emph{iff} $\left( c_{J^{\flat
}},\ell \left( J^{\flat }\right) \right) \in \mathbf{T}\left( K\right) $. We
can now rewrite the augmented size functional of $\mathcal{P}$ in Definition
\ref{augs}\ a
\begin{equation}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}\equiv \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert
_{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus
K}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( K\right) \right) .
\label{def P stop energy' 3}
\end{equation
It will be convenient to writ
\begin{equation*}
\Psi ^{\alpha }\left( K;\mathcal{P}\right) ^{2}\equiv \left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert
K\right\vert }\right) ^{2}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left(
K\right) \right) ,
\end{equation*
so that we have simpl
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}=\sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap \mathcal{C
_{A}^{\limfunc{restrict}}}\frac{\Psi ^{\alpha }\left( K;\mathcal{P}\right)
^{2}}{\left\vert K\right\vert _{\sigma }}.
\end{equation*}
\begin{remark}
The functional $\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right)
\right) $ is increasing in $K$, while the functional $\frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert
K\right\vert }$ is `almost decreasing' in $K$: if $K_{0}\subset K$ the
\begin{eqnarray*}
\frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }
\left\vert K\right\vert } &=&\int_{A\setminus K}\frac{d\sigma \left(
y\right) }{\left( \left\vert K\right\vert +\left\vert y-c_{K}\right\vert
\right) ^{2-\alpha }} \\
&\lesssim &\int_{A\setminus K}\frac{d\sigma \left( y\right) }{\left(
\left\vert K_{0}\right\vert +\left\vert y-c_{K_{0}}\right\vert \right)
^{2-\alpha }} \\
&\leq &C_{\alpha }\int_{A\setminus K_{0}}\frac{d\sigma \left( y\right) }
\left( \left\vert K_{0}\right\vert +\left\vert y-c_{K_{0}}\right\vert
\right) ^{2-\alpha }}=C_{\alpha }\frac{\mathrm{P}^{\alpha }\left( K_{0}
\mathbf{1}_{A\setminus K_{0}}\sigma \right) }{\left\vert K_{0}\right\vert },
\end{eqnarray*
since $\left\vert K_{0}\right\vert +\left\vert y-c_{K_{0}}\right\vert \leq
\left\vert K\right\vert +\left\vert y-c_{K}\right\vert +\frac{1}{2}\limfunc
diam}\left( K\right) $ for $y\in A\setminus K$.
\end{remark}
Recall that if $\mathcal{P}$ is an admissible collection for a dyadic
interval $A$, the corresponding sublinear form in (\ref{def mod B}) and (\re
{First inequality}) is given b
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) &\equiv &\sum_{J\in \Pi _{2}\mathcal{P}
\frac{\mathrm{P}^{\alpha }\left( J,\left\vert \varphi _{J}^{\mathcal{P
}\right\vert \mathbf{1}_{A\setminus I_{\mathcal{P}}\left( J\right) }\sigma
\right) }{\left\vert J\right\vert }\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
}\left\Vert \square _{J}^{\omega ,\mathbf{b}^{\ast }}g\right\Vert
_{L^{2}\left( \omega \right) }^{\bigstar }; \\
\text{where }\varphi _{J}^{\mathcal{P}} &\equiv &\sum_{I\in \mathcal{C}_{A}^
\limfunc{restrict}}:\ \left( I,J\right) \in \mathcal{P}}b_{A}E_{I}^{\sigma
}\left( \widehat{\square }_{\pi I}^{\sigma ,\flat ,\mathbf{b}}f\right) \
\mathbf{1}_{A\setminus I}\ .
\end{eqnarray*
In the notation for $\left\vert \mathsf{B}\right\vert _{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}$, we are omitting dependence on
the parameter $\alpha $, and to avoid clutter, we will often do so from now
on when the dependence on $\alpha $ is inconsequential.
Recall further that the `size testing collection' of intervals $\Pi _{1}^
\limfunc{below}}\mathcal{P}$ for the initial size testing functional
\mathcal{S}_{\limfunc{init}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) $ is the collection of all subintervals of intervals in $\Pi _{1
\mathcal{P}$, and moreover, by Key Fact \#1 in (\ref{later use}), that we
can restrict the collection to $\Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}$. This latter set is used for the
augmented size functional.
\begin{description}
\item[Assumption] We may assume that the corona $\mathcal{C}_{A}$ is finite,
and that each $A$-admissible collection $\mathcal{P}$ is a finite
collection, and hence so are $\Pi _{1}\mathcal{P}$, $\Pi _{1}^{\limfunc{belo
}}\mathcal{P}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ and $\Pi _{2
\mathcal{P}$, provided all of the bounds we obtain are independent of the
cardinality of these latter collections.
\end{description}
Consider $0<\varepsilon <1$, where $\rho =1+\varepsilon $ will be chosen
later in (\ref{choose rho}). Begin by defining the collection $\mathcal{L
_{0}$ to consist of the \emph{minimal} dyadic intervals $K$ in $\Pi _{1}^
\limfunc{below}}\mathcal{P}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ such
tha
\begin{equation*}
\frac{\Psi ^{\alpha }\left( K;\mathcal{P}\right) ^{2}}{\left\vert
K\right\vert _{\sigma }}\geq \varepsilon \mathcal{S}_{\limfunc{aug}\limfunc
size}}^{\alpha ,A}\left( \mathcal{P}\right) ^{2}.
\end{equation*
where we recall tha
\begin{equation*}
\Psi ^{\alpha }\left( K;\mathcal{P}\right) ^{2}\equiv \left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert
K\right\vert }\right) ^{2}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left(
K\right) \right) .
\end{equation*
Note that such minimal intervals exist when $0<\varepsilon <1$ because
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2}$ is the supremum over $K\in \Pi _{1}^{\limfunc{below}}\mathcal{
}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ of $\frac{\Psi ^{\alpha }\left(
K;\mathcal{P}\right) ^{2}}{\left\vert K\right\vert _{\sigma }}$. A key
property of the minimality requirement is tha
\begin{equation}
\frac{\Psi ^{\alpha }\left( K^{\prime };\mathcal{P}\right) ^{2}}{\left\vert
K^{\prime }\right\vert _{\sigma }}<\varepsilon \mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}\right) ^{2},
\label{key property 3}
\end{equation
whenever there is $K^{\prime }\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}$ with $K^{\prime }\varsubsetneqq K$ and
$K\in \mathcal{L}_{0}$.
We now perform a stopping time argument `from the bottom up' with respect to
the atomic measure $\omega _{\mathcal{P}}$ in the upper half space. This
construction of a stopping time `from the bottom up', together with the
subsequent applications of the Orthogonality Lemma and the Straddling Lemma,
comprise the key innovations in Lacey's argument \cite{Lac}. However, in our
situation the intervals $I$ belonging to $\Pi _{1}^{\limfunc{below}}\mathcal
P}$ are no longer `good' in any sense, and we must include an additional
top/down stopping criterion in the next subsection to accommodate this lack
of `goodness'. The argument in \cite{Lac} will apply to these special
stopping intervals, called `indented' intervals, and the remaining intervals
form towers with a common endpoint, that are controlled using all three
straddling lemmas.
We refer to $\mathcal{L}_{0}$ as the initial or level $0$ generation of
stopping intervals. Se
\begin{equation}
\rho =1+\varepsilon . \label{def rho}
\end{equation
As in \cite{SaShUr7}, \cite{SaShUr9} and \cite{SaShUr10}, we follow Lacey
\cite{Lac} by recursively defining a finite sequence of generations $\left\{
\mathcal{L}_{m}\right\} _{m\geq 0}$ by letting $\mathcal{L}_{m}$ consist of
the \emph{minimal} dyadic intervals $L$ in $\Pi _{1}^{\limfunc{below}
\mathcal{P}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ that contain an
interval from some previous level $\mathcal{L}_{\ell }$, $\ell <m$, such tha
\begin{equation}
\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( L\right) \right) \geq
\rho \omega _{\flat \mathcal{P}}\left( \dbigcup\limits_{L^{\prime }\in
\dbigcup\limits_{\ell =0}^{m-1}\mathcal{L}_{\ell }:\ L^{\prime }\subset L
\mathbf{T}\left( L^{\prime }\right) \right) . \label{up stopping condition}
\end{equation
Since $\mathcal{P}$ is finite this recursion stops at some level $M$. We
then let $\mathcal{L}_{M+1}$ consist of all the maximal intervals in $\Pi
_{1}^{\limfunc{below}}\mathcal{P}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$
that are not already in some $\mathcal{L}_{m}$ with $m\leq M$. Thus
\mathcal{L}_{M+1}$ will contain either none, some, or all of the maximal
intervals in $\Pi _{1}^{\limfunc{below}}\mathcal{P}$. We do not of course
have (\ref{up stopping condition}) for $A^{\prime }\in \mathcal{L}_{M+1}$ in
this case, but we do have that (\ref{up stopping condition}) fails for
subintervals $K$ of $A^{\prime }\in \mathcal{L}_{M+1}$ that are not
contained in any other $L\in \mathcal{L}_{m}$ with $m\leq M$, and this is
sufficient for the arguments below.
We now decompose the collection of pairs $\left( I,J\right) $ in $\mathcal{P}
$ into collections $\mathcal{P}^{\flat small}$ and $\mathcal{P}^{\flat big}$
according to the location of $I$ and $J^{\flat }$, but only after
introducing below the indented corona $\mathcal{H}$. The collection
\mathcal{P}^{\flat big}$ will then essentially consist of those pairs
\left( I,J\right) \in \mathcal{P}$ for which there are $L^{\prime },L\in
\mathcal{H}$ with $L^{\prime }\varsubsetneqq L$ and such that $J^{\flat }\in
\mathcal{C}_{L^{\prime }}^{\mathcal{H}}$ and $I\in \mathcal{C}_{L}^{\mathcal
H}}$. The collection $\mathcal{P}^{\flat small}$ will consist of the
remaining pairs $\left( I,J\right) \in \mathcal{P}$ for which there is $L\in
\mathcal{H}$ such that $J^{\flat },I\in \mathcal{C}_{L}^{\mathcal{H}}$,
along with the pairs $\left( I,J\right) \in \mathcal{P}$ such that $I\subset
I_{0}$ for some $I_{0}\in \mathcal{L}_{0}$. This will cover all pairs
\left( I,J\right) $ in $\mathcal{P}\subset \mathcal{P}_{A}$, since for such
pairs, $I\in \mathcal{C}_{A}^{\limfunc{restrict}}$\ and $J\in \mathcal{C
_{A}^{\mathcal{G}\func{shift}}$, which in turn implies $I\in \mathcal{C
_{L}^{\mathcal{H}}$ and $J^{\flat }\in \mathcal{C}_{L^{\prime }}^{\mathcal{H
}$ for some $L,L^{\prime }\in \mathcal{H}$. But a considerable amount of
further analysis is required to prove (\ref{First inequality}).
First recall that $\mathcal{L}\equiv \dbigcup\limits_{m=0}^{M+1}\mathcal{L
_{m}$ is the tree of stopping $\omega _{\mathcal{P}}$-energy intervals
defined above. By the construction above, the maximal elements in $\mathcal{
}$ are the maximal intervals in $\Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}$. For $L\in \mathcal{L}$, denote by
\mathcal{C}_{L}^{\mathcal{L}}$ the \emph{corona} associated with $L$ in the
tree $\mathcal{L}$
\begin{equation*}
\mathcal{C}_{L}^{\mathcal{L}}\equiv \left\{ K\in \mathcal{D}:K\subset L\text{
and there is no }L^{\prime }\in \mathcal{L}\text{ with }K\subset L^{\prime
}\subsetneqq L\right\} ,
\end{equation*
and define the \emph{shifted} $\mathcal{L}$-corona and the $\flat $\emph
shifted} $\mathcal{L}$-corona b
\begin{eqnarray*}
\mathcal{C}_{L}^{\mathcal{L},\limfunc{shift}} &\equiv &\left\{ J\in \mathcal
G}:J^{\maltese }\in \mathcal{C}_{L}^{\mathcal{L}}\text{ }\right\} , \\
\mathcal{C}_{L}^{\mathcal{L},\flat \limfunc{shift}} &\equiv &\left\{ J\in
\mathcal{G}:J^{\flat }\in \mathcal{C}_{L}^{\mathcal{L}}\text{ }\right\} .
\end{eqnarray*
It is the second flat shifted corona $\mathcal{C}_{L}^{\mathcal{L},\flat
\limfunc{shift}}$ that will be used in our decompositions below, but we
retain the more natural shifted coronas $\mathcal{C}_{L}^{\mathcal{L}
\limfunc{shift}}$ in order to make useful comparisons. Now the parameter $m$
in $\mathcal{L}_{m}$ refers to the level at which the stopping construction
was performed, but for\thinspace $L\in \mathcal{L}_{m}$, the corona children
$L^{\prime }$ of $L$ are \emph{not} all necessarily in $\mathcal{L}_{m-1}$,
but may be in $\mathcal{L}_{m-t}$ for $t$ large.
At this point we introduce the notion of geometric depth $d$ in the tree
\mathcal{L}$ by definin
\begin{eqnarray}
\mathcal{G}_{0} &\equiv &\left\{ L\in \mathcal{L}:L\text{ is maximal
\right\} , \label{geom depth} \\
\mathcal{G}_{1} &\equiv &\left\{ L\in \mathcal{L}:L\text{ is maximal wrt
L\subsetneqq L_{0}\text{ for some }L_{0}\in \mathcal{G}_{0}\right\} , \notag
\\
&&\vdots \notag \\
\mathcal{G}_{d+1} &\equiv &\left\{ L\in \mathcal{L}:L\text{ is maximal wrt
L\subsetneqq L_{d}\text{ for some }L_{d}\in \mathcal{G}_{d}\right\} , \notag
\\
&&\vdots \notag
\end{eqnarray
We refer to $\mathcal{G}_{d}$ as the $d^{th}$ generation of intervals in the
tree $\mathcal{L}$, and say that the intervals in $\mathcal{G}_{d}$ are at
depth $d$ in the tree $\mathcal{L}$ (the generations $\mathcal{G}_{d}$ here
are \emph{not} related to the grid $\mathcal{G}$), and we write $d_{\limfunc
geom}}\left( L\right) $ for the geometric depth of $L$. Thus the intervals
in $\mathcal{G}_{d}$ are the stopping intervals in $\mathcal{L}$ that are $d$
levels in the \emph{geometric} sense below the top level. While the
geometric depth $d_{\limfunc{geom}}$ is about to be superceded by the
`indented' depth $d_{\limfunc{indent}}$ defined in the next subsection, we
will return to the geometric depth in order to iterate Lacey's bottom/up
stopping criterion when proving the second line in (\ref{rest bounds}) in
Proposition \ref{bottom up 3} below.
\subsection{The indented corona construction}
Now we address the lack of goodness in $\Pi _{1}^{\limfunc{below}}\mathcal{P
\cap \mathcal{C}_{A}^{\limfunc{restrict}}$. For this we introduce an
additional top/down stopping time $\mathcal{H}$ over the collection
\mathcal{L}$. Given the initial generation
\begin{equation*}
\mathcal{H}_{0}\equiv \mathcal{L}_{M+1}=\left\{ \text{maximal }L\in \mathcal
L}\right\} =\left\{ \text{maximal }I\in \Pi _{1}^{\limfunc{below}}\mathcal{P
\right\} ,
\end{equation*
define subsequent generations $\mathcal{H}_{k}$ as follows. For $k\geq 1$
and each $H\in \mathcal{H}_{k-1}$, let
\begin{equation*}
\mathcal{H}_{k}\left( H\right) \equiv \left\{ \text{maximal }L\in \mathcal{L
:3L\subset H\right\}
\end{equation*
consist of the next $\mathcal{H}$-generation of $\mathcal{L}$-intervals
below $H$, and set $\mathcal{H}_{k}\equiv \dbigcup\limits_{H\in \mathcal{H
_{k-1}}\mathcal{H}_{k}\left( H\right) $. Finally set $\mathcal{H}\equiv
\dbigcup\limits_{k=0}^{\infty }\mathcal{H}_{k}$. We refer to the stopping
intervals $H\in \mathcal{H}$ as \emph{indented} stopping intervals since
3H\subset \pi _{\mathcal{H}}H$ for all $H\in \mathcal{H}$ at indented
generation one or more, i.e. each successive such $H$ is `indented' in its
\mathcal{H}$-parent. This property of indentation is precisely what is
required in order to generate geometric decay in indented generations at the
end of the proof. We refer to $k$ as the \emph{indented depth} of the
stopping interval $H\in \mathcal{H}_{k}$, written $k=d_{\limfunc{indent
}\left( H\right) $, which is a refinement of the geometric depth $d_
\limfunc{geom}}$ introduced above. We will often revert to writing the dummy
variable for intervals in $\mathcal{H}$ as $L$ instead of $H$. For $L\in
\mathcal{H}$ define the $\mathcal{H}$-corona $\mathcal{C}_{L}^{\mathcal{H}
\limfunc{shift}}$ and the $\mathcal{H}$-shifted corona $\mathcal{C}_{L}^
\mathcal{H},\limfunc{shift}}$ and $\mathcal{H}$-$\flat $shifted corona
\mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}}$ b
\begin{eqnarray*}
\mathcal{C}_{L}^{\mathcal{H}} &\equiv &\left\{ I\in \mathcal{D}:I\subset
\text{ and }I\not\subset L^{\prime }\text{ for any }L^{\prime }\in \mathfrak
C}_{\mathcal{H}}\left( L\right) \right\} , \\
\mathcal{C}_{L}^{\mathcal{H},\limfunc{shift}} &\equiv &\left\{ J\in \mathcal
G}:J^{\maltese }\in \mathcal{C}_{L}^{\mathcal{H}}\right\} , \\
\mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}} &\equiv &\left\{ J\in
\mathcal{G}:J^{\flat }\in \mathcal{C}_{L}^{\mathcal{H}}\right\} .
\end{eqnarray*
We will also need recourse to the coronas $\mathcal{C}_{L}^{\mathcal{H}}$
restricted to intervals in $\mathcal{L}$, i.e
\begin{equation*}
\mathcal{C}_{L}^{\mathcal{H}}\left( \mathcal{L}\right) \equiv \mathcal{C
_{L}^{\mathcal{H}}\cap \mathcal{L}=\left\{ T\in \mathcal{L}:T\subset L\text{
and }T\not\subset L^{\prime }\text{ for any }L^{\prime }\in \mathcal{H}\text{
with }L^{\prime }\subsetneqq L\right\} .
\end{equation*}
Then for $L\in \mathcal{H}$ and $t\geq 0$ defin
\begin{equation}
\mathcal{P}_{L,t}^{\mathcal{H}}\equiv \left\{ \left( I,J\right) \in \mathcal
P}:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and }J\in \mathcal{C}_{L^{\prime
}}^{\mathcal{H},\limfunc{shift}}\text{ for some }L^{\prime }\in \mathcal{H
_{d_{\limfunc{indent}}\left( L\right) +t}\text{ with }L^{\prime }\subset
L\right\} . \label{def PHLt}
\end{equation
In particular, $\left( I,J\right) \in \mathcal{P}_{L,t}^{\mathcal{H}}$
implies that $I$ is in the corona $\mathcal{C}_{L}^{\mathcal{H}}$, and that
J$ is in a shifted corona $\mathcal{C}_{L^{\prime }}^{\mathcal{H},\limfunc
shift}}$ that is $t$ levels of indented generation \emph{below} $\mathcal{C
_{L}^{\mathcal{H}}$ (when $t=0$ we have $L^{\prime }=L$). We emphasize the
distinction `indented generation' as this refers to the indented depth
rather than either the level of initial stopping construction of $\mathcal{L}
$, or the geometric depth. The point of introducing the tree $\mathcal{H}$
of indented stopping intervals, is that the inclusion $3L\subset \pi _
\mathcal{H}}L$ for all $L\in \mathcal{H}$ with $d_{\limfunc{indent}}\left(
L\right) \geq 1$ turns out to be an adequate substitute for the standard
`goodness' lost in the process of infusing the weak goodness of Hyt\"{o}nen
and Martikainen in\ Subsection \ref{Subsec HM} above.
Now within the $\mathcal{H}$-corona $\mathcal{C}_{L}^{\mathcal{H}}\left(
\mathcal{L}\right) $, there are in general further intervals $T\in \mathcal{
}$ in addition to $L\in \mathcal{H}$ itself, but all of these further
intervals are contained in the two endpoint $\mathcal{L}$-tower
\begin{eqnarray*}
\mathcal{T}_{\limfunc{left}}\left( L\right) &\equiv &\left\{ L^{\prime }\in
\mathcal{L}:L^{\prime }\subsetneqq L\text{ and }\limfunc{left}\func{end
\left( L^{\prime }\right) =\limfunc{left}\func{end}\left( L\right) \right\}
\\
\mathcal{T}_{\limfunc{right}}\left( L\right) &\equiv &\left\{ L^{\prime }\in
\mathcal{L}:L^{\prime }\subsetneqq L\text{ and }\limfunc{right}\func{end
\left( L^{\prime }\right) =\limfunc{right}\func{end}\left( L\right) \right\}
\end{eqnarray*
where $\limfunc{left}\func{end}\left( I\right) $ and $\limfunc{right}\func
end}\left( I\right) $ denote the left and right hand endpoints of $I$
respectively. Thus $\mathcal{C}_{L}^{\mathcal{H},\limfunc{restrict}}\left(
\mathcal{L}\right) \equiv \mathcal{C}_{L}^{\mathcal{H}}\left( \mathcal{L
\right) \setminus \left\{ L\right\} $ consists of two `connected' $\mathcal{
}$-towers (possibly one or both empty), one in $\mathcal{T}_{\limfunc{left
}\left( L\right) $ and the other in $\mathcal{T}_{\limfunc{right}}\left(
L\right) $. Set $\mathcal{T}\left( L\right) \equiv \mathcal{T}_{\limfunc{lef
}}\left( L\right) \dot{\cup}\mathcal{T}_{\limfunc{right}}\left( L\right)
\dot{\cup}\left\{ L\right\} $. See Figure \ref{ind}.
\FRAME{ftbpFU}{6.8393in}{3.4006in}{0pt}{\Qcb{Line segments (not to scale)
are the bottom/up stopping intervals in Lacey's tree $\mathcal{L}$. Red
segments are the \emph{indented} intervals and green segments are the
intervals in \emph{towers}. The top indented interval is boxed in purple,
the first generation of indented intervals are boxed in orange, and the
second generation in blue. Vertical lines indicate common endpoints.}}{\Qlb
ind}}{indented.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width
6.8393in;height 3.4006in;depth 0pt;original-width 7.1805in;original-height
10.469in;cropleft "0.0901";croptop "0.6533";cropright "1";cropbottom
"0.3462";filename 'Indented.wmf';file-properties "XNPEU";}}
\subsubsection{Decomposition of coronas}
Here we describe the decomposition of admissible collections of pairs
according to the regular shifted coronas $\mathcal{C}_{L}^{\mathcal{H}
\limfunc{shift}}$ and $\mathcal{C}_{L}^{\mathcal{L},\limfunc{shift}}$.
Strictly speaking, these decompositions will not be used in the sequel, but
they do help to provide insight via comparison with the flat shifted
decompositions introduced in the next subsubsection, which will be used to
finish the proof. For $L\in \mathcal{H}$ and $t=0$ we further decompose
\mathcal{P}_{L,0}^{\mathcal{H}}$ in (\ref{def PHLt}) with $t=0$, i.e.
\begin{equation*}
\mathcal{P}_{L,0}^{\mathcal{H}}=\left\{ \left( I,J\right) \in \mathcal{P
:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and }J\in \mathcal{C}_{L}^
\mathcal{H},\limfunc{shift}}\right\} ,
\end{equation*
a
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\mathcal{H}} &=&\mathcal{P}_{L,0}^{\mathcal{H}-small}\dot
\cup}\mathcal{P}_{L,0}^{\mathcal{H}-big}; \\
\mathcal{P}_{L,0}^{\mathcal{H}-small} &\equiv &\left\{ \left( I,J\right) \in
\mathcal{P}_{L,0}^{\mathcal{H}}:\text{there is no }L^{\prime }\in \mathcal{T
\left( L\right) \text{ with }J^{\maltese }\subset L^{\prime }\subset
I\right\} \\
&=&\left\{ \left( I,J\right) \in \mathcal{P}_{L,0}^{\mathcal{H}}:I\in
\mathcal{C}_{L^{\prime }}^{\mathcal{L}}\setminus \left\{ L^{\prime }\right\}
\text{ and }J\in \mathcal{C}_{L^{\prime }}^{\mathcal{L},\limfunc{shift}
\text{ for some }L^{\prime }\in \mathcal{T}\left( L\right) \right\} , \\
\mathcal{P}_{L,0}^{\mathcal{H}-big} &\equiv &\left\{ \left( I,J\right) \in
\mathcal{P}_{L,0}^{\mathcal{H}}:\text{there is }L^{\prime }\in \mathcal{T
\left( L\right) \text{ with }J^{\maltese }\subset L^{\prime }\subset
I\right\} ,
\end{eqnarray*
with one exception: if $L\in \mathcal{H}_{0}=\mathcal{L}_{M+1}$ we set
\mathcal{P}_{L,0}^{\mathcal{H}-small}\equiv \mathcal{P}_{L,0}^{\mathcal{H}}$
and $\mathcal{P}_{L,0}^{\mathcal{H}-big}\equiv \emptyset $ since in this
case $L$ fails to satisfy (\ref{up stopping condition}) as pointed out
above. Finally, for $L\in \mathcal{H}$ we further decompose $\mathcal{P
_{L,0}^{\mathcal{H}-small}$ a
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\mathcal{H}-small} &=&\overset{\cdot }{\dbigcup
_{L^{\prime }\in \mathcal{T}\left( L\right) }\mathcal{P}_{L^{\prime },0}^
\mathcal{L}-small}; \\
\mathcal{P}_{L^{\prime },0}^{\mathcal{L}-small} &\equiv &\left\{ \left(
I,J\right) \in \mathcal{P}:I\in \mathcal{C}_{L^{\prime }}^{\mathcal{L
}\setminus \left\{ L^{\prime }\right\} \text{ and }J\in \mathcal{C
_{L^{\prime }}^{\mathcal{L},\limfunc{shift}}\right\} .
\end{eqnarray*
Then we se
\begin{eqnarray}
\mathcal{P}^{big} &\equiv &\left\{ \dbigcup\limits_{L\in \mathcal{H}
\mathcal{P}_{L,0}^{\mathcal{H}-big}\right\} \dbigcup \left\{
\dbigcup\limits_{t\geq 1}\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P
_{L,t}^{\mathcal{H}}\right\} ; \label{def big small} \\
\mathcal{P}^{small} &\equiv &\dbigcup\limits_{L\in \mathcal{L}}\mathcal{P
_{L,0}^{\mathcal{L}-small}\text{ }. \notag
\end{eqnarray
Note that every pair $\left( I,J\right) \in \mathcal{P}$ is included in
either $\mathcal{P}^{small}$ or $\mathcal{P}^{big}$ since every $I\in \Pi
_{1}^{\limfunc{below}}\mathcal{P}$ is contained in some $L\in \mathcal{H
_{0}=\mathcal{L}_{M+1}$.
\subsubsection{Flat shifted coronas}
More importantly, we now define the corresponding $\flat $shifted admissible
collections of pairs $\mathcal{P}_{L,t}^{\flat \mathcal{H}}$, etc., in which
we replace $\mathcal{C}_{L}^{\mathcal{H},\limfunc{shift}}$ and $\mathcal{C
_{L}^{\mathcal{L},\limfunc{shift}}$ wit
\begin{equation*}
\mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}}\equiv \left\{ J\in \Pi
_{2}\mathcal{P}:J^{\flat }\in \mathcal{C}_{L}^{\mathcal{H}}\right\} \text{
and }\mathcal{C}_{L}^{\mathcal{L},\flat \limfunc{shift}}\equiv \left\{ J\in
\Pi _{2}\mathcal{P}:J^{\flat }\in \mathcal{C}_{L}^{\mathcal{L}}\right\} .
\end{equation*
In these flat shifted $\mathcal{H}$ and $\mathcal{L}$ coronas, we have
effectively shift the intervals $J$ two levels `up' by requiring $J^{\flat
}\in \mathcal{C}_{L}^{\mathcal{L}}$ instead of $J^{\maltese }\in \mathcal{C
_{L}^{\mathcal{L}}$, etc., but because $\mathcal{P}$ is admissible, we
always have $J^{\maltese }\in \mathcal{C}_{A}^{\mathcal{A},\limfunc{restrict
}$. We define
\begin{eqnarray*}
\mathcal{P}_{L,t}^{\flat \mathcal{H}} &\equiv &\left\{ \left( I,J\right) \in
\mathcal{P}:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and }J\in \mathcal{C
_{L^{\prime }}^{\mathcal{H},\flat \limfunc{shift}}\text{ for some }L^{\prime
}\in \mathcal{H}_{d_{\limfunc{indent}}\left( L\right) +t}\text{ with
L^{\prime }\subset L\right\} , \\
\mathcal{P}_{L,0}^{\flat \mathcal{H}} &=&\left\{ \left( I,J\right) \in
\mathcal{P}:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and }J\in \mathcal{C
_{L}^{\mathcal{H},\flat \limfunc{shift}}\right\} ,
\end{eqnarray*
an
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\flat \mathcal{H}} &=&\mathcal{P}_{L,0}^{\flat \mathcal{H
-small}\dot{\cup}\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}; \\
\mathcal{P}_{L,0}^{\flat \mathcal{H}-small} &\equiv &\left\{ \left(
I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H}}:\text{there is no
L^{\prime }\in \mathcal{T}\left( L\right) \text{ with }J^{\flat }\subset
L^{\prime }\subset I\right\} \\
&=&\left\{ \left( I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H}}:I\in
\mathcal{C}_{L^{\prime }}^{\mathcal{L}}\setminus \left\{ L^{\prime }\right\}
\text{ and }J\in \mathcal{C}_{L^{\prime }}^{\mathcal{L},\flat \limfunc{shift
}\text{ for some }L^{\prime }\in \mathcal{T}\left( L\right) \right\} , \\
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big} &\equiv &\left\{ \left( I,J\right)
\in \mathcal{P}_{L,0}^{\flat \mathcal{H}}:\text{there is }L^{\prime }\in
\mathcal{T}\left( L\right) \text{ with }J^{\flat }\subset L^{\prime }\subset
I\right\} ,
\end{eqnarray*
with one exception: if $L\in \mathcal{H}_{0}=\mathcal{L}_{M+1}$ we set
\mathcal{P}_{L,0}^{\flat \mathcal{H}-small}\equiv \mathcal{P}_{L,0}^{\flat
\mathcal{H}}$ and $\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}\equiv \emptyset
$ since in this case $L$ fails to satisfy (\ref{up stopping condition}) as
pointed out above. Finally, for $L\in \mathcal{H}$ we further decompose
\mathcal{P}_{L,0}^{\flat \mathcal{H}-small}$ a
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-small} &=&\overset{\cdot }{\dbigcup
_{L^{\prime }\in \mathcal{T}\left( L\right) }\mathcal{P}_{L^{\prime
},0}^{\flat \mathcal{L}-small}; \\
\mathcal{P}_{L^{\prime },0}^{\flat \mathcal{L}-small} &\equiv &\left\{
\left( I,J\right) \in \mathcal{P}:I\in \mathcal{C}_{L^{\prime }}^{\mathcal{L
}\setminus \left\{ L^{\prime }\right\} \text{ and }J\in \mathcal{C
_{L^{\prime }}^{\mathcal{L},\flat \limfunc{shift}}\right\} .
\end{eqnarray*
Then we se
\begin{eqnarray}
\mathcal{P}^{\flat big} &\equiv &\left\{ \dbigcup\limits_{L\in \mathcal{H}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}\right\} \dbigcup \left\{
\dbigcup\limits_{t\geq 1}\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}\right\} ; \label{def big small flat} \\
\mathcal{P}^{\flat small} &\equiv &\dbigcup\limits_{L\in \mathcal{L}
\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\text{ }. \notag
\end{eqnarray
We observed above that every pair $\left( I,J\right) \in \mathcal{P}$ is
included in either $\mathcal{P}^{small}$ or $\mathcal{P}^{big}$, and it
follows that every pair $\left( I,J\right) \in \mathcal{P}$ is thus included
in either $\mathcal{P}^{\flat small}$ or $\mathcal{P}^{\flat big}$, simply
because the pairs $\left( I,J\right) $ have been shifted up by two dyadic
levels in the interval $J$. Thus the coronas $\mathcal{P}_{L,0}^{\flat
\mathcal{L}-small}$ are now even \emph{smaller} than the regular coronas
\mathcal{P}_{L,0}^{\mathcal{L}-small}$, which permits the estimate (\re
{small claim' 3}) below to hold for the larger augmented size functional. On
the other hand, the coronas $\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}$ and
\mathcal{P}_{L,t}^{\flat \mathcal{H}}$ are now bigger than before, requiring
the stronger straddling lemmas above in order to obtain the estimates (\re
{rest bounds}) below. More specifically, we will see that stopping forms
with pairs in $\mathcal{P}^{\flat big}$ will be estimated using the $\flat
Straddling and Substraddling Lemmas (Substraddling applies to part of
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}$ and $\flat $Straddling applies to
the remaining part of $\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}$ and to
\mathcal{P}_{L,t}^{\flat \mathcal{H}}$), and it is here that the removal of
the corona-straddling collection $\mathcal{P}_{\func{cor}}^{A}$ is
essential, while forms with pairs in $\mathcal{P}^{\flat small}$ will be
absorbed.
\subsection{Size estimates}
Now we turn to proving the \emph{size estimates} we need for these
collections. Recall that the \emph{restricted} norm $\widehat{\mathfrak{N}}_
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P}}$ is the best
constant in the inequalit
\begin{equation*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) \leq \widehat{\mathfrak{N}}_{\limfunc
stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P}}\left\Vert \mathsf{P}_{\Pi
_{1}\mathcal{P}}^{\sigma ,\mathbf{b}}f\right\Vert _{L^{2}\left( \sigma
\right) }^{\bigstar }\left\Vert \mathsf{P}_{\Pi _{2}\mathcal{P}}^{\omega
\mathbf{b}^{\ast }}g\right\Vert _{L^{2}\left( \omega \right) }^{\bigstar },
\end{equation*
where $f\in L^{2}\left( \sigma \right) $ satisfies $E_{I}^{\sigma
}\left\vert f\right\vert \leq \alpha _{\mathcal{A}}\left( A\right) $ for all
$I\in \mathcal{C}_{A}$, and $g\in L^{2}\left( \omega \right) $.
\begin{proposition}
\label{bottom up 3}Suppose $\rho $ in (\ref{def rho}) is greater than $1$,
and $\mathcal{P}$ is a \emph{reduced admissible} collection of pairs for a
dyadic interval $A$. Let $\mathcal{P}=\mathcal{P}^{\flat big}\dot{\cup
\mathcal{P}^{\flat small}$ be the decomposition satisfying\ (\ref{def big
small}) above, i.e
\begin{equation*}
\mathcal{P}=\left\{ \dbigcup\limits_{L\in \mathcal{H}}\mathcal{P
_{L,0}^{\flat \mathcal{H}-big}\right\} \dbigcup \left\{
\dbigcup\limits_{t\geq 1}\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}\right\} \ \cup \ \left( \dbigcup_{L\in \mathcal{L
}\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\right) \ .
\end{equation*
Then all of these collections $\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}$,
$\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}$ and $\mathcal{P}_{L,t}^{\flat
\mathcal{H}}$ are reduced admissible, and we have the estimate
\begin{equation}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2}\leq \left( \rho -1\right)
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2},\ \ \ \ \ L\in \mathcal{L}, \label{small claim' 3}
\end{equation
and the localized norm bounds
\begin{eqnarray}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P}_{L,0}^{\flat \mathcal{H
-big}} &\leq &C\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}\right) , \label{rest bounds} \\
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}} &\leq &C\rho ^{-\frac{t}{2}}\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) ,\ \ \ \ \ t\geq 1. \notag
\end{eqnarray}
\end{proposition}
Using this proposition on size estimates, we can finish the proof of (\re
{First inequality}), and hence the proof of (\ref{B stop form 3}).
\begin{corollary}
The sublinear stopping form inequality (\ref{First inequality}) holds.
\end{corollary}
\begin{proof}
Recall that $\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}$ is the best constant in the inequality (\ref{best hat}).
Since $\left\{ \mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\right\} _{L\in
\mathcal{L}}$ is a mutually orthogonal family of $A$-admissible pairs, the
Orthogonality Lemma \ref{mut orth} implies tha
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\dbigcup\limits_{L\in \mathcal{L}}\mathcal{P}_{L,0}^{\flat \mathcal{L
-small}}\leq \sup_{L\in \mathcal{L}}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}}.
\end{equation*
Using this, together with the decomposition of $\mathcal{P}$ and (\ref{rest
bounds}) above, we obtai
\begin{eqnarray*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}} &\leq &\sup_{L\in \mathcal{H}}\widehat{\mathfrak{N}}_{\limfunc
stop},\bigtriangleup ^{\omega }}^{A,\dbigcup\limits_{L\in \mathcal{H}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}}+\sum_{t=1}^{M+1}\sup_{L\in
\mathcal{H}}\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}+\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\dbigcup\limits_{L\in \mathcal{L}}\mathcal{P}_{L,0}^{\flat \mathcal{L
-small}} \\
&\lesssim &\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}\right) +\left( \sum_{t=1}^{M+1}\rho ^{-\frac{t}{2}}\right)
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) +\sup_{L\in \mathcal{L}}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}}\
.
\end{eqnarray*
Since the admissible collection $\mathcal{P}^{A}$ in (\ref{initial P}) that
arises in the stopping form is finite, we can define $\mathfrak{L}$ to be
the best constant in the inequalit
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}}\leq \mathfrak{L}\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) \text{ for all }A\text{-admissible
collections }\mathcal{P}.
\end{equation*
Now choose $\mathcal{P}$ so that
\begin{equation*}
\frac{\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}}}{\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}\right) }>\frac{1}{2}\mathfrak{L=}\frac{1}{2}\sup_{\mathcal{Q
\text{ is }A\text{-admissible}}\frac{\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}}{\mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q}\right) }\ .
\end{equation*
Then using $\sum_{t=1}^{M+1}\rho ^{-\frac{t}{2}}\leq \frac{1}{\sqrt{\rho }-1}
$ we hav
\begin{eqnarray*}
\mathfrak{L} &<&2\frac{\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup
^{\omega }}^{A,\mathcal{P}}}{\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) }\leq \frac{C\frac{1}{\sqrt{\rho }-1
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) +C\sup_{L\in \mathcal{L}}\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}}}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) } \\
&\leq &C\frac{1}{\sqrt{\rho }-1}+C\sup_{L\in \mathcal{L}}\mathfrak{L}\frac
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) }{\mathcal{S}_{\limfunc{aug}\limfunc
size}}^{\alpha ,A}\left( \mathcal{P}\right) }\leq C\frac{1}{\sqrt{\rho }-1}+
\mathfrak{L}\sqrt{\rho -1}\ ,
\end{eqnarray*
where we have used (\ref{small claim' 3}) in the last line. If we choose
\rho >1$ so that
\begin{equation}
C\sqrt{\rho -1}<\frac{1}{2}, \label{choose rho}
\end{equation
then we obtain $\mathfrak{L}\leq 2C\frac{1}{\sqrt{\rho }-1}$. Together with
Lemma \ref{energy control}, this yield
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}}\leq \mathfrak{L}\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) \leq 2C\frac{1}{\sqrt{\rho }-1}\left(
\mathcal{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha }}+\sqrt{A_{2}^{\alpha
\limfunc{punct}}}\right)
\end{equation*
as desired, and completes the proof of inequality (\ref{First inequality}).
\end{proof}
Thus, in view of Conclusion \ref{assume}, it remains only to prove
Proposition \ref{bottom up 3} using the Orthogonality and Straddling and
Substraddling Lemmas above, and we now turn to this task.
\begin{proof}[Proof of Proposition \protect\ref{bottom up 3}]
We split the proof into three parts.
\textbf{Proof of (\ref{small claim' 3})}: To prove the inequality (\re
{small claim' 3}), suppose first that $L\notin \mathcal{L}_{M+1}$. In the
case that $L\in \mathcal{L}_{0}$ is an initial generation interval, then
from (\ref{key property 3}) and the fact that every $I\in \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}$ satisfies $I\subsetneqq L$, we obtain tha
\begin{eqnarray*}
&&\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2}=\sup_{K^{\prime }\in \Pi _{1}^
\limfunc{below}}\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\cap \mathcal{C
_{A}^{\limfunc{restrict}}}\frac{\Psi ^{\alpha }\left( K^{\prime };\mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2}}{\left\vert K^{\prime
}\right\vert _{\sigma }} \\
&\leq &\sup_{K^{\prime }\in \Pi _{1}^{\limfunc{below}}\mathcal{P}\cap
\mathcal{C}_{A}^{\limfunc{restrict}}:\ K^{\prime }\varsubsetneqq L}\frac
\Psi ^{\alpha }\left( K^{\prime };\mathcal{P}_{L,0}^{\flat \mathcal{L
-small}\right) ^{2}}{\left\vert K^{\prime }\right\vert _{\sigma }}\leq
\varepsilon \mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}\right) ^{2}.
\end{eqnarray*
Now suppose that $L\not\in \mathcal{L}_{0}$ in addition to $L\notin \mathcal
L}_{M+1}$. Pick a pair $\left( I,J\right) \in \mathcal{P}_{L,0}^{\flat
\mathcal{L}-small}$. Then $I$ is in the restricted corona $\mathcal{C}_{L}^
\mathcal{L},\limfunc{restrict}}$ and $J$ is in the $\flat $\emph{shifted}
corona $\mathcal{C}_{L}^{\mathcal{L},\flat \limfunc{shift}}$. Since
\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}$ is a finite collection, the
definition of $\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\right) $ shows that there is an
interval $K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}_{L,0}^{\flat \mathcal{L
-small}\cap \mathcal{C}_{A}^{\limfunc{restrict}}$ so tha
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2}=\frac{1}{\left\vert
K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1
_{A\setminus K}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\omega
_{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \right) .
\end{equation*
Note that $K\subsetneqq L$ by definition of $\mathcal{P}_{L,0}^{\flat
\mathcal{L}-small}$. Now let $t$ be such that $L\in \mathcal{L}_{t}$, and
define
\begin{equation*}
t^{\prime }=t^{\prime }\left( K\right) \equiv \max \left\{ s:\text{there is
L^{\prime }\in \mathcal{L}_{s}\text{ with }L^{\prime }\subset K\right\} ,
\end{equation*
and note that $t^{\prime }<t$. First, suppose that $t^{\prime }=0$ so that
K $ does not contain any $L^{\prime }\in \mathcal{L}$. Then it follows from
the construction at level $\ell =0$ tha
\begin{equation*}
\frac{1}{\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha
}\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert K\right\vert
\right) ^{2}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right)
\right) <\varepsilon \mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha
,A}\left( \mathcal{P}\right) ^{2},
\end{equation*
and hence from $\rho =1+\varepsilon $ we obtain
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2}<\varepsilon \mathcal{S}_
\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}\right)
^{2}=\left( \rho -1\right) \mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha
,A}\left( \mathcal{P}\right) ^{2}.
\end{equation*
Now suppose that $t^{\prime }\geq 1$. Then $K$ fails the stopping condition
\ref{up stopping condition}) with $m=t^{\prime }+1$, since otherwise it
would contain an interval $L^{\prime \prime }\in \mathcal{L}_{t^{\prime }+1}$
contradicting our definition of $t^{\prime }$, and s
\begin{equation*}
\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \right) <\rho
\omega _{\flat \mathcal{P}}\left( \mathbf{V}\left( K\right) \right) \text{
where }\mathbf{V}\left( K\right) \equiv \dbigcup\limits_{L^{\prime }\in
\dbigcup\limits_{\ell =0}^{t^{\prime }}\mathcal{L}_{\ell }:\ L^{\prime
}\subset K}\mathbf{T}\left( L^{\prime }\right) .
\end{equation*}
Now we use the crucial fact that the positive measure $\omega _{\flat
\mathcal{P}}$ is \emph{additive} and finite to obtain from this tha
\begin{equation}
\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \setminus
\mathbf{V}\left( K\right) \right) =\omega _{\flat \mathcal{P}}\left( \mathbf
T}\left( K\right) \right) -\omega _{\flat \mathcal{P}}\left( \mathbf{V
\left( K\right) \right) \leq \left( \rho -1\right) \omega _{\flat \mathcal{P
}\left( \mathbf{V}\left( K\right) \right) . \label{additive}
\end{equation
Now recall tha
\begin{equation*}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q
\right) ^{2}\equiv \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{Q}\cap
C_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert _{\sigma }
\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{\Pi
_{2}^{K,\limfunc{aug}}\mathcal{Q}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}.
\end{equation*
We claim it follows that for each $J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P
_{L,0}^{\flat \mathcal{L}-small}$, the support $\left( c_{J^{\flat }},\ell
\left( J^{\flat }\right) \right) $ of the atom $\delta _{\left( c_{J^{\flat
}},\ell \left( J^{\flat }\right) \right) }$ is contained in the set $\mathbf
T}\left( K\right) $, but not in the set
\begin{equation*}
\mathbf{V}\left( K\right) \equiv \dbigcup \left\{ \mathbf{T}\left( L^{\prime
}\right) :L^{\prime }\in \dbigcup\limits_{\ell =0}^{t^{\prime }}\mathcal{L
_{\ell }:\ L^{\prime }\subset K\right\} .
\end{equation*
Indeed, suppose in order to derive a contradiction, that $\left( c_{J^{\flat
}},\ell \left( J^{\flat }\right) \right) \in \mathbf{T}\left( L^{\prime
}\right) $ for some $L^{\prime }\in \mathcal{L}_{\ell }$ with $0\leq \ell
\leq t^{\prime }$. Recall that $L\in \mathcal{L}_{t}$ with $t^{\prime }<t$
so that $L^{\prime }\subsetneqq L$. Thus $\left( c_{J^{\flat }},\ell \left(
J^{\flat }\right) \right) \in \mathbf{T}\left( L^{\prime }\right) $ implies
J^{\flat }\subset L^{\prime }$, which contradicts the fact that
\begin{equation*}
J\in \Pi _{2}^{K}\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}\subset \Pi _{2
\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}=\left\{ \left( I,J\right) \in
\mathcal{P}:I\in \mathcal{C}_{L}^{\mathcal{L}}\setminus \left\{ L\right\}
\text{ and }J\in \mathcal{C}_{L}^{\mathcal{L},\flat \limfunc{shift}}\right\}
\end{equation*
implies $J^{\flat }\in \mathcal{C}_{L}^{\mathcal{L}}$ - because $L^{\prime
}\notin \mathcal{C}_{L}^{\mathcal{L}}$.
Thus from the definition of $\omega _{\flat \mathcal{P}}$ in (\ref{def
atomic}), the `energy' $\left\Vert \mathsf{Q}_{\Pi _{2}^{K,\limfunc{aug}
\mathcal{P}_{L,0}^{\flat \mathcal{L}-small}}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}$ is at most the
$\omega _{\flat \mathcal{P}}$-measure of $\mathbf{T}\left( K\right)
\setminus \mathbf{V}\left( K\right) $. Using
\begin{equation*}
\omega _{\flat \mathcal{P}_{L,0}^{\flat \mathcal{L}-small}}\left( \mathbf{T
\left( K\right) \right) \leq \omega _{\flat \mathcal{P}}\left( \mathbf{T
\left( K\right) \setminus \mathbf{V}\left( K\right) \right) ,
\end{equation*
and (\ref{additive}), we then hav
\begin{eqnarray*}
&&\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2} \\
&\leq &\sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P}_{L,0}^{\flat
\mathcal{L}-small}\cap \mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}
\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K
\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert K\right\vert }\right)
^{2}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \setminus
\mathbf{V}\left( K\right) \right) \\
&\leq &\left( \rho -1\right) \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\cap \mathcal{C}_{A}^{\limfunc{restrict}}
\frac{1}{\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha
}\left( K,\mathbf{1}_{A\setminus K}\sigma \right) }{\left\vert K\right\vert
\right) ^{2}\omega _{\flat \mathcal{P}}\left( \mathbf{V}\left( K\right)
\right) ,
\end{eqnarray*
and we can continue with
\begin{eqnarray*}
&&\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{L}-small}\right) ^{2} \\
&\leq &\left( \rho -1\right) \sup_{K\in \Pi _{1}^{\limfunc{below}}\mathcal{P
\cap \mathcal{C}_{A}^{\limfunc{restrict}}}\frac{1}{\left\vert K\right\vert
_{\sigma }}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus
K}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( K\right) \right) \\
&\leq &\left( \rho -1\right) \mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) ^{2}.
\end{eqnarray*}
In the remaining case where $L\in \mathcal{L}_{M+1}$ we can include $L$ as a
testing interval $K$ and the same reasoning applies. This completes the
proof of (\ref{small claim' 3}).
\bigskip
To prove the other inequality (\ref{rest bounds}) in Proposition \ref{bottom
up 3}, we will use the Straddling and Substraddling Lemmas to bound the norm
of certain `straddled' stopping forms by the augmented size functional
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}$, and the
Orthogonality Lemma to bound sums of `mutually orthogonal' stopping forms.
Recall that
\begin{eqnarray*}
\mathcal{P}^{\flat big} &=&\left\{ \dbigcup\limits_{L\in \mathcal{H}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}\right\} \dbigcup \left\{
\dbigcup\limits_{t\geq 1}\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}\right\} \equiv \mathcal{Q}_{0}^{\flat \mathcal{H
-big}\dbigcup \mathcal{Q}_{1}^{\flat \mathcal{H}-big}; \\
\mathcal{Q}_{0}^{\flat \mathcal{H}-big} &\equiv &\dbigcup\limits_{L\in
\mathcal{L}}\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}\ ,\ \ \ \ \ \mathcal{Q
_{1}^{\flat \mathcal{H}-big}\equiv \dbigcup\limits_{t\geq 1}\mathcal{P
_{t}^{\flat \mathcal{H}-big},\ \ \ \ \ \mathcal{P}_{t}^{\flat \mathcal{H
-big}\equiv \dbigcup\limits_{L\in \mathcal{H}}\mathcal{P}_{L,t}^{\flat
\mathcal{H}}.
\end{eqnarray*}
\bigskip
\textbf{Proof of the second line in (\ref{rest bounds})}: We first turn to
the collectio
\begin{eqnarray*}
\mathcal{Q}_{1}^{\flat \mathcal{H}-big} &=&\dbigcup\limits_{t\geq
1}\dbigcup\limits_{L\in \mathcal{H}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}=\dbigcup\limits_{t\geq 1}\mathcal{P}_{t}^{\flat \mathcal{H}-big}; \\
\mathcal{P}_{t}^{\flat \mathcal{H}-big} &\equiv &\dbigcup\limits_{L\in
\mathcal{L}}\mathcal{P}_{L,t}^{\flat \mathcal{H}}\ ,\ \ \ \ \ t\geq 1,
\end{eqnarray*
wher
\begin{equation*}
\mathcal{P}_{L,t}^{\flat \mathcal{H}}=\left\{ \left( I,J\right) \in \mathcal
P}:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and }J\in \mathcal{C}_{L^{\prime
}}^{\mathcal{H},\flat \limfunc{shift}}\text{ for some }L^{\prime }\in
\mathcal{H}_{d_{\limfunc{indent}}\left( L\right) +t}\text{ with }L^{\prime
}\subset L\right\} .
\end{equation*
We now claim that the second line in (\ref{rest bounds}) holds, i.e
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{t}^{\flat \mathcal{H}-big}}\leq C\rho ^{-\frac{t}{2}}\mathcal{S
_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}\right) ,\ \ \ \
\ t\geq 1, \label{S big t 3}
\end{equation
which recovers the key geometric gain obtained by Lacey in \cite{Lac},
except that here we are only gaining this decay relative to the indented
subtree $\mathcal{H}$ of the tree $\mathcal{L}$.
The case $t=1$ can be handled with relative ease since decay is not relevant
here. Indeed, $\mathcal{P}_{L,1}^{\flat \mathcal{H}}$ straddles the
collection $\mathfrak{C}_{\mathcal{H}}\left( L\right) $ of $\mathcal{H}
-children of $L$, and so the localized $\flat $Straddling Lemma \re
{straddle 3 ref} applies to giv
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{L,1}^{\flat \mathcal{H}}}\leq C\mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}_{L,1}^{\flat \mathcal{H
}\right) \leq C\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{P}\right) ,
\end{equation*
and then the Orthogonality Lemma \ref{mut orth} applies to giv
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{1}^{\flat \mathcal{H}-big}}\leq \sup_{L\in \mathcal{H}
\mathfrak{N}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P
_{L,1}^{\flat \mathcal{H}}}\leq C\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) ,
\end{equation*
since $\left\{ \mathcal{P}_{L,1}^{\flat \mathcal{H}}\right\} _{L\in \mathcal
L}}$ is mutually orthogonal as $\mathcal{P}_{L,1}^{\flat \mathcal{H}}\subset
\mathcal{C}_{L}^{\mathcal{H}}\times \mathcal{C}_{L^{\prime }}^{\mathcal{H
,\flat \limfunc{shift}}$ with $L\in \mathcal{H}_{k}$ and $L^{\prime }\in
\mathcal{H}_{k+1}$ for indented depth $k=k\left( L\right) $. The case $t=2$
is equally easy.
Now we consider the case $t\geq 2$, where it is essential to obtain
geometric decay in $t$. We remind the reader that all of our admissible
collections $\mathcal{P}_{L,t}^{\flat \mathcal{H}}$ are \emph{reduced} by
Conclusion \ref{assume}. We again apply Lemma \ref{straddle 3 ref} to
\mathcal{P}_{L,t}^{\flat \mathcal{H}}$ with $\mathcal{S}=\mathfrak{C}_
\mathcal{H}}\left( L\right) $, so that for any $\left( I,J\right) \in
\mathcal{P}_{L,t}^{\flat \mathcal{H}}$, there is $H^{\prime }\in \mathfrak{C
_{\mathcal{H}}\left( L\right) $ with $J^{\flat }\subset H^{\prime
}\subsetneqq I\in \mathcal{C}_{L}^{\mathcal{H}}$. But this time we must use
the stronger localized bounds $\mathcal{S}_{\limfunc{loc}\limfunc{size
}^{\alpha ,A;S}$ with an $S$-hole, that giv
\begin{eqnarray}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{L,t}^{\flat \mathcal{H}}} &\leq &C\sup_{H^{\prime }\in
\mathfrak{C}_{\mathcal{H}}\left( L\right) }\mathcal{S}_{\limfunc{loc
\limfunc{size}}^{\alpha ,A;H^{\prime }}\left( \mathcal{P}_{L,t}^{\flat
\mathcal{H}}\right) ,\ \ \ \ \ t\geq 0; \label{t,n 3} \\
\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;H^{\prime }}\left(
\mathcal{P}_{L,t}^{\flat \mathcal{H}}\right) ^{2} &=&\sup_{K\in \mathcal{W
^{\ast }\left( H^{\prime }\right) \cap \mathcal{C}_{A}^{\limfunc{restrict}}
\frac{1}{\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha
}\left( K,\mathbf{1}_{A\setminus H^{\prime }}\sigma \right) }{\left\vert
K\right\vert }\right) ^{2}\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\ .
\notag
\end{eqnarray}
It remains to show tha
\begin{eqnarray}
&&\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \rho ^{-\left( t-2\right)
}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \right) ,
\label{rem} \\
&&\text{for}\ t\geq 2,\ K\in \mathcal{W}^{\ast }\left( H^{\prime }\right)
\cap \mathcal{C}_{A}^{\limfunc{restrict}},\ H^{\prime }\in \mathfrak{C}_
\mathcal{H}}\left( L\right) . \notag
\end{eqnarray
so that we then hav
\begin{eqnarray*}
&&\frac{1}{\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{A\setminus H^{\prime }}\sigma \right) }
\left\vert K\right\vert }\right) ^{2}\sum_{J\in \Pi _{2}^{K,\limfunc{aug}
\mathcal{P}_{L,t}^{\flat \mathcal{H}}}\left\Vert \bigtriangleup _{J}^{\omega
,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} \\
&\leq &\rho ^{-\left( t-2\right) }\frac{1}{\left\vert K\right\vert _{\sigma
}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{A\setminus K}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\omega _{\flat \mathcal{P
}\left( \mathbf{T}\left( K\right) \right) \leq \rho ^{-\left( t-2\right)
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) ^{2},
\end{eqnarray*
by (\ref{def P stop energy' 3}), and hence conclude the required bound for
\mathfrak{N}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P
_{L,t}^{\flat \mathcal{H}}}$, namely tha
\begin{eqnarray}
&& \label{N_L} \\
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{L,t}^{\flat \mathcal{H}}} &\leq &C\sup_{H^{\prime }\in
\mathfrak{C}_{\mathcal{H}}\left( L\right) }\sup_{K\in \mathcal{W}^{\ast
}\left( H^{\prime }\right) \cap \mathcal{C}_{A}^{\limfunc{restrict}}}\sqrt
\frac{1}{\left\vert K\right\vert _{\sigma }}\left( \frac{\mathrm{P}^{\alpha
}\left( K,\mathbf{1}_{A\setminus H^{\prime }}\sigma \right) }{\left\vert
K\right\vert }\right) ^{2}\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}} \notag
\\
&\leq &C\sqrt{\rho ^{-\left( t-2\right) }}\mathcal{S}_{\limfunc{aug}\limfunc
size}}^{\alpha ,A}\left( \mathcal{P}\right) =C^{\prime }\rho ^{-\frac{t}{2}
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
\right) . \notag
\end{eqnarray}
\medskip
\textbf{Remark on lack of usual goodness}: To prove (\ref{rem}), it is
essential that the intervals $H^{k+2}\in \mathcal{H}_{k+2}$ at the next
indented level down from $H^{k+1}\in \mathfrak{C}_{\mathcal{H}}\left(
L\right) $ are each contained in one of the Whitney intervals $K\in \mathcal
W}\left( H^{k+1}\right) \cap \mathcal{C}_{A}^{\limfunc{restrict}}$ for some
H^{k+1}\in \mathfrak{C}_{\mathcal{H}}\left( L\right) $. And this is the
reason we introduced the indented corona - namely so that $3H^{k+2}\subset
H^{k+1}$ for some $H^{k+1}\in \mathfrak{C}_{\mathcal{H}}\left( L\right) $,
and hence $H^{k+2}\subset K$ for some $K\in \mathcal{W}\left( H^{k+1}\right)
$. In the argument of Lacey in \cite{Lac}, the corresponding intervals were
good in the usual sense, and so the above triple property was automatic.
\medskip
So we begin by fixing $K\in \mathcal{W}^{\ast }\left( H^{k+1}\right) \cap
\mathcal{C}_{A}^{\limfunc{restrict}}$ with $H^{k+1}\in \mathfrak{C}_
\mathcal{H}}\left( L\right) $, and noting from the above that each $J\in \Pi
_{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H}}$ satisfies
\begin{equation*}
J^{\flat }\subset H^{k+t}\subset H^{k+t-1}\subset ...\subset H^{k+2}\subset K
\end{equation*
for $H^{k+j}\in \mathcal{H}_{k+j}$ uniquely determined by $J^{\flat }$. Thus
for $t\geq 2$ we have
\begin{eqnarray*}
\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &=&\sum_{H^{k+t}\in \mathcal{H
_{k+t}:\ H^{k+t}\subset K}\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P
_{L,t}^{\flat \mathcal{H}}:\ J^{\flat }\subset H^{k+t}}\left\Vert
\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2} \\
&\leq &\sum_{H^{k+t}\in \mathcal{H}_{k+t}:\ H^{k+t}\subset K}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( H^{k+t}\right) \right) .
\end{eqnarray*
In the case $t=2$ we are done since the final sum above is at most $\omega
_{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \right) $.
Now suppose $t\geq 3$. In order to obtain geometric gain in $t$, we will
apply the stopping criterion (\ref{up stopping condition}) in the following
form
\begin{equation}
\sum_{L^{\prime }\in \mathfrak{C}_{\mathcal{L}}\left( L_{0}\right) }\omega
_{\flat \mathcal{P}}\left( \mathbf{T}\left( L^{\prime }\right) \right)
=\omega _{\flat \mathcal{P}}\left( \dbigcup\limits_{L^{\prime }\in \mathfrak
C}_{\mathcal{L}}\left( L_{0}\right) }\mathbf{T}\left( L^{\prime }\right)
\right) \leq \frac{1}{\rho }\omega _{\flat \mathcal{P}}\left( \mathbf{T
\left( L_{0}\right) \right) ,\ \ \ \ \ \text{for all }L_{0}\in \mathcal{L},
\label{foll form}
\end{equation
where we have used the fact that the \emph{maximal} intervals $L^{\prime }$
in the collection $\dbigcup\limits_{\ell =0}^{m-1}\left\{ L^{\prime }\in
\mathcal{L}_{\ell }:\ L^{\prime }\subset L_{0}\right\} $ for $L_{0}\in
\mathcal{L}_{m}$ (that appears in (\ref{up stopping condition})) are
precisely the $\mathcal{L}$-children of $L_{0}$ in the tree $\mathcal{L}$
(the intervals $L^{\prime }$ above are strictly contained in $L_{0}$ since
\rho >1$ in (\ref{up stopping condition})), so tha
\begin{equation*}
\dbigcup\limits_{L^{\prime }\in \Gamma }L^{\prime
}=\dbigcup\limits_{L^{\prime }\in \mathfrak{C}_{\mathcal{L}}\left(
L_{0}\right) }L^{\prime }\text{ where }\Gamma \equiv \dbigcup\limits_{\ell
=0}^{m-1}\left\{ L^{\prime }\in \mathcal{L}_{\ell }:\ L^{\prime }\subset
L_{0}\right\} .
\end{equation*}
In order to apply (\ref{foll form}), we collect the pairwise disjoint
intervals $H^{k+t}\in \mathcal{H}_{k+t}$ such that$\ H^{k+t}\subset
H^{k+2}\subset K$, into groups according to which interval $L^{k^{\prime
}+t-2}\in \mathcal{G}_{k^{\prime }+t-2}$ they are contained in, where
k^{\prime }=d_{\limfunc{geom}}\left( H^{k+2}\right) $ is the geometric depth
of $H^{k+2}$ in the tree $\mathcal{L}$ introduced in (\ref{geom depth}). It
follows that each interval $H^{k+t}\in \mathcal{H}_{k+t}$ is contained in a
unique interval $L^{d_{\limfunc{geom}}\left( H^{k+2}\right) +t-2}\in
\mathcal{G}_{d_{\limfunc{geom}}\left( H^{k+2}\right) +t-2}$. Thus we\ obtain
from the previous inequality tha
\begin{eqnarray*}
\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &\leq &\sum_{H^{k+t}\in
\mathcal{H}_{k+t}:\ H^{k+t}\subset K}\omega _{\flat \mathcal{P}}\left(
\mathbf{T}\left( H^{k+t}\right) \right) \\
&\leq &\sum_{\substack{ H^{k+2}\in \mathcal{H}_{k+2} \\ H^{k+2}\subset K}
\sum_{\substack{ L^{k^{\prime }+t-2}\in \mathcal{G}_{k^{\prime }+t-2}:\
L^{k^{\prime }+t-2}\subset H^{k+2} \\ \text{where }k^{\prime }=d_{\limfunc
geom}}\left( H^{k+2}\right) }}\omega _{\flat \mathcal{P}}\left( \mathbf{T
\left( L^{k^{\prime }+t-2}\right) \right) .
\end{eqnarray*
In the case $t=2$ we are done since the final sum above is dominated by
\begin{equation*}
\sum_{H^{k+2}\in \mathcal{H}_{k+2}:\ H^{k+2}\subset K}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( H^{k+2}\right) \right) \leq \omega
_{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right) \right) .
\end{equation*}
For $t\geq 3$, we have
\begin{eqnarray*}
&&\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \sum_{\substack{
H^{k+2}\in \mathcal{H}_{k+2} \\ H^{k+2}\subset K}}\sum_{\substack{
L^{k^{\prime }+t-2}\in \mathcal{G}_{k^{\prime }+t-2}:\ L^{k^{\prime
}+t-2}\subset H^{k+2} \\ \text{where }k^{\prime }=d_{\limfunc{geom}}\left(
H^{k+2}\right) }}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left(
L^{k^{\prime }+t-2}\right) \right) \\
&=&\sum_{\substack{ H^{k+2}\in \mathcal{H}_{k+2} \\ H^{k+2}\subset K}}\sum
_{\substack{ L^{k^{\prime }+t-3}\in \mathcal{G}_{k^{\prime }+t-3}:\
L^{k^{\prime }+t-3}\subset H^{k+2} \\ \text{where }k^{\prime }=d_{\limfunc
geom}}\left( H^{k+2}\right) }}\left\{ \sum_{\substack{ L^{k^{\prime
}+t-2}\in \mathcal{G}_{k^{\prime }+t-2}:\ L^{k^{\prime }+t-2}\subset
L^{k^{\prime }+t-3} \\ \text{where }k^{\prime }=d_{\limfunc{geom}}\left(
H^{k+2}\right) }}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left(
L^{k^{\prime }+t-2}\right) \right) \right\} \\
&\leq &\sum_{\substack{ H^{k+2}\in \mathcal{H}_{k+2} \\ H^{k+2}\subset K}
\sum_{\substack{ L^{k^{\prime }+t-3}\in \mathcal{G}_{k^{\prime }+t-3}:\
L^{k^{\prime }+t-3}\subset H^{k+2} \\ \text{where }k^{\prime }=d_{\limfunc
geom}}\left( H^{k+2}\right) }}\left\{ \frac{1}{\rho }\omega _{\flat \mathcal
P}}\left( \mathbf{T}\left( L^{k^{\prime }+t-3}\right) \right) \right\} ,
\end{eqnarray*
where in the last line we have used (\ref{foll form}) with
L_{0}=L^{k^{\prime }+t-3}$ on the sum in braces. We then continue (if
necessary) wit
\begin{eqnarray*}
\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} &\leq &\frac{1}{\rho }\sum
_{\substack{ H^{k+2}\in \mathcal{H}_{k+2} \\ H^{k+2}\subset K}}\sum
_{\substack{ L^{k^{\prime }+t-3}\in \mathcal{G}_{k^{\prime }+t-3}:\
L^{k^{\prime }+t-3}\subset H^{k+2} \\ \text{where }k^{\prime }=d_{\limfunc
geom}}\left( H^{k+2}\right) }}\omega _{\flat \mathcal{P}}\left( \mathbf{T
\left( L^{k^{\prime }+t-3}\right) \right) \\
&\leq &\frac{1}{\rho ^{2}}\sum_{\substack{ H^{k+2}\in \mathcal{H}_{k+2} \\
H^{k+2}\subset K}}\sum_{\substack{ L^{k^{\prime }+t-4}\in \mathcal{G
_{k^{\prime }+t-4}:\ L^{k^{\prime }+t-4}\subset H^{k+2} \\ \text{where
k^{\prime }=d_{\limfunc{geom}}\left( H^{k+2}\right) }}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( L^{k^{\prime }+t-4}\right) \right) \\
&&\vdots \\
&\leq &\frac{1}{\rho ^{t-2}}\sum_{\substack{ H^{k+2}\in \mathcal{H}_{k+2}
\\ H^{k+2}\subset K}}\sum_{\substack{ L^{k^{\prime }}\in \mathcal{G
_{k^{\prime }}:\ L^{k^{\prime }}\subset H^{k+2} \\ \text{where }k^{\prime
}=d_{\limfunc{geom}}\left( H^{k+2}\right) }}\omega _{\flat \mathcal{P
}\left( \mathbf{T}\left( L^{k^{\prime }}\right) \right) .
\end{eqnarray*
Since $L^{k^{\prime }}\subset H^{k+2}$ implies $L^{k^{\prime }}=H^{k+2}$, we
now obtai
\begin{equation*}
\sum_{J\in \Pi _{2}^{K,\limfunc{aug}}\mathcal{P}_{L,t}^{\flat \mathcal{H
}}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \frac{1}{\rho ^{t-2}
\sum_{H^{k+2}\in \mathcal{H}_{k+2}:\ H^{k+2}\subset K}\omega _{\flat
\mathcal{P}}\left( \mathbf{T}\left( H^{k+2}\right) \right) \leq \frac{1}
\rho ^{t-2}}\omega _{\flat \mathcal{P}}\left( \mathbf{T}\left( K\right)
\right) ,
\end{equation*
which completes the proof of (\ref{rem}), and hence that of (\ref{N_L}).
Finally, an application of the Orthogonality Lemma \ref{mut orth}\ proves
\ref{S big t 3}).
\bigskip
\textbf{Proof of the first line in (\ref{rest bounds})}: At last we turn to
proving the first line in (\ref{rest bounds}). Recalling that $\mathcal{T
\left( L\right) =\mathcal{T}_{\limfunc{left}}\left( L\right) \dot{\cup
\mathcal{T}_{\limfunc{right}}\left( L\right) \dot{\cup}\left\{ L\right\} $,
we consider the collection
\begin{eqnarray*}
&&\mathcal{Q}_{0}^{\flat \mathcal{H}-big}=\dbigcup\limits_{L\in \mathcal{H}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}, \\
\text{where } &&\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}=\left\{ \left(
I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H}}:\text{there is
L^{\prime }\in \mathcal{T}\left( L\right) \text{ with }J^{\flat }\subset
L^{\prime }\subset I\right\} ,\ \ \ L\in \mathcal{H}, \\
\text{and } &&\mathcal{P}_{L,0}^{\flat \mathcal{H}}=\left\{ \left(
I,J\right) \in \mathcal{P}:I\in \mathcal{C}_{L}^{\mathcal{H}}\text{ and
J\in \mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}}\text{ for some
L\in \mathcal{H}\right\} ,\ \ \ L\in \mathcal{H},
\end{eqnarray*
and begin by claiming tha
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big}}\leq C\mathcal{S}_{\limfunc{aug
\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}_{L,0}^{\flat \mathcal{H
-big}\right) \leq C\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha
,A}\left( \mathcal{P}\right) ,\ \ \ \ \ L\in \mathcal{H}. \label{big t 3}
\end{equation
To see this, we fix $L\in \mathcal{H}$ and order the `left' tower of
intervals $\mathcal{T}_{\limfunc{left}}\left( L\right) =\left\{
L^{k}\right\} _{k=1}^{\infty }$ that lie in the restricted corona $\mathcal{
}_{L}^{\mathcal{H},\limfunc{restricted}}$ by decreasing side length, i.e.
\ell \left( L^{k+1}\right) \leq \ell \left( L^{k}\right) $ for all $k\geq 1
, and set $L^{0}=L$ (of course the tower may be finite, but for convenience
in notation, we won't reflect this in the notation). The `right' tower of
intervals $\mathcal{T}_{\limfunc{right}}\left( L\right) $ is handled
similarly and so not considered further here. Then $\mathcal{P}_{L,0}^{\flat
\mathcal{H}-big}$ can be decomposed as follows, remembering that $J^{\flat
}\subset I\subset L$ for $\left( I,J\right) \in \mathcal{P}_{L,0}^{\flat
\mathcal{H}-big}\subset \mathcal{P}_{L,0}^{\flat \mathcal{H}}$
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big} &=&\overset{\cdot }{\bigcup
_{k=1}^{\infty }\left\{ \mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal
L}}\dot{\cup}\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat \mathcal{L}}\dot
\cup}\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}\right\} \\
&=&\left( \overset{\cdot }{\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc
left}}^{k}}^{\flat \mathcal{L}}\right) \dot{\cup}\left( \overset{\cdot }
\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat
\mathcal{L}}\right) \dot{\cup}\left( \overset{\cdot }{\bigcup
_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L
}\right) \ ; \\
\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat \mathcal{L}} &\equiv &\left\{
\left( I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H}-big}:I\in
\mathcal{C}_{L^{k-1}}^{\mathcal{L}}\text{ and }J^{\flat }\subset L_{\limfunc
right}}^{k}\right\} , \\
\mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal{L}} &\equiv &\left\{
\left( I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H}-big}:I\in
\mathcal{C}_{L^{k-1}}^{\mathcal{L}}\text{ and }J^{\flat }\subset L_{\limfunc
left}}^{k}\right\} \\
\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}} &\equiv
&\left\{ \left( I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H
-big}:I\in \mathcal{C}_{L^{k-1}}^{\mathcal{L}}\text{ and }J^{\flat }\in
\mathcal{C}_{L^{k-1}}^{\mathcal{L}}\text{ and }J^{\flat }\cap
L^{k}=\emptyset \right\} \\
&=&\left\{ \left( I,J\right) \in \mathcal{P}_{L,0}^{\flat \mathcal{H
-big}:I=L^{k-1}\text{ and }J^{\flat }\in \mathcal{C}_{L^{k-1}}^{\mathcal{L}
\text{ and }J^{\flat }\cap L^{k}=\emptyset \right\} ,
\end{eqnarray*
and where in the last line we have used the fact that if $I,J^{\flat }\in
\mathcal{C}_{L^{k-1}}^{\mathcal{L}}$and there is $L^{\prime }\in \mathcal{T
\left( L\right) $ with $J^{\flat }\subset L^{\prime }\subset I$, then we
must have $I=L^{k-1}$. All of the pairs $\left( I,J\right) \in \mathcal{P
_{L,0}^{\flat \mathcal{H}-big}$ are included in either $\mathcal{R}_{L_
\limfunc{right}}^{k}}^{\flat \mathcal{L}}$, $\mathcal{R}_{L_{\limfunc{left
}^{k}}^{\flat \mathcal{L}}$ or $\mathcal{R}_{L_{\limfunc{disjoint
}^{k}}^{\flat \mathcal{L}}$ for some $k$, since if $J^{\flat }\supset L^{k}
, then $J^{\flat }$ shares an endpoint with $L$, which contradicts the fact
that $3J^{\flat }\subset J^{\maltese }\subset I\subset L$.
We can easily deal with the `disjoint' collection $\mathcal{Q}^{\limfunc
disjoint}}\equiv \overset{\cdot }{\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_
\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}$ by applying a $\emph{trivial}$
case of the $\flat $Straddling Lemma to $\mathcal{R}_{L_{\limfunc{disjoint
}^{k}}^{\flat \mathcal{L}}$ with a single straddling interval, followed by
an application of the Orthogonality Lemma to $\mathcal{Q}^{\limfunc{disjoint
}$. More precisely, every pair $\left( I,J\right) \in \mathcal{R}_{L_
\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}$ satisfies $J^{\flat }\subset
L^{k-1}=I$, so that the reduced admissible collection $\mathcal{R}_{L_
\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}$ $\flat $straddles the trivial
choice $\mathcal{S}=\left\{ L^{k-1}\right\} $, the singleton consisting of
just the interval $L^{k-1}$. Then the inequalit
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}}\leq C\mathcal{
}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{R}_{L_{\limfunc
disjoint}}^{k}}^{\flat \mathcal{L}}\right) ,
\end{equation*
follows from $\flat $Straddling Lemma \ref{straddle 3 ref}. The collection
\left\{ \mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}\right\}
_{k=1}^{\infty }$ is mutually orthogonal sinc
\begin{eqnarray*}
\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}} &\subset
\mathcal{C}_{L^{k-1}}^{\mathcal{L}}\times \mathcal{C}_{L^{k-1}}^{\mathcal{L
,\flat \func{shift}}\ , \\
\dsum\limits_{k=1}^{\infty }\mathbf{1}_{\mathcal{C}_{L^{k-1}}^{\mathcal{L}}}
&\leq &\mathbf{1}\text{ and }\dsum\limits_{k=1}^{\infty }\mathbf{1}_
\mathcal{C}_{L^{k-1}}^{\mathcal{L},\flat \limfunc{shift}}}\leq \mathbf{1}.
\end{eqnarray*
Since $\overset{\cdot }{\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc
disjoint}}^{k}}^{\flat \mathcal{L}}$ is reduced and admissible (each $J\in
\Pi _{2}\left( \overset{\cdot }{\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_
\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}\right) $ is paired with a
single $I$, namely the top of the $\mathcal{L}$-corona to which $J^{\flat }$
belongs), the Orthogonality Lemma \ref{mut orth} applies to obtain the
estimat
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\overse
{\cdot }{\bigcup }_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{disjoint
}^{k}}^{\flat \mathcal{L}}}=\widehat{\mathfrak{N}}_{\limfunc{stop
,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}^{\limfunc{disjoint}}}\leq
\sup_{k\geq 1}\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}}\leq
C\sup_{k\geq 1}\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left(
\mathcal{R}_{L_{\limfunc{disjoint}}^{k}}^{\flat \mathcal{L}}\right) \leq
\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P
_{L,0}^{\flat \mathcal{H}-big}\right) . \label{disjoint bound}
\end{equation}
Now we turn to estimating the norm of the `right' collection $\mathcal{Q}^
\limfunc{right}}\equiv \bigcup_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{right
}^{k}}^{\flat \mathcal{L}}$. First we note that $L_{\limfunc{right}}^{k}\in
\mathcal{C}_{A}^{\mathcal{A},\limfunc{restrict}}$ if $\left( I,J\right) \in
\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat \mathcal{L}}$ since $\mathcal{R
_{L_{\limfunc{right}}^{k}}^{\flat \mathcal{L}}$ is reduced, i.e. doesn't
contain any pairs $\left( I,J\right) $ with $J^{\flat }\subset A^{\prime }$
for some $A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right) $. Next
we note that $\mathcal{Q}^{\limfunc{right}}$ is admissible since if $J\in
\Pi _{2}\mathcal{Q}^{\limfunc{right}}$, then $J\in \Pi _{2}\mathcal{R}_{L_
\limfunc{right}}^{k}}^{\flat \mathcal{L}}$ for a unique index $k$, and of
course $\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat \mathcal{L}}$ is
admissible, so that the intervals $I$ that are paired with $J$ are
tree-connected. Thus we can apply the Straddling Lemma \ref{straddle 3 ref}
to the reduced admissible collection $\mathcal{Q}^{\limfunc{right}}$, with
the `straddling' set $\mathcal{S}\equiv \left\{ L_{\limfunc{right
}^{k}\right\} _{k=1}^{\infty }\cap \mathcal{C}_{A}^{\mathcal{A},\limfunc
restrict}}$, to obtain the estimat
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\bigcup_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{right}}^{k}}^{\flat
\mathcal{L}}}=\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}^{\limfunc{right}}}\leq C\mathcal{S}_{\limfunc{aug}\limfunc
size}}^{\alpha ,A}\left( \mathcal{Q}^{\limfunc{right}}\right) \leq C\mathcal
S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}_{L,0}^{\flat
\mathcal{H}-big}\right) \ . \label{right bound}
\end{equation}
As for the remaining `left' form $\left\vert \mathsf{B}\right\vert _
\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\bigcup_{k=0}^{\infty }\mathcal
R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal{L}}}\left( f,g\right) $, if the
interval pair $\left( I,J\right) \in \mathcal{R}_{L_{\limfunc{left
}^{k}}^{\flat \mathcal{L}}$, then either $J^{\flat }\subset L_{\limfunc{left
}^{k}\subsetneqq J^{\maltese }$ or $J^{\maltese }\subset L_{\limfunc{left
}^{k}$. But $J^{\flat }\subset L_{\limfunc{left}}^{k}\subsetneqq J^{\maltese
}$ implies that either $J^{\flat }=L_{\limfunc{left}}^{k}\subsetneqq
J^{\maltese }\subset I\subset L$, which is impossible since $J^{\flat }$
cannot share an endpoint with $L$, or that $J^{\flat }=L_{-/+}^{k}=\left( L_
\limfunc{left}}^{k}\right) _{\limfunc{right}}\ $and $J^{\maltese }=L^{k}$.
So we conclude that if $\left( I,J\right) \in \mathcal{R}_{L_{\limfunc{left
}^{k}}^{\flat \mathcal{L}}$, then
\begin{equation}
\text{either }J^{\maltese }\subset L_{\limfunc{left}}^{k}\text{ or "
J^{\maltese }=L^{k}\text{ and }J\subset L_{\limfunc{left}}^{k}\text{"}.
\label{either or}
\end{equation}
In either case in (\ref{either or}), there is a unique interval $K=K\left[
\right] \in \mathcal{W}\left( L\right) $ that contains $J$. It follows that
there are now two remaining cases:
\textbf{Case 1}: $K\left[ J\right] \in \mathcal{C}_{A}^{\limfunc{restrict}}$,
\textbf{Case 2}: $K\left[ J\right] \subset A^{\prime }\subsetneqq I$ for
some $A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right) $.
However, by Key Fact \#2 in (\ref{indentation}), and the fact that the \emph
great} grandparent $\pi _{\mathcal{D}}^{\left( 3\right) }A^{\prime }$ of
A^{\prime }$ contains $A^{\prime }$ inside its leftmost grandchild $\left(
\pi _{\mathcal{D}}^{\left( 3\right) }A^{\prime }\right) _{-/-}$, the pairs
\left( I,J\right) $ in \textbf{Case 2} lie in the `corona straddling'
collection $P_{\func{cor}}^{A}$ that was removed from all $A$-admissible
collections in (\ref{empty assumption}) of Conclusion \ref{assume}\ above,
and thus there are no pairs in \textbf{Case 2} here. (We note in passing
that a given $A^{\prime }\in \mathfrak{C}_{A}\left( A\right) $ can occur as
one of the Whitney intervals $K$ in $\mathcal{W}\left( L\right) $ for at
most one $L\in \mathcal{H}$, the indented corona.) Thus we conclude that $
\left[ J\right] \in \mathcal{C}_{A}^{\limfunc{restrict}}$.
We now claim that $3K\left[ J\right] \subset I$ for all pairs $\left(
I,J\right) \in \bigcup_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{left
}^{k}}^{\flat \mathcal{L}}$. To see this, suppose that $\left( I,J\right)
\in \mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal{L}}$ for some
k\geq 1$. Then by (\ref{either or}) we have both that $K\left[ J\right]
\subset L_{\limfunc{left}}^{k}$ and $L^{k}\subsetneqq I$. But then $K\left[
\right] \subset L_{\limfunc{left}}^{k}$ implies that $3K\left[ J\right]
\subset L^{k}\subset I$ as claimed. See Figure \ref{tow}.
\FRAME{ftbpFU}{6.714in}{2.7422in}{0pt}{\Qcb{The case when $\left( I,J\right)
\in \mathcal{R}_{L_{\limfunc{right}}^{k}}^{\mathcal{L}}$ and $J^{\maltese }$
shares an endpoint with $L$ and $K\in \mathcal{W}\left( L\right) $ equals
J_{-/+}^{\maltese }$. In this case $3K\subset I$.}}{\Qlb{tow}}{tower.wmf}
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TRUE;display "USEDEF";valid_file "F";width 6.714in;height 2.7422in;depth
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Now the `left' collection $\mathcal{Q}^{\limfunc{left}}\equiv
\bigcup_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal{
}}$ is admissible, since if $J\in \Pi _{2}\mathcal{Q}^{\limfunc{left}}$ and
I_{j}\in \Pi _{1}\mathcal{Q}^{\limfunc{left}}$ with $\left( I_{j},J\right)
\in \mathcal{Q}^{\limfunc{left}}$ for $j=1,2$, then $I_{j}\in \mathcal{C
_{L^{k_{j}-1}}^{\mathcal{L}}$ for some $k_{j}$ and all of the intervals
I\in \left[ I_{1},I_{2}\right] \ $lie in one of the coronas $\mathcal{C
_{L^{k-1}}^{\mathcal{L}}$ for $k$ between $k_{1}$ and $k_{2}$. Thus $\left(
I,J\right) \in \mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat \mathcal{L
}\subset \mathcal{Q}^{\limfunc{left}}$, and we have proved the required
connectedness. From the containment $3K\left[ J\right] \subset I\subset L$
for all $\left( I,J\right) \in \bigcup_{k=1}^{\infty }\mathcal{R}_{L_
\limfunc{left}}^{k}}^{\flat \mathcal{L}}$, we now see that the reduced
admissible collection $\mathcal{Q}^{\limfunc{left}}$ \emph{substraddles} the
interval $L$. Hence the Substraddling Lemma \ref{substraddle ref} yields the
boun
\begin{equation}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\bigcup_{k=1}^{\infty }\mathcal{R}_{L_{\limfunc{left}}^{k}}^{\flat
\mathcal{L}}}=\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}^{\limfunc{left}}}\leq C\mathcal{S}_{\limfunc{aug}\limfunc
size}}^{\alpha ,A}\left( \mathcal{Q}^{\limfunc{left}}\right) \leq C\mathcal{
}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{P}_{L,0}^{\flat
\mathcal{H}-big}\right) . \label{left bound}
\end{equation
Combining the bounds (\ref{disjoint bound}), (\ref{right bound}) and (\re
{left bound}), we obtain (\ref{big t 3}).
Finally, we observe that the collections $\mathcal{P}_{L,0}^{\flat \mathcal{
}-big}$ themselves are \emph{mutually orthogonal}, namely
\begin{eqnarray*}
\mathcal{P}_{L,0}^{\flat \mathcal{H}-big} &\subset &\mathcal{C}_{L}^
\mathcal{H}}\times \mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}}\ ,\ \
\ \ \ L\in \mathcal{H}\ , \\
\dsum\limits_{L\in \mathcal{H}}\mathbf{1}_{\mathcal{C}_{L}^{\mathcal{H}}}
&\leq &\mathbf{1}\text{ and }\dsum\limits_{L\in \mathcal{H}}\mathbf{1}_
\mathcal{C}_{L}^{\mathcal{H},\flat \limfunc{shift}}}\leq \mathbf{1}.
\end{eqnarray*
Thus an application of the Orthogonality Lemma \ref{mut orth} shows tha
\begin{equation*}
\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{Q}_{0}^{\flat \mathcal{H}-big}}\leq \sup_{L\in \mathcal{L}}\widehat
\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A,\mathcal{P
_{L,0}^{\flat \mathcal{H}-big}}\leq C\mathcal{S}_{\limfunc{aug}\limfunc{size
}^{\alpha ,A}\left( \mathcal{P}\right) .
\end{equation*
Altogether, the proof of Proposition \ref{bottom up 3} is now complete.
\end{proof}
This finishes the proofs of the inequalities (\ref{First inequality}) and
\ref{B stop form 3}).
\section{Wrapup of the proof\label{Sub wrapup}}
At this point we have controlled, either directly or probabilistically, the
norms of all of the forms in our decompositions - namely the disjoint,
nearby, far below, paraproduct, neighbour, broken and stopping forms - in
terms of the Muckenhoupt, energy and \emph{functional energy} conditions,
along with an arbitrarily small multiple of the operator norm. Thus it only
remains to control the functional energy condition by the Muckenhoupt and
energy conditions, since then, using $\int \left( T_{\sigma }^{\alpha
}f\right) gd\omega =\Theta \left( f,g\right) +\Theta ^{\ast }\left(
f,g\right) $ with the further decompositions above, we will have shown that
for any fixed tangent line truncation of the operator $T_{\sigma }^{\alpha }
, as defined in Definition \ref{truncated op}, we have
\begin{eqnarray*}
&&\left\vert \int \left( T_{\sigma }^{\alpha }f\right) gd\omega \right\vert
\boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^{\mathcal{G
}\left\vert \int \left( T_{\sigma }^{\alpha }f\right) gd\omega \right\vert
\leq \boldsymbol{E}_{\Omega }^{\mathcal{D}}\boldsymbol{E}_{\Omega }^
\mathcal{G}}\sum_{i=1}^{3}\left( \left\vert \Theta _{i}\left( f,g\right)
\right\vert +\left\vert \Theta _{i}^{\ast }\left( f,g\right) \right\vert
\right) \\
&&\ \ \ \ \ \leq \left( C_{\eta }\mathcal{NTV}_{\alpha }+\eta \mathfrak{N
_{T^{\alpha }}\right) \left\Vert f\right\Vert _{L^{2}\left( \sigma \right)
}\left\Vert g\right\Vert _{L^{2}\left( \omega \right) },\ \ \ f\in
L^{2}\left( \sigma \right) \text{ and }g\in L^{2}\left( \omega \right) ,
\end{eqnarray*
for an arbitarily small positive constant $\eta >0$, and a correspondingly
large finite constant $C_{\eta }$. Note that the testing constants
\mathfrak{T}_{T^{\alpha }}$ and $\mathfrak{T}_{T^{\alpha ,\ast }}$ in
\mathcal{NTV}_{\alpha }$ already include the supremum over all tangent line
truncations of $T^{\alpha }$, while the operator norm $\mathfrak{N
_{T^{\alpha }}$ on the left refers to a \emph{fixed} tangent line truncation
of $T^{\alpha }$. This give
\begin{equation*}
\mathfrak{N}_{T^{\alpha }}=\sup_{\left\Vert f\right\Vert _{L^{2}(\sigma
)}=1}\sup_{\left\Vert g\right\Vert _{L^{2}(\omega )}=1}\left\vert \int
\left( T_{\sigma }^{\alpha }f\right) gd\omega \right\vert \leq C_{\eta
\mathcal{NTV}_{\alpha }+\eta \mathfrak{N}_{T^{\alpha }},
\end{equation*
and since the truncated operators have finite operator norm $\mathfrak{N
_{T^{\alpha }}$, we can absorb the term $\eta \mathfrak{N}_{T^{\alpha }}$
into the left hand side for $\eta <1$ and obtain $\mathfrak{N}_{T^{\alpha
}}\leq C_{\eta }^{\prime }\mathcal{NTV}_{\alpha }$ for each tangent line
truncation of $T^{\alpha }$. Taking the supremum over all such truncations
of $T^{\alpha }$ finishes the proof of Theorem \ref{dim one}.
The task of controlling functional energy is taken up in Appendix B, after
first establishing weak frame and weak Riesz inequalities for martingale and
dual martingale differences in Appendix A (except for the lower weak Riesz
inequality for the martingale difference $\bigtriangleup _{Q}^{\mu ,\mathbf{
}}$).
\section{Appendix A: Martingale differences}
Most of the material in this appendix is known, see e.g. \cite{NTV3} and
\cite{HyMa}. First, we recall the construction in \cite{SaShUr9} of a Haar
basis in $\mathbb{R}$ that is adapted to a measure $\mu $ (c.f. \cite{NTV2}
where this type of construction is made explicit). Given a dyadic interval
Q\in \mathcal{D}$, where $\mathcal{D}$ is a dyadic grid of intervals from
\mathcal{P}$, let $\bigtriangleup _{Q}^{\mu }$ denote orthogonal projection
onto the \textbf{one}-dimensional subspace $L_{Q}^{2}\left( \mu \right) $ of
$L^{2}\left( \mu \right) $ that consists of linear combinations of the
indicators of\ the children $\mathfrak{C}\left( Q\right) $ of $Q$ that have
\mu $-mean zero over $Q$
\begin{equation*}
L_{Q}^{2}\left( \mu \right) \equiv \left\{ f=\dsum\limits_{Q^{\prime }\in
\mathfrak{C}\left( Q\right) }a_{Q^{\prime }}\mathbf{1}_{Q^{\prime
}}:a_{Q^{\prime }}\in \mathbb{R},\int_{Q}fd\mu =0\right\} .
\end{equation*
Then we have the important telescoping property for dyadic intervals
Q_{1}\subset Q_{2}$
\begin{equation}
\mathbf{1}_{Q_{0}}\left( x\right) \left( \dsum\limits_{Q\in \left[
Q_{1},Q_{2}\right] }\bigtriangleup _{Q}^{\mu }f\left( x\right) \right)
\mathbf{1}_{Q_{0}}\left( x\right) \left( \mathbb{E}_{Q_{0}}^{\mu }f-\mathbb{
}_{Q_{2}}^{\mu }f\right) ,\ \ \ \ \ Q_{0}\in \mathfrak{C}\left( Q_{1}\right)
,\ f\in L^{2}\left( \mu \right) , \label{telescope}
\end{equation}
\begin{notation}
Here $\mathbb{E}_{Q}^{\mu }f\left( x\right) \equiv \mathbf{1}_{Q}\left(
x\right) \frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}fd\mu $ denotes
the projection of $f$ onto $\limfunc{Span}\left\{ \mathbf{1}_{Q}\right\} $,
the one-dimensional subspace of multiples of the indicator of $Q$. We will
also denote the average value itself by $E_{Q}^{\mu }f\equiv \frac{1}
\left\vert Q\right\vert _{\mu }}\int_{Q}fd\mu $.
\end{notation}
It is convenient at times to use a fixed normalized basis $\left\{
h_{Q}^{\mu }\right\} $ of $L_{Q}^{2}\left( \mu \right) $. Then $\left\{
h_{Q}^{\mu }\right\} _{Q\in \mathcal{D}}$ is an orthonormal basis for
L^{2}\left( \mu \right) $, with the understanding that we add the constant
function $\mathbf{1}$ if $\mu $ is a finite measure. In particular we hav
\begin{equation*}
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2}=\sum_{Q\in \mathcal{
}}\left\Vert \bigtriangleup _{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu
\right) }^{2}=\sum_{Q\in \mathcal{D}}\left\vert \widehat{f}\left( Q\right)
\right\vert ^{2},\ \ \ \ \ \widehat{f}\left( Q\right) \equiv \left\langle
f,h_{Q}^{\mu }\right\rangle _{\mu }\ ,
\end{equation*
where the measure is suppressed in the notation $\widehat{f}$. Indeed, this
follows from (\ref{telescope}) and Lebesgue's differentiation theorem for
dyadic intervals. We also record the following useful estimate. If
I^{\prime }$ is either of the two $\mathcal{D}$-children of $I$, then
\begin{equation}
\left\vert \mathbb{E}_{I^{\prime }}^{\mu }h_{I}^{\mu }\right\vert \leq \sqrt
\mathbb{E}_{I^{\prime }}^{\mu }\left( h_{I}^{\mu }\right) ^{2}}\leq \frac{1}
\sqrt{\left\vert I^{\prime }\right\vert _{\mu }}}. \label{useful Haar}
\end{equation}
In the next subsection, we introduce martingale and dual martingale
differences for various $p$-weakly $\mu $-accretive families $\mathbf{b
=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$, and establish convergence
properties for their expansions. Then in later subsections, we turn to frame
inequalities, and weak Riesz inequalities.
\subsection{Convergence for weakly and controlled accretive families}
Supposes that $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{P}}$ is a $p
-weakly $\mu $-accretive family on $\mathbb{R}$. Define the $\mathbf{b}
-expectation operator $\mathbb{E}_{Q}^{\mu ,\mathbf{b}}$ and the dual
\mathbf{b}$-expectation operator $\mathbb{F}_{Q}^{\mu ,\mathbf{b}}$ by
\begin{eqnarray}
\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\mathbf{1
_{Q}\left( x\right) \frac{1}{\int_{Q}b_{Q}d\mu }\int_{Q}fb_{Q}d\mu ,\ \ \ \
\ Q\in \mathcal{P}\ , \label{def expectation} \\
\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\mathbf{1
_{Q}\left( x\right) b_{Q}\left( x\right) \frac{1}{\int_{Q}b_{Q}d\mu
\int_{Q}fd\mu ,\ \ \ \ \ Q\in \mathcal{P}\ . \notag
\end{eqnarray
Occasionally we will use the modification of $\mathbb{F}_{Q}^{\mu ,\mathbf{b
}$ given by `dividing out' the factor $b_{Q}$
\begin{equation}
\widehat{\mathbb{F}}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) \equiv \mathbf{1
_{Q}\left( x\right) \frac{1}{\int_{Q}b_{Q}d\mu }\int_{Q}fd\mu ,\ \ \ \ \
Q\in \mathcal{P}\ . \label{F hat}
\end{equation
Then define the corresponding martingale and dual martingale differences b
\begin{eqnarray}
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left(
\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E}_{Q^{\prime
}}^{\mu ,\mathbf{b}}f\left( x\right) \right) -\mathbb{E}_{Q}^{\mu ,\mathbf{b
}f\left( x\right) =\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbf
1}_{Q^{\prime }}\left( x\right) \left( \mathbb{E}_{Q^{\prime }}^{\mu
\mathbf{b}}f\left( x\right) -\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left(
x\right) \right) , \label{def diff} \\
\square _{Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left(
\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{F}_{Q^{\prime
}}^{\mu ,\mathbf{b}}f\left( x\right) \right) -\mathbb{F}_{Q}^{\mu ,\mathbf{b
}f\left( x\right) =\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbf
1}_{Q^{\prime }}\left( x\right) \left( \mathbb{F}_{Q^{\prime }}^{\mu
\mathbf{b}}f\left( x\right) -\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left(
x\right) \right) . \notag
\end{eqnarray}
\begin{description}
\item[Exception] In the special case that $\mathfrak{C}\left( Q\right)
=\left\{ Q_{1},Q_{2}\right\} $ where $\left\vert Q_{1}\right\vert _{\mu }=0$
and $\left\vert Q\right\vert _{\mu }>0$, we set both $\bigtriangleup
_{Q}^{\mu ,\mathbf{b}}\equiv 0$ and $\square _{Q}^{\mu ,\mathbf{b}}\equiv 0
, and redefine the parent differences $\bigtriangleup _{\pi Q}^{\mu ,\mathbf
b}}$ and $\square _{\pi Q}^{\mu ,\mathbf{b}}$ as follows. Let $\mathfrak{C
\left( \pi Q\right) =\left\{ Q,\widetilde{Q}\right\} $ and set
\begin{eqnarray*}
\bigtriangleup _{\pi Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left(
\mathbb{E}_{Q}^{\mu ,b_{Q_{2}}}f\left( x\right) +\mathbb{E}_{\widetilde{Q
}^{\mu ,\mathbf{b}}f\left( x\right) \right) -\mathbb{E}_{\pi Q}^{\mu
\mathbf{b}}f\left( x\right) , \\
\square _{\pi Q}^{\mu ,\mathbf{b}}f\left( x\right) &\equiv &\left( \mathbb{F
_{Q}^{\mu ,b_{Q_{2}}}f\left( x\right) +\mathbb{F}_{\widetilde{Q}}^{\mu
\mathbf{b}}f\left( x\right) \right) -\mathbb{F}_{\pi Q}^{\mu ,\mathbf{b
}f\left( x\right) ,
\end{eqnarray*
where we have used the test function $b_{Q_{2}}$ in place of the expected
b_{Q}$ (because $\left\vert Q_{1}\right\vert _{\mu }=0$). With the analogous
modification when only one grandchild at level $k$ below $Q$ is charged by
\mu $, the telescoping property holds for these differences, and the reader
can easily verify all of the convergence statements and formulas below. For
the sake of convenience only, we will ignore these exceptions in the sequel,
and proceed under the assumption that all intervals are charged by $\mu $.
\end{description}
Note that in \cite{NTV3} and \cite{HyMa} this notation is reversed - they
use $\bigtriangleup _{Q}^{\mu ,\mathbf{b}}$ for our $\square _{Q}^{\mu
\mathbf{b}}$. Finally, define the dual $\mathbf{b}$-expectation operator
\mathbb{F}_{m}^{\mu ,\mathbf{b}}$ on a function $f$ at level $m$ b
\begin{equation*}
\mathbb{F}_{m}^{\mu ,\mathbf{b}}f\left( x\right) \equiv \sum_{Q\in \mathcal{
}_{m}}\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) =\sum_{Q\in \mathcal{
}_{m}}\mathbf{1}_{Q}\left( x\right) b_{Q}\left( x\right) \frac{1}
\int_{Q}b_{Q}d\mu }\int_{Q}fd\mu ,
\end{equation*
where $\mathcal{D}_{m}\equiv \left\{ Q\in \mathcal{D}:\ell \left( Q\right)
=2^{-m}\right\} $, and define the operators $\square _{m}^{\mu ,\mathbf{b}}$
b
\begin{equation*}
\square _{m}^{\mu ,\mathbf{b}}\equiv \mathbb{F}_{m}^{\mu ,\mathbf{b}}
\mathbb{F}_{m-1}^{\mu ,\mathbf{b}}\ .
\end{equation*}
\begin{definition}
\label{controlled accretive}Let $\mu $ be a locally finite positive Borel
measure on $\mathbb{R}$, let $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in
\mathcal{P}}$ be a $p$-weakly $\mu $-accretive family on $\mathbb{R}$ with
reverse H\"{o}lder control (\ref{rev Hol con}) on children, and let
\mathcal{A}$ be\ subset of a dyadic grid $\mathcal{D}$. We say that the
subfamily $\mathbf{b}=\left\{ b_{Q}\right\} _{Q\in \mathcal{D}}$ is a $p
\emph{-weakly }$\mu $\emph{-controlled accretive} family on $\mathcal{D}\
(note we omit dependence $\mathcal{A}$ on in this notation) i
\begin{equation*}
b_{Q}=\mathbf{1}_{Q}b_{A}\ ,\ \ \ \ \ Q\in \mathcal{C}_{A}\mathcal{\ },A\in
\mathcal{A\ },
\end{equation*
and the set $\mathcal{A}$ satisfies a Carleson condition
\begin{equation*}
\sum_{Q\in \mathcal{A}:\ Q\subset K}\left\vert Q\right\vert _{\mu }\leq
C\left\vert K\right\vert _{\mu }\ ,\ \ \ \ \ \text{for all }K\in \mathcal{D},
\end{equation*
equivalentl
\begin{equation*}
\sum_{Q\in \mathcal{A}:\ Q\subset \Omega }\left\vert Q\right\vert _{\mu
}\leq C^{\prime }\left\vert \Omega \right\vert _{\mu }\ ,\ \ \ \ \ \text{for
all open sets }\Omega \subset \mathbb{R}.
\end{equation*}
\end{definition}
Denote the coronas associated to $\mathcal{A}$ by $\left\{ \mathcal{C
_{A}\right\} _{A\in \mathcal{A}}$, and now suppose that $\mathbf{b}=\left\{
b_{Q}\right\} _{Q\in \mathcal{P}}$ is $p$-weakly $\mu $-controlled
accretive, i.e.
\begin{equation*}
0<1\leq \left\vert \frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}b_{A}d\mu \right\vert \leq \left\Vert b_{A}\right\Vert _{L^{\infty
}\left( \mu \right) }\leq C_{\mathbf{b}}<\infty ,\ \ \ \ \ Q\in \mathcal{C
_{A}\mathcal{\ },A\in \mathcal{A}\ ,
\end{equation*
where in addition we have reverse H\"{o}lder control (\ref{rev Hol con}) on
children, and $\mathcal{A}$ satisfies a $\mu $-Carleson condition. Now
decompose the children $Q^{\prime }\in \mathfrak{C}\left( Q\right) $ into
the collection $\mathfrak{C}_{\limfunc{broken}}\left( Q\right) $ of \emph
broken} children $Q^{\prime }\in \mathcal{A}$ and the remaining collection
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) $ of \emph{natural} children
$Q^{\prime }\not\in \mathcal{A}$, i.e
\begin{equation*}
\mathfrak{C}_{\limfunc{broken}}\left( Q\right) \equiv \mathfrak{C}\left(
Q\right) \cap \mathcal{A}\text{ and }\mathfrak{C}_{\limfunc{natural}}\left(
Q\right) \equiv \mathfrak{C}\left( Q\right) \setminus \mathcal{A}.
\end{equation*
Let $\left\{ \square _{I}^{\mu ,\mathbf{b}}\right\} _{I\in \mathcal{D}}$ be
associated to the coronas $\left\{ \mathcal{C}_{A}\right\} _{A\in \mathcal{A
}$ and the $p$-weakly $\mu $-controlled accretive family $\mathbf{b}=\left\{
b_{A}\right\} _{A\in \mathcal{A}}$ as in Definition \ref{controlled
accretive} above. We will refer to the collection of dual martingale
differences $\left\{ \square _{I}^{\mu ,\mathbf{b}}\right\} _{I\in \mathcal{
}}$ as a `broken corona decomposition' in light of the fact that the testing
functions $b_{Q}$ `break' when passing from one corona to another. For $A\in
\mathcal{A}$, define the corona `pseudoprojections' $\mathsf{P}_{\mathcal{C
_{A}}^{\mu ,\mathbf{b}}$ b
\begin{equation*}
\mathsf{P}_{\mathcal{C}_{A}}^{\mu ,\mathbf{b}}f=\sum_{I\in \mathcal{C
_{A}}\square _{I}^{\mu ,\mathbf{b}}f\ ,
\end{equation*
We have the \emph{broken corona decomposition}
\begin{equation*}
f=\sum_{I\in \mathcal{D}}\square _{I}^{\mu ,\mathbf{b}}f=\sum_{A\in \mathcal
A}}\mathsf{P}_{\mathcal{C}_{A}}^{\mu ,\mathbf{b}}f\ ,
\end{equation*
whose convergence properties we investigate in the next subsubsection.
\subsubsection{Convergence of controlled martingale differences}
As shown by Hyt\"{o}nen and Martikainen \cite{HyMa}, in the setting of a $2
-weakly $\mu $-controlled accretive\emph{\ }family, we have strong
convergence in $L^{2}\left( \mu \right) $ for the dual martingale
differences - and also for the martingale differences under the stronger
assumption of a $p$-weakly $\mu $-controlled accretive\emph{\ }family for
some $p>2$ (the proofs given there carry over to general measures). We only
use the case $p=\infty $.
\begin{lemma}
\label{conv prop}(\cite{NTV3}, \cite[Lemma 3.5]{HyMa}) Suppose $\mathbf{b}$
is an $\infty $-weakly $\mu $-controlled accretive family on a grid
\mathcal{D}$. Then we have the dual martingale and martingale identitie
\begin{eqnarray*}
f &=&\sum_{I\in \mathcal{D}_{N}}\mathbb{F}_{I}^{\mu ,\mathbf{b}}f+\sum_{I\in
\mathcal{D}:\ \ell \left( I\right) \geq N+1}\square _{I}^{\mu ,\mathbf{b}}f\
,\ \ \ \ \ N\in \mathbb{Z}, \\
f &=&\sum_{I\in \mathcal{D}_{N}}\mathbb{E}_{I}^{\mu ,\mathbf{b}}f+\sum_{I\in
\mathcal{D}:\ \ell \left( I\right) \geq N+1}\bigtriangleup _{I}^{\mu
\mathbf{b}}f\ ,\ \ \ \ \ N\in \mathbb{Z},
\end{eqnarray*
in the sense of pointwise $\mu $-almost everywhere convergence, and also in
the sense of strong convergence in $L^{2}\left( \mu \right) $.
\end{lemma}
\subsection{Frame inequalities}
Define the positive sublinear operator
\begin{eqnarray}
\bigtriangledown _{Q}^{\mu }f &\equiv &\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( Q\right) }\left( \frac{1}{\left\vert Q^{\prime
}\right\vert _{\mu }}\int_{Q^{\prime }}\left\vert f\right\vert d\mu \right)
\mathbf{1}_{Q^{\prime }}, \label{Carleson avg op} \\
\widehat{\bigtriangledown }_{Q}^{\mu }f &\equiv &\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( Q\right) }\left( \frac{1}{\left\vert
Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime }}\left\vert f\right\vert
d\mu +\frac{1}{\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert
f\right\vert d\mu \right) \mathbf{1}_{Q^{\prime }}, \notag
\end{eqnarray
where we are suppressing here the dependence of both $\bigtriangledown
_{Q}^{\mu }$ and its larger version $\widehat{\bigtriangledown }_{Q}^{\mu }$
on the breaking intervals. Note also that $\widehat{\bigtriangledown
_{Q}^{\mu }=0$ if $Q$ has no broken children. We also set $Q_{\limfunc{broke
}}\equiv \bigcup_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
Q\right) }Q^{\prime }$. We now show that the Carleson condition on broken
children gives the inequalit
\begin{equation}
\sum_{Q\in \mathcal{D}}\left\Vert \widehat{\bigtriangledown }_{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ . \label{Car embed}
\end{equation
Indeed
\begin{eqnarray*}
\sum_{Q\in \mathcal{D}}\left\Vert \widehat{\bigtriangledown }_{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2} &=&\sum_{Q\in \mathcal{D}}\int
\left\vert \sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
Q\right) }\mathbf{1}_{Q^{\prime }}\left( \frac{1}{\left\vert Q^{\prime
}\right\vert _{\mu }}\int_{Q^{\prime }}\left\vert f\right\vert d\mu +\frac{
}{\left\vert Q\right\vert _{\mu }}\int_{Q}\left\vert f\right\vert d\mu
\right) \right\vert ^{2}d\mu \\
&\lesssim &\sum_{Q\in \mathcal{D}}\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( Q\right) }\left\vert Q^{\prime }\right\vert _{\mu
}\left( \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime
}}\left\vert f\right\vert d\mu +\frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}\left\vert f\right\vert d\mu \right) ^{2} \\
&\lesssim &\sum_{A\in \mathcal{A}}\left\vert A\right\vert _{\mu }\left(
\frac{1}{\left\vert A\right\vert _{\mu }}\int_{A}\left\vert f\right\vert
d\mu \right) ^{2}+\sum_{A\in \mathcal{A}}\left\vert A\right\vert _{\mu
}\left( \frac{1}{\left\vert \pi _{\mathcal{D}}A\right\vert _{\mu }}\int_{\pi
_{\mathcal{D}}A}\left\vert f\right\vert d\mu \right) ^{2},
\end{eqnarray*
where $\mathcal{A}$ is a collection of stopping times that satisfy a
Carleson condition as in Definition \ref{controlled accretive}. The first
term in the last line above is at most $C\left\Vert f\right\Vert
_{L^{2}\left( \mu \right) }^{2}$ by the Carleson embedding theorem. We claim
that the second term is controlled by a variant of the Carleson embedding
theorem,
\begin{equation}
\sum_{A\in \mathcal{A}}\left\vert A\right\vert _{\mu }\left( \frac{1}
\left\vert \pi _{\mathcal{D}}A\right\vert _{\mu }}\int_{\pi _{\mathcal{D
}A}\left\vert f\right\vert d\mu \right) ^{2}\lesssim \int \left\vert
f\right\vert ^{2}d\mu ,\ \ \ \ \ f\in L^{2}\left( \mu \right) .
\label{parent CET}
\end{equation
Indeed, with the measure $\nu \left( A\right) \equiv \left\vert A\right\vert
_{\mu }$ on $\mathcal{A}$, the sublinear map $T$ defined by $Tf\left(
A\right) \equiv \frac{1}{\left\vert \pi _{\mathcal{D}}A\right\vert _{\mu }
\int_{\pi _{\mathcal{D}}A}\left\vert f\right\vert d\mu $ takes $L^{\infty
}\left( \mu \right) $ to $L^{\infty }\left( \nu \right) $, and also
L^{1}\left( \mu \right) $ to $L^{1,\infty }\left( \nu \right) $, since if
\left\{ M\right\} $ are the maximal dyadic intervals $\pi _{\mathcal{D}}A$
such tha
\begin{equation*}
\frac{1}{\left\vert \pi _{\mathcal{D}}A\right\vert _{\mu }}\int_{\pi _
\mathcal{D}}A}\left\vert f\right\vert d\mu >\lambda >0,
\end{equation*
the
\begin{eqnarray*}
\left\vert \left\{ A\in \mathcal{A}:Tf\left( A\right) >\lambda \right\}
\right\vert _{\nu } &=&\sum_{A:\ Tf\left( A\right) >\lambda }\left\vert
A\right\vert _{\mu }\leq \sum_{M}\sum_{A:\ A\subset M}\left\vert
A\right\vert _{\mu }\lesssim \sum_{M}\left\vert M\right\vert _{\mu } \\
&<&\sum_{M}\frac{1}{\lambda }\int_{M}\left\vert f\right\vert d\mu \leq \frac
1}{\lambda }\int \left\vert f\right\vert d\mu ,
\end{eqnarray*
since the maximal intervals $M$ are pairwise disjoint. Now interpolation
shows that $T$ takes $L^{2}\left( \mu \right) $ to $L^{2}\left( \nu \right)
, which is (\ref{parent CET}). These inequalities provide the reason for
referring to $\bigtriangledown _{Q}^{\mu }$ and $\widehat{\bigtriangledown
_{Q}^{\mu }$\ as \emph{Carleson averaging operators}.
From \cite{NTV3} we have that in the case $\mathbf{b}$ is an $\infty
-weakly $\mu $-controlled accretive family on a grid $\mathcal{D}$, and the
underlying measure $\mu $ is upper doubling, then the following frame
equivalences hold
\begin{eqnarray*}
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\approx &\sum_{Q\in
\mathcal{D}}\left\{ \left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}+\left\Vert \bigtriangledown _{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right\} \\
&\approx &\sum_{Q\in \mathcal{D}}\left\{ \left\Vert \bigtriangleup _{Q}^{\mu
,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right) }^{2}+\left\Vert
\bigtriangledown _{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}\right\} \ .
\end{eqnarray*
It appears however that the arguments in \cite{NTV3} hold for more general
measures\footnote
But in \cite{NTV3}, a proof of their inequality (3.3), which is our lower
frame inequality for $\bigtriangleup _{Q}^{\mu ,\mathbf{b}}$, seems not to
be explicitly given.}, and in any case, we will extend most of these frame
inequalities to certain of the weak Riesz inequalities below for arbitrary
positive Borel measures $\mu $. For this it will be convenient to refer
directly to the proofs of the frame inequalities, and since notation in \cit
{NTV3} is very different from that used in this paper, we will instead
extend a similar argument of Hyt\"{o}nen and Martikainen \cite[Proposition
3.10]{HyMa} to general measures, adding a small additional argument for the
martingale differences in term $III_{A}$ in the proof of Proposition \re
{dual frame} below. This uses the following unweighted square function
estimate, which is essentially just the orthogonality of the standard Haar
projections $\bigtriangleup _{I}^{\mu }$ adapted to the measure $\mu $. For
this, we recall the general Haar projections $\mathsf{P}_{\mathcal{C
_{A}}^{\mu }h=\sum_{Q\in \mathcal{C}_{A}}\bigtriangleup _{Q}^{\mu }h$.
\begin{lemma}
\label{unweighted square
\begin{equation*}
\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert Q^{\prime
}\right\vert _{\mu }\left\vert E_{Q^{\prime }}^{\mu }h-E_{Q}^{\mu
}h\right\vert ^{2}\lesssim \left\Vert h\right\Vert _{L^{2}\left( \mu \right)
}^{2},\ \ \ \ \ h\in L^{2}\left( \mu \right) .
\end{equation*}
\end{lemma}
\begin{proof}
Recall that the Haar projection $\bigtriangleup _{I}^{\mu }$ is given by
\begin{equation*}
\bigtriangleup _{I}^{\mu }h=\left( \sum_{I^{\prime }\in \mathfrak{C}\left(
I\right) }\mathbb{E}_{I^{\prime }}^{\mu }h\right) -\mathbb{E}_{I}^{\mu
}h=\sum_{I^{\prime }\in \mathfrak{C}\left( I\right) }\mathbf{1}_{I^{\prime
}}\left( \mathbb{E}_{I^{\prime }}^{\mu }h-\mathbb{E}_{I}^{\mu }h\right) ,
\end{equation*
and s
\begin{equation*}
\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\left\vert Q^{\prime
}\right\vert _{\mu }\left\vert \mathbb{E}_{Q^{\prime }}^{\mu }h-\mathbb{E
_{Q}^{\mu }h\right\vert ^{2}=\int \left\vert \sum_{Q^{\prime }\in \mathfrak{
}\left( Q\right) }\mathbf{1}_{Q^{\prime }}\left( \mathbb{E}_{Q^{\prime
}}^{\mu }h-\mathbb{E}_{Q}^{\mu }h\right) \right\vert ^{2}d\mu =\left\Vert
\bigtriangleup _{Q}^{\mu }h\right\Vert _{L^{2}\left( \mu \right) }^{2}.
\end{equation*
Thus we hav
\begin{eqnarray*}
\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert Q^{\prime
}\right\vert _{\mu }\left\vert \mathbb{E}_{Q^{\prime }}^{\mu }h-\mathbb{E
_{Q}^{\mu }h\right\vert ^{2} &\leq &\sum_{A\in \mathcal{A}}\sum_{Q\in
\mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\left\vert
Q^{\prime }\right\vert _{\mu }\left\vert \mathbb{E}_{Q^{\prime }}^{\mu }h
\mathbb{E}_{Q}^{\mu }h\right\vert ^{2} \\
&=&\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\left\Vert
\bigtriangleup _{Q}^{\mu }h\right\Vert _{L^{2}\left( \mu \right)
}^{2}=\left\Vert h\right\Vert _{L^{2}\left( \mu \right) }^{2}\ .
\end{eqnarray*}
\end{proof}
\begin{proposition}
\label{dual frame}(see \cite[Remark 3.11]{HyMa} for the case of a doubling
measure with $p>2$) Suppose that $\mathbf{b}$ is an $\infty $-weakly $\mu
-controlled accretive family on a grid $\mathcal{D}$ with corona tops
\mathcal{A\subset D}$. Then we have the lower frame inequalit
\begin{equation*}
\sum_{I\in \mathcal{D}}\left\Vert \bigtriangleup _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ .
\end{equation*}
\end{proposition}
\begin{proof}
Given $A\in \mathcal{A}$, we begin\ wit
\begin{eqnarray*}
\sum_{Q\in \mathcal{C}_{A}}\left\Vert \bigtriangleup _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2} &=&\sum_{Q\in \mathcal{C
_{A}}\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\int_{Q^{\prime
}}\left\vert \mathbb{E}_{Q^{\prime }}^{\mu ,\mathbf{b}}f\left( x\right)
\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) \right\vert ^{2}d\mu \left(
x\right) \\
&=&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( Q\right) }\int_{Q^{\prime }}\left\vert \frac{\int_{Q^{\prime
}}fb_{A}d\mu }{\int_{Q^{\prime }}b_{A}d\mu }-\frac{\int_{Q}fb_{A}d\mu }
\int_{Q}b_{A}d\mu }\right\vert ^{2}d\mu \left( x\right) \\
&&+\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
broken}}\left( Q\right) }\int_{Q^{\prime }}\left\vert \frac{\int_{Q^{\prime
}}fb_{Q^{\prime }}d\mu }{\int_{Q^{\prime }}b_{Q^{\prime }}d\mu }-\frac
\int_{Q}fb_{A}d\mu }{\int_{Q}b_{A}d\mu }\right\vert ^{2}d\mu \left( x\right)
\\
&=&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( Q\right) }\left\vert \frac{\int_{Q^{\prime }}fb_{A}d\mu }
\int_{Q^{\prime }}b_{A}d\mu }-\frac{\int_{Q}fb_{A}d\mu }{\int_{Q}b_{A}d\mu
\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&&+\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
broken}}\left( Q\right) }\left\vert \frac{\int_{Q^{\prime }}fb_{Q^{\prime
}}d\mu }{\int_{Q^{\prime }}b_{Q^{\prime }}d\mu }-\frac{\int_{Q}fb_{A}d\mu }
\int_{Q}b_{A}d\mu }\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu }
\\
&\equiv &I_{A}+II_{A}.
\end{eqnarray*
To estimate term $II_{A}$ we writ
\begin{eqnarray}
&&\left\vert \frac{\int_{Q^{\prime }}fb_{Q^{\prime }}d\mu }{\int_{Q^{\prime
}}b_{Q^{\prime }}d\mu }-\frac{\int_{Q}fb_{A}d\mu }{\int_{Q}b_{A}d\mu
\right\vert \label{analogue} \\
&=&\frac{\left\vert \left( \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu
}\int_{Q^{\prime }}fb_{Q^{\prime }}d\mu \right) \left( \frac{1}{\left\vert
Q\right\vert _{\mu }}\int_{Q}b_{A}d\mu \right) -\left( \frac{1}{\left\vert
Q\right\vert _{\mu }}\int_{Q}fb_{A}d\mu \right) \left( \frac{1}{\left\vert
Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime }}b_{Q^{\prime }}d\mu \right)
\right\vert }{\left\vert \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }
\int_{Q^{\prime }}b_{Q^{\prime }}d\mu \right\vert \left\vert \frac{1}
\left\vert Q\right\vert _{\mu }}\int_{Q}b_{A}d\mu \right\vert } \notag \\
&\lesssim &\left\vert E_{Q^{\prime }}^{\mu }\left( fb_{Q^{\prime }}\right)
E_{Q}^{\mu }\left( b_{A}\right) -E_{Q}^{\mu }\left( fb_{A}\right)
E_{Q^{\prime }}^{\mu }\left( b_{Q^{\prime }}\right) \right\vert \notag \\
&=&\left\vert E_{Q^{\prime }}^{\mu }\left( fb_{Q^{\prime }}\right)
E_{Q}^{\mu }\left( b_{A}\right) -E_{Q^{\prime }}^{\mu }\left( fb_{A}\right)
E_{Q^{\prime }}^{\mu }\left( b_{Q^{\prime }}\right) -\left[ E_{Q}^{\mu
}\left( fb_{A}\right) -E_{Q^{\prime }}^{\mu }\left( fb_{A}\right) \right]
E_{Q^{\prime }}^{\mu }\left( b_{Q^{\prime }}\right) \right\vert \notag \\
&\lesssim &\left\vert E_{Q^{\prime }}^{\mu }\left( fb_{Q^{\prime }}\right)
\right\vert +\left\vert E_{Q^{\prime }}^{\mu }\left( fb_{A}\right)
\right\vert +\left\vert E_{Q}^{\mu }\left( fb_{A}\right) -E_{Q^{\prime
}}^{\mu }\left( fb_{A}\right) \right\vert , \notag
\end{eqnarray
and then
\begin{eqnarray*}
II_{A} &\lesssim &\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{
}_{\limfunc{broken}}\left( Q\right) }\left( \left\vert E_{Q^{\prime }}^{\mu
}\left( fb_{Q^{\prime }}\right) \right\vert ^{2}+\left\vert E_{Q^{\prime
}}^{\mu }\left( fb_{A}\right) \right\vert ^{2}+\left\vert E_{Q}^{\mu }\left(
fb_{A}\right) \right\vert ^{2}\right) \left\vert Q^{\prime }\right\vert
_{\mu } \\
&\lesssim &\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( Q\right) }\left( \left( E_{Q^{\prime }}^{\mu
}\left\vert f\right\vert \right) ^{2}+\left( E_{Q}^{\mu }\left\vert
f\right\vert \right) ^{2}\right) \left\vert Q^{\prime }\right\vert _{\mu }\ .
\end{eqnarray*}
Now we turn to term $I_{A}$ and use the analogue of (\ref{analogue}) for
natural children, along with the natural child bounds $\left\vert
\int_{Q^{\prime }}b_{A}d\mu \right\vert \gtrsim \left\vert Q^{\prime
}\right\vert _{\mu }$\ and $\left\vert \int_{Q}b_{A}d\mu \right\vert \gtrsim
\left\vert Q\right\vert _{\mu }$, to obtai
\begin{eqnarray*}
I_{A} &=&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{natural}}\left( Q\right) }\left\vert \frac{\int_{Q^{\prime
}}fb_{A}d\mu }{\int_{Q^{\prime }}b_{A}d\mu }-\frac{\int_{Q}fb_{A}d\mu }
\int_{Q}b_{A}d\mu }\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu }
\\
&=&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( Q\right) }\left\vert \frac{\left( \int_{Q^{\prime
}}fb_{A}d\mu \right) \left( \int_{Q}b_{A}d\mu \right) -\left(
\int_{Q^{\prime }}b_{A}d\mu \right) \left( \int_{Q}fb_{A}d\mu \right) }
\left( \int_{Q^{\prime }}b_{A}d\mu \right) \left( \int_{Q}b_{A}d\mu \right)
\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&\lesssim &\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu }\left(
b_{A}\right) \right\vert ^{2}\left\vert E_{Q^{\prime }}^{\mu }\left(
fb_{A}\right) -E_{Q}^{\mu }\left( fb_{A}\right) \right\vert ^{2}\left\vert
Q^{\prime }\right\vert _{\mu } \\
&&+\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( Q\right) }\left\vert E_{Q}^{\mu }\left( b_{A}\right)
-E_{Q^{\prime }}^{\mu }b_{A}\right\vert ^{2}\left\vert E_{Q}^{\mu }\left(
fb_{A}\right) \right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&\equiv &III_{A}+IV_{A}.
\end{eqnarray*}
Now we hav
\begin{equation*}
III_{A}\lesssim \sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C
_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q^{\prime }}^{\mu }\left(
fb_{A}\right) -E_{Q}^{\mu }\left( fb_{A}\right) \right\vert ^{2}\left\vert
Q^{\prime }\right\vert _{\mu }\ ,
\end{equation*
and for term $IV_{A}$, we introduce the quantities
\begin{equation*}
\gamma _{Q}\equiv \sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{natural
}\left( Q\right) }\left\vert E_{Q}^{\mu }b_{A}-E_{Q^{\prime }}^{\mu
}b_{A}\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu },\ \ \ \ \
\text{for }Q\in \mathcal{C}_{A},A\in \mathcal{A\ },
\end{equation*
so tha
\begin{eqnarray*}
IV_{A} &=&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu }\left(
b_{A}\right) -E_{Q^{\prime }}^{\mu }b_{A}\right\vert ^{2}\left\vert
Q^{\prime }\right\vert _{\mu }\left\vert E_{Q}^{\mu }\left( fb_{A}\right)
\right\vert ^{2} \\
&=&\sum_{Q\in \mathcal{C}_{A}}\gamma _{Q}\left\vert E_{Q}^{\mu }\left(
fb_{A}\right) \right\vert ^{2}\lesssim \sum_{Q\in \mathcal{C}_{A}}\gamma
_{Q}\left( E_{Q}^{\mu }\left\vert f\right\vert \right) ^{2}.
\end{eqnarray*}
Now note that the coefficients $\left\{ \gamma _{Q}\right\} _{Q\in \mathcal{
}}$ satisfy the Carleson conditio
\begin{equation}
\dsum\limits_{Q\subset B}\gamma _{Q}\lesssim \left\vert B\right\vert
_{\sigma }. \label{Car cond}
\end{equation
Indeed, if $B\in \mathcal{C}_{A}$, then using $E_{Q}^{\mu }b_{A}=E_{Q}^{\mu
}\left( \mathbf{1}_{B}b_{A}\right) $ for $Q\subset B$ and the unweighted
square function estimate, and denoting by $\mathfrak{G}_{\mathcal{A
}^{t}\left( A\right) $ the $\mathcal{A}$-grandchildren at level $t$ below $A
, we have
\begin{eqnarray*}
\dsum\limits_{Q\subset B}\gamma _{Q} &=&\sum_{A\in \mathcal{A
}\dsum\limits_{Q\in \mathcal{C}_{A}:\ Q\subset B}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu
}b_{A}-E_{Q^{\prime }}^{\mu }b_{A}\right\vert ^{2}\left\vert Q^{\prime
}\right\vert _{\mu } \\
&&+\sum_{t=1}^{\infty }\sum_{H\in \mathfrak{G}_{\mathcal{A}}^{t}\left(
A\right) :\ H\subset B}\dsum\limits_{Q\in \mathcal{C}_{H}}\sum_{Q^{\prime
}\in \mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu
}b_{H}-E_{Q^{\prime }}^{\mu }b_{H}\right\vert ^{2}\left\vert Q^{\prime
}\right\vert _{\mu } \\
&\lesssim &\int_{B}\left\vert b_{A}\right\vert ^{2}d\mu +\sum_{t=1}^{\infty
}\sum_{H\in \mathfrak{G}_{\mathcal{A}}^{t}\left( A\right) :\ H\subset B}\int
\left\vert b_{H}\right\vert ^{2}d\mu \\
&\lesssim &\left\vert B\right\vert _{\mu }+\sum_{t=1}^{\infty }\sum_{H\in
\mathfrak{G}_{\mathcal{A}}^{t}\left( A\right) :\ H\subset B}\left\vert
H\right\vert _{\mu }\lesssim \left\vert B\right\vert _{\mu }\ .
\end{eqnarray*}
Altogether then, using the above inequalities for $II_{A},III_{A}$ and
IV_{A}$, and then using the Carleson embedding theorem on the sum of the
terms $II_{A}$, and the Carleson embedding theorem again on the sum of the
terms $IV_{A}$ with the Carleson condition (\ref{Car cond}), we conclude tha
\begin{eqnarray*}
\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\left\Vert \bigtriangleup
_{Q}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\lesssim
&\sum_{A\in \mathcal{A}}\left( II_{A}+III_{A}+IV_{A}\right) \\
&\lesssim &\sum_{A\in \mathcal{A}}\left\vert A\right\vert _{\mu }\left\{
\left( \frac{1}{\left\vert A\right\vert _{\mu }}\int_{A}\left\vert
f\right\vert d\mu \right) ^{2}+\left( \frac{1}{\left\vert \pi A\right\vert
_{\mu }}\int_{\pi A}\left\vert f\right\vert d\mu \right) ^{2}\right\} \\
&&+\sum_{A\in \mathcal{A}}III_{A}+\sum_{Q\in \mathcal{C}_{A}}\gamma
_{Q}\left( E_{Q}^{\mu }\left\vert f\right\vert \right) ^{2}\lesssim \int
\left\vert f\right\vert ^{2}d\mu +\sum_{A\in \mathcal{A}}III_{A}.
\end{eqnarray*
Thus it remains only to estimate the final square function expression, which
requires an additional argument due to the fact that the functions $fb_{A}$
differ from one corona to the next. For this we defin
\begin{equation}
b\equiv \sum_{A\in \mathcal{A}}b_{A}\mathbf{1}_{A\setminus
\bigcup_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right) }}
\label{def b}
\end{equation
and writ
\begin{eqnarray*}
&&\sum_{A\in \mathcal{A}}III_{A}\lesssim \sum_{A\in \mathcal{A}}\sum_{Q\in
\mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{natural}}\left(
Q\right) }\left\vert E_{Q^{\prime }}^{\mu }\left( fb_{A}\right) -E_{Q}^{\mu
}\left( fb_{A}\right) \right\vert ^{2}\left\vert Q^{\prime }\right\vert
_{\mu } \\
&\lesssim &\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime
}\in \mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert
E_{Q^{\prime }}^{\mu }\left( fb_{A}-fb\right) -E_{Q}^{\mu }\left(
fb_{A}-fb\right) \right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&&+\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q^{\prime
}}^{\mu }\left( fb\right) -E_{Q}^{\mu }\left( fb\right) \right\vert
^{2}\left\vert Q^{\prime }\right\vert _{\mu }\ .
\end{eqnarray*
Now the unweighted square function estimate applies to the second sum to giv
\begin{equation*}
\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q^{\prime
}}^{\mu }\left( fb\right) -E_{Q}^{\mu }\left( fb\right) \right\vert
^{2}\left\vert Q^{\prime }\right\vert _{\mu }\lesssim \int \left\vert
fb\right\vert ^{2}d\mu \lesssim \int \left\vert f\right\vert ^{2}d\mu .
\end{equation*
To handle the first sum we write bot
\begin{eqnarray}
E_{Q^{\prime }}^{\mu }\left( fb_{A}-fb\right) &=&E_{Q^{\prime }}^{\mu
}\left( \sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right)
\mathbf{1}_{A^{\prime }}\left( fb_{A}-fb\right) \right) =E_{Q^{\prime
}}^{\mu }\left( \sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left(
A\right) }\mathbb{E}_{A^{\prime }}^{\mu }\left( fb_{A}-fb\right) \right) ,
\label{write both} \\
E_{Q}^{\mu }\left( fb_{A}-fb\right) &=&E_{Q}^{\mu }\left( \sum_{A^{\prime
}\in \mathfrak{C}_{\mathcal{A}}\left( A\right) }\mathbf{1}_{A^{\prime
}}\left( fb_{A}-fb\right) \right) =E_{Q}^{\mu }\left( \sum_{A^{\prime }\in
\mathfrak{C}_{\mathcal{A}}\left( A\right) }\mathbb{E}_{A^{\prime }}^{\mu
}\left( fb_{A}-fb\right) \right) , \notag
\end{eqnarray
and use the unweighted square function estimate on each corona $\mathcal{C
_{A}$, applied to the function $fb_{A}-fb$, to obtai
\begin{eqnarray*}
&&\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc
natural}}\left( Q\right) }\left\vert E_{Q^{\prime }}^{\mu }\left(
fb_{A}-fb\right) -E_{Q}^{\mu }\left( fb_{A}-fb\right) \right\vert
^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&\lesssim &\int_{A}\left\vert \sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A
}\left( A\right) }\mathbb{E}_{A^{\prime }}^{\mu }\left( fb_{A}-fb\right)
\right\vert ^{2}d\mu =\sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left(
A\right) }\left\vert E_{A^{\prime }}^{\mu }\left( fb_{A}-fb\right)
\right\vert ^{2}\left\vert A^{\prime }\right\vert _{\mu }\lesssim
\sum_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left( A\right) }\left(
E_{A^{\prime }}^{\mu }\left\vert f\right\vert \right) ^{2}\left\vert
A^{\prime }\right\vert _{\mu }\ ,
\end{eqnarray*
We can now sum over $A\in \mathcal{A}$ to obtai
\begin{eqnarray*}
&&\sum_{A\in \mathcal{A}}\sum_{Q\in \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q^{\prime
}}^{\mu }\left( fb_{A}-fb\right) -E_{Q}^{\mu }\left( fb_{A}-fb\right)
\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&\lesssim &\sum_{A\in \mathcal{A}}\sum_{A^{\prime }\in \mathfrak{C}_
\mathcal{A}}\left( A\right) }\left( E_{A^{\prime }}^{\mu }\left\vert
f\right\vert \right) ^{2}\left\vert A^{\prime }\right\vert _{\mu }\lesssim
\int \left\vert f\right\vert ^{2}d\mu
\end{eqnarray*
by the Carleson embedding theorem yet again.
\end{proof}
Essentially the same proof gives the lower frame inequality for the dual
martingale difference $\square _{Q}^{\mu ,\mathbf{b}}f=\left( \bigtriangleup
_{Q}^{\mu ,\mathbf{b}}\right) ^{\ast }f$:
\begin{equation}
\sum_{Q\in \mathcal{D}}\left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\ , \label{low frame}
\end{equation
but matters are made slightly simpler by the fact that we do not need the
function $b$ introduced in (\ref{def b}) above because in the definition of
\square _{Q}^{\mu ,\mathbf{b}}f$ the function $b_{Q}$ doesn't multiply $f$,
rather it sits outside the integral where it can be estimated crudely,
leaving the unweighted square function applied to $f$ alone. We leave this
for the reader, pointing out that the general idea of the proof can be found
in the proof given below in Proposition \ref{reverse half Riesz dual} for
the lower weak Riesz inequality for $\square _{Q}^{\mu ,\mathbf{b}}$.
The corresponding upper weak frame inequalities
\begin{eqnarray}
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\lesssim
&\sum_{Q\in \mathcal{D}}\left\Vert \bigtriangleup _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}+\sum_{Q\in \mathcal{D
}\left\Vert \nabla _{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ ,
\label{corr upper} \\
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2} &\lesssim
&\sum_{Q\in \mathcal{D}}\left\Vert \square _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}+\sum_{Q\in \mathcal{D
}\left\Vert \nabla _{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ ,
\notag
\end{eqnarray
are proved by duality using the `Calderon reproducing formulas'
\begin{equation*}
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}=\left( \bigtriangleup _{Q}^{\mu ,\pi
\mathbf{b}}\right) ^{2}+\bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\mathbf{b
}\text{ and }\square _{Q}^{\mu ,\mathbf{b}}=\left( \square _{Q}^{\mu ,\pi
\mathbf{b}}\right) ^{2}+\square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b}}
\end{equation*
introduced in the following subsection. We will actually prove stronger
inequalities below, namely upper weak \emph{Riesz} inequalities.
\subsection{Unbroken projections, broken differences and a Calder\'{o}n
reproducing formula}
Here we assume that $\int_{Q^{\prime }}b_{Q}d\mu \neq 0$, noting that in
applications we will have the stronger inequality $\left\vert
\int_{Q^{\prime }}b_{Q}d\omega \right\vert \gtrsim \mathbf{1}_{Q^{\prime
}}\left\Vert b_{Q}\right\Vert _{L^{\infty }\left( \omega \right) }>0$ due to
the assumed reverse H\"{o}lder control on children (\ref{rev Hol con}).
Define
\begin{equation}
\mathbb{E}_{Q}^{\mu ,\pi ,\mathbf{b}}f\equiv \mathbf{1}_{Q}\frac{1}
\int_{Q}b_{\pi Q}d\mu }\int_{Q}b_{\pi Q}fd\mu \text{ and }\mathbb{F
_{Q}^{\mu ,\pi ,\mathbf{b}}f\equiv \mathbf{1}_{Q}\frac{b_{\pi Q}}
\int_{Q}b_{\pi Q}d\mu }\int_{Q}fd\mu \label{def pi exp}
\end{equation
an
\begin{eqnarray}
\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}f &=&\left[ \sum_{Q^{\prime }\in
\mathfrak{C}\left( Q\right) }\mathbb{E}_{Q^{\prime }}^{\mu ,\pi ,\mathbf{b}}
\right] -\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f=\sum_{Q^{\prime }\in \mathfrak{C
\left( Q\right) }\mathbb{E}_{Q^{\prime }}^{\mu ,b_{Q}}f-\mathbb{E}_{Q}^{\mu
,b_{Q}}f, \label{def pi box} \\
\square _{Q}^{\mu ,\pi ,\mathbf{b}}f &=&\left[ \sum_{Q^{\prime }\in
\mathfrak{C}\left( Q\right) }\mathbb{F}_{Q^{\prime }}^{\mu ,\pi ,\mathbf{b}}
\right] -\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f=\sum_{Q^{\prime }\in \mathfrak{C
\left( Q\right) }\mathbb{F}_{Q^{\prime }}^{\mu ,b_{Q}}f-\mathbb{F}_{Q}^{\mu
,b_{Q}}f, \notag
\end{eqnarray
where on the far right of (\ref{def pi box}) we are using the notation
\mathbb{E}_{Q^{\prime }}^{\mu ,b}=\mathbf{1}_{Q^{\prime }}\frac{1}
\int_{Q^{\prime }}bd\mu }\int_{Q^{\prime }}bfd\mu $ when $b$ is simply a
function, rather than a family of functions $\mathbf{b}$, in order to
specify the testing function $b$ we use if it differs from the function
b_{Q^{\prime }}$ that is selected in the notation $\mathbb{E}_{Q^{\prime
}}^{\mu ,\mathbf{b}}$ when the family $\mathbf{b}$ appears in boldface as an
exponent. Similarly for other pseudoprojections.
For convenience of notation, we set $\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{
}}=\bigtriangleup _{Q}^{\mu ,b}$ and $\square _{Q}^{\mu ,\pi ,\mathbf{b
}=\square _{Q}^{\mu ,b}$ with $b=b_{Q}$. Note tha
\begin{eqnarray*}
\mathbb{E}_{Q}^{\mu ,b}\left( 1\right) &=&\mathbf{1}_{Q}\frac{1}
\int_{Q}bd\mu }\int_{Q}\left( 1\right) bd\mu =\mathbf{1}_{Q}\ , \\
\mathbb{F}_{Q}^{\mu ,b}\left( b\right) &=&\mathbf{1}_{Q}\frac{b}
\int_{Q}bd\mu }\int_{Q}\left( b\right) d\mu =\mathbf{1}_{Q}\ b, \\
\triangle _{Q}^{\mu ,b}\left( 1\right) &=&\left[ \sum_{Q^{\prime \prime }\in
\mathfrak{C}\left( Q\right) }\mathbb{E}_{Q^{\prime \prime }}^{\mu ,b}1\right]
-\mathbb{E}_{Q}^{\mu ,b}1=\left[ \sum_{Q^{\prime \prime }\in \mathfrak{C
\left( Q\right) }\mathbf{1}_{Q^{\prime \prime }}\right] -\mathbf{1}_{Q}=0, \\
\square _{Q}^{\mu ,b}b &=&\left[ \sum_{Q^{\prime \prime }\in \mathfrak{C
\left( Q\right) }\mathbb{F}_{Q^{\prime \prime }}^{\mu ,b}b\right] -\mathbb{F
_{Q}^{\mu ,b}b=\left[ \sum_{Q^{\prime \prime }\in \mathfrak{C}\left(
Q\right) }\mathbf{1}_{Q^{\prime \prime }}b\right] -\mathbf{1}_{Q}b=0.
\end{eqnarray*
The next two lemmas are in \cite[see page 193]{NTV2}.
\begin{lemma}
\label{b orth}For dyadic cubes $R$ and $Q$ we have
\begin{equation*}
\mathbb{E}_{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}=\left\{
\begin{array}{ccc}
0 & \text{ if } & R\supset Q\text{ or }R\cap Q=\emptyset \\
\mathbf{1}_{R}E_{Q_{R}}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f & \text{ if }
& R\subsetneqq
\end{array
\right. .
\end{equation*}
\end{lemma}
\begin{proof}
If $R\supset Q$, then since $\square _{Q}^{\mu ,b}b=0$, we have
\begin{equation*}
\mathbb{E}_{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f=\mathbf{1}_{R}\frac{1}
\int_{R}bd\mu }\int_{R}\left( \bigtriangleup _{Q}^{\mu ,b}f\right) bd\mu
\mathbf{1}_{R}\frac{\left\langle \bigtriangleup _{Q}^{\mu
,b}f,b\right\rangle _{L^{2}\left( \mu \right) }}{\int_{R}bd\mu }=\mathbf{1
_{R}\frac{\left\langle f,\square _{Q}^{\mu ,b}b\right\rangle _{L^{2}\left(
\mu \right) }}{\int_{R}bd\mu }=0.
\end{equation*
On the other hand, if $R\subsetneqq Q$, then $R\subset Q^{\prime }$ for some
$Q^{\prime }\in \mathfrak{C}\left( Q\right) $, and since $\bigtriangleup
_{Q}^{\mu ,b}f$ equals the constant $A=E_{Q_{R}}^{\mu ,b}\left(
\bigtriangleup _{Q}^{\mu ,b}f\right) $ on $Q^{\prime }$, and $E_{I}^{\mu
,b}1=1$ for all cubes $I$, we hav
\begin{equation*}
\mathbb{E}_{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f=\mathbb{E}_{R}^{\mu
,b}A=\mathbf{1}_{R}A=\mathbf{1}_{R}E_{Q_{R}}^{\mu ,b}\left( \bigtriangleup
_{Q}^{\mu ,b}f\right) .
\end{equation*}
\end{proof}
\begin{lemma}
\label{b proj}For dyadic cubes $R$ and $Q$ we hav
\begin{equation*}
\bigtriangleup _{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}=\left\{
\begin{array}{ccc}
\bigtriangleup _{Q}^{\mu ,b} & \text{ if } & R=Q \\
0 & \text{ if } & R\not=
\end{array
\right. .
\end{equation*}
\end{lemma}
\begin{proof}
By the top line in Lemma \ref{b orth}, it suffices to consider the case
R\cap Q\neq \emptyset $. First we suppose that $R\subsetneqq Q$. Then
R\subset Q^{\prime }$ for some $Q^{\prime }\in \mathfrak{C}\left( Q\right)
, and since $\bigtriangleup _{Q}^{\mu ,b}f=A$ is constant on $Q^{\prime }$,
and $\mathbb{E}_{I}^{\mu ,b}1=\mathbf{1}_{I}$ for any cube $I$, we obtain
\begin{equation*}
\bigtriangleup _{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f=\left[
\sum_{R^{\prime }\in \mathfrak{C}\left( R\right) }\mathbb{E}_{R^{\prime
}}^{\mu ,b}\left\{ \bigtriangleup _{Q}^{\mu ,b}f\right\} \right] -\mathbb{E
_{R}^{\mu ,b}\left\{ \bigtriangleup _{Q}^{\mu ,b}f\right\} =\left[
\sum_{R^{\prime }\in \mathfrak{C}\left( R\right) }\mathbf{1}_{R^{\prime }}
\right] -\mathbf{1}_{R}A=0.
\end{equation*}
Next we suppose that $R=Q$ and obtai
\begin{eqnarray*}
&&\bigtriangleup _{Q}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f \\
&=&\left[ \sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E
_{Q^{\prime }}^{\mu ,b}\left\{ \bigtriangleup _{Q}^{\mu ,b}f\right\} \right]
-\mathbb{E}_{Q}^{\mu ,b}\left\{ \bigtriangleup _{Q}^{\mu ,b}f\right\} \\
&=&\left[ \sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E
_{Q^{\prime }}^{\mu ,b}\left\{ \left[ \sum_{Q^{\prime \prime }\in \mathfrak{
}\left( Q\right) }\mathbb{E}_{Q^{\prime \prime }}^{\mu ,b}f\right] -\mathbb{
}_{Q}^{\mu ,b}f\right\} \right] -\mathbb{E}_{Q}^{\mu ,b}\left[ \left\{ \left[
\sum_{Q^{\prime \prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E
_{Q^{\prime \prime }}^{\mu ,b}f\right] -\mathbb{E}_{Q}^{\mu ,b}f\right\}
\right] \\
&=&\left\{ \left[ \sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{
}_{Q^{\prime }}^{\mu ,b}f\right] -\mathbb{E}_{Q}^{\mu ,b}f\right\} -\mathbb{
}_{Q}^{\mu ,b}\left[ \left\{ \left[ \sum_{Q^{\prime \prime }\in \mathfrak{C
\left( Q\right) }\mathbb{E}_{Q^{\prime \prime }}^{\mu ,b}f\right] -\mathbb{E
_{Q}^{\mu ,b}f\right\} \right] \\
&=&\bigtriangleup _{Q}^{\mu ,b}f-\mathbb{E}_{Q}^{\mu ,b}\bigtriangleup
_{Q}^{\mu ,b}f=\bigtriangleup _{Q}^{\mu ,b}f,
\end{eqnarray*
where we have used Lemma \ref{b orth} with $R=Q$ for the final equality.
Finally we suppose that $R\supsetneqq Q$. Then $R_{Q}\supset Q$, and so by
the top line in Lemma \ref{b orth} we have
\begin{equation*}
\bigtriangleup _{R}^{\mu ,b}\bigtriangleup _{Q}^{\mu ,b}f=\sum_{R^{\prime
}\in \mathfrak{C}\left( R\right) }\mathbb{E}_{R^{\prime }}^{\mu
,b}\bigtriangleup _{Q}^{\mu ,b}f-\mathbb{E}_{R}^{\mu ,b}\bigtriangleup
_{Q}^{\mu ,b}f=0-0=0.
\end{equation*}
\end{proof}
Now since we are assuming that $\int_{Q^{\prime }}b_{Q}\neq 0$, we can defin
\begin{eqnarray*}
\bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f
&=&\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f-\bigtriangleup _{Q}^{\mu ,\pi
\mathbf{b}}f \\
&=&\left( \sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E
_{Q^{\prime }}^{\mu ,b_{Q^{\prime }}}f-\mathbb{E}_{Q}^{\mu ,b_{Q}}f\right)
-\left( \sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbb{E
_{Q^{\prime }}^{\mu ,b_{Q}}f-\mathbb{E}_{Q}^{\mu ,b_{Q}}f\right) \\
&=&\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right)
\mathbb{E}_{Q^{\prime }}^{\mu ,b_{Q^{\prime }}}f-\mathbb{E}_{Q^{\prime
}}^{\mu ,b_{Q}}f\ ,
\end{eqnarray*
with a similar definition for $\square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b
}f$. Altogether, with $\square _{Q}^{\mu ,\mathbf{b}}=\left( \bigtriangleup
_{Q}^{\mu ,\mathbf{b}}\right) ^{\ast }$, $\square _{Q}^{\mu ,\pi ,\mathbf{b
}=\left( \bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}\right) ^{\ast }$ and
\square _{Q,\limfunc{broken}}^{\mu ,\mathbf{b}}=\left( \bigtriangleup _{Q
\limfunc{broken}}^{\mu ,\mathbf{b}}\right) ^{\ast }$, we hav
\begin{equation}
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}=\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf
b}}+\bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}\text{ and
\square _{Q}^{\mu ,\mathbf{b}}=\square _{Q}^{\mu ,\pi ,\mathbf{b}}+\square
_{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}} \label{box pi equals}
\end{equation
where $\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}$ and $\square _{Q}^{\mu
,\pi ,\mathbf{b}}$ are projections and
\begin{eqnarray*}
\bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f
&=&\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right)
}\left( \mathbb{E}_{Q^{\prime }}^{\mu ,b_{Q^{\prime }}}f-\mathbb{E
_{Q^{\prime }}^{\mu ,b_{Q}}f\right) , \\
\left\vert \bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b
}f\right\vert &\leq &\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken
}\left( Q\right) }\left( \frac{\left\Vert \mathbf{1}_{Q^{\prime
}}b_{Q}\right\Vert _{L^{\infty }\left( \mu \right) }}{\left\vert
\int_{Q^{\prime }}b_{Q}d\mu \right\vert }+\left\Vert b_{Q^{\prime
}}\right\Vert _{L^{\infty }\left( \mu \right) }\right) \frac{1}{\left\vert
Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime }}\left\vert f\right\vert
d\mu \leq C\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
Q\right) }\frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime
}}\left\vert f\right\vert d\mu , \\
\square _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f &=&\sum_{Q^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right) }\left( \mathbb{F
_{Q^{\prime }}^{\mu ,b_{Q^{\prime }}}f-\mathbb{F}_{Q^{\prime }}^{\mu
,b_{Q}}f\right) , \\
\left\vert \square _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f\right\vert
&\leq &\sum_{Q^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right)
}\left( \frac{\left\Vert \mathbf{1}_{Q^{\prime }}b_{Q}\right\Vert
_{L^{\infty }\left( \mu \right) }}{\left\vert \int_{Q^{\prime }}b_{Q}d\mu
\right\vert }+\left\Vert b_{Q^{\prime }}\right\Vert _{L^{\infty }\left( \mu
\right) }\right) \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }
\int_{Q^{\prime }}\left\vert f\right\vert d\mu \leq C\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{broken}}\left( Q\right) }\frac{1}{\left\vert
Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime }}\left\vert f\right\vert
d\mu ,
\end{eqnarray*
where $C$ depends on both $C_{\mathbf{b}}$ and the constant in the reverse
\"{o}lder condition on children in (\ref{rev Hol con}).
Altogether then we have when $\int_{Q_{i}}b_{Q}d\sigma \neq 0$ for both
children $Q_{i}$ of $Q$, the `Calder\'{o}n reproducing formula',
\begin{equation}
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f=\bigtriangleup _{Q}^{\mu ,\pi
\mathbf{b}}\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}f+\bigtriangleup _{Q
\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f\text{ and }\square _{Q}^{\mu
\mathbf{b}}f=\square _{Q}^{\mu ,\pi ,\mathbf{b}}\square _{Q}^{\mu ,\pi
\mathbf{b}}f+\square _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f,
\label{square of delta}
\end{equation
and the pointwise estimate
\begin{equation*}
\left\vert \bigtriangleup _{Q}^{\mu ,\mathbf{b}}f\right\vert ,\left\vert
\square _{Q}^{\mu ,\mathbf{b}}f\right\vert \leq C_{\delta ,\mathbf{b
}\sum_{Q^{\prime }\in \mathfrak{C}\left( Q\right) }\mathbf{1}_{Q^{\prime
}}\left( \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime
}}\left\vert f\right\vert d\mu +\frac{1}{\left\vert Q\right\vert _{\mu }
\int_{Q}\left\vert f\right\vert d\mu \right) \ ,
\end{equation*
an
\begin{equation}
\left\vert \bigtriangleup _{Q,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b
}f\right\vert \lesssim \left\vert \widehat{\bigtriangledown }_{Q}^{\mu
}f\right\vert \text{ and }\left\vert \square _{Q,\limfunc{broken}}^{\mu ,\pi
,\mathbf{b}}f\right\vert \lesssim \left\vert \widehat{\bigtriangledown
_{Q}^{\mu }f\right\vert , \label{F est}
\end{equation
which follow from the reverse H\"{o}lder property in Lemma \ref{prelim
control of corona} of the children of $Q$
\begin{equation}
\left\Vert \mathbf{1}_{Q_{i}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma
\right) }<\frac{16C_{b_{Q}}}{\delta }\left\vert \frac{1}{\left\vert
Q_{i}\right\vert _{\sigma }}\int_{Q_{i}}b_{Q}d\sigma \right\vert \ ,\ \ \ \
\ Q_{i}\in \mathfrak{C}\left( Q\right) . \label{rev Hold}
\end{equation
Note again that the formulas in (\ref{square of delta}) always hold because
our reverse H\"{o}lder assumption (\ref{rev Hol con}) in the triple corona
construction implies in particular that $\left\Vert \mathbf{1
_{Q_{i}}b_{Q}\right\Vert _{L^{\infty }\left( \sigma \right) }>0$.
\subsubsection{Another modified dual martingale difference}
Define another modified dual martingale difference by
\begin{equation}
\square _{I}^{\sigma ,\flat ,\mathbf{b}}f\equiv \square _{I}^{\sigma
\mathbf{b}}f-\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{b}}f=\left(
\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{natural}}\left( I\right)
\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{b}}f\right) -\mathbb{F
_{I}^{\sigma ,\mathbf{b}}f, \label{flat box}
\end{equation
where we have removed the averages over broken children from $\square
_{I}^{\sigma ,\mathbf{b}}f$, but left the average over $I$ intact. On any
child $I^{\prime }$ of $I$, the function $\square _{I}^{\sigma ,\flat
\mathbf{b}}f$ is thus a constant multiple of $b_{I}$, and so we hav
\begin{eqnarray}
\square _{I}^{\sigma ,\flat ,\mathbf{b}}f &=&b_{I}\sum_{I^{\prime }\in
\mathfrak{C}\left( I\right) }\mathbf{1}_{I^{\prime }}E_{I^{\prime }}^{\sigma
}\left( \frac{1}{b_{I}}\square _{I}^{\sigma ,\flat ,\mathbf{b}}f\right)
=b_{I}\ \sum_{I^{\prime }\in \mathfrak{C}\left( I\right) }\mathbf{1
_{I^{\prime }}E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f\right) ; \label{flat box hat} \\
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f &\equiv &\sum_{I^{\prime
}\in \mathfrak{C}\left( I\right) }\mathbf{1}_{I^{\prime }}\ E_{I^{\prime
}}^{\sigma }\left( \frac{1}{b_{I}}\square _{I}^{\sigma ,\flat ,\mathbf{b
}f\right) , \notag
\end{eqnarray
where we have denoted the constants in question by the expressions
E_{I^{\prime }}^{\sigma }\left( \widehat{\square }_{I}^{\sigma ,\flat
\mathbf{b}}f\right) $, and then defined $\widehat{\square }_{I}^{\sigma
,\flat ,\mathbf{b}}f$ to be the corresponding operator. We record the
precise formula
\begin{equation*}
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f=\sum_{I^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( I\right) }\mathbf{1}_{I^{\prime }}\
\left[ \frac{1}{\int_{I^{\prime }}b_{I}d\mu }\int_{I^{\prime }}fd\mu -\frac{
}{\int_{I}b_{I}d\mu }\int_{I}fd\mu \right] -\sum_{I^{\prime }\in \mathfrak{C
_{\limfunc{broken}}\left( I\right) }\mathbf{1}_{I^{\prime }}\ \left[ \frac{
}{\int_{I}b_{I}d\mu }\int_{I}fd\mu \right] .
\end{equation*
Thus for $I\in \mathcal{C}_{A}$ we have
\begin{equation}
\square _{I}^{\sigma ,\flat ,\mathbf{b}}f=b_{A}\sum_{I^{\prime }\in
\mathfrak{C}\left( I\right) }\mathbf{1}_{I^{\prime }}E_{I^{\prime }}^{\sigma
}\left( \widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) =b_{A
\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f, \label{factor b_A}
\end{equation
where the averages $E_{I^{\prime }}^{\sigma }\left( \widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) $ satisfy the following telescoping
property for all $K\in \left( \mathcal{C}_{A}\setminus \left\{ A\right\}
\right) \cup \left( \bigcup_{A^{\prime }\in \mathfrak{C}_{\mathcal{A}}\left(
A\right) }A^{\prime }\right) $ and $L\in \mathcal{C}_{A}$ with $K\subset L$
\begin{equation}
\sum_{I:\ \pi K\subset I\subset L}E_{I_{K}}^{\sigma }\left( \widehat{\square
}_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) =\left\{
\begin{array}{ccc}
-E_{L}^{\sigma }\widehat{\mathbb{F}}_{L}^{\sigma }f & \text{ if } & K\in
\mathfrak{C}_{\mathcal{A}}\left( A\right) \\
E_{K}^{\sigma }\widehat{\mathbb{F}}_{K}^{\sigma }f-E_{L}^{\sigma }\widehat
\mathbb{F}}_{L}^{\sigma }f & \text{ if } & K\in \mathcal{C}_{A
\end{array
\right. , \label{telescoping}
\end{equation
where $\widehat{\mathbb{F}}_{K}^{\sigma }$ is defined in (\ref{F hat})
above. Indeed, recalling that $I_{K}$ denotes the child of $I$ that contains
$K$, this is evident if we writ
\begin{equation*}
\mathbf{1}_{K}\ b_{A}E_{I_{K}}^{\sigma }\left( \widehat{\square
_{I}^{\sigma ,\flat ,\mathbf{b}}f\right) =\mathbf{1}_{K}\ \square
_{I}^{\sigma ,\flat ,\mathbf{b}}f=\mathbf{1}_{K}\ \left( \square
_{I}^{\sigma ,\mathbf{b}}f-\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broke
}}\left( I\right) }\mathbf{1}_{I^{\prime }}\mathbb{F}_{I^{\prime }}^{\sigma
\mathbf{b}}f\right) ,
\end{equation*
and use the telescoping properties of $\left\{ \square _{I}^{\sigma ,\mathbf
b}}\right\} _{I\in \mathcal{D}}$, together with the fact that the set
\begin{equation*}
\left\{ I^{\prime }:I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left(
I\right) \text{ for some }I\in \mathcal{C}_{A}\right\} =\mathfrak{C}_
\mathcal{A}}\left( A\right)
\end{equation*
is pairwise disjoint and lies beneath the corona $\mathcal{C}_{A}$. Of
course we can similarly define $\bigtriangleup _{I}^{\sigma ,\flat ,\mathbf{
}}$.
Finally, in analogy with the broken differences $\bigtriangleup _{Q,\limfunc
broken}}^{\mu ,\pi ,\mathbf{b}}$ and $\square _{Q,\limfunc{broken}}^{\mu
,\pi ,\mathbf{b}}$ introduced above, we defin
\begin{equation}
\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\flat ,\mathbf{b}}f\equiv
\sum_{I^{\prime }\in \mathfrak{C}_{\limfunc{broken}}\left( I\right) }\mathbb
E}_{I^{\prime }}^{\sigma ,\mathbf{b}}f\text{ and }\square _{I,\limfunc{broke
}}^{\mu ,\flat ,\mathbf{b}}f\equiv \sum_{I^{\prime }\in \mathfrak{C}_
\limfunc{broken}}\left( I\right) }\mathbb{F}_{I^{\prime }}^{\sigma ,\mathbf{
}}f\ , \label{def flat broken}
\end{equation
so tha
\begin{equation}
\bigtriangleup _{I}^{\mu ,\mathbf{b}}=\bigtriangleup _{I}^{\mu ,\flat
\mathbf{b}}+\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\flat ,\mathbf{b}
\text{ and }\square _{I}^{\mu ,\mathbf{b}}=\square _{I}^{\mu ,\flat ,\mathbf
b}}+\square _{I,\limfunc{broken}}^{\mu ,\flat ,\mathbf{b}}\ .
\label{flat broken}
\end{equation
These modified differences and the identities (\ref{factor b_A}) and (\re
{telescoping}) play a useful role in the analysis of the nearby and
paraproduct forms.
\subsection{Weak Riesz inequalities\label{two sided Riesz}}
We begin with a strengthening of the upper frame inequality for dual
martingale pseudoprojections
\begin{equation*}
\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \sum_{Q\in
\mathcal{D}}\left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}+\sum_{Q\in \mathcal{D}}\left\Vert \widehat
\bigtriangledown }_{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2},
\end{equation*
and then proceed to consider the corresponding lower inequalities and
martingale versions as well. We refer to these strengthened inequalities as
weak Riesz inequalities for the following reason. A family of
pseudoprojections $\left\{ \Psi _{Q}^{\mu ,\mathbf{b}}f\right\} _{Q\in
\mathcal{D}}$ is said to be a \emph{Riesz basis} for $L^{2}\left( \mu
\right) $ if for all subsets $\mathcal{B}\subset \mathcal{D}$ of the dyadic
grid we hav
\begin{equation*}
\sum_{Q\in \mathcal{B}}\left\Vert \Psi _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert \sum_{Q\in \mathcal{B
}\Psi _{Q}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right)
}^{2}\lesssim \sum_{Q\in \mathcal{B}}\left\Vert \Psi _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ ,\ \ \ \ \ f\in L^{2}\left(
\mu \right) .
\end{equation*
We refer to the left (respectively right) hand inequality above as the lower
(respectively upper) Riesz inequality. The families $\left\{ \square
_{Q}^{\mu ,\mathbf{b}}f\right\} _{Q\in \mathcal{D}}$ and $\left\{
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f\right\} _{Q\in \mathcal{D}}$ can
\emph{fail} to be a Riesz basis for $L^{2}\left( \mu \right) $, but in the
case $p=\infty $, we show that each of these families is a Riesz basis in a
certain weak sense,\ involving Carleson averaging operators, that is made
precise below.
\begin{definition}
\label{Psi op}For any subset $\mathcal{B}$ of the grid $\mathcal{D}$, and
any sequence of real numbers $\mathbf{\lambda }=\left\{ \lambda _{I}\right\}
_{I\in \mathcal{B}}$, define the linear operato
\begin{equation*}
\Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}f\equiv \sum_{I\in
\mathcal{B}}\lambda _{I}\square _{I}^{\mu ,\mathbf{b}}f,
\end{equation*
which, with abuse of notation, we will refer to as a `pseudoprojection'. In
the event that all $\lambda _{I}=1$, we denote the operator by $\Psi _
\mathcal{B}}^{\mu ,\mathbf{b}}$, and note that, despite the fact it is a sum
of dual martingale averages, it is typically \textbf{not} a projection on
L^{2}\left( \mu \right) $.
\end{definition}
Note that the failure of $\square _{I}^{\mu ,\mathbf{b}}$ to be a projection
in general is what motivated the introduction of the projections $\square
_{I}^{\mu ,\pi ,\mathbf{b}}$ above. These projections have already played a
significant role in the proof of the Monotonicity Lemma earlier, and will
continue to play a role in other `duality' situations below.
\subsubsection{An upper weak Riesz inequality}
\begin{proposition}
\label{half Riesz}Suppose that $\mathbf{b}$ is an $\infty $-weakly $\mu
-controlled accretive family on a grid $\mathcal{D}$. Then we have the
following `$\square _{I}^{\mu ,\mathbf{b}}$-upper weak Riesz' inequality
\begin{eqnarray*}
&&\left\Vert \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq C\left\Vert \mathbf
\lambda }\right\Vert _{\infty }^{2}\left( \sum_{I\in \mathcal{B}}\left\Vert
\square _{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right)
}^{2}+\sum_{I\in \mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) \\
&&\ \ \ \ \ \text{for all }f\in L^{2}\left( \mu \right) \text{ and all
subsets }\mathcal{B}\text{ of the grid }\mathcal{D}\text{ and all sequences
\mathbf{\lambda },
\end{eqnarray*
where $\left\Vert \mathbf{\lambda }\right\Vert _{\infty }\equiv \sup_{I\in
\mathcal{B}}\left\vert \lambda _{I}\right\vert $ and the positive constant
C $ is independent of the subset $\mathcal{B}$. In particular, the
pseudoprojection $\Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}$
is a bounded linear operator on $L^{2}\left( \mu \right) $ if $\left\Vert
\mathbf{\lambda }\right\Vert _{\infty }<\infty $
\begin{equation}
\left\Vert \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq C\left\Vert \mathbf
\lambda }\right\Vert _{\infty }^{2}\left\Vert f\right\Vert _{L^{2}\left( \mu
\right) }^{2}\ . \label{Psi bound}
\end{equation}
\end{proposition}
\begin{proof}
We may suppose that the subset $\mathcal{B}$ is finite provided the
estimates we get are independent of the size of $\mathcal{B}$. Now let
g=\Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}f=\sum_{I\in
\mathcal{B}}\lambda _{I}\square _{I}^{\mu ,\mathbf{b}}f$. Then from (\re
{square of delta}) we hav
\begin{eqnarray*}
\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }^{2} &=&\int \left(
\sum_{I\in \mathcal{B}}\lambda _{I}\square _{I}^{\mu ,\mathbf{b}}f\right) g\
d\mu =\int \left( \sum_{I\in \mathcal{B}}\lambda _{I}\left[ \square
_{I}^{\mu ,\pi ,\mathbf{b}}\square _{I}^{\mu ,\pi ,\mathbf{b}}f+\square _{I
\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f\right] \right) g\ d\mu \\
&=&\sum_{I\in \mathcal{B}}\lambda _{I}\int \left( \square _{I}^{\mu ,\pi
\mathbf{b}}f\right) \left( \square _{I}^{\mu ,\pi ,\mathbf{b}}\right) ^{\ast
}g\ d\mu +\sum_{I\in \mathcal{B}}\lambda _{I}\int \left( \square _{I
\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f\right) g\ d\mu \\
&\lesssim &\left\Vert \mathbf{\lambda }\right\Vert _{\infty }\left(
\sum_{I\in \mathcal{B}}\left\Vert \square _{I}^{\mu ,\pi ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}}\left(
\sum_{I\in \mathcal{B}}\left\Vert \bigtriangleup _{I}^{\mu ,\pi ,\mathbf{b
}g\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}} \\
&&+\left\Vert \mathbf{\lambda }\right\Vert _{\infty }\left( \sum_{I\in
\mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu }f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}}\left( \sum_{I\in
\mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu }g\right\Vert
_{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}},
\end{eqnarray*
where we have used (\ref{F est}) in the last line. Now using $\bigtriangleup
_{I}^{\mu ,\pi ,\mathbf{b}}=\bigtriangleup _{I}^{\mu ,\mathbf{b
}-\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}$ and
\left\vert \bigtriangleup _{I,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b
}g\right\vert \leq C_{\mathbf{b}}\widehat{\nabla }_{I}^{\mu }\left\vert
g\right\vert $, and $\square _{I}^{\mu ,\pi ,\mathbf{b}}=\square _{I}^{\mu
\mathbf{b}}-\square _{I,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}$, together
with the lower frame inequalities in Proposition \ref{dual frame}\ (note
that we do \emph{not} use lower \emph{Riesz} inequalities for
\bigtriangleup $!
\begin{eqnarray*}
\sum_{I\in \mathcal{B}}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b
}g\right\Vert _{L^{2}\left( \mu \right) }^{2} &\leq &\sum_{I\in \mathcal{D
}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b}}g\right\Vert _{L^{2}\left(
\mu \right) }^{2}\lesssim \left\Vert g\right\Vert _{L^{2}\left( \mu \right)
}^{2}\ , \\
\sum_{I\in \mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu
}\left\vert g\right\vert \right\Vert _{L^{2}\left( \mu \right) }^{2} &\leq
&\sum_{I\in \mathcal{D}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu
}\left\vert g\right\vert \right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim
\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }^{2}\ ,
\end{eqnarray*
we obtain
\begin{equation*}
\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
\mathbf{\lambda }\right\Vert _{\infty }\left( \sum_{I\in \mathcal{B
}\left\Vert \square _{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu
\right) }^{2}+\sum_{I\in \mathcal{B}}\left\Vert \widehat{\bigtriangledown
_{I}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2
}\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }\ ,
\end{equation*
which gives the desired upper weak Riesz inequality upon dividing through by
the finite positive number $\left\Vert g\right\Vert _{L^{2}\left( \mu
\right) }$, and then squaring the resulting inequality.
\end{proof}
An analogous argument yields the next proposition. Recall that $\Psi _
\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}f\equiv \sum_{I\in \mathcal{
}}\lambda _{I}\square _{I}^{\mu ,\mathbf{b}}f$ and so
\begin{equation*}
\left( \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}\right) ^{\ast
}f\equiv \sum_{I\in \mathcal{B}}\lambda _{I}\left( \square _{I}^{\mu
\mathbf{b}}\right) ^{\ast }f=\sum_{I\in \mathcal{B}}\lambda
_{I}\bigtriangleup _{I}^{\mu ,\mathbf{b}}f.
\end{equation*}
\begin{proposition}
\label{half Riesz dual}Suppose that $\mathbf{b}$ is an $\infty $-weakly $\mu
$-controlled accretive family. Then we have the `$\bigtriangleup _{I}^{\mu
\mathbf{b}}$-upper weak Riesz' inequality
\begin{eqnarray*}
&&\left\Vert \left( \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b
}\right) ^{\ast }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq
C\left\Vert \mathbf{\lambda }\right\Vert _{\infty }^{2}\left( \sum_{I\in
\mathcal{B}}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}+\sum_{I\in \mathcal{B}}\left\Vert \widehat
\bigtriangledown }_{I}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}\right) , \\
&&\ \ \ \ \ \text{for all }f\in L^{2}\left( \mu \right) \text{ and all
subsets }\mathcal{B}\text{ of the grid }\mathcal{D}\text{ and all sequences
\mathbf{\lambda },
\end{eqnarray*
and where the positive constant $C$ is independent of the subset $\mathcal{B}
$ and the sequence $\mathbf{\lambda }$. In particular, the pseudoprojection
\left( \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}\right) ^{\ast
}$ is a bounded linear operator on $L^{2}\left( \mu \right) $
\begin{equation}
\left\Vert \left( \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b
}\right) ^{\ast }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq
C\left\Vert \mathbf{\lambda }\right\Vert _{\infty }^{2}\left\Vert
f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ . \label{Psi * bound}
\end{equation}
\end{proposition}
\begin{proof}
We may again suppose that the subset $\mathcal{B}$ is finite provided the
estimates we get are independent of the size of $\mathcal{B}$. Now let
g=\sum_{I\in \mathcal{B}}\lambda _{I}\bigtriangleup _{I}^{\mu ,\mathbf{b}}f
. Recall from (\ref{square of delta}) that $\bigtriangleup _{Q}^{\mu
\mathbf{b}}=\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}\bigtriangleup
_{Q}^{\mu ,\pi ,\mathbf{b}}+\bigtriangleup _{Q,\limfunc{broken}}^{\mu
\mathbf{b}}$, and hence we hav
\begin{eqnarray*}
\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }^{2} &=&\int \left(
\sum_{I\in \mathcal{B}}\lambda _{I}\bigtriangleup _{I}^{\mu ,\mathbf{b
}f\right) g\ d\mu =\int \left( \sum_{I\in \mathcal{B}}\lambda _{I}\left[
\bigtriangleup _{I}^{\mu ,\pi ,\mathbf{b}}\bigtriangleup _{I}^{\mu ,\pi
\mathbf{b}}f+\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}
\right] \right) g\ d\mu \\
&=&\sum_{I\in \mathcal{B}}\lambda _{I}\int \left( \bigtriangleup _{I}^{\mu
,\pi ,\mathbf{b}}f\right) \left( \bigtriangleup _{I}^{\mu ,\pi ,\mathbf{b
}\right) ^{\ast }g\ d\mu +\sum_{I\in \mathcal{B}}\lambda _{I}\int \left(
\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\pi ,\mathbf{b}}f\right) g\ d\mu
\\
&\lesssim &\left\Vert \mathbf{\lambda }\right\Vert _{\infty }\left(
\sum_{I\in \mathcal{B}}\left\Vert \bigtriangleup _{I}^{\mu ,\pi ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}}\left(
\sum_{I\in \mathcal{B}}\left\Vert \square _{I}^{\mu ,\pi ,\mathbf{b
}g\right\Vert _{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}} \\
&&+\left\Vert \mathbf{\lambda }\right\Vert _{\infty }\left( \sum_{I\in
\mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{I}^{\mu }f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\right) ^{\frac{1}{2}}\left( \sum_{I\in
\mathcal{B}}\left\Vert \nabla _{I}^{\mu }g\right\Vert _{L^{2}\left( \mu
\right) }^{2}\right) ^{\frac{1}{2}}.
\end{eqnarray*
Now we continue as in the proof of Propostion \ref{half Riesz}, but using
instead the lower frame inequality (\ref{low frame}), to obtai
\begin{equation*}
\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
\mathbf{\lambda }\right\Vert _{\infty }\left( \sum_{I\in \mathcal{B
}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left(
\mu \right) }^{2}+\sum_{I\in \mathcal{B}}\left\Vert \widehat
\bigtriangledown }_{I}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}\right) \left\Vert g\right\Vert _{L^{2}\left( \mu \right) }\ ,
\end{equation*
which completes the proof of Propostion \ref{half Riesz dual} upon dividing
through by $\left\Vert g\right\Vert _{L^{2}\left( \mu \right) }$ and
squaring.
\end{proof}
\begin{remark}
The boundedness of the pseudoprojections $\Psi _{\mathcal{B}}^{\mu ,\mathbf{
}}$ and $\left( \Psi _{\mathcal{B}}^{\mu ,\mathbf{b}}\right) ^{\ast }$ on
L^{2}\left( \mu \right) $ (where the absence of the sequence $\mathbf
\lambda }$ in the subscript implies all $\lambda _{I}=1$) given by (\ref{Psi
bound}) and (\ref{Psi * bound}), can fail for a $2$-weakly $\mu $-controlled
accretive family on a grid $\mathcal{D}$. Indeed, if (\ref{Psi * bound})
holds, then for a $2$-weakly $\mu $-controlled accretive family
\begin{equation*}
\left\Vert \sum_{I\in \mathcal{B}_{+}}\bigtriangleup _{I}^{\mu ,\mathbf{b
}f-\sum_{I\in \mathcal{B}_{-}}\bigtriangleup _{I}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq 2\left\Vert \sum_{I\in
\mathcal{B}_{+}}\bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}+2\left\Vert \sum_{I\in \mathcal{B
_{-}}\bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu
\right) }^{2}\leq C\left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2}
\end{equation*
holds for all decompositions of $\mathcal{B}$ into a disjoint union
\mathcal{B=B}_{+}\overset{\cdot }{\cup }\mathcal{B}_{-}$. The
\begin{equation*}
\sum_{I\in \mathcal{B}}\left\Vert \bigtriangleup _{I}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}=\mathbb{E}_{\pm }\left\Vert
\sum_{I\in \mathcal{B}}\pm \bigtriangleup _{I}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\leq \mathbb{E}_{\pm }C\left\Vert
f\right\Vert _{L^{2}\left( \mu \right) }^{2}=C\left\Vert f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\ ,
\end{equation*
which contradicts the example of Hyt\"{o}nen and Martikainen in \cite
Section 3.9]{HyMa} if $p=2$. Thus in order to prove a two weight local $Tb$
theorem for $p=2$, one \textbf{cannot} appeal in general to the boundedness
of pseudoprojections $\left( \Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu
\mathbf{b}}\right) ^{\ast }=\sum_{I\in \mathcal{B}}\lambda
_{I}\bigtriangleup _{I}^{\mu ,\mathbf{b}}$ on $L^{2}\left( \mu \right) $
even when $\lambda _{I}=1$ for all $I\in \mathcal{B}$.
\end{remark}
\subsubsection{A lower weak Riesz inequality}
The next proposition also assumes an $\infty $-weakly $\mu $-controlled
accretive family on a grid $\mathcal{D}$. For a subset $\mathcal{B}$ of a
dyadic grid $\mathcal{D}$, let
\begin{equation}
\mathsf{P}_{\mathcal{B}}^{\mu }f=\sum_{Q\in \mathcal{B}}\bigtriangleup
_{Q}^{\mu }f \label{Haar proj}
\end{equation
denote the orthogonal projection of $f$ onto the closed linear span of the
collection $\left\{ \bigtriangleup _{Q}^{\mu }\right\} _{Q\in \mathcal{B}}$
of Haar projections $\bigtriangleup _{Q}^{\mu }$ with $Q\in \mathcal{B}$. We
obtain an appropriate form of a weak lower Riesz inequality for the dual
martingale differences $\square _{Q}^{\mu ,\mathbf{b}}$ since for these
operators we no longer need the disruptive device of introducing the
function $b$ in (\ref{def b}) above.
\begin{proposition}
\label{reverse half Riesz dual}Suppose that $\mathbf{b}$ is an $\infty
-weakly $\mu $-controlled accretive family. Then we have a `weak lower
Riesz' inequality for dual martingale differences
\begin{eqnarray*}
&&\sum_{Q\in \mathcal{B}}\left\Vert \square _{Q}^{\mu ,\mathbf{b
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\leq C\left( \left\Vert \mathsf
P}_{\mathcal{B}}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}+\sum_{Q\in \mathcal{B}}\left\Vert \widehat{\bigtriangledown }_{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}+\sum_{Q\in \mathcal{B}}\gamma
_{Q}\left\vert E_{Q}^{\mu }f\right\vert ^{2}\right) , \\
&&\text{for all }f\in L^{2}\left( \mu \right) \text{ and all subsets
\mathcal{B}\text{ of the grid }\mathcal{D}\text{, and where }\left\Vert
\mathsf{P}_{\mathcal{B}}^{\mu }f\right\Vert _{L^{2}\left( \mu \right)
}^{2}=\sum_{Q\in \mathcal{B}}\left\Vert \bigtriangleup _{Q}^{\mu
}f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ .
\end{eqnarray*
and where the positive constant $C$ depends only on the accretivity
constants, but is \emph{independent} of the subset $\mathcal{B}$ and the
testing family $\mathbf{b}$. Here the coefficients $\left\{ \gamma
_{Q}\right\} _{Q\in \mathcal{D}}$ form a Carleson sequence indexed by
\mathcal{D}$, i.e
\begin{equation*}
\sum_{Q\in \mathcal{D}:\ Q\subset P}\gamma _{Q}\leq C\left\vert P\right\vert
_{\mu }\ ,\ \ \ \ \ \text{for all }P\in \mathcal{D}.
\end{equation*}
\end{proposition}
The third term on the right hand side above is additive in $\mathcal{B}$
and, by the Carleson embedding theorem, satisfie
\begin{equation*}
\sum_{Q\in \mathcal{D}}\gamma _{Q}\left\vert E_{Q}^{\mu }f\right\vert
^{2}\lesssim \left\Vert f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ .
\end{equation*}
\begin{proof}
The main modifications in the proof of Proposition \ref{dual frame} that are
needed here, are that we use $\square _{Q}^{\mu ,\mathbf{b}}$ in place of
\bigtriangleup _{Q}^{\mu ,\mathbf{b}}$, which results in the testing
functions $b_{Q}$ appearing outside the integrals rather than inside the
integrals, and that we restrict the sums over $Q$ to $\mathcal{B}$, which
results in the presence of the term $\sum_{Q\in \mathcal{B}}\gamma
_{Q}\left\vert E_{Q}^{\mu }f\right\vert ^{2}$ on the right hand side above.
Since we are working with $\square _{Q}^{\mu ,\mathbf{b}}$ we will not need
the extra complications arising from the introduction of the function $b$ in
(\ref{def b}). With these modifications in mind, we now describe the
estimates we obtain for the terms analogous to $II_{A}$, $III_{A}$ and
IV_{A}$ in the proof of Proposition \ref{dual frame}. Given $A\in \mathcal{A}
$, we begin\ wit
\begin{eqnarray*}
\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\left\Vert \square _{Q}^{\mu
\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right) }^{2} &=&\sum_{Q\in
\mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C}\left(
Q\right) }\int_{Q^{\prime }}\left\vert \mathbb{F}_{Q^{\prime }}^{\mu
\mathbf{b}}f\left( x\right) -\mathbb{F}_{Q}^{\mu ,\mathbf{b}}f\left(
x\right) \right\vert ^{2}d\mu \left( x\right) \\
&=&\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak
C}_{\limfunc{natural}}\left( Q\right) }\left\vert \frac{b_{A}\int_{Q^{\prime
}}fd\mu }{\int_{Q^{\prime }}b_{A}d\mu }-\frac{b_{A}\int_{Q}fd\mu }
\int_{Q}b_{A}d\mu }\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu }
\\
&&+\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak
C}_{\limfunc{broken}}\left( Q\right) }\left\vert \frac{b_{Q^{\prime
}}\int_{Q^{\prime }}fd\mu }{\int_{Q^{\prime }}b_{Q^{\prime }}d\mu }-\frac
b_{A}\int_{Q}fd\mu }{\int_{Q}b_{A}d\mu }\right\vert ^{2}\left\vert Q^{\prime
}\right\vert _{\mu } \\
&\equiv &I_{A}+II_{A}.
\end{eqnarray*
To estimate term $II_{A}$ we writ
\begin{eqnarray}
&&\left\vert \frac{b_{Q^{\prime }}\int_{Q^{\prime }}fd\mu }{\int_{Q^{\prime
}}b_{Q^{\prime }}d\mu }-\frac{b_{A}\int_{Q}fd\mu }{\int_{Q}b_{A}d\mu
\right\vert \label{analogue'} \\
&=&\frac{\left\vert \left( \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu
}b_{Q^{\prime }}\int_{Q^{\prime }}fd\mu \right) \left( \frac{1}{\left\vert
Q\right\vert _{\mu }}\int_{Q}b_{A}d\mu \right) -\left( \frac{1}{\left\vert
Q\right\vert _{\mu }}b_{A}\int_{Q}fd\mu \right) \left( \frac{1}{\left\vert
Q^{\prime }\right\vert _{\mu }}\int_{Q^{\prime }}b_{Q^{\prime }}d\mu \right)
\right\vert }{\left\vert \frac{1}{\left\vert Q^{\prime }\right\vert _{\mu }
\int_{Q^{\prime }}b_{Q^{\prime }}d\mu \right\vert \left\vert \frac{1}
\left\vert Q\right\vert _{\mu }}\int_{Q}b_{A}d\mu \right\vert } \notag \\
&\lesssim &\left\vert b_{Q^{\prime }}\left( E_{Q^{\prime }}^{\mu }f\right)
\left( E_{Q}^{\mu }b_{A}\right) -b_{A}\left( E_{Q}^{\mu }f\right) \left(
E_{Q^{\prime }}^{\mu }b_{Q^{\prime }}\right) \right\vert \notag \\
&=&\left\vert b_{Q^{\prime }}\left( E_{Q^{\prime }}^{\mu }f\right) \left(
E_{Q}^{\mu }b_{A}\right) -b_{A}\left( E_{Q^{\prime }}^{\mu }f\right) \left(
E_{Q^{\prime }}^{\mu }b_{Q^{\prime }}\right) -\left[ b_{A}\left( E_{Q}^{\mu
}f\right) -b_{A}\left( E_{Q^{\prime }}^{\mu }f\right) \right] \left(
E_{Q^{\prime }}^{\mu }b_{Q^{\prime }}\right) \right\vert \notag \\
&\lesssim &\left\vert b_{Q^{\prime }}\left( E_{Q^{\prime }}^{\mu }f\right)
\right\vert +\left\vert b_{A}\left( E_{Q^{\prime }}^{\mu }f\right)
\right\vert +\left\vert b_{A}\left( E_{Q}^{\mu }f\right) -b_{A}\left(
E_{Q^{\prime }}^{\mu }f\right) \right\vert , \notag \\
&\lesssim &\left\vert E_{Q^{\prime }}^{\mu }f\right\vert +\left\vert
E_{Q^{\prime }}^{\mu }f\right\vert +\left\vert \left( E_{Q}^{\mu }f\right)
-\left( E_{Q^{\prime }}^{\mu }f\right) \right\vert , \notag
\end{eqnarray
and then
\begin{equation*}
II_{A}\lesssim \sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime
}\in \mathfrak{C}_{\limfunc{broken}}\left( Q\right) }\left( \left\vert
E_{Q^{\prime }}^{\mu }f\right\vert ^{2}+\left\vert E_{Q}^{\mu }f\right\vert
^{2}\right) \left\vert Q^{\prime }\right\vert _{\mu }\approx \sum_{Q\in
\mathcal{B}\cap \mathcal{C}_{A}}\left\Vert \widehat{\bigtriangledown
_{Q}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\ .
\end{equation*
Now we writ
\begin{eqnarray*}
I_{A} &=&\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert \frac
b_{A}\int_{Q^{\prime }}fd\mu }{\int_{Q^{\prime }}b_{A}d\mu }-\frac
b_{A}\int_{Q}fd\mu }{\int_{Q}b_{A}d\mu }\right\vert ^{2}\left\vert Q^{\prime
}\right\vert _{\mu } \\
&=&\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak
C}_{\limfunc{natural}}\left( Q\right) }\left\vert \frac{\left(
b_{A}\int_{Q^{\prime }}fd\mu \right) \left( \int_{Q}b_{A}d\mu \right)
-\left( \int_{Q^{\prime }}b_{A}d\mu \right) \left( b_{A}\int_{Q}fd\mu
\right) }{\left( \int_{Q^{\prime }}b_{A}d\mu \right) \left(
\int_{Q}b_{A}d\mu \right) }\right\vert ^{2}\left\vert Q^{\prime }\right\vert
_{\mu } \\
&\lesssim &\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in
\mathfrak{C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu
}b_{A}\right\vert ^{2}\left\vert E_{Q^{\prime }}^{\mu }f-E_{Q}^{\mu
}f\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&&+\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak
C}_{\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu
}b_{A}-E_{Q^{\prime }}^{\mu }b_{A}\right\vert ^{2}\left\vert E_{Q}^{\mu
}f\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu } \\
&\equiv &III_{A}+IV_{A}.
\end{eqnarray*
Then we use the following stronger form of an inequality used in the proof
of the unweighted square function estimate in Lemma \ref{unweighted square},
namely
\begin{equation}
\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\sum_{Q^{\prime }\in \mathfrak{C
_{\limfunc{natural}}\left( Q\right) }\left\vert Q^{\prime }\right\vert _{\mu
}\left\vert E_{Q^{\prime }}^{\mu }h-E_{Q}^{\mu }h\right\vert ^{2}\lesssim
\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\left\Vert \bigtriangleup
_{Q}^{\mu }h\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
\mathsf{P}_{\mathcal{B}\cap \mathcal{C}_{A}}^{\mu }h\right\Vert
_{L^{2}\left( \mu \right) }^{2}, \label{stronger unweighted}
\end{equation
to dominate term $III_{A}$ b
\begin{equation*}
III_{A}\lesssim \left\Vert b_{A}\mathsf{P}_{\mathcal{B}\cap \mathcal{C
_{A}}^{\mu }f\right\Vert _{L^{2}\left( \mu \right) }^{2}\lesssim \left\Vert
\mathsf{P}_{\mathcal{B}\cap \mathcal{C}_{A}}^{\mu }f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\ ,
\end{equation*
and we write term $IV_{A}$ a
\begin{eqnarray*}
IV_{A} &=&\sum_{Q\in \mathcal{B}\cap \mathcal{C}_{A}}\gamma _{Q}\left\vert
E_{Q}^{\mu }f\right\vert ^{2}\ , \\
\text{where }\gamma _{Q} &\equiv &\sum_{Q^{\prime }\in \mathfrak{C}_
\limfunc{natural}}\left( Q\right) }\left\vert E_{Q}^{\mu }b_{A}-E_{Q^{\prime
}}^{\mu }b_{A}\right\vert ^{2}\left\vert Q^{\prime }\right\vert _{\mu },\ \
\ \ \ \text{for }Q\in \mathcal{C}_{A},A\in \mathcal{A\ }.
\end{eqnarray*
Now we sum over $A\in \mathcal{A}$ to obtai
\begin{eqnarray*}
\sum_{Q\in \mathcal{B}}\left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2} &=&\sum_{A\in \mathcal{A}}\sum_{Q\in
\mathcal{B\cap C}_{A}}\left\Vert \square _{Q}^{\mu ,\mathbf{b}}f\right\Vert
_{L^{2}\left( \mu \right) }^{2}\lesssim \sum_{A\in \mathcal{A}}\left(
II_{A}+III_{A}+IV_{A}\right) \\
&\lesssim &\sum_{Q\in \mathcal{B}}\left\Vert \widehat{\bigtriangledown
_{Q}^{\mu ,\mathbf{b}}f\right\Vert _{L^{2}\left( \mu \right)
}^{2}+\left\Vert \mathsf{P}_{\mathcal{B}}^{\mu }f\right\Vert _{L^{2}\left(
\mu \right) }^{2}+\sum_{Q\in \mathcal{B}}\gamma _{Q}\left\vert E_{Q}^{\mu
}f\right\vert ^{2}\ ,
\end{eqnarray*
where the sequence $\left\{ \gamma _{Q}\right\} _{Q\in \mathcal{D}}$
satisfies the Carleson condition by (\ref{Car cond}) in the proof of
Proposition \ref{dual frame}.
\end{proof}
\begin{remark}
We are unable to obtain a corresponding lower weak Riesz inequality for the
martingale differences $\bigtriangleup _{Q}^{\mu ,\mathbf{b}}$ due to the
need for introducing the function $b$ in (\ref{def b}) as in the proof of
Proposition \ref{dual frame}, which does not interact well with $\mathcal{B}$
- see the argument surrounding (\ref{write both}) in the proof of
Proposition \ref{dual frame}. However, lower weak Riesz inequalities for the
martingale differences $\bigtriangleup _{Q}^{\mu ,\mathbf{b}}$ are not
needed in this paper - in fact, only upper weak Riesz inequalities are
needed for both $\square _{Q}^{\mu ,\mathbf{b}}$ and $\bigtriangleup
_{Q}^{\mu ,\mathbf{b}}$.
\end{remark}
\section{Appendix B:\ Control of functional energy\label{equiv}}
Now we arrive at one of the main propositions used in the proof of our
theorem. This result is proved \emph{independently} of the main theorem, and
only using the results on dual martingale differences established in the
previous appendix. The organization of the proof is almost identical to that
of the corresponding result in\ \cite[pages 128-151]{SaShUr7}, together with
the modifications in \cite[pages 348-360]{SaShUr9} to accommodate common
point masses, but we repeat the organization here with modifications
required for the use of two independent grids, and the appearance of weak
goodness entering through the intervals $J^{\maltese }$. Recall that the
functional energy constant $\mathfrak{F}_{\alpha }=\mathfrak{F}_{\alpha }^
\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G}\right) $ in (\re
{e.funcEnergy n}), $0\leq \alpha <n$, namely the best constant in the
inequality (see (\ref{def M_r-deep}) below for the definition of $\mathcal{W
\left( F\right) $),
\begin{equation}
\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) }\left( \frac
\mathrm{P}^{\alpha }\left( M,h\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}};M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\leq \mathfrak{F}_{\alpha }\lVert h\rVert
_{L^{2}\left( \sigma \right) }\,, \label{fec}
\end{equation
depends on the grids $\mathcal{D}$ and $\mathcal{G}$, the goodness parameter
$\varepsilon >0$ used in the definition of $J^{\maltese }$ through the
shifted corona $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$, and on the
family of martingale differences $\left\{ \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}\right\} _{J\in \mathcal{G}}$ associated with $x\in
L_{loc}^{2}\left( \omega \right) $, but not on the family of dual martingale
differences $\left\{ \square _{I}^{\sigma ,\mathbf{b}}\right\} _{I\in
\mathcal{D}}$, since the function $h\in L^{2}\left( \sigma \right) $
appearing in the definition of functional energy is not decomposed as a sum
of pseudoprojections $\square _{I}^{\sigma ,\mathbf{b}}h$. Finally, we
emphasize that the pseudoprojection
\begin{equation}
\mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};M}^{\omega
\mathbf{b}^{\ast }}\equiv \sum_{J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc
shift}}:\ J\subset M}\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}
\label{def pseudo rest}
\end{equation
here uses the shifted restricted corona i
\begin{eqnarray}
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}} &=&\left\{ J\in \mathcal{G
:J^{\maltese }\in \mathcal{C}_{F}^{\mathcal{D}}\right\} ,
\label{def shift cor rest} \\
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};K &\equiv &\left\{ J\in
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}:J\subset K\right\} , \notag
\end{eqnarray
where $J^{\maltese }$ is defined using the $\limfunc{body}$ of an interval
as in Definition \ref{def sharp cross}, and where we have defined here the
`restriction' $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};K$ to the
interval $K$ of the corona $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$
(c.f. $\Pi _{2}^{K}\mathcal{P}$\ in Definition \ref{rest K}, which uses the
stronger requirement $J^{\maltese }\subset K$). Moreover, recall from
Notation \ref{nonstandard norm} and the definition of $\nabla _{J}^{\omega }$
in (\ref{Carleson avg op}), that for any subset $\mathcal{H}$ of the grid
\mathcal{G}$
\begin{equation*}
\left\Vert \mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\equiv \sum_{J\in \mathcal{H
}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}=\sum_{J\in \mathcal{H}}\left(
\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\inf_{z\in \mathbb{R}}\left\Vert \widehat
\nabla }_{J}^{\omega }\left( x-z\right) \right\Vert _{L^{2}\left( \omega
\right) }^{2}\right) ,
\end{equation*
so that we never need to consider the norm squared $\left\Vert \mathsf{Q}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{2}$ of the pseudoprojection
\mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};M}^{\omega
\mathbf{b}^{\ast }}x$, something for which we have no lower Riesz
inequality. Note moreover that for $J\in \mathcal{G}$ and an arbitrary
interval $K$, we have by the frame inequality in Proposition \ref{dual frame
,
\begin{eqnarray}
\sum_{J\in \mathcal{G}:\ J\subset K}\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{2} &\lesssim
&\left\Vert x-m_{K}^{\omega }\right\Vert _{L^{2}\left( \mathbf{1}_{K}\omega
\right) }^{2}, \label{note more} \\
\sum_{J\in \mathcal{G}:\ J\subset K}\inf_{z\in \mathbb{R}}\left\Vert
\widehat{\nabla }_{J}^{\omega }\left( x-z\right) \right\Vert _{L^{2}\left(
\omega \right) }^{2} &\leq &\sum_{J\in \mathcal{G}:\ J\subset K}\left\Vert
\widehat{\nabla }_{J}^{\omega }\left\{ \left( x-p\right) \mathbf{1
_{K}\left( x\right) \right\} \right\Vert _{L^{2}\left( \omega \right)
}^{2}\lesssim \left\Vert \left( x-p\right) \right\Vert _{L^{2}\left( \mathbf
1}_{K}\omega \right) }^{2},\ \ \ p\in K, \notag
\end{eqnarray
where the second line follows from (\ref{Car embed}).
\begin{description}
\item[Important note] If $J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}
, then in particular $J\Subset _{\mathbf{\rho },\varepsilon }F$ with
\mathbf{\rho }=\left[ \frac{3}{\varepsilon }\right] $ by Lemma \ref{good
scale}, and so $J\cap M\neq \emptyset $ for a \emph{unique} $M\in \mathcal{W
\left( F\right) $.
\end{description}
We will show that, uniformly in pairs of grids $\mathcal{D}$ and $\mathcal{G}
$, the functional energy constants $\mathfrak{F}_{\alpha }\left( \mathcal{D}
\mathcal{G}\right) $ in (\ref{e.funcEnergy n}) are controlled by $\mathcal{A
_{2}^{\alpha }$, $A_{2}^{\alpha ,\limfunc{punct}}$ and the large energy
constant $\mathfrak{E}_{2}^{\alpha }$ - actually the proof shows that we
have control by the Whitney plugged energy constant as defined in (\ref{def
deep plug}) below. More precisely this is our control of functional energy
proposition.
\begin{proposition}
\label{func ener control}For all grids $\mathcal{D}$ and $\mathcal{G}$, and
\varepsilon >0$ sufficiently small, we hav
\begin{eqnarray*}
\mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G
\right) &\lesssim &\mathfrak{E}_{2}^{\alpha }+\sqrt{\mathcal{A}_{2}^{\alpha
}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}+\sqrt{A_{2}^{\alpha ,\limfunc{punct
}}\ , \\
\mathfrak{F}_{\alpha }^{\mathbf{b},\ast }\left( \mathcal{G},\mathcal{D
\right) &\lesssim &\mathfrak{E}_{2}^{\alpha ,\ast }+\sqrt{\mathcal{A
_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}+\sqrt{A_{2}^{\alpha
,\ast ,\limfunc{punct}}}\ ,
\end{eqnarray*
with implied constants independent of the grids $\mathcal{D}$ and $\mathcal{
}$.
\end{proposition}
In order to prove this proposition, we first turn to recalling these more
refined notions of energy constants.
\subsection{Various energy conditions\label{sub various energy cond}}
In this subsection we recall various refinements of the strong energy
conditions appearing in the main theorem above. Variants of this material
already appear in earlier papers, but we repeat it here both for convenience
and in order to introduce some arguments we will use repeatedly later on.
These refinements represent the `weakest' energy side conditions that
suffice for use in our proof, but despite this, we will usually use the
large energy constant $\mathfrak{E}_{2}^{\alpha }$ in estimates to avoid
having to pay too much attention to which of the energy conditions we need
to use - leaving the determination of the weakest conditions in such
situations to the interested reader. We begin with the notion of `deeply
embedded'. Recall that the goodness parameter $\mathbf{r}\in \mathbb{N}$ is
determined by $\varepsilon >0$ in (\ref{choice of r}), and that
0<\varepsilon <\frac{1}{2}<\frac{1}{2-\alpha }$.
For arbitrary intervals in $\,J,K\in \mathcal{P}$, we say that $J$ is
\left( \mathbf{\rho },\varepsilon \right) $-\emph{deeply embedded} in $K$,
which we write as $J\Subset _{\mathbf{\rho },\varepsilon }K$, when $J\subset
K$ and both
\begin{eqnarray}
\ell \left( J\right) &\leq &2^{-\mathbf{\rho }}\ell \left( K\right) ,
\label{def deep embed} \\
d\left( J,\partial K\right) &\geq &2\ell \left( J\right) ^{\varepsilon }\ell
\left( K\right) ^{1-\varepsilon }. \notag
\end{eqnarray
Note that we use the \emph{boundary} of $K$ for the definition of $J\Subset
_{\mathbf{\rho },\varepsilon }K$, rather than the \emph{skeleton} or \emph
body} of $K$, which would result in a more restrictive notion of $\left(
\mathbf{\rho },\varepsilon \right) $-deeply embedded. We will use this
notion for the purpose of grouping $\varepsilon -\limfunc{good}$ intervals
into the following collections. Fix grids $\mathcal{D}$ and $\mathcal{G}$.
For $K\in \mathcal{D}$, define the collections
\begin{eqnarray}
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{G}}\left( K\right) &\equiv &\left\{ J\in \mathcal{G}:J\text{ is
maximal w.r.t }J\Subset _{\mathbf{\rho },\varepsilon }K\right\} ,
\label{def M_r-deep} \\
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{D}}\left( K\right) &\equiv &\left\{ M\in \mathcal{D}:M\text{ is
maximal w.r.t }M\Subset _{\mathbf{\rho },\varepsilon }K\right\} , \notag \\
\mathcal{W}\left( K\right) &\equiv &\left\{ M\in \mathcal{D}:M\text{ is
maximal w.r.t }3M\subset K\right\} \notag
\end{eqnarray
where the first two consist of \emph{maximal} $\left( \mathbf{\rho
,\varepsilon \right) $-deeply embedded dyadic $\mathcal{G}$-subintervals $J
, respectively $\mathcal{D}$-subintervals $M$, of a $\mathcal{D}$-interval
K $, and the third consists of the maximal $\mathcal{D}$-subintervals $M$
whose triples are contained in $K$.
Let $\gamma >1$. Then the following bounded overlap property holds where
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep
}\left( K\right) $ can be taken to be either $\mathcal{M}_{\left( \mathbf
\rho },\varepsilon \right) -\limfunc{deep},\mathcal{G}}\left( K\right) $ or
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep}
\mathcal{D}}\left( K\right) $ or $\mathcal{W}\left( K\right) $ throughout.
\begin{lemma}
Let $0<\varepsilon \leq 1<\gamma \leq 1+4\cdot 2^{\mathbf{\rho }\left(
1-\varepsilon \right) }$. The
\begin{equation}
\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc
deep}}\left( K\right) }\mathbf{1}_{\gamma J}\leq \beta \mathbf{1}_{\left[
\dbigcup\limits_{J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right)
-\limfunc{deep}}\left( K\right) }\gamma J\right] } \label{bounded overlap}
\end{equation
holds for some positive constant $\beta $ depending only on $\gamma ,\mathbf
\rho }$ and $\varepsilon $. In addition $\gamma J\subset K$ for all $J\in
\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep
}\left( K\right) $, and consequentl
\begin{equation}
\sum_{J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc
deep}}\left( K\right) }\mathbf{1}_{\gamma J}\leq \beta \mathbf{1}_{K}\ .
\label{bounded overlap in K}
\end{equation
A similar result holds for $\mathcal{W}\left( K\right) $.
\end{lemma}
\begin{proof}
We suppose $0<\varepsilon <1$ and leave the simpler case $\varepsilon =1$
for the reader. To prove (\ref{bounded overlap}), we first note that there
are at most $2^{\mathbf{\rho }+1}$ intervals $J$ contained in $K$ for which
\ell \left( J\right) >2^{-\mathbf{\rho }}\ell \left( K\right) $. On the
other hand, the maximal $\left( \mathbf{\rho },\varepsilon \right) $-deeply
embedded subintervals $J$ of $K$ also satisfy the comparability conditio
\begin{equation*}
2\ell \left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon
}\leq d\left( J,\partial K\right) \leq d\left( \pi J,\partial K\right) -\ell
\left( J\right) \leq 2\left( 2\ell \left( J\right) \right) ^{\varepsilon
}\ell \left( K\right) ^{1-\varepsilon }-\ell \left( J\right) \leq 4\ell
\left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon }-\ell
\left( J\right) .
\end{equation*
Now with $0<\varepsilon <1$ and $\gamma >1$ fixed, let $y\in K$. Then if
y\in \gamma J$, we hav
\begin{eqnarray*}
2\ell \left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon }
&\leq &d\left( J,\partial K\right) \leq \gamma \ell \left( J\right) +d\left(
\gamma J,\partial K\right) \\
&\leq &\gamma \ell \left( J\right) +d\left( y,\partial K\right) .
\end{eqnarray*
Now assume that $\frac{\ell \left( J\right) }{\ell \left( K\right) }\leq
\left( \frac{1}{\gamma }\right) ^{\frac{1}{1-\varepsilon }}$. Then we have
\gamma \ell \left( J\right) \leq \ell \left( J\right) ^{\varepsilon }\ell
\left( K\right) ^{1-\varepsilon }$ and so
\begin{equation*}
\ell \left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon
}\leq d\left( y,\partial K\right) .
\end{equation*
But we also have
\begin{equation*}
d\left( y,\partial K\right) \leq \ell \left( J\right) +d\left( J,\partial
K\right) \leq \ell \left( J\right) +4\ell \left( J\right) ^{\varepsilon
}\ell \left( K\right) ^{1-\varepsilon }-\ell \left( J\right) \leq 4\ell
\left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon },
\end{equation*
and so altogether, under the assumption that $\frac{\ell \left( J\right) }
\ell \left( K\right) }\leq \left( \frac{1}{\gamma }\right) ^{\frac{1}
1-\varepsilon }}$, we hav
\begin{eqnarray*}
\frac{1}{4}d\left( y,\partial K\right) &\leq &\ell \left( J\right)
^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon }\leq d\left( y,\partial
K\right) , \\
\text{i.e. }\left( \frac{1}{4}\frac{d\left( y,\partial K\right) }{\ell
\left( K\right) ^{1-\varepsilon }}\right) ^{\frac{1}{\varepsilon }} &\leq
&\ell \left( J\right) \leq \left( \frac{d\left( y,\partial K\right) }{\ell
\left( K\right) ^{1-\varepsilon }}\right) ^{\frac{1}{\varepsilon }},
\end{eqnarray*
which shows that the number of $J^{\prime }s$ satisfying $y\in \gamma J$ and
$\frac{\ell \left( J\right) }{\ell \left( K\right) }\leq \left( \frac{1}
\gamma }\right) ^{\frac{1}{1-\varepsilon }}$ is at most $C^{\prime }\frac{1}
\varepsilon }$. On the other hand, the number of $J^{\prime }s$ contained in
$K$ satisfying $y\in \gamma J$ and $\frac{\ell \left( J\right) }{\ell \left(
K\right) }>\left( \frac{1}{\gamma }\right) ^{\frac{1}{1-\varepsilon }}$ is
at most $C^{\prime }\frac{1}{1-\varepsilon }\left( 1+\log _{2}\gamma \right)
$. This proves (\ref{bounded overlap}) with
\begin{equation*}
\beta =2^{\mathbf{\rho }+1}+C^{\prime }\frac{1}{\varepsilon }+C^{\prime
\frac{1}{1-\varepsilon }\left( 1+\log _{2}\gamma \right) .
\end{equation*}
In order to prove (\ref{bounded overlap in K}) it suffices, by (\ref{bounded
overlap}), to prove $\gamma J\subset K$ for all $J\in \mathcal{M}_{\left(
\mathbf{\rho },\varepsilon \right) -\limfunc{deep}}\left( K\right) $. But
J\in \mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep
}\left( K\right) $ implie
\begin{equation*}
2\ell \left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon
}\leq d\left( J,\partial K\right) =d\left( c_{J},\partial K\right) +\frac{1}
2}\ell \left( J\right) .
\end{equation*
We wish to show $\gamma J\subset K$, which is implied by
\begin{equation*}
\gamma \frac{1}{2}\ell \left( J\right) \leq d\left( c_{J},K^{c}\right)
=d\left( J,\partial K\right) +\frac{1}{2}\ell \left( J\right) .
\end{equation*
But we hav
\begin{equation*}
d\left( J,\partial K\right) +\frac{1}{2}\ell \left( J\right) \geq 2\ell
\left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon }+\frac{
}{2}\ell \left( J\right) ,
\end{equation*
and so it suffices to show tha
\begin{equation*}
2\ell \left( J\right) ^{\varepsilon }\ell \left( K\right) ^{1-\varepsilon }
\frac{1}{2}\ell \left( J\right) \geq \gamma \frac{1}{2}\ell \left( J\right) ,
\end{equation*
which is equivalent t
\begin{equation*}
\gamma -1\leq 4\ell \left( J\right) ^{\varepsilon -1}\ell \left( K\right)
^{1-\varepsilon }.
\end{equation*
But the smallest that $\ell \left( J\right) ^{\varepsilon -1}\ell \left(
K\right) ^{1-\varepsilon }$ can get for $J\in \mathcal{M}_{\left( \mathbf
\rho },\varepsilon \right) -\limfunc{deep}}\left( K\right) $ is $2^{\mathbf
\rho }\left( 1-\varepsilon \right) }\geq 1$, and so $\gamma \leq 1+4\cdot 2^
\mathbf{\rho }\left( 1-\varepsilon \right) }$ implies $\gamma -1\leq 4\ell
\left( J\right) ^{\varepsilon -1}\ell \left( K\right) ^{1-\varepsilon }$,
which completes the proof.
The reader can easily verify the same argument works for the Whitney
collection $\mathcal{W}\left( K\right) $.
\end{proof}
Now we recall the notion of \emph{alternate} dyadic intervals from \cit
{SaShUr7}, which we rename \emph{augmented} dyadic intervals here.
\begin{definition}
\label{def dyadic}Given a dyadic grid $\mathcal{D}$, the \emph{augmented
dyadic grid} $\mathcal{AD}$ consists\ of those intervals $I$ whose dyadic
children $I^{\prime }$ belong to the grid $\mathcal{D}$.
\end{definition}
Of course an augmented grid is not actually a grid because the nesting
property fails, but this terminology should cause no confusion. These
augmented grids will be needed in order to use the `prepare to puncture'
argument (introduced in \cite{SaShUr9}) at several places below.
Now we proceed to recall certain of the definitions of various energy
conditions from \cite{SaShUr5} and \cite{SaShUr7}. While these definitions
are not explicitly used in the proof of functional energy, some of the
arguments we give to control them will be appealed to later, and so we take
the time to develop these definitions in detail.
\subsubsection{Whitney energy conditions}
The following definition of Whitney energy condition uses the \emph{Whitney}
decomposition $\mathcal{M}_{\left( \mathbf{\rho },1\right) -\limfunc{deep}
\mathcal{D}}\left( I_{r}\right) $ into $\mathcal{D}$-dyadic intervals in
which $\varepsilon =1$, as well as the `large' pseudoprojection
\begin{equation}
\mathsf{Q}_{K}^{\omega ,\mathbf{b}^{\ast }}\equiv \sum_{J\in \mathcal{G}:\
J\subset K}\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}.
\label{large pseudo}
\end{equation}
\begin{definition}
\label{energy condition}Suppose $\sigma $ and $\omega $ are locally finite
positive Borel measures on $\mathbb{R}$ and fix $\gamma >1$. Then the\
Whitney energy condition constant $\mathcal{E}_{2}^{\alpha ,\func{Whitney}}$
is given b
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha ,\func{Whitney}}\right) ^{2}\equiv \sup_
\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}{\left\vert
I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W}\left(
I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{I\setminus \gamma M}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2},
\end{equation*
where $\sup_{\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}$ is taken over
\begin{enumerate}
\item all dyadic grids $\mathcal{D}$ and $\mathcal{G}$,
\item all $\mathcal{D}$-dyadic intervals $I$,
\item and all partitions $\left\{ I_{r}\right\} _{r=1}^{N\text{ or }\infty }$
of the interval $I$ into $\mathcal{D}$-dyadic subintervals $I_{r}$.
\end{enumerate}
\end{definition}
If the parameter $\gamma >1$ above is chosen sufficiently close to $1$, then
the collection of intervals $\left\{ \gamma M\right\} _{M\in \mathcal{W
\left( I_{r}\right) }$ has bounded overlap $\beta $ by (\ref{bounded overlap
in K}), and the Whitney energy constant $\mathcal{E}_{2}^{\alpha ,\func
Whitney}}$ is controlled by the strong energy constant $\mathcal{E
_{2}^{\alpha }$ in (\ref{strong b* energy})
\begin{equation}
\mathcal{E}_{2}^{\alpha ,\func{Whitney}}\lesssim \mathcal{E}_{2}^{\alpha }.
\label{en con}
\end{equation
Indeed, to see this, fix a decomposition of an interval
\begin{equation}
I=\overset{\cdot }{\dbigcup }_{1\leq r<\infty }\overset{\cdot }{\dbigcup
_{M\in \mathcal{W}\left( I_{r}\right) }M \label{decomp int}
\end{equation
as in Definition \ref{energy condition}. Then consider the \emph{sub
decomposition
\begin{equation*}
I\supset \overset{\cdot }{\dbigcup }_{1\leq r<\infty }\overset{\cdot }
\dbigcup }_{M\in \mathcal{W}\left( I_{r}\right) }M
\end{equation*
of the interval $I$ given by the collection of intervals
\begin{equation*}
\mathcal{I}\equiv \overset{\cdot }{\dbigcup }_{1\leq r<\infty }\mathcal{W
\left( I_{r}\right) .
\end{equation*
We then hav
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\geq \frac{1}{\left\vert
I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W}\left(
I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma
\right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert x-m_{M}^{\omega
}\right\Vert _{L^{2}\left( \mathbf{1}_{M}\omega \right) }^{2}\ .
\end{equation*
Now $\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) \geq \mathrm{
}^{\alpha }\left( M,\mathbf{1}_{I\setminus \gamma M}\sigma \right) $ and
from (\ref{note more})
\begin{equation*}
\left\Vert x-m_{M}^{\omega }\right\Vert _{L^{2}\left( \mathbf{1}_{M}\omega
\right) }^{2}\gtrsim \left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2},
\end{equation*}
and combining these two inequalities, we obtain tha
\begin{equation*}
\left( \mathcal{E}_{2}^{\alpha }\right) ^{2}\geq c\frac{1}{\left\vert
I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W}\left(
I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{I\setminus \gamma M}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\ .
\end{equation*
Thus we conclude tha
\begin{equation*}
\frac{1}{\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in
\mathcal{W}\left( I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M
\mathbf{1}_{I\setminus \gamma M}\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \frac{C}{c
\beta \left( \mathcal{E}_{2}^{\alpha }\right) ^{2},
\end{equation*
and taking the supremum over all decompositions (\ref{decomp int}) as in
Definition \ref{energy condition}, we obtain (\ref{en con}).
There is a similar definition for the dual (backward) Whitney energy
conditions that simply interchanges $\sigma $ and $\omega $ everywhere.
These definitions of\ the Whitney energy conditions depend on the choice of
\gamma >1$.
\begin{description}
\item[Commentary on proofs] We now introduce a number of results concerning
partial plugging of the hole for Whitney energy conditions.
\end{description}
Note that we can `partially' plug the $\gamma $-hole in the Poisson integral
$\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{I\setminus \gamma J}\sigma \right)
$ for $\mathcal{E}_{2}^{\alpha ,\func{Whitney}}$ using the offset
A_{2}^{\alpha }$ condition and the bounded overlap property (\ref{bounded
overlap in K}). Indeed, define
\begin{eqnarray}
&& \label{plug} \\
&&\left( \mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{partial}}\right)
^{2}\equiv \sup_{\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}
\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W
\left( I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{I\setminus M}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\ . \notag
\end{eqnarray
Recall from (\ref{bounded overlap in K}) tha
\begin{equation*}
\gamma M\subset I_{r}\text{ for all }M\in \mathcal{W}\left( I_{r}\right)
\text{ provided }\gamma \leq 5.
\end{equation*
At this point we need the following analogues of the `energy $A_{2}^{\alpha
} $ conditions' from \cite{SaShUr9}, which we denote by $A_{2}^{\alpha
\limfunc{energy}}$ and $A_{2}^{\alpha ,\ast ,\limfunc{energy}}$, and define
b
\begin{eqnarray}
A_{2}^{\alpha ,\limfunc{energy}}\left( \sigma ,\omega \right) &\equiv
&\sup_{Q\in \mathcal{P}}\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b
^{\ast }}\frac{x}{\ell \left( Q\right) }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}}{\left\vert Q\right\vert ^{1-\alpha }}\frac
\left\vert Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{1-\alpha }},
\label{def energy A2} \\
A_{2}^{\alpha ,\ast ,\limfunc{energy}}\left( \sigma ,\omega \right) &\equiv
&\sup_{Q\in \mathcal{P}}\frac{\left\vert Q\right\vert _{\omega }}{\left\vert
Q\right\vert ^{1-\alpha }}\frac{\left\Vert \mathsf{Q}_{Q}^{\sigma ,\mathbf{b
}\frac{x}{\ell \left( Q\right) }\right\Vert _{L^{2}\left( \sigma \right)
}^{\spadesuit 2}}{\left\vert Q\right\vert ^{1-\alpha }}. \notag
\end{eqnarray
Then if $\gamma \leq 5$, we hav
\begin{eqnarray}
&&\left( \mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{partial}}\right)
^{2} \label{plug the hole deep} \\
&\lesssim &\sup_{\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}
\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W
\left( I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{I\setminus \gamma M}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} \notag \\
&&+\sup_{\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}{\left\vert
I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W}\left(
I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{\gamma
M\setminus M}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} \notag \\
&\lesssim &\left( \mathcal{E}_{2}^{\alpha ,\func{Whitney}}\right) ^{2}+\sup_
\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}{\left\vert
I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W}\left(
I_{r}\right) }A_{2}^{\alpha ,\func{energy}}\left\vert \gamma M\right\vert
_{\sigma }\lesssim \left( \mathcal{E}_{2}^{\alpha ,\limfunc{deep}}\right)
^{2}+\beta A_{2}^{\alpha ,\func{energy}}\ , \notag
\end{eqnarray
by (\ref{bounded overlap in K}).
\subsubsection{Plugged energy conditions}
We continue to recall some results from \cite{SaShUr9} and \cite{SaShUr10}
that we will use repeatedly here. For example, we will use the punctured
Muckenhoupt conditions $A_{2}^{\alpha ,\limfunc{punct}}$ and $A_{2}^{\alpha
,\ast ,\limfunc{punct}}$ introduced earlier in (\ref{puncture}) to control
the \emph{plugged }energy conditions, where the hole in the argument of the
Poisson term $\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I\setminus M}\sigma
\right) $ in the partially plugged energy condition above, is replaced with
the `plugged' term $\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma
\right) $, for exampl
\begin{equation}
\left( \mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{plug}}\right)
^{2}\equiv \sup_{\mathcal{D},\mathcal{G}}\sup_{I=\dot{\cup}I_{r}}\frac{1}
\left\vert I\right\vert _{\sigma }}\sum_{r=1}^{\infty }\sum_{M\in \mathcal{W
\left( I_{r}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{I}\sigma \right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf
Q}_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\ . \label{def deep plug}
\end{equation
By an argument similar to that in (\ref{plug the hole deep}), we obtai
\begin{equation}
\mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{plug}}\lesssim \mathcal{E
_{2}^{\alpha ,\func{Whitney}\limfunc{partial}}+A_{2}^{\alpha ,\func{energy}}.
\label{plug the hole deep'}
\end{equation}
We first show that the punctured Muckenhoupt conditions $A_{2}^{\alpha
\limfunc{punct}}$ and $A_{2}^{\alpha ,\ast ,\limfunc{punct}}$ control
respectively the `energy $A_{2}^{\alpha }$ conditions' in (\ref{def energy
A2}). We will make reference to the proof of the next lemma (for the $T1$
theorem this is from \cite[Lemma 3.2 on page 328.]{SaShUr9}) several times
in the sequel. We repeat the proof from \cite[Lemma 3.2 on page 328.
{SaShUr9} but with modifications to accommodate the differences that arise
here in the setting of a local $Tb$ theorem. Recall that $\mathfrak{P
_{\left( \sigma ,\omega \right) }$ is defined in (\ref{def common point mass
) above.
\begin{lemma}
\label{energy A2}For any positive locally finite Borel measures $\sigma
,\omega $ we hav
\begin{eqnarray*}
A_{2}^{\alpha ,\limfunc{energy}}\left( \sigma ,\omega \right) &\lesssim
&A_{2}^{\alpha ,\limfunc{punct}}\left( \sigma ,\omega \right) , \\
A_{2}^{\alpha ,\ast ,\limfunc{energy}}\left( \sigma ,\omega \right)
&\lesssim &A_{2}^{\alpha ,\ast ,\limfunc{punct}}\left( \sigma ,\omega
\right) .
\end{eqnarray*}
\end{lemma}
\begin{proof}
Fix an interval $Q\in \mathcal{D}$. Recall the definition of $\omega \left(
Q,\mathfrak{P}_{\left( \sigma ,\omega \right) }\right) $ in (\ref{puncture
). If $\omega \left( Q,\mathfrak{P}_{\left( \sigma ,\omega \right) }\right)
\geq \frac{1}{2}\left\vert Q\right\vert _{\omega }$, then we trivially hav
\begin{eqnarray*}
\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell
\left( Q\right) }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left\vert Q\right\vert ^{1-\alpha }}\frac{\left\vert Q\right\vert _{\sigma
}{\left\vert Q\right\vert ^{1-\alpha }} &\lesssim &\frac{\left\vert
Q\right\vert _{\omega }}{\left\vert Q\right\vert ^{1-\alpha }}\frac
\left\vert Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{1-\alpha }} \\
&\leq &2\frac{\omega \left( Q,\mathfrak{P}_{\left( \sigma ,\omega \right)
}\right) }{\left\vert Q\right\vert ^{1-\alpha }}\frac{\left\vert
Q\right\vert _{\sigma }}{\left\vert Q\right\vert ^{1-\alpha }}\leq
2A_{2}^{\alpha ,\limfunc{punct}}\left( \sigma ,\omega \right) .
\end{eqnarray*
On the other hand, if $\omega \left( Q,\mathfrak{P}_{\left( \sigma ,\omega
\right) }\right) <\frac{1}{2}\left\vert Q\right\vert _{\omega }$ then there
is a point $p\in Q\cap \mathfrak{P}_{\left( \sigma ,\omega \right) }$ such
tha
\begin{equation*}
\omega \left( \left\{ p\right\} \right) >\frac{1}{2}\left\vert Q\right\vert
_{\omega }\ ,
\end{equation*
and consequently, $p$ is the largest $\omega $-point mass in $Q$. Thus if we
define $\widetilde{\omega }=\omega -\omega \left( \left\{ p\right\} \right)
\delta _{p}$, then we hav
\begin{equation*}
\omega \left( Q,\mathfrak{P}_{\left( \sigma ,\omega \right) }\right)
=\left\vert Q\right\vert _{\widetilde{\omega }}\ .
\end{equation*
Now we observe from the construction of martingale differences tha
\begin{equation*}
\bigtriangleup _{J}^{\widetilde{\omega },\mathbf{b}^{\ast }}=\bigtriangleup
_{J}^{\omega ,\mathbf{b}^{\ast }},\ \ \ \ \ \text{for all }J\in \mathcal{D
\text{ with }p\notin J.
\end{equation*
So for each $s\geq 0$ there is a unique interval $J_{s}\in \mathcal{D}$ with
$\ell \left( J_{s}\right) =2^{-s}\ell \left( Q\right) $ that contains the
point $p$. Now observe that, just as for the Haar projection, the
one-dimensional projection $\bigtriangleup _{J_{s}}^{\omega ,\mathbf{b
^{\ast }}$ is given by $\bigtriangleup _{J_{s}}^{\omega ,\mathbf{b}^{\ast
}}f=\left\langle h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }},f\right\rangle
_{\omega }h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }}$ for a unique up to $\pm $
unit vector $h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }}$. For this interval we
then hav
\begin{eqnarray*}
\left\Vert \bigtriangleup _{J_{s}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2} &=&\left\vert \left\langle
h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }},x\right\rangle _{\omega }\right\vert
^{2}=\left\vert \left\langle h_{J_{s}}^{\omega ,\mathbf{b}^{\ast
}},x-p\right\rangle _{\omega }\right\vert ^{2} \\
&=&\left\vert \int_{J_{s}}h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }}\left(
x\right) \left( x-p\right) d\omega \left( x\right) \right\vert
^{2}=\left\vert \int_{J_{s}}h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }}\left(
x\right) \left( x-p\right) d\widetilde{\omega }\left( x\right) \right\vert
^{2} \\
&\leq &\left\Vert h_{J_{s}}^{\omega ,\mathbf{b}^{\ast }}\right\Vert
_{L^{2}\left( \widetilde{\omega }\right) }^{2}\left\Vert \mathbf{1
_{J_{s}}\left( x-p\right) \right\Vert _{L^{2}\left( \widetilde{\omega
\right) }^{2}\leq \left\Vert h_{J_{s}}^{\omega ,\mathbf{b}^{\ast
}}\right\Vert _{L^{2}\left( \omega \right) }^{2}\left\Vert \mathbf{1
_{J_{s}}\left( x-p\right) \right\Vert _{L^{2}\left( \widetilde{\omega
\right) }^{2} \\
&\leq &\ell \left( J_{s}\right) ^{2}\left\vert J_{s}\right\vert _{\widetilde
\omega }}\leq 2^{-2s}\ell \left( Q\right) ^{2}\left\vert Q\right\vert _
\widetilde{\omega }},
\end{eqnarray*
as well a
\begin{equation*}
\inf_{z\in \mathbb{R}}\left\Vert \widehat{\nabla }_{J_{s}}^{\omega }\left(
x-z\right) \right\Vert _{L^{2}\left( \omega \right) }^{2}\lesssim \left\Vert
\left( x-p\right) \right\Vert _{L^{2}\left( \mathbf{1}_{J_{s}}\omega \right)
}^{2}=\left\Vert \left( x-p\right) \right\Vert _{L^{2}\left( \mathbf{1
_{J_{s}}\widetilde{\omega }\right) }^{2}\leq \ell \left( J_{s}\right)
^{2}\left\vert J_{s}\right\vert _{\widetilde{\omega }}\leq 2^{-2s}\ell
\left( Q\right) ^{2}\left\vert Q\right\vert _{\widetilde{\omega }}\ ,
\end{equation*
from (\ref{note more}). Thus we can estimat
\begin{eqnarray}
&& \label{omega tilda} \\
\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell \left(
Q\right) }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} &\leq
\frac{1}{\ell \left( Q\right) ^{2}}\left( \sum_{J\in \mathcal{D}:\ J\subset
Q}\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}+\inf_{z\in \mathbb{R}}\left\Vert \widehat
\nabla }_{J_{s}}^{\omega }\left( x-z\right) \right\Vert _{L^{2}\left( \omega
\right) }^{2}\right) \notag \\
&=&\frac{1}{\ell \left( Q\right) ^{2}}\left( \sum_{J\in \mathcal{D}:\
p\notin J\subset Q}\left\Vert \bigtriangleup _{J}^{\widetilde{\omega }
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \widetilde{\omega }\right)
}^{2}+\sum_{s=0}^{\infty }\left\Vert \bigtriangleup _{J_{s}}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{2}+\inf_{z\in \mathbb{R}}\left\Vert \widehat{\nabla }_{J_{s}}^{\omega
}\left( x-z\right) \right\Vert _{L^{2}\left( \omega \right) }^{2}\right)
\notag \\
&\lesssim &\frac{1}{\ell \left( Q\right) ^{2}}\left( \left\Vert \mathsf{Q
_{Q}^{\widetilde{\omega },\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\widetilde{\omega }\right) }^{\spadesuit 2}+\sum_{s=0}^{\infty }2^{-2s}\ell
\left( Q\right) ^{2}\left\vert Q\right\vert _{\widetilde{\omega }}\right)
\notag \\
&\lesssim &\frac{1}{\ell \left( Q\right) ^{2}}\left( \ell \left( Q\right)
^{2}\left\vert Q\right\vert _{\widetilde{\omega }}+\sum_{s=0}^{\infty
}2^{-2s}\ell \left( Q\right) ^{2}\left\vert Q\right\vert _{\widetilde{\omega
}}\right) \notag \\
&\leq &3\left\vert Q\right\vert _{\widetilde{\omega }}=3\omega \left( Q
\mathfrak{P}_{\left( \sigma ,\omega \right) }\right) , \notag
\end{eqnarray
and so
\begin{equation*}
\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell
\left( Q\right) }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left\vert Q\right\vert ^{1-\alpha }}\frac{\left\vert Q\right\vert _{\sigma
}{\left\vert Q\right\vert ^{1-\alpha }}\lesssim \frac{3\omega \left( Q
\mathfrak{P}_{\left( \sigma ,\omega \right) }\right) }{\left\vert
Q\right\vert ^{1-\alpha }}\frac{\left\vert Q\right\vert _{\sigma }}
\left\vert Q\right\vert ^{1-\alpha }}\leq 3A_{2}^{\alpha ,\limfunc{punct
}\left( \sigma ,\omega \right) .
\end{equation*
Now take the supremum over $Q\in \mathcal{D}$ to obtain $A_{2}^{\alpha
\limfunc{energy}}\left( \sigma ,\omega \right) \lesssim A_{2}^{\alpha
\limfunc{punct}}\left( \sigma ,\omega \right) $. The dual inequality follows
upon interchanging the measures $\sigma $ and $\omega $.
\end{proof}
We isolate a simple but key fact that will be used repeatedly in what
follows
\begin{equation}
\sum_{Q\in \mathcal{D}:\ Q\subset P}\ell \left( Q\right) ^{2}\left\vert
Q\right\vert _{\mu }\lesssim \ell \left( P\right) ^{2}\left\vert
P\right\vert _{\mu }\ ,\ \ \ \ \ \text{for }P\in \mathcal{D}\text{ and }\mu
\text{ a positive measure}. \label{key fact}
\end{equation
Indeed, to see (\ref{key fact}), simply pigeonhole the length of $Q$
relative to that of $P$ and sum. The next corollary follows immediately from
Lemma \ref{energy A2}, (\ref{plug the hole deep}) and (\ref{plug the hole
deep'}).
\begin{corollary}
\label{all plugged}Provided $1<\gamma \leq 5$
\begin{equation*}
\mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{plug}}\lesssim \mathcal{E
_{2}^{\alpha ,\func{Whitney}\limfunc{partial}}+A_{2}^{\alpha ,\limfunc{punct
}\lesssim \mathcal{E}_{2}^{\alpha ,\func{Whitney}}+A_{2}^{\alpha ,\limfunc
punct}}\ ,
\end{equation*
and similarly for the dual plugged energy condition.
\end{corollary}
\subsubsection{Plugged $\mathcal{A}_{2}^{\protect\alpha ,\limfunc{energy
\limfunc{plug}}$ conditions}
Using Lemma \ref{energy A2} we can control the `plugged' energy $\mathcal{A
_{2}^{\alpha }$ conditions
\begin{eqnarray*}
\mathcal{A}_{2}^{\alpha ,\limfunc{energy}\limfunc{plug}}\left( \sigma
,\omega \right) &\equiv &\sup_{Q\in \mathcal{P}}\frac{\left\Vert \mathsf{Q
_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell \left( Q\right) }\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left\vert Q\right\vert
^{1-\alpha }}\mathcal{P}^{\alpha }\left( Q,\sigma \right) , \\
\mathcal{A}_{2}^{\alpha ,\ast ,\limfunc{energy}\limfunc{plug}}\left( \sigma
,\omega \right) &\equiv &\sup_{Q\in \mathcal{P}}\mathcal{P}^{\alpha }\left(
Q,\omega \right) \frac{\left\Vert \mathsf{Q}_{Q}^{\sigma ,\mathbf{b}}\frac{
}{\ell \left( Q\right) }\right\Vert _{L^{2}\left( \sigma \right)
}^{\spadesuit 2}}{\left\vert Q\right\vert ^{1-\alpha }}.
\end{eqnarray*}
\begin{lemma}
\label{energy A2 plugged}We have
\end{lemma}
\begin{eqnarray*}
\mathcal{A}_{2}^{\alpha ,\limfunc{energy}\limfunc{plug}}\left( \sigma
,\omega \right) &\mathcal{\lesssim }&\mathcal{A}_{2}^{\alpha }\left( \sigma
,\omega \right) +A_{2}^{\alpha ,\limfunc{energy}}\left( \sigma ,\omega
\right) , \\
\mathcal{A}_{2}^{\alpha ,\ast ,\limfunc{energy}\limfunc{plug}}\left( \sigma
,\omega \right) &\mathcal{\lesssim }&\mathcal{A}_{2}^{\alpha ,\ast }\left(
\sigma ,\omega \right) +A_{2}^{\alpha ,\ast ,\limfunc{energy}}\left( \sigma
,\omega \right) .
\end{eqnarray*}
\begin{proof}
We hav
\begin{eqnarray*}
\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell
\left( Q\right) }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left\vert Q\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left( Q,\sigma
\right) &=&\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac
x}{\ell \left( Q\right) }\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}}{\left\vert Q\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha
}\left( Q,\mathbf{1}_{Q^{c}}\sigma \right) +\frac{\left\Vert \mathsf{Q
_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell \left( Q\right) }\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left\vert Q\right\vert
^{1-\alpha }}\mathcal{P}^{\alpha }\left( Q,\mathbf{1}_{Q}\sigma \right) \\
&\lesssim &\frac{\left\vert Q\right\vert _{\omega }}{\left\vert Q\right\vert
^{1-\alpha }}\mathcal{P}^{\alpha }\left( Q,\mathbf{1}_{Q^{c}}\sigma \right)
\frac{\left\Vert \mathsf{Q}_{Q}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\ell
\left( Q\right) }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left\vert Q\right\vert ^{1-\alpha }}\frac{\left\vert Q\right\vert _{\sigma
}{\left\vert Q\right\vert ^{1-\alpha }} \\
&\lesssim &\mathcal{A}_{2}^{\alpha }\left( \sigma ,\omega \right)
+A_{2}^{\alpha ,\limfunc{energy}}\left( \sigma ,\omega \right) .
\end{eqnarray*}
\end{proof}
\subsection{The Poisson formulation}
Recall from Definitions \ref{def sharp cross} and \ref{shifted corona} tha
\begin{equation*}
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}=\left\{ J\in \mathcal{G
:J^{\maltese }\in \mathcal{C}_{F}\right\} ,
\end{equation*
where $F\in \mathcal{F}$ is a stopping interval in the dyadic grid $\mathcal
D}$. For convenience we repeat here the main result of this section,
Proposition \ref{func ener control}.
\begin{proposition}
\label{func ener control'}For all grids $\mathcal{D}$ and $\mathcal{G}$, and
$\varepsilon >0$ sufficiently small, we hav
\begin{eqnarray*}
\mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast }}\left( \mathcal{D},\mathcal{G
\right) &\lesssim &\mathfrak{E}_{2}^{\alpha }+\sqrt{\mathcal{A}_{2}^{\alpha
}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}+\sqrt{A_{2}^{\alpha ,\limfunc{punct
}}\ , \\
\mathfrak{F}_{\alpha }^{\mathbf{b},\ast }\left( \mathcal{G},\mathcal{D
\right) &\lesssim &\mathfrak{E}_{2}^{\alpha ,\ast }+\sqrt{\mathcal{A
_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}+\sqrt{A_{2}^{\alpha
,\ast ,\limfunc{punct}}}\ ,
\end{eqnarray*
with implied constants independent of the grids $\mathcal{D}$ and $\mathcal{
}$.
\end{proposition}
To prove Proposition \ref{func ener control'}, we fix grids $\mathcal{D}$
and $\mathcal{G}$ and a subgrid $\mathcal{F}$ of $\mathcal{D}$\ as in (\re
{e.funcEnergy n}), and set
\begin{equation}
\mu \equiv \sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right)
}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\cdot \delta _{\left(
c_{M},\ell \left( M\right) \right) }\text{ and }d\overline{\mu }\left(
x,t\right) \equiv \frac{1}{t^{2}}d\mu \left( x,t\right) \ , \label{def mu n}
\end{equation
where $\mathcal{W}\left( F\right) $ consists of the maximal $\mathcal{D}
-subintervals of $F$ whose triples are contained in $F$, and where $\delta
_{\left( c_{M},\ell \left( M\right) \right) }$ denotes the Dirac unit mass
at the point $\left( c_{M},\ell \left( M\right) \right) $ in the upper
half-space $\mathbb{R}_{+}^{2}$. Here $M\in \mathcal{D}$ is a dyadic
interval with center $c_{M}$ and side length $\ell \left( M\right) $, and
for any interval $K\in \mathcal{P}$, the shorthand notation $\mathsf{P
_{F,K}^{\omega ,\mathbf{b}^{\ast }}$ (resp. $\mathsf{Q}_{F,K}^{\omega
\mathbf{b}^{\ast }}$) is used for the localized pseudoprojection $\mathsf{P
_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};K}^{\omega ,\mathbf{b}^{\ast
}}$ (resp. $\mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
};K}^{\omega ,\mathbf{b}^{\ast }}$) given in (\ref{def localization})
\begin{equation}
\mathsf{P}_{F,K}^{\omega ,\mathbf{b}^{\ast }}\equiv \mathsf{P}_{\mathcal{C
_{F}^{\mathcal{G},\limfunc{shift}};K}^{\omega ,\mathbf{b}^{\ast
}}=\sum_{J\subset K:\ J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}}\square _{J}^{\omega ,\mathbf{b}^{\ast }}\text{ }\left( \text{resp.
\mathsf{Q}_{F,K}^{\omega ,\mathbf{b}^{\ast }}\equiv \mathsf{Q}_{\mathcal{C
_{F}^{\mathcal{G},\limfunc{shift}};K}^{\omega ,\mathbf{b}^{\ast
}}=\sum_{J\subset K:\ J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}}\bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast }}\right) . \label{def F,K}
\end{equation
We emphasize that all the subintervals $J$ that arise in the projection
\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ are good inside the intervals
$F$ and beyond since $J^{\maltese }\subset F$. Here $J^{\maltese }$ is
defined in Definition \ref{def sharp cross} using the body of an interval.
Thus every $J\in \mathsf{Q}_{F}^{\omega ,\mathbf{b}^{\ast }}$ is contained
in a unique $M\in \mathcal{W}\left( F\right) $, so that $\mathsf{Q
_{F}^{\omega ,\mathbf{b}^{\ast }}=\overset{\cdot }{\dbigcup }_{M\in \mathcal
W}\left( F\right) }\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$. We can
replace $x$ by $x-c$ inside the projection for any choice of $c$ we wish;
the projection is unchanged. More generally, $\delta _{q}$ denotes a Dirac
unit mass at a point $q$ in the upper half-space $\mathbb{R}_{+}^{2}$.
We will prove the two-weight inequality
\begin{equation}
\left\Vert \mathbb{P}^{\alpha }\left( f\sigma \right) \right\Vert _{L^{2}
\mathbb{R}_{+}^{2},\overline{\mu })}\lesssim \left( \mathfrak{E}_{2}^{\alpha
}+\sqrt{\mathcal{A}_{2}^{\alpha }}+\sqrt{\mathcal{A}_{2}^{\alpha ,\ast }}
\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\right) \lVert f\rVert _{L^{2}\left(
\sigma \right) }\,, \label{two weight Poisson n}
\end{equation
for all nonnegative $f$ in $L^{2}\left( \sigma \right) $, noting that
\mathcal{F}$ and $f$ are \emph{not} related here. Above, $\mathbb{P}^{\alpha
}(\cdot )$ denotes the $\alpha $-fractional Poisson extension to the upper
half-space $\mathbb{R}_{+}^{2}$,
\begin{equation*}
\mathbb{P}^{\alpha }\rho \left( x,t\right) \equiv \int_{\mathbb{R}}\frac{t}
\left( t^{2}+\left\vert x-y\right\vert ^{2}\right) ^{\frac{2-\alpha }{2}}
d\rho \left( y\right) ,
\end{equation*
so that in particular
\begin{equation*}
\left\Vert \mathbb{P}^{\alpha }(f\sigma )\right\Vert _{L^{2}(\mathbb{R
_{+}^{2},\overline{\mu })}^{2}=\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W
\left( F\right) }\mathbb{P}^{\alpha }\left( f\sigma \right) (c(M),\ell
\left( M\right) )^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }
\frac{x}{\left\vert M\right\vert }\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\,,
\end{equation*
and so (\ref{two weight Poisson n}) proves the first line in Proposition \re
{func ener control} upon inspecting (\ref{e.funcEnergy n}). Note also that
we can equivalently write $\left\Vert \mathbb{P}^{\alpha }\left( f\sigma
\right) \right\Vert _{L^{2}(\mathbb{R}_{+}^{2},\overline{\mu })}=\left\Vert
\widetilde{\mathbb{P}}^{\alpha }\left( f\sigma \right) \right\Vert _{L^{2}
\mathbb{R}_{+}^{2},\mu )}$ where $\widetilde{\mathbb{P}}^{\alpha }\nu \left(
x,t\right) \equiv \frac{1}{t}\mathbb{P}^{\alpha }\nu \left( x,t\right) $ is
the renormalized Poisson operator. Here we have simply shifted the factor
\frac{1}{t^{2}}$ in $\overline{\mu }$ to $\left\vert \widetilde{\mathbb{P}
^{\alpha }\left( f\sigma \right) \right\vert ^{2}$ instead, and we will do
this shifting often throughout the proof when it is convenient to do so.
One version of the characterization of the two-weight inequality for
fractional and Poisson integrals in \cite{Saw3} was stated in terms of a
fixed dyadic grid $\mathcal{D}$ of intervals in $\mathbb{R}$ with sides
parallel to the coordinate axes. Using this theorem for the two-weight
Poisson inequality, but adapted to the $\alpha $-fractional Poisson integral
$\mathbb{P}^{\alpha }$\footnote
The proof for $0\leq \alpha <1$ is essentially identical to that for $\alpha
=0$ given in \cite{Saw3}.}, we see that inequality (\ref{two weight Poisson
n}) requires checking these two inequalities for dyadic intervals $I\in
\mathcal{D}$ and boxes $\widehat{I}=I\times \left[ 0,\ell \left( I\right)
\right) $ in the upper half-space $\mathbb{R}_{+}^{2}$:
\begin{equation}
\int_{\mathbb{R}_{+}^{2}}\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) \left( x,t\right) ^{2}d\overline{\mu }\left( x,t\right) \equiv
\left\Vert \mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma \right)
\right\Vert _{L^{2}(\overline{\mu })}^{2}\lesssim \left( \left( \mathfrak{E
_{2}^{\alpha }\right) ^{2}+\mathcal{A}_{2}^{\alpha }+\mathcal{A}_{2}^{\alpha
,\ast }+A_{2}^{\alpha ,\limfunc{punct}}\right) \sigma (I)\,, \label{e.t1 n}
\end{equation
\begin{equation}
\int_{\mathbb{R}}[\mathbb{Q}^{\alpha }(t\mathbf{1}_{\widehat{I}}\overline
\mu })]^{2}d\sigma (x)\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha
}\right) ^{2}+\mathcal{A}_{2}^{\alpha }+A_{2}^{\alpha ,\limfunc{punct
}\right) \int_{\widehat{I}}t^{2}d\overline{\mu }(x,t), \label{e.t2 n}
\end{equation
for all \emph{dyadic} intervals $I\in \mathcal{D}$, and where the dual
Poisson operator $\mathbb{Q}^{\alpha }$ is given by
\begin{equation*}
\mathbb{Q}^{\alpha }(t\mathbf{1}_{\widehat{I}}\overline{\mu })\left(
x\right) =\int_{\widehat{I}}\frac{t^{2}}{\left( t^{2}+\lvert x-y\rvert
^{2}\right) ^{\frac{2-\alpha }{2}}}d\overline{\mu }\left( y,t\right) \,.
\end{equation*
It is important to note that we can choose for $\mathcal{D}$ any fixed
dyadic grid, the compensating point being that the integrations on the left
sides of (\ref{e.t1 n}) and (\ref{e.t2 n}) are taken over the entire spaces
\mathbb{R}_{+}^{2}$ and $\mathbb{R}$ respectively\footnote
There is a gap in the proof of the Poisson inequality at the top of page 542
in \cite{Saw3}. However, this gap can be fixed as in \cite{SaWh} or \cit
{LaSaUr1}.}.
\subsection{Poisson testing}
We now turn to proving the Poisson testing conditions (\ref{e.t1 n}) and
\ref{e.t2 n}). Similar testing conditions have been considered in \cit
{SaShUr5}, \cite{SaShUr7}, \cite{SaShUr9} and \cite{SaShUr10}, and the
proofs there essentially carry over to the situation here, but careful
attention must now be paid to the changed definition of functional energy
and the weaker notion of goodness. We continue to circumvent the difficulty
of permitting common point masses here by using the energy Muckenhoupt
constants $A_{2}^{\alpha ,\limfunc{energy}}$ and $A_{2}^{\alpha ,\ast
\limfunc{energy}}$, which require control by the punctured Muckenhoupt
constants $A_{2}^{\alpha ,\limfunc{punct}}$ and $A_{2}^{\alpha ,\ast
\limfunc{punct}}$. The following elementary Poisson inequalities (see e.g.
\cite{Vol}) will be used extensively.
\begin{lemma}
\label{Poisson inequalities}Suppose that $J,K,I$ are intervals in $\mathbb{R}
$, and that $\mu $ is a positive measure supported in $\mathbb{R}\setminus I
. If $J\subset K\subset \beta K\subset I$ for some $\beta >1$, the
\begin{equation*}
\frac{\mathrm{P}^{\alpha }\left( J,\mu \right) }{\left\vert J\right\vert
\approx \frac{\mathrm{P}^{\alpha }\left( K,\mu \right) }{\left\vert
K\right\vert },
\end{equation*
while if $J\subset \beta K$, the
\begin{equation*}
\frac{\mathrm{P}^{\alpha }\left( K,\mu \right) }{\left\vert K\right\vert
\lesssim \frac{\mathrm{P}^{\alpha }\left( J,\mu \right) }{\left\vert
J\right\vert }.
\end{equation*}
\end{lemma}
\begin{proof}
We hav
\begin{equation*}
\frac{\mathrm{P}^{\alpha }\left( J,\mu \right) }{\left\vert J\right\vert }
\frac{1}{\left\vert J\right\vert }\int \frac{\left\vert J\right\vert }
\left( \left\vert J\right\vert +\left\vert x-c_{J}\right\vert \right)
^{2-\alpha }}d\mu \left( x\right) ,
\end{equation*
where $J\subset K\subset \beta K\subset I$ implies tha
\begin{equation*}
\left\vert J\right\vert +\left\vert x-c_{J}\right\vert \approx \left\vert
K\right\vert +\left\vert x-c_{K}\right\vert ,\ \ \ \ \ x\in \mathbb{R
\setminus I,
\end{equation*
and where $J\subset \beta K$ implies tha
\begin{equation*}
\left\vert J\right\vert +\left\vert x-c_{J}\right\vert \lesssim \left\vert
J\right\vert +\left\vert c_{K}-c_{J}\right\vert +\left\vert
x-c_{K}\right\vert \lesssim \left\vert K\right\vert +\left\vert
x-c_{K}\right\vert ,\ \ \ \ \ x\in \mathbb{R}.
\end{equation*}
\end{proof}
Recall that in the case\ of the $T1$ theorem in \cite{SaShUr7}, where we
assumed \emph{traditional} goodness in a single family of grids $\mathcal{D}
, we had a \emph{strong} bounded overlap property associated with the
projections $\mathsf{P}_{F,J}^{\omega ,\mathbf{b}^{\ast }}$ defined there;
namely, that for each interval $I_{0}\in \mathcal{D}$, there were a bounded
number of intervals $F\in \mathcal{F}$ with the property that $F\supsetneqq
I_{0}\supset J$ for some $J\in \mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep}}\left( F\right) $ with $\mathsf{P
_{F,J}^{\omega ,\mathbf{b}^{\ast }}\neq 0$ (see the first part of Lemma 10.4
in \cite{SaShUr7}). However, we no longer have this strong bounded overlap
property when ordinary goodness is replaced with the \emph{weak} goodness of
Hyt\"{o}nen and Martikainen. Indeed, there may now be an \emph{unbounded}
number of intervals $F\in \mathcal{F}$ with $F\supsetneqq I_{0}\supset J$
and $\mathsf{P}_{F,J}^{\omega ,\mathbf{b}^{\ast }}\neq 0$, simply because
there can be $J^{\prime }\in \mathcal{G}$ with both $J^{\prime }\subset
I_{0} $ and $\left( J^{\prime }\right) ^{\maltese }$ \emph{arbitrarily}
large.
What will save us in obtaining the following lemma is that the Whitney
intervals $M$ in $\mathcal{W}\left( F\right) $ that happen to lie in some
I\in \mathcal{D}$ with $I\subset F$ have one of just two different forms: if
$I$ shares an endpoint with $F$ then the intervals $M$ near that endpoint
are the same as those in $\mathcal{W}\left( I\right) $ - note that $F$ has
been replaced with $I$ here - while otherwise there are a bounded number of
Whitney intervals $M$ in $I$, and each such $M$ has side length comparable
to $\ell \left( I\right) $.
The next lemma will be used in bounding both of the local Poisson testing
conditions. Recall from Definition \ref{def dyadic}\ that $\mathcal{AD}$
consists of all augmented $\mathcal{D}$-dyadic intervals where $K$ is an
augmented dyadic interval if it is a union of $2$ $\mathcal{D}$-dyadic
intervals $K^{\prime }$ with $\ell \left( K^{\prime }\right) =\frac{1}{2
\ell \left( K\right) $.
\begin{lemma}
\label{refined lemma}Let $\mathcal{D}$ and $\mathcal{G}$ and $\mathcal
F\subset D}$ be grids and let $\left\{ \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}\right\} _{\substack{ F\in \mathcal{F} \\ M\in \mathcal{W}\left(
F\right) }}$ be as in (\ref{def F,K}) above. For any augmented interval
I\in \mathcal{AD}$ defin
\begin{equation}
B\left( I\right) \equiv \sum_{F\in \mathcal{F}:\ F\supsetneqq I^{\prime
\text{ for some }I^{\prime }\in \mathfrak{C}\left( I\right) }\sum_{M\in
\mathcal{W}\left( F\right) :\ M\subset I}\left( \frac{\mathrm{P}^{\alpha
}\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\ . \label{term B}
\end{equation
The
\begin{equation}
B\left( I\right) \lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}\right) \left\vert I\right\vert
_{\sigma }\ . \label{B bound}
\end{equation}
\end{lemma}
\begin{proof}
We first prove the bound (\ref{B bound}) for $B\left( I\right) $ ignoring
for the moment the possible case when $M=I$ in the sum defining $B\left(
I\right) $. So suppose that $I\in \mathcal{AD}$ is an augmented $\mathcal{D}
-dyadic interval. Defin
\begin{equation*}
\Lambda ^{\ast }\left( I\right) \equiv \left\{ M\subsetneqq I:M\in \mathcal{
}\left( F\right) \text{ for some }F\supsetneqq I^{\prime }\text{, }I^{\prime
}\in \mathfrak{C}\left( I\right) \text{ with }\mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\neq 0\right\} ,
\end{equation*
and pigeonhole this collection as $\Lambda ^{\ast }\left( I\right)
=\dbigcup\limits_{I^{\prime }\in \mathfrak{C}\left( I\right) }\Lambda \left(
I^{\prime }\right) $, where for each $I^{\prime }\in \mathfrak{C}\left(
I\right) $ we define
\begin{equation*}
\Lambda \left( I^{\prime }\right) \equiv \left\{ M\subset I^{\prime }:M\in
\mathcal{W}\left( F\right) \text{ for some }F\supsetneqq I^{\prime }\text{
with }\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\neq 0\right\} .
\end{equation*
Consider first the case when $3I^{\prime }\subset F$, so that $d\left(
I^{\prime },\partial F\right) \geq \ell \left( I^{\prime }\right) $. Then if
$M\in \mathcal{W}\left( F\right) $ for some $F\supsetneqq I^{\prime }$ we
have $\ell \left( M\right) =d\left( M,\partial F\right) $, and if in
addition $M\subset I^{\prime }$, then $M=I^{\prime }$. Consider the sum over
all $F\supsetneqq I^{\prime }=M$
\begin{eqnarray*}
B_{M}\left( I\right) &\equiv &\sum_{F\in \mathcal{F}:\ F\supsetneqq M\text{
for some }M\in \mathfrak{C}\left( I\right) \cap \mathcal{W}\left( F\right)
}\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }
\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}
\\
&\leq &\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma
\right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q
_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim \left( \frac{\mathrm{P}^{\alpha }\left( I,\mathbf{1
_{I}\sigma \right) }{\left\vert I\right\vert }\right) ^{2}\left\Vert \mathsf
Q}_{I}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim A_{2}^{\alpha ,\limfunc{energy}}\left\vert
I\right\vert _{\sigma }\ ,
\end{eqnarray*
where we have used the definitions (\ref{def F,K}) and (\ref{large pseudo}).
Thus we have obtained the boun
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\supsetneqq M\text{ for some }M\in \mathfrak{C
\left( I\right) \cap \mathcal{W}\left( F\right) }\left( \frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\lesssim
A_{2}^{\alpha ,\limfunc{energy}}\left\vert I\right\vert _{\sigma }\ .
\end{equation*}
Now we turn to the case $3I^{\prime }\not\subset F$, i.e. when $\partial
I^{\prime }\cap \partial F$ consists of exactly one boundary point. In this
case, if both $M\subset I^{\prime }$ and $M\in \mathcal{W}\left( F\right) $
for some $F\supsetneqq I^{\prime }$, then we must have either $M\in \mathcal
W}\left( I^{\prime }\right) $ or $M\in \mathfrak{C}\left( I^{\prime }\right)
$, since both $M$ and $I^{\prime }$ are then close to the same boundary
point in $\partial F$. Note that it is here that we use the Whitney
decompositions to full advantage. So again we can estimat
\begin{eqnarray*}
&&\sum_{\substack{ F\in \mathcal{F}:\ F\supsetneqq I^{\prime }\text{ for
some }I^{\prime }\in \mathfrak{C}\left( I\right) \\ 3I^{\prime }\not\subset
F}}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset I^{\prime }}\left(
\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\leq &\sum_{M\in \left\{ \mathcal{W}\left( I^{\prime }\right) \cup
\mathfrak{C}\left( I^{\prime }\right) \right\} \cap \mathcal{W}\left(
F\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma
\right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q
_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim \left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}\left\vert I\right\vert _{\sigma }\ .
\end{eqnarray*}
Finally, we consider the case $M=I$. In this case $I\in \mathcal{D}$ and so
F\supsetneqq I^{\prime }$ implies $F\supset I$ and we can estimat
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\supset I}\left( \frac{\mathrm{P}^{\alpha }\left(
I,\mathbf{1}_{I}\sigma \right) }{\left\vert I\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{F,I}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \left( \frac{\mathrm{P
^{\alpha }\left( I,\mathbf{1}_{I}\sigma \right) }{\left\vert I\right\vert
\right) ^{2}\left\Vert \mathsf{Q}_{I}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\lesssim
A_{2}^{\alpha ,\limfunc{energy}}\left\vert I\right\vert _{\sigma }\ .
\end{equation*
This completes the proof of Lemma \ref{refined lemma}.
\end{proof}
\subsection{The forward Poisson testing inequality}
Fix $I\in \mathcal{D}$. We split the integration on the left side of (\re
{e.t1 n}) into a local and global piece
\begin{equation*}
\int_{\mathbb{R}_{+}^{2}}\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) ^{2}d\overline{\mu }=\int_{\widehat{I}}\mathbb{P}^{\alpha }\left(
\mathbf{1}_{I}\sigma \right) ^{2}d\overline{\mu }+\int_{\mathbb{R
_{+}^{2}\setminus \widehat{I}}\mathbb{P}^{\alpha }\left( \mathbf{1
_{I}\sigma \right) ^{2}d\overline{\mu }\equiv \mathbf{Local}\left( I\right)
\mathbf{Global}\left( I\right) ,
\end{equation*
where more explicitly
\begin{eqnarray}
&&\mathbf{Local}\left( I\right) \equiv \int_{\widehat{I}}\left[ \mathbb{P
^{\alpha }\left( \mathbf{1}_{I}\sigma \right) \left( x,t\right) \right] ^{2}
\overline{\mu }\left( x,t\right) ;\ \ \ \ \ \overline{\mu }\equiv \frac{1}
t^{2}}\mu , \label{def local forward} \\
\text{i.e. }\overline{\mu } &\equiv &\ \sum_{F\in \mathcal{F}}\sum_{M\in
\mathcal{W}\left( F\right) }\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}\frac{x}{\ell \left( M\right) }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\cdot \delta _{\left( c_{M},\ell \left( M\right)
\right) }, \notag
\end{eqnarray
where we recall $\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ is defined
in (\ref{def F,K}) above. Here is a brief schematic diagram of the
decompositions, with bounds in $\fbox{}$, used in this subsection
\begin{equation*}
\fbox{
\begin{array}{ccc}
\mathbf{Local}\left( I\right) & & \\
\downarrow & & \\
\mathbf{Local}^{\limfunc{plug}}\left( I\right) & + & \mathbf{Local}^
\limfunc{hole}}\left( I\right) \\
\downarrow & & \fbox{$\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}$} \\
\downarrow & & \\
A & + & B \\
\fbox{$\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc
energy}}$} & & \fbox{$\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}$
\end{array
$}
\end{equation*
an
\begin{equation*}
\fbox{
\begin{array}{ccccccc}
\mathbf{Global}\left( I\right) & & & & & & \\
\downarrow & & & & & & \\
A & + & B & + & C & + & D \\
\fbox{$A_{2}^{\alpha }$} & & \fbox{$\left( \mathfrak{E}_{2}^{\alpha
}\right) ^{2}+A_{2}^{\alpha }+A_{2}^{\alpha ,\limfunc{energy}}$} & & \fbox{
\mathcal{A}_{2}^{\alpha ,\ast }$} & & \fbox{$\mathcal{A}_{2}^{\alpha ,\ast
}+A_{2}^{\alpha ,\limfunc{punct}}$
\end{array
$}.
\end{equation*}
As in our earlier papers \cite{SaShUr2}-\cite{SaShUr10} that used a single
family of random grids, we have the useful equivalence tha
\begin{equation}
\left( c\left( M\right) ,\ell \left( M\right) \right) \in \widehat{I}\text{
\textbf{if and only if} }M\subset I, \label{tent consequence}
\end{equation
since $M$ and $I$ live in the common grid $\mathcal{D}$. We thus have
\begin{eqnarray*}
&&\mathbf{Local}\left( I\right) =\int_{\widehat{I}}\mathbb{P}^{\alpha
}\left( \mathbf{1}_{I}\sigma \right) \left( x,t\right) ^{2}d\overline{\mu
\left( x,t\right) \\
&=&\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset
I}\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma \right) \left( c_{M},\ell
\left( M\right) \right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}\frac{x}{\left\vert M\right\vert }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2} \\
&\approx &\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) :\
M\subset I}\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right)
^{2}\lVert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\left\vert
M\right\vert }\rVert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\approx &\mathbf{Local}^{\limfunc{plug}}\left( I\right) +\mathbf{Local}^
\func{hole}}\left( I\right) ,
\end{eqnarray*
wher
\begin{eqnarray*}
\mathbf{Local}^{\limfunc{plug}}\left( I\right) &\equiv &\sum_{F\in \mathcal{
}}\sum_{M\in \mathcal{W}\left( F\right) ):\ M\subset I}\left( \frac{\mathrm{
}^{\alpha }\left( M,\mathbf{1}_{F\cap I}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{2}, \\
\mathbf{Local}^{\func{hole}}\left( I\right) &\equiv &\sum_{F\in \mathcal{F
}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset I}\left( \frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{I\setminus F}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}.
\end{eqnarray*
The `plugged' local sum $\mathbf{Local}^{\limfunc{plug}}\left( I\right) $
can be further decomposed into
\begin{align*}
& \mathbf{Local}^{\limfunc{plug}}\left( I\right) =\left\{ \sum_{F\in
\mathcal{F}:\ F\subset I}+\sum_{F\in \mathcal{F}:\ F\supsetneqq I}\right\}
\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset I}\left( \frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{F\cap I}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
& =A+B.
\end{align*
Then an application of the Whitney plugged energy condition gives
\begin{eqnarray*}
A &=&\sum_{F\in \mathcal{F}:\ F\subset I}\sum_{M\in \mathcal{W}\left(
F\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{F\cap
I}\sigma \right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&\leq &\sum_{F\in \mathcal{F}:\ F\subset I}\left( \mathfrak{E}_{2}^{\alpha }
\sqrt{A_{2}^{\alpha ,\limfunc{energy}}}\right) ^{2}\left\vert F\right\vert
_{\sigma }\lesssim \left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha
\limfunc{energy}}}\right) ^{2}\left\vert I\right\vert _{\sigma }\,,
\end{eqnarray*
since $\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\leq \left\Vert \mathsf{Q
_{M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}$. We also used here that the stopping intervals $\mathcal{F}
$ satisfy a $\sigma $-Carleson measure estimate,
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\subset F_{0}}\left\vert F\right\vert _{\sigma
}\lesssim \left\vert F_{0}\right\vert _{\sigma }.
\end{equation*
Lemma \ref{refined lemma} applies to the remaining term $B$ to obtain the
boun
\begin{equation*}
B\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
,\limfunc{energy}}\right) \left\vert I\right\vert _{\sigma }\ .
\end{equation*}
Next we show the inequality with `holes', where the support of $\sigma $ is
restricted to the complement of the interval $F$.
\begin{lemma}
\label{local hole}We have
\begin{equation}
\mathbf{Local}^{\func{hole}}\left( I\right) \lesssim \left( \mathfrak{E
_{2}^{\alpha }\right) ^{2}\left\vert I\right\vert _{\sigma }\,.
\label{RTS n}
\end{equation}
\end{lemma}
\begin{proof}
Fix $I\in \mathcal{D}$ and defin
\begin{equation*}
\mathcal{F}_{I}\equiv \left\{ F\in \mathcal{F}:F\subset I\right\} \cup
\left\{ I\right\} ,
\end{equation*
and denote by $\pi F$, for this proof only, the parent of $F$ in the tree
\mathcal{F}_{I}$. Also denote by $d\left( F,F^{\prime }\right) \equiv d_
\mathcal{F}_{I}}\left( F,F^{\prime }\right) $ the distance from $F$ to
F^{\prime }$ in the tree $\mathcal{F}_{I}$, and denote by $d\left( F\right)
\equiv d_{\mathcal{F}_{I}}\left( F,I\right) $ the distance of $F$ from the
root $I$. Since $I\setminus F$ appears in the argument of the Poisson
integral, those $F\in \mathcal{F}\setminus \mathcal{F}_{I}$ do not
contribute to the sum and so we estimat
\begin{equation*}
S\equiv \mathbf{Local}^{\func{hole}}\left( I\right) =\sum_{F\in \mathcal{F
_{I}}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset I}\left( \frac
\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I\setminus F}\sigma \right) }
\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}
\end{equation*
by using $\sum_{F^{\prime }\in \mathcal{F}:\ F\subset F^{\prime }\subsetneqq
I}\frac{1}{d\left( F^{\prime }\right) ^{2}}\leq C$ to obtain\footnote
In \cite{SaShUr7} and \cite{SaShUr6} the first line of this display
incorrectly avoided the use of the Cauchy-Schwarz inequality. In the earlier
versions \cite{SaShUr5} and version \#2 of \cite{SaShUr6}, the argument was
correctly given by duality. The fix used here is taken from pages 94-95 of
version \#4 of \cite{SaShUr5}.}
\begin{eqnarray*}
S &=&\sum_{F\in \mathcal{F}_{I}}\sum_{M\in \mathcal{W}\left( F\right) :\
M\subset I}\left( \sum_{F^{\prime }\in \mathcal{F}:\ F\subset F^{\prime
}\subsetneqq I}\frac{d\left( F^{\prime }\right) }{d\left( F^{\prime }\right)
}\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{\pi F^{\prime }\setminus
F^{\prime }}\sigma \right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert
\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2} \\
&\leq &\sum_{F\in \mathcal{F}_{I}}\sum_{M\in \mathcal{W}\left( F\right) :\
M\subset I}\left( \sum_{F^{\prime }\in \mathcal{F}:\ F\subset F^{\prime
}\subsetneqq I}\frac{1}{d\left( F^{\prime }\right) ^{2}}\right) \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \left( \sum_{F^{\prime }\in
\mathcal{F}:\ F\subset F^{\prime }\subsetneqq I}d\left( F^{\prime }\right)
^{2}\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{\pi F^{\prime
}\setminus F^{\prime }}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\right) \left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\leq &C\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{F\in \mathcal{F}:\ F\subset F^{\prime }}\sum_{M\in \mathcal{W
\left( F\right) :\ M\subset I}\left( \frac{\mathrm{P}^{\alpha }\left( M
\mathbf{1}_{\pi F^{\prime }\setminus F^{\prime }}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&=&C\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\sum_{F\in \mathcal{F
:\ F\subset F^{\prime }}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset
I}\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{\pi F^{\prime
}\setminus F^{\prime }}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{F,M\cap K}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\lesssim &\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{\pi F^{\prime }\setminus F^{\prime }}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\sum_{F\in \mathcal{F}:\
F\subset F^{\prime }}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset
I}\left\Vert \mathsf{Q}_{F,M\cap K}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2},
\end{eqnarray*
where in the fifth line we have used that each $J^{\prime }$ appearing in
\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ occurs in one of the $\mathsf
Q}_{F,M\cap K}^{\omega ,\mathbf{b}^{\ast }}$ since each $M$ is contained in
a unique $K$. We have also used there the Poisson inequalities in Lemma \re
{Poisson inequalities}.
We now use the lower frame inequality from Appendix A applied to the
function $\mathbf{1}_{K}\left( x-m_{K}^{\omega }\right) $ to obtai
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\subset F^{\prime }}\sum_{M\in \mathcal{W}\left(
F\right) :\ M\subset I}\left\Vert \mathsf{Q}_{F,M\cap K}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\lesssim
\left\Vert \mathbf{1}_{K}\left( x-m_{K}^{\omega }\right) \right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\ .
\end{equation*}
Since the collection $\mathcal{F}_{I}$ satisfies a Carleson condition,
namely $\sum_{F\in \mathcal{F}_{I}}\left\vert F\cap I^{\prime }\right\vert
_{\sigma }\leq C\left\vert I^{\prime }\right\vert _{\sigma }$ for all
intervals $I^{\prime }$, we have geometric decay in generations
\begin{equation}
\sum_{F\in \mathcal{F}_{I}:\ d\left( F\right) =k}\left\vert F\right\vert
_{\sigma }\lesssim 2^{-\delta k}\left\vert I\right\vert _{\sigma }\ ,\ \ \ \
\ k\geq 0. \label{geometric decay}
\end{equation
Indeed, with $m>2C$ we have for each $F^{\prime }\in \mathcal{F}_{I}$
\begin{equation}
\sum_{F\in \mathcal{F}_{I}:\ F\subset F^{\prime }\text{ and }d\left(
F,F^{\prime }\right) =m}\left\vert F\cap F^{\prime }\right\vert _{\sigma }
\frac{1}{2}\left\vert F^{\prime }\right\vert _{\sigma }\ , \label{half}
\end{equation
since otherwis
\begin{equation*}
\sum_{F\in \mathcal{F}_{I}:\ F\subset F^{\prime }\text{ and }d\left(
F,F^{\prime }\right) \leq m}\left\vert F\cap F^{\prime }\right\vert _{\sigma
}\geq m\frac{1}{2}\left\vert F^{\prime }\right\vert _{\sigma }\ ,
\end{equation*
a contradiction. Now iterate (\ref{half}) to obtain (\ref{geometric decay}).
Thus we can writ
\begin{eqnarray*}
S &\lesssim &\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{\pi F^{\prime }\setminus F^{\prime }}\sigma
\right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathbf{1
_{K}\left( x-m_{K}^{\omega }\right) \right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&=&\sum_{k=1}^{\infty }k^{2}\sum_{F^{\prime }\in \mathcal{F}_{I}:\ d\left(
F^{\prime }\right) =k}\sum_{K\in \mathcal{W}\left( F^{\prime }\right)
}\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{\pi F^{\prime
}\setminus F^{\prime }}\sigma \right) }{\left\vert K\right\vert }\right)
^{2}\left\Vert \mathbf{1}_{K}\left( x-m_{K}^{\omega }\right) \right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\equiv \sum_{k=1}^{\infty
}A_{k}\ ,
\end{eqnarray*
where $A_{k}$ is defined at the end of the above display. Hence using the
strong energy condition
\begin{eqnarray*}
A_{k} &=&k^{2}\sum_{F^{\prime }\in \mathcal{F}_{I}:\ d\left( F^{\prime
}\right) =k}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\left( \frac
\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{\pi F^{\prime }\setminus F^{\prime
}}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathbf{1
_{K}\left( x-m_{K}^{\omega }\right) \right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&\lesssim &k^{2}\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}\sum_{F^{\prime
\prime }\in \mathcal{F}_{I}:\ d\left( F^{\prime \prime }\right)
=k-1}\left\vert F^{\prime \prime }\right\vert _{\sigma }\lesssim \left(
\mathfrak{E}_{2}^{\alpha }\right) ^{2}k^{2}2^{-\delta k}\left\vert
I\right\vert _{\sigma }\ ,
\end{eqnarray*
where we have applied the strong energy condition for each $F^{\prime \prime
}\in \mathcal{F}_{I}$ with $d\left( F^{\prime \prime }\right) =k-1$ to obtai
\begin{equation}
\sum_{F^{\prime }\in \mathcal{F}_{I}:\ \pi F^{\prime }=F^{\prime \prime
}}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{F^{\prime \prime }\setminus F^{\prime
}}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathbf{1
_{K}\left( x-m_{K}^{\omega }\right) \right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\leq \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}\left\vert
F^{\prime \prime }\right\vert _{\sigma }\ . \label{to obtain}
\end{equation
Finally then we obtai
\begin{equation*}
S\lesssim \sum_{k=1}^{\infty }\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}k^{2}2^{-\delta k}\left\vert I\right\vert _{\sigma }\lesssim \left(
\mathfrak{E}_{2}^{\alpha }\right) ^{2}\left\vert I\right\vert _{\sigma }\ ,
\end{equation*
which is (\ref{RTS n}).
\end{proof}
Altogether we have now proved the estimate $\mathbf{Local}\left( I\right)
\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
\limfunc{energy}}\right) \left\vert I\right\vert _{\sigma }$ when $I\in
\mathcal{D}$, i.e. for every dyadic interval $I\in \mathcal{D}$
\begin{eqnarray}
&& \label{local} \\
\mathbf{Local}\left( I\right) &\approx &\sum_{F\in \mathcal{F}}\sum_{M\in
\mathcal{W}\left( F\right) :\ M\subset I}\left( \frac{\mathrm{P}^{\alpha
}\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} \notag \\
&\lesssim &\left( \left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}\right) \left\vert I\right\vert
_{\sigma },\ \ \ I\in \mathcal{D}. \notag
\end{eqnarray}
\subsubsection{The augmented local estimate}
For future use in the `prepare to puncture' arguments below, we prove a
strengthening of the local estimate $\mathbf{Local}\left( I\right) $ to
\emph{augmented} intervals $L\in \mathcal{AD}$.
\begin{lemma}
\label{shifted}With notation as above and $L\in \mathcal{AD}$ an augmented
interval, we have
\begin{eqnarray}
&& \label{shifted local} \\
\mathbf{Local}\left( L\right) &\equiv &\sum_{F\in \mathcal{F}}\sum_{M\in
\mathcal{W}\left( F\right) :\ M\subset L}\left( \frac{\mathrm{P}^{\alpha
}\left( M,\mathbf{1}_{L}\sigma \right) }{\left\vert M\right\vert }\right)
^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} \notag \\
&\lesssim &\left( \left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}\right) \left\vert L\right\vert
_{\sigma },\ \ \ L\in \mathcal{AD}. \notag
\end{eqnarray}
\end{lemma}
\begin{proof}
We prove (\ref{shifted local}) by repeating the above proof of (\ref{local})
and noting the points requiring change. First we decompose
\begin{equation*}
\mathbf{Local}\left( L\right) \lesssim \mathbf{Local}^{\limfunc{plug}}\left(
L\right) +\mathbf{Local}^{\func{hole}}\left( L\right) +\mathbf{Local}^
\limfunc{offset}}\left( L\right)
\end{equation*
where $\mathbf{Local}^{\limfunc{plug}}\left( L\right) $ and $\mathbf{Local}^
\func{hole}}\left( L\right) $ are analogous to $\mathbf{Local}^{\limfunc{plu
}}\left( I\right) $ and $\mathbf{Local}^{\func{hole}}\left( I\right) $
above, and where $\mathbf{Local}^{\limfunc{offset}}\left( L\right) $ is an
additional term arising because $L\setminus F$ need not be empty when $L\cap
F\neq \emptyset $ and $F$ is not contained in $L$
\begin{eqnarray*}
\mathbf{Local}^{\limfunc{plug}}\left( L\right) &\equiv &\sum_{F\in \mathcal{
}}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset L}\left( \frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{L\cap F}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\ , \\
\mathbf{Local}^{\func{hole}}\left( L\right) &\equiv &\sum_{F\in \mathcal{F
:\ F\subset L}\sum_{M\in \mathcal{W}\left( F\right) :\ M\subset L}\left(
\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{L\setminus F}\sigma \right) }
\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}\ , \\
\mathbf{Local}^{\limfunc{offset}}\left( L\right) &\equiv &\sum_{F\in
\mathcal{F}:\ F\not\subset L}\sum_{M\in \mathcal{W}\left( F\right) :\
M\subset L}\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{L\setminus
F}\sigma \right) }{\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\ .
\end{eqnarray*
We hav
\begin{align*}
& \mathbf{Local}^{\limfunc{plug}}\left( L\right) =\left\{ \sum_{F\in
\mathcal{F}:\ F\subset \text{ some }L^{\prime }\in \mathfrak{C}\left(
L\right) }+\sum_{F\in \mathcal{F}:\ F\supsetneqq \text{ some }L^{\prime }\in
\mathfrak{C}_{\mathcal{D}}\left( L\right) }\right\} \sum_{M\in \mathcal{W
\left( F\right) :\ M\subset L} \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \left(
\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{F\cap L}\sigma \right) }
\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}
\\
& =A+B.
\end{align*}
Term $A$ satisfie
\begin{equation*}
A\lesssim \left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha ,\limfunc
energy}}}\right) ^{2}\left\vert L\right\vert _{\sigma }\ ,
\end{equation*
just as above using $\left\Vert \mathsf{Q}_{F,M}^{\omega }x\right\Vert
_{L^{2}\left( \omega \right) }^{2}\leq \left\Vert \mathsf{Q}_{M}^{\omega
}x\right\Vert _{L^{2}\left( \omega \right) }^{2}$, and the fact that the
stopping intervals $\mathcal{F}$ satisfy a $\sigma $-Carleson measure
estimate,
\begin{equation*}
\sum_{F\in \mathcal{F}:\ F\subset L}\left\vert F\right\vert _{\sigma
}\lesssim \left\vert L\right\vert _{\sigma }.
\end{equation*}
Term $B$ is handled directly by Lemma \ref{refined lemma} with the augmented
interval $I=L$ to obtai
\begin{equation*}
B\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
,\limfunc{energy}}\right) \left\vert L\right\vert _{\sigma }\ .
\end{equation*}
To handle $\mathbf{Local}^{\func{hole}}\left( L\right) $, we defin
\begin{equation*}
\mathcal{F}_{L}\equiv \left\{ F\in \mathcal{F}:F\subset L\right\} \cup
\left\{ L\right\} ,
\end{equation*
and follow along the proof there with only trivial changes. The analogue of
\ref{to obtain}) is no
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}_{L}:\ \pi F^{\prime }=F^{\prime \prime
}}\sum_{K\in \mathcal{W}\left( F^{\prime }\right) }\left( \frac{\mathrm{P
^{\alpha }\left( K,\mathbf{1}_{F^{\prime \prime }\setminus F^{\prime
}}\sigma \right) }{\left\vert K\right\vert }\right) ^{2}\left\Vert \mathbf{1
_{K}\left( x-m_{K}^{\omega }\right) \right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\leq \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}\left\vert
F^{\prime \prime }\right\vert _{\sigma }\ ,
\end{equation*
the only change being that $\mathcal{F}_{L}$ now appears in place of
\mathcal{F}_{I}$, so that the energy condition still applies. We conclude
that
\begin{equation*}
\mathbf{Local}^{\func{hole}}\left( L\right) \lesssim \left( \mathfrak{E
_{2}^{\alpha }\right) ^{2}\left\vert L\right\vert _{\sigma }\ .
\end{equation*}
Finally, the additional term $\mathbf{Local}^{\limfunc{offset}}\left(
L\right) $ is handled directly by Lemma \ref{refined lemma}, and this
completes the proof of the estimate (\ref{shifted local}) in Lemma \re
{shifted}.
\end{proof}
\subsubsection{The global estimate}
Now we turn to proving the following estimate for the global part of the
first testing condition \eqref{e.t1 n}
\begin{equation*}
\mathbf{Global}\left( I\right) =\int_{\mathbb{R}_{+}^{2}\setminus \widehat{I
}\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma \right) ^{2}d\overline{\mu
\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+\mathcal{A
_{2}^{\alpha ,\ast }+A_{2}^{\alpha ,\limfunc{punct}}\right) \left\vert
I\right\vert _{\sigma }.
\end{equation*
We begin by decomposing the integral above into four pieces. We have from
\ref{tent consequence})
\begin{eqnarray*}
&&\int_{\mathbb{R}_{+}^{2}\setminus \widehat{I}}\mathbb{P}^{\alpha }\left(
\mathbf{1}_{I}\sigma \right) ^{2}d\overline{\mu }=\sum_{M:\ \left(
c_{M},\ell \left( M\right) \right) \in \mathbb{R}_{+}^{2}\setminus \widehat{
}}\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma \right) \left( c_{M},\ell
\left( M\right) \right) ^{2}\sum_{\substack{ F\in \mathcal{F}: \\ M\in
\mathcal{W}\left( F\right) }}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}\frac{x}{\left\vert M\right\vert }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2} \\
&=&\left\{ \sum_{\substack{ M\cap 3I=\emptyset \\ \ell \left( M\right) \leq
\ell \left( I\right) }}+\sum_{M\subset 3I\setminus I}+\sum_{\substack{ M\cap
I=\emptyset \\ \ell \left( M\right) >\ell \left( I\right) }
+\sum_{M\supsetneqq I}\right\} \mathbb{P}^{\alpha }\left( \mathbf{1
_{I}\sigma \right) \left( c_{M},\ell \left( M\right) \right) ^{2}\sum
_{\substack{ F\in \mathcal{F}: \\ M\in \mathcal{W}\left( F\right) }
\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\left\vert
M\right\vert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&=&A+B+C+D.
\end{eqnarray*}
We further decompose term $A$ according to the length of $M$ and its
distance from $I$, and then use the pairwise disjointedness of the
projections $\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ in $F$ (see the
definition in (\ref{def F,K})) to obtain
\begin{eqnarray*}
A &\lesssim &\sum_{m=0}^{\infty }\sum_{k=1}^{\infty }\sum_{\substack{
M\subset 3^{k+1}I\setminus 3^{k}I \\ \ell \left( M\right) =2^{-m}\ell
\left( I\right) }}\left( \frac{2^{-m}\left\vert I\right\vert }{d\left(
M,I\right) ^{2-\alpha }}\left\vert I\right\vert _{\sigma }\right)
^{2}\left\vert M\right\vert _{\omega } \\
&\lesssim &\sum_{m=0}^{\infty }2^{-2m}\sum_{k=1}^{\infty }\frac{\left\vert
I\right\vert ^{2}\left\vert I\right\vert _{\sigma }\left\vert
3^{k+1}I\setminus 3^{k}I\right\vert _{\omega }}{\left\vert 3^{k}I\right\vert
^{2\left( 2-\alpha \right) }}\left\vert I\right\vert _{\sigma } \\
&\lesssim &\sum_{m=0}^{\infty }2^{-2m}\sum_{k=1}^{\infty }3^{-2k}\left\{
\frac{\left\vert 3^{k+1}I\setminus 3^{k}I\right\vert _{\omega }\left\vert
3^{k}I\right\vert _{\sigma }}{\left\vert 3^{k}I\right\vert ^{2\left(
1-\alpha \right) }}\right\} \left\vert I\right\vert _{\sigma }\lesssim
A_{2}^{\alpha }\left\vert I\right\vert _{\sigma },
\end{eqnarray*
where the offset Muckenhoupt constant $A_{2}^{\alpha }$ applies because
3^{k+1}I$ has only three times the side length of $3^{k}I$.
\medskip
For term $B$ we first dispose of the nearby sum $B_{\limfunc{nearby}}$ that
consists of the sum over those $M$ which satisfy in addition $2^{-\mathbf
\rho }}\ell \left( I\right) \leq \ell \left( M\right) \leq \ell \left(
I\right) $. But it is a straightforward task to bound $B_{\limfunc{nearby}}$
by $CA_{2}^{\alpha ,\limfunc{energy}}\left\vert I\right\vert _{\sigma }$ as
there are at most $2^{\mathbf{\rho }+1}$ such intervals $M$. To bound $B_
\func{away}}\equiv B-B_{\limfunc{nearby}}$, we further decompose the sum
over $F\in \mathcal{F}$ according to whether or not $F\subset 3I\setminus I$
\begin{eqnarray*}
B_{\func{away}} &\approx &\sum_{M\subset 3I\setminus I\text{ and }\ell
\left( M\right) <2^{-\mathbf{\rho }}\ell \left( I\right) }\left( \frac
\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert
M\right\vert }\right) ^{2}\sum_{\substack{ F\in \mathcal{F}:\ F\subset
3I\setminus I \\ M\in \mathcal{W}\left( F\right) }}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&&+\sum_{M\subset 3I\setminus I\text{ and }\ell \left( M\right) <2^{-\mathbf
\rho }}\ell \left( I\right) }\left( \frac{\mathrm{P}^{\alpha }\left( M
\mathbf{1}_{I}\sigma \right) }{\left\vert M\right\vert }\right) ^{2}\sum
_{\substack{ F\in \mathcal{F}:\ F\not\subset 3I\setminus I \\ M\in \mathcal
W}\left( F\right) }}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\equiv &B_{\func{away}}^{1}+B_{\func{away}}^{2}\ .
\end{eqnarray*
\ \
To estimate $B_{\func{away}}^{1}$, let
\begin{equation}
\mathcal{J}^{\ast }\equiv \dbigcup\limits_{\substack{ F\in \mathcal{F} \\
F\subset 3I\setminus I}}\dbigcup\limits_{\substack{ M\in \mathcal{W}\left(
F\right) \\ M\subset 3I\setminus I\text{ and }\ell \left( M\right) <2^{
\mathbf{\rho }}\ell \left( I\right) }}\left\{ J\in \mathcal{C}_{F}^{\mathcal
G},\limfunc{shift}}:J\subset M\right\} \label{def J*}
\end{equation
consist of all intervals $J\in \mathcal{G}$ for which the projection
\triangle _{J}^{\omega ,\mathbf{b}^{\ast }}$ occurs in one of the
projections $\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ in term $B_
\func{away}}^{1}$. In order to use $\mathcal{J}^{\ast }$ in the estimate for
$B_{\func{away}}^{1}$ we need the following inequality. For any interval
M\in \mathcal{W}\left( F\right) $ we hav
\begin{eqnarray}
\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }
\left\vert M\right\vert }\right) ^{2}\left\Vert \mathsf{Q}_{F;M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} &=&\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right)
}{\left\vert M\right\vert }\right) ^{2}\sum_{J\in \mathcal{C}_{F}^{\mathcal{
},\func{shift}}:\ J\subset M}\left\Vert \triangle _{J}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}
\label{accomplished} \\
&\lesssim &\sum_{J\in \mathcal{C}_{F}^{\mathcal{G},\func{shift}}:\ J\subset
M}\left( \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{I}\sigma \right) }
\left\vert J\right\vert }\right) ^{2}\left\Vert \triangle _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}, \notag
\end{eqnarray
sinc
\begin{eqnarray*}
\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert
M\right\vert } &=&\int_{I}\frac{1}{\left( \ell \left( M\right) +\left\vert
x-c_{M}\right\vert \right) ^{2-\alpha }}d\sigma \left( x\right) \\
&\lesssim &\int_{I}\frac{1}{\left( \ell \left( J\right) +\left\vert
x-c_{J}\right\vert \right) ^{2-\alpha }}d\sigma \left( x\right) =\frac
\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{I}\sigma \right) }{\left\vert
J\right\vert }
\end{eqnarray*
for $J\subset M$ becaus
\begin{equation*}
\ell \left( J\right) +\left\vert x-c_{J}\right\vert \lesssim \ell \left(
M\right) +\left\vert x-c_{M}\right\vert ,\ \ \ \ \ J\subset M\text{ and
x\in \mathbb{R}.
\end{equation*
We now use (\ref{accomplished}) to replace the sum over $M\in \mathcal{W
\left( F\right) $ in $B_{\func{away}}^{1}$, with a sum over $J\in \mathcal{J
^{\ast }$
\begin{eqnarray*}
B_{\func{away}}^{1} &=&\sum_{M\subset 3I\setminus I\text{ and }\ell \left(
M\right) <2^{-\mathbf{\rho }}\ell \left( I\right) }\left( \frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{I}\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\sum_{\substack{ F\in \mathcal{F}:\ F\subset 3I\setminus I \\
M\in \mathcal{W}\left( F\right) }}\left\Vert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}
\\
&\lesssim &\sum_{M\subset 3I\setminus I\text{ and }\ell \left( M\right) <2^{
\mathbf{\rho }}\ell \left( I\right) }\sum_{\substack{ F\in \mathcal{F}:\
F\subset 3I\setminus I \\ M\in \mathcal{W}\left( F\right) }}\sum_{J\in
\mathcal{C}_{F}^{\mathcal{G},\func{shift}}:\ J\subset M}\left( \frac{\mathrm
P}^{\alpha }\left( J,\mathbf{1}_{I}\sigma \right) }{\left\vert J\right\vert
\right) ^{2}\left\Vert \triangle _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\lesssim &\sum_{J\in \mathcal{J}^{\ast }}\left( \frac{\mathrm{P}^{\alpha
}\left( J,\mathbf{1}_{I}\sigma \right) }{\left\vert J\right\vert }\right)
^{2}\mathbf{\ }\left\Vert \bigtriangleup _{J}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\ ,
\end{eqnarray*
where the final line follows since for each $J\in \mathcal{J}^{\ast }$ there
is a unique pair $\left( F,M\right) $ satisfying the conditions in the
second line.\bigskip
We will now exploit the smallness of $\varepsilon >0$ in the weak goodness
condition by decomposing the sum over $J\in \mathcal{J}^{\ast }$ according
to the length of $J$, and then using the fractional Poisson inequality (\re
{e.Jsimeq}) in Lemma \ref{Poisson inequality} on the neighbour $I^{\prime }
\ of $I$ containing $J$. Indeed, for $J\subset I^{\prime }\subset \mathbb{R}$
and $I\subset \mathbb{R}\setminus I^{\prime }$, we have
\begin{equation}
\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{I}\sigma \right) ^{2}\lesssim
\left( \frac{\ell \left( J\right) }{\ell \left( I\right) }\right)
^{2-2\left( 2-\alpha \right) \varepsilon }\mathrm{P}^{\alpha }\left( I
\mathbf{1}_{I}\sigma \right) ^{2},\ \ \ \ \ J\in \mathcal{J}^{\ast },
\label{Poisson inequalities 2}
\end{equation
where we have used that $\ell \left( I^{\prime }\right) =\ell \left(
I\right) $ and $\mathrm{P}^{\alpha }\left( I^{\prime },\mathbf{1}_{I}\sigma
\right) \approx \mathrm{P}^{\alpha }\left( I,\mathbf{1}_{I}\sigma \right) $,
and that the intervals $J\in \mathcal{J}^{\ast }$ are good in $I^{\prime }$
and beyond, and have side length at most $2^{-\mathbf{\rho }}\ell \left(
I\right) $, all because $J^{\maltese }\subset F\subset 3I\setminus I$ and we
have already dealt with the term $B_{\limfunc{nearby}}$. Moreover, we may
also assume here that the exponent $2-2\left( 2-\alpha \right) \varepsilon $
is positive, i.e.$\ \varepsilon <\frac{1}{2-\alpha }$, which is of course
implied by $0<\varepsilon <\frac{1}{2}$. We then obtain from (\ref{Poisson
inequalities 2}), the inequality $\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}\lesssim \left\vert J\right\vert ^{2}\left\vert J\right\vert _{\omega }$,
the pairwise disjointedness of the $M\in \mathcal{W}\left( F\right) $, the
uniqueness of $F$ with $J\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$,
and since $F\subset 3I\setminus I$ in the sum over $J\in \mathcal{J}^{\ast }
, tha
\begin{eqnarray*}
B_{\func{away}}^{1} &\lesssim &\sum_{J\in \mathcal{J}^{\ast }}\left( \frac
\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{I}\sigma \right) }{\left\vert
J\right\vert }\right) ^{2}\mathbf{\ }\left\Vert \bigtriangleup _{J}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}\lesssim \sum_{m=\mathbf{\rho }}^{\infty }\sum_{\substack{ J\in \mathcal{J
^{\ast } \\ \ell \left( J\right) =2^{-m}\ell \left( I\right) }}\left(
2^{-m}\right) ^{2-2\left( 2-\alpha \right) \varepsilon }\mathrm{P}^{\alpha
}\left( I,\mathbf{1}_{I}\sigma \right) ^{2}\left\vert J\right\vert _{\omega }
\\
&\lesssim &\sum_{m=\mathbf{\rho }}^{\infty }\left( 2^{-m}\right) ^{2-2\left(
2-\alpha \right) \varepsilon }\left( \frac{\left\vert I\right\vert _{\sigma
}{\left\vert I\right\vert ^{1-\alpha }}\right) ^{2}\sum_{\substack{ J\subset
3I\setminus I \\ \ell \left( J\right) =2^{-m}\ell \left( I\right) }
\left\vert J\right\vert _{\omega }\lesssim \sum_{m=\mathbf{\rho }}^{\infty
}\left( 2^{-m}\right) ^{2-2\left( 2-\alpha \right) \varepsilon }\frac
\left\vert I\right\vert _{\sigma }\left\vert 3I\setminus I\right\vert
_{\omega }}{\left\vert 3I\right\vert ^{2\left( 1-\alpha \right) }}\left\vert
I\right\vert _{\sigma }\lesssim A_{2}^{\alpha }\left\vert I\right\vert
_{\sigma }\ ,
\end{eqnarray*
since $2-2\left( 2-\alpha \right) \varepsilon >0$.
To complete the bound for term $B=B_{\limfunc{nearby}}+B_{\func{away
}^{1}+B_{\func{away}}^{2}$, it remains to estimate term $B_{\func{away}}^{2}$
in which we sum over $F\not\subset 3I\setminus I$. In this case
F\varsupsetneqq I^{\prime }$ for one of the two neighbours $I^{\prime }$ of
I$, and so we can apply Lemma \ref{refined lemma}, with $I$ there replaced
by the augmented intervals $I^{\prime }\cup I$, to obtain the estimat
\begin{equation*}
B_{\func{away}}^{2}\lesssim \left( \left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}\right) \left\vert I\right\vert
_{\sigma }\ .
\end{equation*}
\medskip
Next we turn to term $D$. The intervals $M$ occurring here are included in
the set of ancestors $A_{k}\equiv \pi _{\mathcal{D}}^{\left( k\right) }I$ of
$I$, $1\leq k<\infty $
\begin{eqnarray*}
D &=&\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) \left( c\left( A_{k}\right) ,\left\vert A_{k}\right\vert \right)
^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in \mathcal{W}\left(
F\right) }}\left\Vert \mathsf{Q}_{F,A_{k}}^{\omega ,\mathbf{b}^{\ast }}\frac
x}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&=&\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) \left( c\left( A_{k}\right) ,\left\vert A_{k}\right\vert \right)
^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in \mathcal{W}\left(
F\right) }}\sum_{J^{\prime }\in \mathcal{C}_{F}^{\mathcal{G},\limfunc{shift
}:\ J^{\prime }\subset A_{k}\setminus I}\left\Vert \bigtriangleup
_{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\lvert A_{k}\rvert
\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&&+\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) \left( c\left( A_{k}\right) ,\left\vert A_{k}\right\vert \right)
^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in \mathcal{W}\left(
F\right) }}\sum_{\substack{ J^{\prime }\in \mathcal{C}_{F}^{\mathcal{G}
\limfunc{shift}}:\ J^{\prime }\subset A_{k} \\ J^{\prime }\cap I\neq
\emptyset \text{ and }\ell \left( J^{\prime }\right) \leq \ell \left(
I\right) }}\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b
^{\ast }}\frac{x}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2} \\
&&+\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left( \mathbf{1}_{I}\sigma
\right) \left( c\left( A_{k}\right) ,\left\vert A_{k}\right\vert \right)
^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in \mathcal{W}\left(
F\right) }}\sum_{\substack{ J^{\prime }\in \mathcal{C}_{F}^{\mathcal{G}
\limfunc{shift}}:\ J^{\prime }\subset A_{k} \\ J^{\prime }\cap I\neq
\emptyset \text{ and }\ell \left( J^{\prime }\right) >\ell \left( I\right) }
\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\frac{
}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} \\
&\equiv &D_{\limfunc{disjoint}}+D_{\limfunc{descendent}}+D_{\limfunc{ancesto
}}\ .
\end{eqnarray*
We thus have from the pairwise disjointedness of the projections $\mathsf{Q
_{F,A_{k}}^{\omega ,\mathbf{b}^{\ast }}$ in $F$ once again
\begin{eqnarray*}
D_{\limfunc{disjoint}} &=&\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left(
\mathbf{1}_{I}\sigma \right) \left( c\left( A_{k}\right) ,\left\vert
A_{k}\right\vert \right) ^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in
\mathcal{W}\left( F\right) }}\sum_{J^{\prime }\in \mathcal{C}_{F}^{\mathcal{
},\limfunc{shift}}:\ J^{\prime }\subset A_{k}\setminus I}\left\Vert
\bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\lvert
A_{k}\rvert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\lesssim &\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}\left\vert A_{k}\right\vert }{\left\vert A_{k}\right\vert ^{2-\alpha }
\right) ^{2}\mathbf{\;}\left\vert A_{k}\setminus I\right\vert _{\omega
}=\left\{ \frac{\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert
^{1-\alpha }}\sum_{k=1}^{\infty }\frac{\left\vert I\right\vert ^{1-\alpha }}
\left\vert A_{k}\right\vert ^{2\left( 1-\alpha \right) }}\left\vert
A_{k}\setminus I\right\vert _{\omega }\right\} \left\vert I\right\vert
_{\sigma } \\
&\lesssim &\left\{ \frac{\left\vert I\right\vert _{\sigma }}{\left\vert
I\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left( I,\mathbf{1
_{I^{c}}\omega \right) \right\} \left\vert I\right\vert _{\sigma }\lesssim
\mathcal{A}_{2}^{\alpha ,\ast }\left\vert I\right\vert _{\sigma },
\end{eqnarray*
sinc
\begin{eqnarray*}
\sum_{k=1}^{\infty }\frac{\left\vert I\right\vert ^{1-\alpha }}{\left\vert
A_{k}\right\vert ^{2\left( 1-\alpha \right) }}\left\vert A_{k}\setminus
I\right\vert _{\omega } &=&\int \sum_{k=1}^{\infty }\frac{\left\vert
I\right\vert ^{1-\alpha }}{\left\vert A_{k}\right\vert ^{2\left( 1-\alpha
\right) }}\mathbf{1}_{A_{k}\setminus I}\left( x\right) d\omega \left(
x\right) \\
&=&\int \sum_{k=1}^{\infty }\frac{1}{2^{2\left( 1-\alpha \right) k}}\frac
\left\vert I\right\vert ^{1-\alpha }}{\left\vert I\right\vert ^{2\left(
1-\alpha \right) }}\mathbf{1}_{A_{k}\setminus I}\left( x\right) d\omega
\left( x\right) \\
&\lesssim &\int_{I^{c}}\left( \frac{\left\vert I\right\vert }{\left[
\left\vert I\right\vert +d\left( x,I\right) \right] ^{2}}\right) ^{1-\alpha
}d\omega \left( x\right) =\mathcal{P}^{\alpha }\left( I,\mathbf{1
_{I^{c}}\omega \right) ,
\end{eqnarray*
upon summing a geometric series with $2\left( 1-\alpha \right) >0$.
The next term $D_{\limfunc{descendent}}$ satisfie
\begin{eqnarray*}
D_{\limfunc{descendent}} &\lesssim &\sum_{k=1}^{\infty }\left( \frac
\left\vert I\right\vert _{\sigma }\left\vert A_{k}\right\vert }{\left\vert
A_{k}\right\vert ^{2-\alpha }}\right) ^{2}\mathbf{\;}\left\Vert \mathsf{Q
_{3I}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{2^{k}\lvert I\rvert }\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&=&\sum_{k=1}^{\infty }2^{-2k\left( 2-\alpha \right) }\left( \frac
\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert ^{1-\alpha }
\right) ^{2}\left\Vert \mathsf{Q}_{3I}^{\omega ,\mathbf{b}^{\ast }}\frac{x}
\lvert I\rvert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\lesssim &\left\{ \frac{\left\vert I\right\vert _{\sigma }\left\Vert
\mathsf{Q}_{3I}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\lvert I\rvert
\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left\vert
I\right\vert ^{2\left( 1-\alpha \right) }}\right\} \left\vert I\right\vert
_{\sigma }\lesssim A_{2}^{\alpha ,\limfunc{energy}}\left\vert I\right\vert
_{\sigma }\ .
\end{eqnarray*}
Lastly, for $D_{\limfunc{ancestor}}$ we note that there are at most two
intervals $K_{1}$ and $K_{2}$ in $\mathcal{G}$ having side length $\ell
\left( I\right) $ and such that $K_{i}\cap I\neq \emptyset $. Then each
J^{\prime }$ occurring in the sum in $D_{\limfunc{ancestor}}$ is of the form
$J^{\prime }=A_{i}^{\ell }\equiv \pi _{\mathcal{G}}^{\left( \ell \right)
}K_{i}$ with $J^{\prime }\subset A_{k}$ for some $1\leq \ell \leq k$ and
i\in \left\{ 1,2\right\} $. Now we writ
\begin{eqnarray*}
D_{\limfunc{ancestor}} &=&\sum_{k=1}^{\infty }\mathbb{P}^{\alpha }\left(
\mathbf{1}_{I}\sigma \right) \left( c\left( A_{k}\right) ,\left\vert
A_{k}\right\vert \right) ^{2}\sum_{\substack{ F\in \mathcal{F}: \\ A_{k}\in
\mathcal{W}\left( F\right) }}\sum_{\substack{ J^{\prime }\in \mathcal{C
_{F}^{\mathcal{G},\limfunc{shift}}:\ J^{\prime }\subset A_{k} \\ J^{\prime
}\cap I\neq \emptyset \text{ and }\ell \left( J^{\prime }\right) >\ell
\left( I\right) }}\left\Vert \bigtriangleup _{J^{\prime }}^{\omega ,\mathbf{
}^{\ast }}\frac{x}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2} \\
&\lesssim &\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}\left\vert A_{k}\right\vert }{\left\vert A_{k}\right\vert ^{2-\alpha }
\right) ^{2}\sum_{i=1}^{2}\sum_{\ell =1}^{k}\left\Vert \bigtriangleup
_{A_{i}^{\ell }}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\lvert A_{k}\rvert
\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\leq &2\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}\left\vert A_{k}\right\vert }{\left\vert A_{k}\right\vert ^{2-\alpha }
\right) ^{2}\left\Vert \mathsf{Q}_{A_{k}}^{\omega ,\mathbf{b}^{\ast }}\frac{
}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}.
\end{eqnarray*
At this point we need a \emph{`prepare to puncture'} argument, as we will
want to derive geometric decay from $\left\Vert \mathsf{Q}_{J^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}$ by dominating it by the `nonenergy' term $\left\vert
J^{\prime }\right\vert ^{2}\left\vert J^{\prime }\right\vert _{\omega }$, as
well as using the Muckenhoupt energy constant. For this we define
\widetilde{\omega }=\omega -\omega \left( \left\{ p\right\} \right) \delta
_{p}$ where $p$ is an atomic point in $I$ for which
\begin{equation*}
\omega \left( \left\{ p\right\} \right) =\sup_{q\in \mathfrak{P}_{\left(
\sigma ,\omega \right) }:\ q\in I}\omega \left( \left\{ q\right\} \right) .
\end{equation*
(If $\omega $ has no atomic point in common with $\sigma $ in $I$ set
\widetilde{\omega }=\omega $.) Then we have $\left\vert I\right\vert _
\widetilde{\omega }}=\omega \left( I,\mathfrak{P}_{\left( \sigma ,\omega
\right) }\right) $ an
\begin{equation*}
\frac{\left\vert I\right\vert _{\widetilde{\omega }}}{\left\vert
I\right\vert ^{1-\alpha }}\frac{\left\vert I\right\vert _{\sigma }}
\left\vert I\right\vert ^{1-\alpha }}=\frac{\omega \left( I,\mathfrak{P
_{\left( \sigma ,\omega \right) }\right) }{\left\vert I\right\vert
^{1-\alpha }}\frac{\left\vert I\right\vert _{\sigma }}{\left\vert
I\right\vert ^{1-\alpha }}\leq A_{2}^{\alpha ,\limfunc{punct}}.
\end{equation*
A key observation, already noted in the proof of Lemma \ref{energy A2}
above, is tha
\begin{equation}
\left\Vert \bigtriangleup _{K}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{2}=\left\{
\begin{array}{ccc}
\left\Vert \bigtriangleup _{K}^{\omega ,\mathbf{b}^{\ast }}\left( x-p\right)
\right\Vert _{L^{2}\left( \omega \right) }^{2} & \text{ if } & p\in K \\
\left\Vert \bigtriangleup _{K}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \widetilde{\omega }\right) }^{2} & \text{ if } & p\notin
\end{array
\right. \leq \ell \left( K\right) ^{2}\left\vert K\right\vert _{\widetilde
\omega }},\ \ \ \ \ \text{for all }K\in \mathcal{D}\ , \label{key obs}
\end{equation
and so, as in the proof of (\ref{omega tilda}) in Lemma \ref{energy A2}
\begin{equation*}
\left\Vert \mathsf{Q}_{A_{k}}^{\omega ,\mathbf{b}^{\ast }}\frac{x}
\left\vert A_{k}\right\vert }\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim \left\vert A_{k}\right\vert _{\widetilde{\omega }}\
.
\end{equation*
Then we continue wit
\begin{eqnarray*}
&&\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}\left\vert A_{k}\right\vert }{\left\vert A_{k}\right\vert ^{2-\alpha }
\right) ^{2}\left\Vert \mathsf{Q}_{A_{k}}^{\omega ,\mathbf{b}^{\ast }}\frac{
}{\lvert A_{k}\rvert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2} \\
&\lesssim &\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}\left\vert A_{k}\right\vert }{\left\vert A_{k}\right\vert ^{2-\alpha }
\right) ^{2}\left\vert A_{k}\right\vert _{\widetilde{\omega }} \\
&=&\sum_{k=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma }}
\left\vert A_{k}\right\vert ^{1-\alpha }}\right) ^{2}\left\vert
A_{k}\setminus I\right\vert _{\omega }+\sum_{k=1}^{\infty }\left( \frac
\left\vert I\right\vert _{\sigma }}{2^{k\left( 1-\alpha \right) }\left\vert
I\right\vert ^{1-\alpha }}\right) ^{2}\left\vert I\right\vert _{\widetilde
\omega }} \\
&\lesssim &\left( \mathcal{A}_{2}^{\alpha ,\ast }+A_{2}^{\alpha ,\limfunc
punct}}\right) \left\vert I\right\vert _{\sigma },
\end{eqnarray*
where the inequality $\sum_{k=1}^{\infty }\left( \frac{\left\vert
I\right\vert _{\sigma }}{\left\vert A_{k}\right\vert ^{1-\alpha }}\right)
^{2}\left\vert A_{k}\setminus I\right\vert _{\omega }\lesssim \mathcal{A
_{2}^{\alpha ,\ast }\left\vert I\right\vert _{\sigma }$ is already proved
above in the display estimating $D_{\limfunc{disjoint}}$.
\medskip
Finally, for term $C$ we will have to group the intervals $M$ into blocks
B_{i}$. We first split the sum according to whether or not $I$ intersects
the triple of $M$
\begin{eqnarray*}
C &\approx &\left\{ \sum_{\substack{ M:\ I\cap 3M=\emptyset \\ \ell \left(
M\right) >\ell \left( I\right) }}+\sum_{\substack{ M:\ I\subset 3M\setminus
M \\ \ell \left( M\right) >\ell \left( I\right) }}\right\} \left( \frac
\left\vert M\right\vert }{\left( \left\vert M\right\vert +d\left( M,I\right)
\right) ^{2-\alpha }}\left\vert I\right\vert _{\sigma }\right) ^{2}\sum
_{\substack{ F\in \mathcal{F}: \\ M\in \mathcal{W}\left( F\right) }
\left\Vert \mathsf{Q}_{F,M}^{\omega \mathbf{b}^{\ast }}\frac{x}{\left\vert
M\right\vert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&=&C_{1}+C_{2}.
\end{eqnarray*
We first consider $C_{1}$. Let $\mathcal{M}$ consist of the maximal dyadic
intervals in the collection $\left\{ Q:3Q\cap I=\emptyset \right\} $, and
then let $\left\{ B_{i}\right\} _{i=1}^{\infty }$ be an enumeration of those
$Q\in \mathcal{M}$ whose side length is at least $\ell \left( I\right) $.
Note in particular that $3B_{i}\cap I=\emptyset $. Now we further decompose
the sum in $C_{1}$ by grouping the intervals $M$ into the `Whitney'
intervals $B_{i}$, and then using the pairwise disjointedness of the
martingale supports of the pseudoprojections $\mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}$ in $F$:
\begin{eqnarray*}
C_{1} &\leq &\sum_{i=1}^{\infty }\sum_{M:\ M\subset B_{i}}\left( \frac{1}
\left( \left\vert M\right\vert +d\left( M,I\right) \right) ^{2-\alpha }
\left\vert I\right\vert _{\sigma }\right) ^{2}\sum_{\substack{ F\in \mathcal
F}: \\ M\in \mathcal{W}\left( F\right) }}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&\lesssim &\sum_{i=1}^{\infty }\left( \frac{1}{\left( \left\vert
B_{i}\right\vert +d\left( B_{i},I\right) \right) ^{2-\alpha }}\left\vert
I\right\vert _{\sigma }\right) ^{2}\sum_{M:\ M\subset B_{i}}\sum_{\substack{
F\in \mathcal{F}: \\ M\in \mathcal{W}\left( F\right) }}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2} \\
&\lesssim &\sum_{i=1}^{\infty }\left( \frac{1}{\left( \left\vert
B_{i}\right\vert +d\left( B_{i},I\right) \right) ^{2-\alpha }}\left\vert
I\right\vert _{\sigma }\right) ^{2}\sum_{M:\ M\subset B_{i}}\left\vert
M\right\vert ^{2}\left\vert M\right\vert _{\omega } \\
&\lesssim &\sum_{i=1}^{\infty }\left( \frac{1}{\left( \left\vert
B_{i}\right\vert +d\left( B_{i},I\right) \right) ^{2-\alpha }}\left\vert
I\right\vert _{\sigma }\right) ^{2}\mathbf{\ }\left\vert B_{i}\right\vert
^{2}\left\vert B_{i}\right\vert _{\omega } \\
&\lesssim &\left\{ \sum_{i=1}^{\infty }\frac{\left\vert B_{i}\right\vert
_{\omega }\left\vert I\right\vert _{\sigma }}{\left\vert B_{i}\right\vert
^{2\left( 1-\alpha \right) }}\right\} \left\vert I\right\vert _{\sigma }\ ,
\end{eqnarray*
Now since $\left\vert B_{i}\right\vert \approx d\left( x,I\right) $ for
x\in B_{i}$,
\begin{eqnarray*}
\sum_{i=1}^{\infty }\frac{\left\vert B_{i}\right\vert _{\omega }\left\vert
I\right\vert _{\sigma }}{\left\vert B_{i}\right\vert ^{2\left( 1-\alpha
\right) }} &=&\frac{\left\vert I\right\vert _{\sigma }}{\left\vert
I\right\vert ^{1-\alpha }}\sum_{i=1}^{\infty }\frac{\left\vert I\right\vert
^{1-\alpha }}{\left\vert B_{i}\right\vert ^{2\left( 1-\alpha \right) }
\left\vert B_{i}\right\vert _{\omega } \\
&\approx &\frac{\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert
^{1-\alpha }}\sum_{i=1}^{\infty }\int_{B_{i}}\frac{\left\vert I\right\vert
^{1-\alpha }}{d\left( x,I\right) ^{2\left( 1-\alpha \right) }}d\omega \left(
x\right) \\
&\approx &\frac{\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert
^{1-\alpha }}\sum_{i=1}^{\infty }\int_{B_{i}}\left( \frac{\left\vert
I\right\vert }{\left[ \left\vert I\right\vert +d\left( x,I\right) \right]
^{2}}\right) ^{1-\alpha }d\omega \left( x\right) \\
&\leq &\frac{\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert
^{1-\alpha }}\mathcal{P}^{\alpha }\left( I,\mathbf{1}_{I^{c}}\omega \right)
\leq \mathcal{A}_{2}^{\alpha ,\ast },
\end{eqnarray*
we obtai
\begin{equation*}
C_{1}\lesssim \mathcal{A}_{2}^{\alpha ,\ast }\left\vert I\right\vert
_{\sigma }\ .
\end{equation*}
Next we turn to estimating term $C_{2}$ where the triple of $M$ contains $I$
but $M$ itself does not. Note that there are at most two such intervals $M$
of a given side length. So with this in mind, we sum over the intervals $M$
according to their lengths to obtai
\begin{eqnarray*}
C_{2} &=&\sum_{m=1}^{\infty }\sum_{\substack{ M:\ I\subset 3M\setminus M \\
\ell \left( M\right) =2^{m}\ell \left( I\right) }}\left( \frac{\left\vert
M\right\vert }{\left( \left\vert M\right\vert +\limfunc{dist}\left(
M,I\right) \right) ^{2-\alpha }}\left\vert I\right\vert _{\sigma }\right)
^{2}\sum_{\substack{ F\in \mathcal{F}: \\ M\in \mathcal{W}\left( F\right) }
\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}\frac{x}{\left\vert
M\right\vert }\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&\lesssim &\sum_{m=1}^{\infty }\left( \frac{\left\vert I\right\vert _{\sigma
}}{\left\vert 2^{m}I\right\vert ^{1-\alpha }}\right) ^{2}\mathbf{\
\left\vert \left( 5\cdot 2^{m}I\right) \setminus I\right\vert _{\omega
}=\left\{ \frac{\left\vert I\right\vert _{\sigma }}{\left\vert I\right\vert
^{1-\alpha }}\sum_{m=1}^{\infty }\frac{\left\vert I\right\vert ^{1-\alpha
}\left\vert \left( 5\cdot 2^{m}I\right) \setminus I\right\vert _{\omega }}
\left\vert 2^{m}I\right\vert ^{2\left( 1-\alpha \right) }}\right\}
\left\vert I\right\vert _{\sigma } \\
&\lesssim &\left\{ \frac{\left\vert I\right\vert _{\sigma }}{\left\vert
I\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left( I,\mathbf{1
_{I^{c}}\omega \right) \right\} \left\vert I\right\vert _{\sigma }\leq
\mathcal{A}_{2}^{\alpha ,\ast }\left\vert I\right\vert _{\sigma },
\end{eqnarray*
since in analogy with the corresponding estimate above
\begin{equation*}
\sum_{m=1}^{\infty }\frac{\left\vert I\right\vert ^{1-\alpha }\left\vert
\left( 5\cdot 2^{m}I\right) \setminus I\right\vert _{\omega }}{\left\vert
2^{m}I\right\vert ^{2\left( 1-\alpha \right) }}=\int \sum_{m=1}^{\infty
\frac{\left\vert I\right\vert ^{1-\alpha }}{\left\vert 2^{m}I\right\vert
^{2\left( 1-\alpha \right) }}\mathbf{1}_{\left( 5\cdot 2^{m}I\right)
\setminus I}\left( x\right) \ d\omega \left( x\right) \lesssim \mathcal{P
^{\alpha }\left( I,\mathbf{1}_{I^{c}}\omega \right) .
\end{equation*}
\subsection{The backward Poisson testing inequality\label{Subsec back test}}
The argument here follows the broad outline of the analogous argument in
\cite{SaShUr7}, but using modifications from \cite{SaShUr9} that involve
`prepare to puncture arguments', using decompositions $\mathcal{W}\left(
F\right) $ in place of $\left( \mathbf{\rho },\varepsilon \right)
-decompositions, and using pseudoprojections $\mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x$ (see (\ref{def F,K}) for the definition). The final
change here is that there is no splitting into local and global parts as in
\cite{SaShUr7} - instead, we follow the treatment in \cite{SaShUr6} in this
regard.
Fix $I\in \mathcal{D}$. It suffices to prov
\begin{equation}
\mathbf{Back}\left( \widehat{I}\right) \equiv \int_{\mathbb{R}}\left[
\mathbb{Q}^{\alpha }\left( t\mathbf{1}_{\widehat{I}}\overline{\mu }\right)
\left( y\right) \right] ^{2}d\sigma (y)\lesssim \left\{ \mathcal{A
_{2}^{\alpha }+\left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha
\limfunc{energy}}}\right) \sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\right\}
\int_{\widehat{I}}t^{2}d\overline{\mu }(x,t). \label{e.t2 n'}
\end{equation
Note that for a `Poisson integral with holes' and a measure $\mu $ built
with Haar projections, Hyt\"{o}nen obtained in \cite{Hyt2} the simpler bound
$A_{2}^{\alpha }$ for a term analogous to, but significantly smaller than,
\ref{e.t2 n'}). Here is a brief schematic diagram of the decompositions,
with bounds in $\fbox{}$, used in this subsection
\begin{equation*}
\fbox{
\begin{array}{ccccc}
\mathbf{Back}\left( \widehat{I}\right) & & & & \\
\downarrow & & & & \\
U_{s} & & & & \\
\downarrow & & & & \\
T_{s}^{\limfunc{proximal}} & + & V_{s}^{\limfunc{remote}} & & \\
\fbox{
\begin{array}{c}
\mathcal{A}_{2}^{\alpha }+ \\
\left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha ,\limfunc{energy}}
\right) \sqrt{A_{2}^{\alpha ,\limfunc{punct}}
\end{array
$} & & \downarrow & & \\
& & \downarrow & & \\
& & T_{s}^{\limfunc{difference}} & + & T_{s}^{\limfunc{intersection}} \\
& & \fbox{
\begin{array}{c}
\mathcal{A}_{2}^{\alpha }+ \\
\left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha ,\limfunc{energy}}
\right) \sqrt{A_{2}^{\alpha ,\limfunc{punct}}
\end{array
$} & & \fbox{$\left( \mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha
\limfunc{energy}}}\right) \sqrt{A_{2}^{\alpha ,\limfunc{punct}}}$
\end{array
$}.
\end{equation*
Using (\ref{tent consequence}) we see that the integral on the right hand
side of (\ref{e.t2 n'}) is
\begin{equation}
\int_{\widehat{I}}t^{2}d\overline{\mu }=\sum_{F\in \mathcal{F}}\sum_{M\in
\mathcal{W}\left( F\right) :\ M\subset I}\lVert \mathsf{Q}_{F,M}^{\omega
\mathbf{b}^{\ast }}x\rVert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\,.
\label{mu I hat}
\end{equation
where $\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}$ was defined in (\re
{def F,K}).
We now compute using (\ref{tent consequence}) again that
\begin{eqnarray}
\mathbb{Q}^{\alpha }\left( t\mathbf{1}_{\widehat{I}}\overline{\mu }\right)
\left( y\right) &=&\int_{\widehat{I}}\frac{t^{2}}{\left( t^{2}+\left\vert
x-y\right\vert ^{2}\right) ^{\frac{2-\alpha }{2}}}d\overline{\mu }\left(
x,t\right) \label{PI hat} \\
&\approx &\sum_{F\in \mathcal{F}}\sum_{M\in \mathcal{W}\left( F\right) :\
M\subset I}\frac{\lVert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\rVert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left( \left\vert
M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2-\alpha }}, \notag
\end{eqnarray
and then expand the square and integrate to obtain that the term $\mathbf
Back}\left( \widehat{I}\right) $ is
\begin{equation*}
\sum_{\substack{ F\in \mathcal{F} \\ M\in \mathcal{W}\left( F\right) \\
M\subset I}}\sum_{\substack{ F^{\prime }\in \mathcal{F}: \\ M^{\prime }\in
\mathcal{W}\left( F^{\prime }\right) \\ M^{\prime }\subset I}}\int_{\mathbb
R}}\frac{\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2-\alpha }
\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) .
\end{equation*}
By symmetry we may assume that $\ell \left( M^{\prime }\right) \leq \ell
\left( M\right) $. We fix a nonnegative integer $s$, and consider those
intervals $M$ and $M^{\prime }$ with $\ell \left( M^{\prime }\right)
=2^{-s}\ell \left( M\right) $. For fixed $s$ we will control the expression
\begin{eqnarray}
U_{s} &\equiv &\sum_{\substack{ F,F^{\prime }\in \mathcal{F}}}\sum
_{\substack{ M\in \mathcal{W}\left( F\right) ,\ M^{\prime }\in \mathcal{W
\left( F^{\prime }\right) \\ M,M^{\prime }\subset I,\ \ell \left( M^{\prime
}\right) =2^{-s}\ell \left( M\right) }} \label{def Us} \\
&&\times \int_{\mathbb{R}}\frac{\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf
b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right)
^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2
}{\left( \left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime
}}\right\vert \right) ^{2-\alpha }}d\sigma \left( y\right) , \notag
\end{eqnarray
by proving tha
\begin{equation}
U_{s}\lesssim 2^{-\delta s}\left\{ \mathcal{A}_{2}^{\alpha }+\left(
\mathfrak{E}_{2}^{\alpha }+\sqrt{A_{2}^{\alpha ,\limfunc{energy}}}\right)
\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\right\} \int_{\widehat{I}}t^{2}
\overline{\mu },\ \ \ \ \ \text{where }\delta =\frac{1}{2}. \label{Us bound}
\end{equation
With this accomplished, we can sum in $s\geq 0$ to control the term $\mathbf
Back}\left( \widehat{I}\right) $. We now decompose $U_{s}=T_{s}^{\limfunc
proximal}}+T_{s}^{\limfunc{difference}}+T_{s}^{\limfunc{intersection}}$ into
three pieces.
Our first decomposition is to writ
\begin{equation}
U_{s}=T_{s}^{\limfunc{proximal}}+V_{s}^{\limfunc{remote}}\ ,
\label{initial decomp}
\end{equation
where in the `proximal' term $T_{s}^{\limfunc{proximal}}$ we restrict the
summation over pairs of intervals $M,M^{\prime }$ to those satisfying
d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) $;
while in the `remote' term $V_{s}^{\limfunc{remote}}$ we restrict the
summation over pairs of intervals $M,M^{\prime }$ to those satisfying the
opposite inequality $d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta
}\ell \left( M\right) $. Then we further decompose
\begin{equation*}
V_{s}^{\limfunc{remote}}=T_{s}^{\limfunc{difference}}+T_{s}^{\limfunc
intersection}},
\end{equation*
where in the `difference' term $T_{s}^{\limfunc{difference}}$ we restrict
integration in $y$ to the difference $\mathbb{R}\setminus B\left(
M,M^{\prime }\right) $ of $\mathbb{R}$ and
\begin{equation}
B\left( M,M^{\prime }\right) \equiv B\left( c_{M},\frac{1}{2}d\left(
c_{M},c_{M^{\prime }}\right) \right) , \label{def BMM'}
\end{equation
the ball centered at $c_{M}$ with radius $\frac{1}{2}d\left(
c_{M},c_{M^{\prime }}\right) $; while in the `intersection' term $T_{s}^
\limfunc{intersection}}$ we restrict integration in $y$ to the intersection
\mathbb{R}\cap B\left( M,M^{\prime }\right) $ of $\mathbb{R}$ with the ball
B\left( M,M^{\prime }\right) $; i.e.
\begin{eqnarray}
T_{s}^{\limfunc{intersection}} &\equiv &\sum_{\substack{ F,F^{\prime }\in
\mathcal{F}}}\sum_{\substack{ M\in \mathcal{W}\left( F\right) ,\ M^{\prime
}\in \mathcal{W}\left( F^{\prime }\right) \\ M,M^{\prime }\subset I,\ \ell
\left( M^{\prime }\right) =2^{-s}\ell \left( M\right) \\ d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\left( 1+\delta \right) }\ell \left(
M^{\prime }\right) }} \label{def Tints} \\
&&\times \int_{B\left( M,M^{\prime }\right) }\frac{\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}}{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2-\alpha }}\frac{\left\Vert \mathsf{Q
_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left( \left\vert M^{\prime
}\right\vert +\left\vert y-c_{M^{\prime }}\right\vert \right) ^{2-\alpha }
d\sigma \left( y\right) . \notag
\end{eqnarray
Here is a schematic reminder of the these decompositions with the
distinguishing points of the definitions boxed:{}
\begin{equation*}
\fbox{
\begin{array}{ccccc}
U_{s} & & & & \\
\downarrow & & & & \\
T_{s}^{\limfunc{proximal}} & + & V_{s}^{\limfunc{remote}} & & \\
\fbox{$d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left(
M\right) $} & & \fbox{$d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta
}\ell \left( M\right) $} & & \\
& & \downarrow & & \\
& & T_{s}^{\limfunc{difference}} & + & T_{s}^{\limfunc{intersection}} \\
& & \fbox{$\int_{\mathbb{R}\setminus B\left( M,M^{\prime }\right) }$} & &
\fbox{$\fbox{$\int_{B\left( M,M^{\prime }\right) }$}$
\end{array
$}.
\end{equation*}
We will exploit the restriction of integration to $B\left( M,M^{\prime
}\right) $, together with the condition
\begin{equation*}
d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\left( 1+\delta \right) }\ell
\left( M^{\prime }\right) =2^{s\delta }\ell \left( M\right) ,
\end{equation*
which will then give an estimate for the term $T_{s}^{\limfunc{intersection
} $ using an argument dual to that used for the other terms $T_{s}^{\limfunc
proximal}}$ and $T_{s}^{\limfunc{difference}}$, to which we now turn.
\subsubsection{The proximal and difference terms}
We hav
\begin{align}
T_{s}^{\limfunc{proximal}}& \equiv \sum_{\substack{ F,F^{\prime }\in
\mathcal{F}}}\sum_{\substack{ M\in \mathcal{W}\left( F\right) ,\ M^{\prime
}\in \mathcal{W}\left( F^{\prime }\right) \\ M,M^{\prime }\subset I,\ \ell
\left( M^{\prime }\right) =2^{-s}\ell \left( M\right) \text{ and }d\left(
c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) }}
\label{def Tproxs} \\
& \times \int_{\mathbb{R}}\frac{\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf
b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right)
^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2
}{\left( \left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime
}}\right\vert \right) ^{2-\alpha }}d\sigma \left( y\right) \notag \\
& \leq \mathcal{M}_{s}^{\limfunc{proximal}}\sum_{F\in \mathcal{F}}\sum
_{\substack{ M\in \mathcal{W}\left( F\right) \\ M\subset I}}\lVert \mathsf{
}_{F,M}^{\omega ,\mathbf{b}^{\ast }}z\rVert _{\omega }^{\spadesuit 2}
\mathcal{M}_{s}^{\limfunc{proximal}}\int_{\widehat{I}}t^{2}d\overline{\mu },
\notag
\end{align
wher
\begin{align*}
\mathcal{M}_{s}^{\limfunc{proximal}}& \equiv \sup_{F\in \mathcal{F}}\sup
_{\substack{ M\in \mathcal{W}\left( F\right) \\ M\subset I}}\mathcal{A
_{s}^{\limfunc{proximal}}\left( M\right) ; \\
\mathcal{A}_{s}^{\limfunc{proximal}}\left( M\right) & \equiv \sum_{F^{\prime
}\in \mathcal{F}}\sum_{\substack{ M^{\prime }\in \mathcal{W}\left( F^{\prime
}\right) \\ M^{\prime }\subset I,\ \ell \left( M^{\prime }\right)
=2^{-s}\ell \left( M\right) \text{ and }d\left( c_{M},c_{M^{\prime }}\right)
<2^{s\delta }\ell \left( M\right) }}\int_{\mathbb{R}}S_{\left( M^{\prime
},M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right) ; \\
S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( x\right) & \equiv \frac
1}{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right)
^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega
}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }},
\end{align*
and similarl
\begin{align}
T_{s}^{\limfunc{difference}}& \equiv \sum_{\substack{ F,F^{\prime }\in
\mathcal{F}}}\sum_{\substack{ M\in \mathcal{W}\left( F\right) ,\ M^{\prime
}\in \mathcal{W}\left( F^{\prime }\right) \\ M,M^{\prime }\subset I,\ \ell
\left( M^{\prime }\right) =2^{-s}\ell \left( M\right) \text{ and }d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }}
\label{def Tdiffs} \\
& \times \int_{\mathbb{R}\setminus B\left( M,M^{\prime }\right) }\frac
\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left( \left\vert
M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2-\alpha }}\frac
\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) \notag \\
& \leq \mathcal{M}_{s}^{\limfunc{difference}}\sum_{F\in \mathcal{F}}\sum
_{\substack{ M\in \mathcal{W}\left( F\right) \\ M\subset I}}\lVert \mathsf{
}_{F,M}^{\omega ,\mathbf{b}^{\ast }}z\rVert _{\omega }^{\spadesuit 2}
\mathcal{M}_{s}^{\limfunc{difference}}\int_{\widehat{I}}t^{2}d\overline{\mu
; \notag
\end{align
wher
\begin{eqnarray*}
\mathcal{M}_{s}^{\limfunc{difference}} &\equiv &\sup_{F\in \mathcal{F}}\sup
_{\substack{ _{\substack{ M\in \mathcal{W}\left( F\right) }} \\ M\subset I}
\mathcal{A}_{s}^{\func{difference}}\left( M\right) ; \\
\mathcal{A}_{s}^{\limfunc{difference}}\left( M\right) &\equiv
&\sum_{F^{\prime }\in \mathcal{F}}\sum_{\substack{ M^{\prime }\in \mathcal{W
\left( F^{\prime }\right) \\ M^{\prime }\subset I,\ \ell \left( M^{\prime
}\right) =2^{-s}\ell \left( M\right) \text{ and }d\left( c_{M},c_{M^{\prime
}}\right) \geq 2^{s\delta }\ell \left( M\right) }}\int_{\mathbb{R}\setminus
B\left( M,M^{\prime }\right) }S_{\left( M^{\prime },M\right) }^{F^{\prime
}}\left( y\right) d\sigma \left( y\right) .
\end{eqnarray*
The restriction of integration in $\mathcal{A}_{s}^{\limfunc{difference}}$
to $\mathbb{R}\setminus B\left( M,M^{\prime }\right) $ will be used to
establish (\ref{vanishing close}) below.
\begin{notation}
\label{Sum *}Since the intervals $F,M,F^{\prime },M^{\prime }$ that arise in
all of the sums here satisfy
\begin{equation*}
M\in \mathcal{W}\left( F\right) ,\ M^{\prime }\in \mathcal{W}\left(
F^{\prime }\right) \text{ and }\ell \left( M^{\prime }\right) =2^{-s}\ell
\left( M\right) \text{ and }M,M^{\prime }\subset I,
\end{equation*
we will often employ the notation $\overset{\ast }{\sum }$ to remind the
reader that, as applicable, these four conditions are in force even when
they are\ not explictly mentioned.
\end{notation}
Now fix $M$ as in $\mathcal{M}_{s}^{\limfunc{proximal}}$ respectively
\mathcal{M}_{s}^{\limfunc{difference}}$, and decompose the sum over
M^{\prime }$ in $\mathcal{A}_{s}^{\limfunc{proximal}}\left( M\right) $
respectively $\mathcal{A}_{s}^{\limfunc{difference}}\left( M\right) $ b
\begin{eqnarray*}
&&\mathcal{A}_{s}^{\limfunc{proximal}}\left( M\right) =\sum_{F^{\prime }\in
\mathcal{F}}\sum_{\substack{ M^{\prime }\in \mathcal{W}\left( F^{\prime
}\right) \\ M^{\prime }\subset I,\ \ell \left( M^{\prime }\right)
=2^{-s}\ell \left( M\right) \text{ and }d\left( c_{M},c_{M^{\prime }}\right)
<2^{s\delta }\ell \left( M\right) }}\int_{\mathbb{R}}S_{\left( M^{\prime
},M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&=&\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2M \\ d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta
}\ell \left( M\right) }}}\int_{\mathbb{R}}S_{\left( M^{\prime },M\right)
}^{F^{\prime }}\left( y\right) d\sigma \left( y\right) +\sum_{F^{\prime }\in
\mathcal{F}}\sum_{\ell =1}^{\infty }\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) }}}\int_
\mathbb{R}}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( y\right)
d\sigma \left( y\right) \\
&\equiv &\sum_{\ell =0}^{\infty }\mathcal{A}_{s}^{\limfunc{proximal},\ell
}\left( M\right) ,
\end{eqnarray*
respectivel
\begin{eqnarray*}
&&\mathcal{A}_{s}^{\limfunc{difference}}\left( M\right) =\sum_{F^{\prime
}\in \mathcal{F}}\sum_{\substack{ M^{\prime }\in \mathcal{W}\left( F^{\prime
}\right) \\ M^{\prime }\subset I,\ \ell \left( M^{\prime }\right)
=2^{-s}\ell \left( M\right) \text{ and }d\left( c_{M},c_{M^{\prime }}\right)
\geq 2^{s\delta }\ell \left( M\right) }}\int_{\mathbb{R}\setminus B\left(
M,M^{\prime }\right) }S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left(
y\right) d\sigma \left( y\right) \\
&=&\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2M \\ d\left( c_{M},c_{M^{\prime }}\right) \geq
2^{s\delta }\ell \left( M\right) }}}\int_{\mathbb{R}\setminus B\left(
M,M^{\prime }\right) }S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left(
y\right) d\sigma \left( y\right) \\
&&+\sum_{\ell =1}^{\infty }\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{\substack{ c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\
d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }
}\int_{\mathbb{R}\setminus B\left( M,M^{\prime }\right) }S_{\left( M^{\prime
},M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&\equiv &\sum_{\ell =0}^{\infty }\mathcal{A}_{s}^{\limfunc{difference},\ell
}\left( M\right) .
\end{eqnarray*
Let $m=2$ so that
\begin{equation}
2^{-m}\leq \frac{1}{3}. \label{smallest m}
\end{equation
Now decompose the integrals over $\mathbb{R}$ in $\mathcal{A}_{s}^{\limfunc
proximal},\ell }\left( M\right) $ b
\begin{eqnarray*}
\mathcal{A}_{s}^{\limfunc{proximal},0}\left( M\right) &=&\sum_{F^{\prime
}\in \mathcal{F}}\overset{\ast }{\sum_{\substack{ c_{M^{\prime }}\in 2M \\
d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) }}
\int_{\mathbb{R}\setminus 4M}S_{\left( M^{\prime },M\right) }^{F^{\prime
}}\left( y\right) d\sigma \left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2M \\ d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta
}\ell \left( M\right) }}}\int_{4M}S_{\left( M^{\prime },M\right)
}^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&\equiv &\mathcal{A}_{s,far}^{\limfunc{proximal},0}\left( M\right) +\mathcal
A}_{s,near}^{\limfunc{proximal},0}\left( M\right) ,
\end{eqnarray*
an
\begin{eqnarray*}
\mathcal{A}_{s}^{\limfunc{proximal},\ell }\left( M\right) &=&\sum_{F^{\prime
}\in \mathcal{F}}\overset{\ast }{\sum_{\substack{ c_{M^{\prime }}\in 2^{\ell
+1}M\setminus 2^{\ell }M \\ d\left( c_{M},c_{M^{\prime }}\right)
<2^{s\delta }\ell \left( M\right) }}}\int_{\mathbb{R}\setminus 2^{\ell
+2}M}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( y\right) d\sigma
\left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) }}
\int_{2^{\ell +2}M\setminus 2^{\ell -m}M}S_{\left( M^{\prime },M\right)
}^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) <2^{s\delta }\ell \left( M\right) }}
\int_{2^{\ell -m}M}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left(
y\right) d\sigma \left( y\right) \\
&\equiv &\mathcal{A}_{s,far}^{\limfunc{proximal},\ell }\left( M\right)
\mathcal{A}_{s,near}^{\limfunc{proximal},\ell }\left( M\right) +\mathcal{A
_{s,close}^{\limfunc{proximal},\ell }\left( M\right) ,\ \ \ \ \ \ell \geq 1.
\end{eqnarray*
Similarly we decompose the integrals over the difference
\begin{equation*}
B^{\ast }\equiv \mathbb{R}\setminus B\left( M,M^{\prime }\right)
\end{equation*
in $\mathcal{A}_{s}^{\limfunc{difference},\ell }\left( M\right) $ b
\begin{eqnarray*}
\mathcal{A}_{s}^{\limfunc{difference},0}\left( M\right) &=&\sum_{F^{\prime
}\in \mathcal{F}}\overset{\ast }{\sum_{\substack{ c_{M^{\prime }}\in 2M \\
d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }
}\int_{B^{\ast }\setminus 4M}S_{\left( M^{\prime },M\right) }^{F^{\prime
}}\left( y\right) d\sigma \left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2M \\ d\left( c_{M},c_{M^{\prime }}\right) \geq
2^{s\delta }\ell \left( M\right) }}}\int_{B^{\ast }\cap 4M}S_{\left(
M^{\prime },M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right)
\\
&\equiv &\mathcal{A}_{s,far}^{\limfunc{difference},0}\left( M\right)
\mathcal{A}_{s,near}^{\limfunc{difference},0}\left( M\right) ,
\end{eqnarray*
an
\begin{eqnarray*}
\mathcal{A}_{s}^{\limfunc{difference},\ell }\left( M\right)
&=&\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }}
\int_{B^{\ast }\setminus 2^{\ell +2}M}S_{\left( M^{\prime },M\right)
}^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }}
\int_{B^{\ast }\cap \left( 2^{\ell +2}M\setminus 2^{\ell -m}M\right)
}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( y\right) d\sigma
\left( y\right) \\
&&+\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{\substack{
c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M \\ d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\delta }\ell \left( M\right) }}
\int_{B^{\ast }\cap 2^{\ell -m}M}S_{\left( M^{\prime },M\right) }^{F^{\prime
}}\left( y\right) d\sigma \left( y\right) \\
&\equiv &\mathcal{A}_{s,far}^{\limfunc{difference},\ell }\left( M\right)
\mathcal{A}_{s,near}^{\limfunc{difference},\ell }\left( M\right) +\mathcal{A
_{s,close}^{\limfunc{difference},\ell }\left( M\right) ,\ \ \ \ \ \ell \geq
1.
\end{eqnarray*}
We now note the important point that the close terms $\mathcal{A}_{s,close}^
\limfunc{proximal},\ell }\left( M\right) $ and $\mathcal{A}_{s,close}^
\limfunc{difference},\ell }\left( M\right) $ both $\emph{vanish}$ for $\ell
>\delta s$ because of the decomposition (\ref{initial decomp})
\begin{equation}
\mathcal{A}_{s,close}^{\limfunc{proximal},\ell }\left( M\right) =\mathcal{A
_{s,close}^{\limfunc{difference},\ell }\left( M\right) =0,\ \ \ \ \ \ell
\geq 1+\delta s. \label{vanishing close}
\end{equation
Indeed, if $c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M$, then we
hav
\begin{equation}
\frac{1}{2}2^{\ell }\ell \left( M\right) \leq d\left( c_{M},c_{M^{\prime
}}\right) . \label{distJJ'}
\end{equation
Now the summands in $\mathcal{A}_{s,close}^{\limfunc{proximal},\ell }\left(
M\right) $ satisfy $d\left( c_{M},c_{M^{\prime }}\right) <2^{\delta s}\ell
\left( M\right) $, which by (\ref{distJJ'}) is impossible if $\ell \geq
1+\delta s$ - indeed, if $\ell \geq 1+\delta s$, we get the contradictio
\begin{equation*}
2^{\delta s}\ell \left( M\right) =\frac{1}{2}2^{1+\delta s}\ell \left(
M\right) \leq \frac{1}{2}2^{\ell }\ell \left( M\right) \leq d\left(
c_{M},c_{M^{\prime }}\right) <2^{\delta s}\ell \left( M\right) .
\end{equation*
It now follows that $\mathcal{A}_{s,close}^{\limfunc{proximal},\ell }\left(
M\right) =0$. Thus we are left to consider the term $\mathcal{A}_{s,close}^
\limfunc{difference},\ell }\left( M\right) $, where the integration is taken
over the set $\mathbb{R}\setminus B\left( M,M^{\prime }\right) $. But we are
also restricted in $\mathcal{A}_{s,close}^{\limfunc{difference},\ell }\left(
M\right) $ to integrating over the interval $2^{\ell -m}M$, which is
contained in $B\left( M,M^{\prime }\right) $ by (\ref{distJJ'}). Indeed, the
smallest\ ball centered at $c_{M}$ that contains $2^{\ell -m}M$ has radius
\frac{1}{2}2^{\ell -m}\ell \left( M\right) $, which by (\ref{smallest m})
and (\ref{distJJ'}) is at most $\frac{1}{4}2^{\ell }\ell \left( M\right)
\leq \frac{1}{2}d\left( c_{M},c_{M^{\prime }}\right) $, the radius of
B\left( M,M^{\prime }\right) $. Thus the range of integration in the term
\mathcal{A}_{s,close}^{\limfunc{difference},\ell }\left( M\right) $ is the
empty set, and so $\mathcal{A}_{s,close}^{\limfunc{difference},\ell }\left(
M\right) =0$ as well as $\mathcal{A}_{s,close}^{\limfunc{proximal},\ell
}\left( M\right) =0$. This proves (\ref{vanishing close}).
From now on we treat $T_{s}^{\limfunc{proximal}}$ and $T_{s}^{\limfunc
difference}}$ in the same way since the terms $\mathcal{A}_{s,close}^
\limfunc{proximal},\ell }\left( M\right) $ and $\mathcal{A}_{s,close}^
\limfunc{difference},\ell }\left( M\right) $ both vanish for $\ell \geq
1+\delta s$. Thus we will suppress the superscripts $\limfunc{proximal}$ and
$\limfunc{difference}$ in the $far$, $near$ and $close$ decomposition of
\mathcal{A}_{s}^{\limfunc{proximal},\ell }\left( M\right) $ and $\mathcal{A
_{s}^{\limfunc{difference},\ell }\left( M\right) $, and we will also
suppress the conditions $d\left( c_{M},c_{M^{\prime }}\right) <2^{s\delta
}\ell \left( M\right) $ and $d\left( c_{M},c_{M^{\prime }}\right) \geq
2^{s\delta }\ell \left( M\right) $ in the proximal and difference terms
since they no longer play a role. Using the pairwise disjointedness of the
shifted coronas $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$, we have
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}}\left\Vert \mathsf{Q}_{F^{\prime
},A}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim \left\vert A\right\vert ^{2}\left\vert A\right\vert
_{\omega }\ ,\ \ \ \ \ \text{for any interval }A.
\end{equation*
Note that if $c_{M^{\prime }}\in 2M$, then $M^{\prime }\subset 3M$. Then
with
\begin{equation}
\mathcal{W}_{M}^{s}\equiv \dbigcup\limits_{F^{\prime }\in \mathcal{F
}\left\{ M^{\prime }\in \mathcal{W}\left( F^{\prime }\right) :M^{\prime
}\subset 3M\text{ and }\ell \left( M^{\prime }\right) =2^{-s}\ell \left(
M\right) \right\} , \label{def WMs}
\end{equation
we hav
\begin{eqnarray*}
\mathcal{A}_{s,far}^{0}\left( M\right) &\leq &\sum_{F^{\prime }\in \mathcal{
}}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2M}}\int_{\mathbb{R}\setminus
4M}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( y\right) d\sigma
\left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{M}^{s}}\sum_{F^{\prime }\in \mathcal{F}:\
A\in \mathcal{W}\left( F^{\prime }\right) }\int_{\mathbb{R}\setminus 4M
\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2\left(
2-\alpha \right) }}d\sigma \left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{M}^{s}}\int_{\mathbb{R}\setminus 4M}\frac
\left\vert A\right\vert ^{2}\left\vert A\right\vert _{\omega }}{\left(
\left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2\left(
2-\alpha \right) }}d\sigma \left( y\right) \\
&=&\left( \sum_{A\in \mathcal{W}_{M}^{s}}\left\vert A\right\vert
^{2}\left\vert A\right\vert _{\omega }\right) \int_{\mathbb{R}\setminus 4M
\frac{1}{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert
\right) ^{2\left( 2-\alpha \right) }}d\sigma \left( y\right) .
\end{eqnarray*
Now we use the standard pigeonholing of side length of $A$ to conclude that
\begin{eqnarray}
\sum_{A\in \mathcal{W}_{M}^{s}}\left\vert A\right\vert ^{2}\left\vert
A\right\vert _{\omega } &=&\sum_{k=s}^{\infty }\sum_{A\in \mathcal{W
_{M}^{s}:\ \ell \left( A\right) =2^{-k}\ell \left( M\right) }\left\vert
A\right\vert ^{2}\left\vert A\right\vert _{\omega }\leq \sum_{k=s}^{\infty
}2^{-2k}\left\vert M\right\vert ^{2}\sum_{A\in \mathcal{W}_{M}^{s}:\ \ell
\left( A\right) =2^{-k}\ell \left( M\right) }\left\vert A\right\vert
_{\omega } \label{stan pig} \\
&\leq &\sum_{k=s}^{\infty }2^{-2k}\left\vert M\right\vert ^{2}\left\vert
3M\right\vert _{\omega }\lesssim 2^{-2s}\left\vert M\right\vert
^{2}\left\vert 3M\right\vert _{\omega }, \notag
\end{eqnarray
so that combining the previous two displays we hav
\begin{eqnarray*}
\mathcal{A}_{s,far}^{0}\left( M\right) &\lesssim &2^{-2s}\left\vert
M\right\vert ^{2}\left\vert 3M\right\vert _{\omega }\int_{\mathbb{R
\setminus 4M}\frac{1}{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma \left(
y\right) \\
&\leq &2^{-2s}\left\vert 4M\right\vert _{\omega }\int_{\mathbb{R}\setminus
4M}\frac{1}{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert
\right) ^{2\left( 1-\alpha \right) }}d\sigma \left( y\right) \\
&\approx &2^{-2s}\frac{\left\vert 4M\right\vert _{\omega }}{\left\vert
4M\right\vert ^{1-\alpha }}\int_{\mathbb{R}\setminus 4M}\left( \frac
\left\vert M\right\vert }{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2}}\right) ^{1-\alpha }d\sigma \left( y\right)
\\
&\lesssim &2^{-2s}\frac{\left\vert 4M\right\vert _{\omega }}{\left\vert
4M\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left( 4M,\mathbf{1}_
\mathbb{R}\setminus 4M}\sigma \right) \lesssim 2^{-2s}\mathcal{A
_{2}^{\alpha }\ .
\end{eqnarray*}
To estimate the near term $\mathcal{A}_{s,near}^{0}\left( M\right) $, we
initially keep the energy $\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}z\right\Vert _{L^{2}\left( \omega \right)
}^{2}$ and write
\begin{eqnarray}
\mathcal{A}_{s,near}^{0}\left( M\right) &\leq &\sum_{F^{\prime }\in \mathcal
F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2M}}\int_{4M}S_{\left( M^{\prime
},M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right)
\label{A0snear} \\
&\approx &\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{c_{M^{\prime }}\in 2M}}\int_{4M}\frac{1}{\left\vert M\right\vert
^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2
}{\left( \left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime
}}\right\vert \right) ^{2-\alpha }}d\sigma \left( y\right) \notag \\
&=&\sum_{F^{\prime }\in \mathcal{F}}\frac{1}{\left\vert M\right\vert
^{2-\alpha }}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2M}}\left\Vert \mathsf
Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\int_{4M}\frac{1}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) \notag \\
&=&\sum_{F^{\prime }\in \mathcal{F}}\frac{1}{\left\vert M\right\vert
^{2-\alpha }}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2M}}\left\Vert \mathsf
Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\frac{\mathrm{P}^{\alpha
}\left( M^{\prime },\mathbf{1}_{4M}\sigma \right) }{\left\vert M^{\prime
}\right\vert }. \notag
\end{eqnarray
In order to estimate the final sum above, we must invoke the `prepare to
puncture' argument above, as we will want to derive geometric decay from
\left\Vert \mathsf{Q}_{M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}$ by dominating it by the
`nonenergy' term $\left\vert M^{\prime }\right\vert ^{2}\left\vert M^{\prime
}\right\vert _{\omega }$, as well as using the Muckenhoupt energy constant.
Choose an augmented interval $\widetilde{M}\in \mathcal{AD}$ satisfying
\dbigcup\limits_{c_{M^{\prime }}\in 2M}M^{\prime }\subset 4M\subset
\widetilde{M}$ and $\ell \left( \widetilde{M}\right) \leq C\ell \left(
M\right) $. Define $\widetilde{\omega }=\omega -\omega \left( \left\{
p\right\} \right) \delta _{p}$ where $p$ is an atomic point in $\widetilde{M}
$ for which
\begin{equation*}
\omega \left( \left\{ p\right\} \right) =\sup_{q\in \mathfrak{P}_{\left(
\sigma ,\omega \right) }:\ q\in \widetilde{M}}\omega \left( \left\{
q\right\} \right) .
\end{equation*
(If $\omega $ has no atomic point in common with $\sigma $ in $\widetilde{M}
, set $\widetilde{\omega }=\omega $.) Then we have $\left\vert \widetilde{M
\right\vert _{\widetilde{\omega }}=\omega \left( \widetilde{M},\mathfrak{P
_{\left( \sigma ,\omega \right) }\right) $ an
\begin{equation*}
\frac{\left\vert \widetilde{M}\right\vert _{\widetilde{\omega }}}{\left\vert
\widetilde{M}\right\vert ^{1-\alpha }}\frac{\left\vert \widetilde{M
\right\vert _{\sigma }}{\left\vert \widetilde{M}\right\vert ^{1-\alpha }}
\frac{\omega \left( \widetilde{M},\mathfrak{P}_{\left( \sigma ,\omega
\right) }\right) }{\left\vert \widetilde{M}\right\vert ^{1-\alpha }}\frac
\left\vert \widetilde{M}\right\vert _{\sigma }}{\left\vert \widetilde{M
\right\vert ^{1-\alpha }}\leq A_{2}^{\alpha ,\limfunc{punct}}.
\end{equation*
From (\ref{key obs}) and (\ref{key fact}) we also hav
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}}\left\Vert \mathsf{Q}_{F^{\prime
},A}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\lesssim \ell \left( A\right) ^{2}\left\vert A\right\vert _
\widetilde{\omega }}\ ,\ \ \ \ \ \text{for any interval }A.
\end{equation*}
Now by Cauchy-Schwarz and the augmented local estimate (\ref{shifted local})
in Lemma \ref{shifted} with $M=\widetilde{M}$ applied to the second line
below, and with $\mathcal{W}_{M}^{s}$ as in (\ref{def WMs}), and noting (\re
{stan pig}), the last sum in (\ref{A0snear}) is dominated b
\begin{eqnarray}
&&\frac{1}{\left\vert M\right\vert ^{2-\alpha }}\left( \sum_{F^{\prime }\in
\mathcal{F}}\overset{\ast }{\sum_{c\left( M^{\prime }\right) \in 2M}
\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\right) ^{\frac{
}{2}} \label{dom by} \\
&&\ \ \ \ \ \ \ \ \ \ \times \left( \sum_{F^{\prime }\in \mathcal{F}}\overse
{\ast }{\sum_{c_{M^{\prime }}\in 2M}}\left\Vert \mathsf{Q}_{F^{\prime
},M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\left( \frac{\mathrm{P}^{\alpha }\left( M^{\prime }
\mathbf{1}_{4M}\sigma \right) }{\left\vert M^{\prime }\right\vert }\right)
^{2}\right) ^{\frac{1}{2}} \notag \\
&\lesssim &\frac{1}{\left\vert M\right\vert ^{2-\alpha }}\left( \sum_{A\in
\mathcal{W}_{M}^{s}}\left\vert A\right\vert ^{2}\left\vert A\right\vert _
\widetilde{\omega }}\right) ^{\frac{1}{2}}\sqrt{\left( \mathfrak{E
_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{\left\vert
\widetilde{M}\right\vert _{\sigma }} \notag \\
&\lesssim &\frac{2^{-s}\left\vert M\right\vert }{\left\vert M\right\vert
^{2-\alpha }}\sqrt{\left\vert 4M\right\vert _{\widetilde{\omega }}}\sqrt
\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy
}}\sqrt{\left\vert \widetilde{M}\right\vert _{\sigma }} \notag \\
&\lesssim &2^{-s}\sqrt{\left( \mathfrak{E}_{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{\frac{\left\vert \widetilde{M
\right\vert _{\widetilde{\omega }}}{\left\vert \widetilde{M}\right\vert
^{2-\alpha }}\frac{\left\vert \widetilde{M}\right\vert _{\sigma }}
\left\vert \widetilde{M}\right\vert ^{2-\alpha }}}\lesssim 2^{-s}\sqrt
\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy
}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\ . \notag
\end{eqnarray}
Similarly, for $\ell \geq 1$, we can estimate the far term $\mathcal{A
_{s,far}^{\ell }\left( M\right) $ by the argument used for $\mathcal{A
_{s,far}^{0}\left( M\right) $ but applied to $2^{\ell }M$ in place of $M$.
For this need the following variant of $\mathcal{W}_{M}^{s}$ in (\ref{def
WMs}) given b
\begin{equation}
\mathcal{W}_{M}^{s,\ell }\equiv \dbigcup\limits_{F^{\prime }\in \mathcal{F
}\left\{ M^{\prime }\in \mathcal{W}\left( F^{\prime }\right) :M^{\prime
}\subset 3\left( 2^{\ell }M\right) \text{ and }\ell \left( M^{\prime
}\right) =2^{-s-\ell }\ell \left( 2^{\ell }M\right) \right\} .
\label{def WMsell}
\end{equation
Then we have
\begin{eqnarray*}
\mathcal{A}_{s,far}^{\ell }\left( M\right) &\leq &\sum_{F^{\prime }\in
\mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in \left( 2^{\ell
+1}M\right) \setminus \left( 2^{\ell }M\right) }}\int_{\mathbb{R}\setminus
2^{\ell +2}M}S_{\left( M^{\prime },M\right) }^{F^{\prime }}\left( y\right)
d\sigma \left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{M}^{s,\ell }}\sum_{F^{\prime }\in
\mathcal{F}:\ A\in \mathcal{W}\left( F^{\prime }\right) }\int_{\mathbb{R
\setminus 4\left( 2^{\ell }M\right) }\frac{\left\Vert \mathsf{Q}_{F^{\prime
},A}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}}{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma \left(
y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{M}^{s,\ell }}\int_{\mathbb{R}\setminus
4\left( 2^{\ell }M\right) }\frac{\left\vert A\right\vert ^{2}\left\vert
A\right\vert _{\omega }}{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma \left(
y\right) \\
&=&\left( \sum_{A\in \mathcal{W}_{M}^{s,\ell }}\left\vert A\right\vert
^{2}\left\vert A\right\vert _{\omega }\right) \int_{\mathbb{R}\setminus
4\left( 2^{\ell }M\right) }\frac{1}{\left( \left\vert M\right\vert
+\left\vert y-c_{M}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma
\left( y\right) ,
\end{eqnarray*
where, just as for the sum over $A\in \mathcal{W}_{M}^{s,0}$, we hav
\begin{eqnarray}
&&\sum_{A\in \mathcal{W}_{M}^{s,\ell }}\left\vert A\right\vert
^{2}\left\vert A\right\vert _{\omega } \label{stan pig'} \\
&=&\sum_{k=s}^{\infty }\sum_{A\in \mathcal{W}_{M}^{s,\ell }:\ \ell \left(
A\right) =2^{-k-\ell }\ell \left( 2^{\ell }M\right) }\left\vert A\right\vert
^{2}\left\vert A\right\vert _{\omega }\leq \sum_{k=s}^{\infty }2^{-2k-2\ell
}\left\vert 2^{\ell }M\right\vert ^{2}\sum_{A\in \mathcal{W}_{M}^{s,\ell }:\
\ell \left( A\right) =2^{-k-\ell }\ell \left( 2^{\ell }M\right) }\left\vert
A\right\vert _{\omega } \notag \\
&\leq &\sum_{k=s}^{\infty }2^{-2k-2\ell }\left\vert 2^{\ell }M\right\vert
^{2}\left\vert 3\left( 2^{\ell }M\right) \right\vert _{\omega }\lesssim
2^{-2s-2\ell }\left\vert 2^{\ell }M\right\vert ^{2}\left\vert 3\left(
2^{\ell }M\right) \right\vert _{\omega }\ . \notag
\end{eqnarray
Now using $\frac{\left\vert 2^{\ell }M\right\vert ^{2}}{\left( \left\vert
M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2\left( 2-\alpha
\right) }}\leq \frac{1}{\left( \left\vert 2^{\ell }M\right\vert +\left\vert
y-c_{2^{\ell }M}\right\vert \right) ^{2\left( 1-\alpha \right) }}$ for
y\notin 2^{\ell +2}M$, we can continue wit
\begin{eqnarray*}
&&\mathcal{A}_{s,far}^{\ell }\left( M\right) \lesssim 2^{-2s}2^{-2\ell
}\left\vert 2^{\ell +2}M\right\vert _{\omega }\int_{\mathbb{R}\setminus
2^{\ell +2}M}\frac{1}{\left( \left\vert 2^{\ell }M\right\vert +\left\vert
y-c_{2^{\ell }M}\right\vert \right) ^{2\left( 1-\alpha \right) }}d\sigma
\left( y\right) \\
&\approx &2^{-2s}2^{-2\ell }\frac{\left\vert 2^{\ell +2}M\right\vert
_{\omega }}{\left\vert 2^{\ell }M\right\vert ^{1-\alpha }}\int_{\mathbb{R
\setminus 2^{\ell +2}M}\left( \frac{\left\vert 2^{\ell }M\right\vert }
\left( \left\vert 2^{\ell }M\right\vert +\left\vert y-c_{2^{\ell
}M}\right\vert \right) ^{2}}\right) ^{1-\alpha }d\sigma \left( y\right) \\
&\lesssim &2^{-2s}2^{-2\ell }\left\{ \frac{\left\vert 2^{\ell
+2}M\right\vert _{\omega }}{\left\vert 2^{\ell }M\right\vert ^{1-\alpha }
\mathcal{P}^{\alpha }\left( 2^{\ell +2}M,1_{\mathbb{R}\setminus 2^{\ell
+2}M}\sigma \right) \right\} \lesssim 2^{-2s}2^{-2\ell }\mathcal{A
_{2}^{\alpha }\ .
\end{eqnarray*}
To estimate the near term $\mathcal{A}_{s,near}^{\ell }\left( M\right) $ we
must again invoke the \emph{`prepare to puncture'} argument. Choose an
augmented interval $\widetilde{M}\in \mathcal{AD}$ such that
\dbigcup\limits_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell
}M}M^{\prime }\subset 2^{\ell +2}M\subset \widetilde{M}$ and $\ell \left(
\widetilde{M}\right) \leq C2^{\ell }\ell \left( M\right) $. Define
\widetilde{\omega }=\omega -\omega \left( \left\{ p\right\} \right) \delta
_{p}$ where $p$ is an atomic point in $\widetilde{M}$ for which
\begin{equation*}
\omega \left( \left\{ p\right\} \right) =\sup_{q\in \mathfrak{P}_{\left(
\sigma ,\omega \right) }:\ q\in \widetilde{M}}\omega \left( \left\{
q\right\} \right) .
\end{equation*
(If $\omega $ has no atomic point in common with $\sigma $ in $\widetilde{M}$
set $\widetilde{\omega }=\omega $.) Then we have $\left\vert \widetilde{M
\right\vert _{\widetilde{\omega }}=\omega \left( \widetilde{M},\mathfrak{P
_{\left( \sigma ,\omega \right) }\right) $, and just as in the argument
above following (\ref{A0snear}), we have from (\ref{key obs}) and (\ref{key
fact}) that bot
\begin{equation*}
\frac{\left\vert \widetilde{M}\right\vert _{\widetilde{\omega }}}{\left\vert
\widetilde{M}\right\vert ^{1-\alpha }}\frac{\left\vert \widetilde{M
\right\vert _{\sigma }}{\left\vert \widetilde{M}\right\vert ^{1-\alpha }
\leq A_{2}^{\alpha ,\limfunc{punct}}\text{ and }\sum_{F^{\prime }\in
\mathcal{F}}\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf
b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}\lesssim \ell \left( M^{\prime }\right) ^{2}\left\vert M^{\prime
}\right\vert _{\widetilde{\omega }}\ .
\end{equation*
Thus using that $m=2$ in the definition of $A_{s,near}^{\ell }\left(
M\right) $, we see that
\begin{eqnarray*}
\mathcal{A}_{s,near}^{\ell }\left( M\right) &\leq &\sum_{F^{\prime }\in
\mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus
2^{\ell }M}}\int_{2^{\ell +2}M\setminus 2^{\ell -m}M}S_{\left( M^{\prime
},M\right) }^{F^{\prime }}\left( y\right) d\sigma \left( y\right) \\
&\approx &\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M}}\int_{2^{\ell
+2}M\setminus 2^{\ell -m}M}\frac{1}{\left\vert 2^{\ell }M\right\vert
^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2
}{\left( \left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime
}}\right\vert \right) ^{2-\alpha }}d\sigma \left( y\right) \\
&\lesssim &\frac{1}{\left\vert 2^{\ell }M\right\vert ^{2-\alpha }
\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in
2^{\ell +1}M\setminus 2^{\ell }M}}\left\Vert \mathsf{Q}_{F^{\prime
},M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \int_{2^{\ell +2}M}\frac{1}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) ,
\end{eqnarray*
is dominated b
\begin{eqnarray*}
&&\frac{1}{\left\vert 2^{\ell }M\right\vert ^{2-\alpha }}\sum_{F^{\prime
}\in \mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2^{\ell
+1}M\setminus 2^{\ell }M}}\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\frac{\mathrm{P}^{\alpha }\left( M^{\prime },\mathbf{1
_{2^{\ell +2}M}\sigma \right) }{\left\vert M^{\prime }\right\vert } \\
&\leq &\frac{1}{\left\vert 2^{\ell }M\right\vert ^{2-\alpha }}\left(
\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in
2^{\ell +1}M\setminus 2^{\ell }M}}\left\Vert \mathsf{Q}_{F^{\prime
},M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\right) ^{\frac{1}{2}} \\
&&\times \left( \sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M}}\left\Vert
\mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\left( \frac
\mathrm{P}^{\alpha }\left( M^{\prime },\mathbf{1}_{2^{\ell +2}M}\sigma
\right) }{\left\vert M^{\prime }\right\vert }\right) ^{2}\right) ^{\frac{1}{
}}.
\end{eqnarray*}
This can now be estimated as for the term $\mathcal{A}_{s,near}^{0}\left(
M\right) $, along with the augmented local estimate (\ref{shifted local}) in
Lemma \ref{shifted} with $M=\widetilde{M}$ applied to the final line above,
to get
\begin{eqnarray*}
\mathcal{A}_{s,near}^{\ell }\left( M\right) &\lesssim &2^{-s}2^{-\ell }\frac
\left\vert 2^{\ell }M\right\vert }{\left\vert 2^{\ell }M\right\vert
^{2-\alpha }}\sqrt{\left\vert \widetilde{M}\right\vert _{\widetilde{\omega }
}\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc
energy}}}\sqrt{\left\vert \widetilde{M}\right\vert _{\sigma }} \\
&\lesssim &2^{-s}2^{-\ell }\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{\frac{\left\vert \widetilde{M
\right\vert _{\widetilde{\omega }}}{\left\vert \widetilde{M}\right\vert
^{1-\alpha }}\frac{\left\vert \widetilde{M}\right\vert _{\sigma }}
\left\vert \widetilde{M}\right\vert ^{1-\alpha }}} \\
&\lesssim &2^{-s}2^{-\ell }\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}
\ .
\end{eqnarray*
Each of the estimates for $\mathcal{A}_{s,far}^{\ell }\left( M\right) $ and
\mathcal{A}_{s,near}^{\ell }\left( M\right) $ is summable in both $s$ and
\ell $.
Now we turn to the terms $\mathcal{A}_{s,close}^{\ell }\left( M\right) $,
and recall from (\ref{vanishing close}) that $\mathcal{A}_{s,close}^{\ell
}\left( M\right) =0$ if $\ell \geq 1+\delta s$. So we now suppose that $\ell
\leq \delta s$. We have, with $m=2$ as in (\ref{smallest m})
\begin{eqnarray*}
&&\mathcal{A}_{s,close}^{\ell }\left( M\right) \leq \sum_{F^{\prime }\in
\mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus
2^{\ell }M}}\int_{2^{\ell -m}M}S_{\left( M^{\prime },M\right) }^{F^{\prime
}}\left( y\right) d\sigma \left( y\right) \\
&\approx &\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M}}\int_{2^{\ell
-m}M}\frac{1}{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert
\right) ^{2-\alpha }}\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}}{\left\vert 2^{\ell }M\right\vert ^{2-\alpha }}d\sigma
\left( y\right) \\
&=&\left( \sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }
\sum_{c_{M^{\prime }}\in 2^{\ell +1}M\setminus 2^{\ell }M}}\left\Vert
\mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\right) \frac{1}
\left\vert 2^{\ell }M\right\vert ^{2-\alpha }} \\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \times \int_{2^{\ell -m}M}\frac{1}
\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right)
^{2-\alpha }}d\sigma \left( y\right) .
\end{eqnarray*
The argument used to prove (\ref{stan pig'}) gives the\ analogous inequality
with a hole $2^{\ell -1}M$,
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}}\overset{\ast }{\sum_{c_{M^{\prime }}\in
2^{\ell +1}M\setminus 2^{\ell }M}}\left\Vert \mathsf{Q}_{F^{\prime
},M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}\lesssim 2^{-2s}\left\vert 2^{\ell }M\right\vert
^{2}\left\vert 2^{\ell +2}M\setminus 2^{\ell -1}M\right\vert _{\omega }\ .
\end{equation*
Thus we ge
\begin{eqnarray*}
&&\mathcal{A}_{s,close}^{\ell }\left( M\right) \\
&\lesssim &2^{-2s}\left\vert 2^{\ell }M\right\vert ^{2}\left\vert 2^{\ell
+2}M\setminus 2^{\ell -1}M\right\vert _{\omega }\frac{1}{\left\vert 2^{\ell
}M\right\vert ^{2-\alpha }}\int_{2^{\ell -m}M}\frac{1}{\left( \left\vert
M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2-\alpha }}d\sigma
\left( y\right) \\
&\lesssim &2^{-2s}\left\vert 2^{\ell }M\right\vert ^{2}\frac{\left\vert
2^{\ell +2}M\setminus 2^{\ell -1}M\right\vert _{\omega }}{\left\vert 2^{\ell
}M\right\vert ^{2-\alpha }}\frac{\left\vert 2^{\ell -m}M\right\vert _{\sigma
}}{\left\vert M\right\vert ^{2-\alpha }} \\
&\lesssim &2^{-2s}2^{\left( 2-\alpha \right) \ell }\frac{\left\vert 2^{\ell
+2}M\setminus 2^{\ell -1}M\right\vert _{\omega }}{\left\vert 2^{\ell
+2}M\right\vert ^{1-\alpha }}\frac{\left\vert 2^{\ell -m}M\right\vert
_{\sigma }}{\left\vert 2^{\ell -m}M\right\vert ^{1-\alpha }}\lesssim
2^{-2s}2^{\left( 2-\alpha \right) \ell }A_{2}^{\alpha },
\end{eqnarray*
provided that $m=2>1$. Note that we can use the offset Muckenhoupt constant
A_{2}^{\alpha }$ here since $2^{\ell +2}M\setminus 2^{\ell -1}M$ and
2^{\ell -m}M$ are disjoint. If $\ell \leq s$, then we have the relatively
crude estimate $\mathcal{A}_{s,close}^{\ell }\left( M\right) \lesssim
2^{-s}A_{2}^{\alpha }\ $without decay in $\ell $. But we are assuming $\ell
\leq \delta s$ here, and so we obtain a suitable estimate for $\mathcal{A
_{s,close}^{\ell }\left( M\right) $ provided we choose $0<\delta \leq \frac{
}{2-\alpha }$. Indeed, we then hav
\begin{equation*}
\sum_{l=1}^{\delta s}2^{-2s}2^{\left( 2-\alpha \right) \ell }A_{2}^{\alpha
}=2^{-2s}\left( \sum_{l=1}^{\delta s}2^{\left( 2-\alpha \right) \ell
}\right) A_{2}^{\alpha }\lesssim 2^{-2s}2^{\left( 2-\alpha \right) \delta
s}A_{2}^{\alpha }\leq 2^{-s}A_{2}^{\alpha }\ ,
\end{equation*
provided $\delta \leq \frac{1}{2-\alpha }$, and in particular we may take
\delta =\frac{1}{2}$. Altogether, the above estimates prov
\begin{equation*}
T_{s}^{\limfunc{proximal}}+T_{s}^{\limfunc{difference}}\lesssim 2^{-s}\left(
\mathcal{A}_{2}^{\alpha }+\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}
\right) ,
\end{equation*
which is summable in $s$.
\subsubsection{The intersection term}
Now we return to the term
\begin{eqnarray*}
T_{s}^{\limfunc{intersection}} &\equiv &\sum_{\substack{ F,F^{\prime }\in
\mathcal{F}}}\sum_{\substack{ M\in \mathcal{W}\left( F\right) ,\ M^{\prime
}\in \mathcal{W}\left( F^{\prime }\right) \\ M,M^{\prime }\subset I,\ \ell
\left( M^{\prime }\right) =2^{-s}\ell \left( M\right) \\ d\left(
c_{M},c_{M^{\prime }}\right) \geq 2^{s\left( 1+\delta \right) }\ell \left(
M^{\prime }\right) }} \\
&&\times \int_{B\left( M,M^{\prime }\right) }\frac{\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}}{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2-\alpha }}\frac{\left\Vert \mathsf{Q
_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left( \left\vert M^{\prime
}\right\vert +\left\vert y-c_{M^{\prime }}\right\vert \right) ^{2-\alpha }
d\sigma \left( y\right) .
\end{eqnarray*
It will suffice to show that $T_{s}^{\limfunc{intersection}}$ satisfies the
estimate
\begin{eqnarray*}
T_{s}^{\limfunc{intersection}} &\lesssim &2^{-s\delta }\sqrt{\left(
\mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy}}
\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\sum_{F^{\prime }\in \mathcal{F
^{\prime }}\sum_{\substack{ M^{\prime }\in \mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep}}\left( F^{\prime }\right) \\ M^{\prime
}\subset I}}\lVert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b
^{\ast }}x\rVert _{L^{2}\left( \omega \right) }^{\spadesuit 2} \\
&=&2^{-s\delta }\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}
\int_{\widehat{I}}t^{2}\overline{\mu }\ .
\end{eqnarray*
Recalling $B\left( M,M^{\prime }\right) =B\left( c_{M},\frac{1}{2}d\left(
c_{M},c_{M^{\prime }}\right) \right) $, we can write (suppressing some
notation for clarity)
\begin{eqnarray*}
&&T_{s}^{\limfunc{intersection}} \\
&=&\sum_{F,F^{\prime }}\sum_{\substack{ M,M^{\prime }}}\int_{B\left(
M,M^{\prime }\right) }\frac{\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right) ^{2-\alpha }
\frac{\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b
^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left(
\left\vert M^{\prime }\right\vert +\left\vert y-c_{M^{\prime }}\right\vert
\right) ^{2-\alpha }}d\sigma \left( y\right) \\
&\approx &\sum_{F,F^{\prime }}\sum_{\substack{ M,M^{\prime }}}\left\Vert
\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2}\left\Vert \mathsf{Q}_{F^{\prime },M^{\prime
}}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\frac{1}{\left\vert c_{M}-c_{M^{\prime }}\right\vert
^{2-\alpha }}\int_{B\left( M,M^{\prime }\right) }\frac{d\sigma \left(
y\right) }{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert
\right) ^{2-\alpha }} \\
&\leq &\sum_{F^{\prime }}\sum_{\substack{ M^{\prime }}}\left\Vert \mathsf{Q
_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\sum_{F}\sum_{\substack{ M}
\frac{1}{\left\vert c_{M}-c_{M^{\prime }}\right\vert ^{2-\alpha }}\left\Vert
\mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left(
\omega \right) }^{\spadesuit 2}\int_{B\left( M,M^{\prime }\right) }\frac
d\sigma \left( y\right) }{\left( \left\vert M\right\vert +\left\vert
y-c_{M}\right\vert \right) ^{2-\alpha }} \\
&\equiv &\sum_{F^{\prime }}\sum_{\substack{ M^{\prime }}}\left\Vert \mathsf{
}_{F^{\prime },M^{\prime }}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}S_{s}\left( M^{\prime }\right) ,
\end{eqnarray*
and since $\int_{B\left( M,M^{\prime }\right) }\frac{d\sigma \left( y\right)
}{\left( \left\vert M\right\vert +\left\vert y-c_{M}\right\vert \right)
^{2-\alpha }}\approx \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{B\left(
M,M^{\prime }\right) }\sigma \right) }{\left\vert M\right\vert }$, it
remains to show that for each fixed $M^{\prime }$
\begin{eqnarray*}
S_{s}\left( M^{\prime }\right) &\approx &\sum_{F}\overset{\ast }{\sum
_{\substack{ M:\ d\left( c_{M},c_{M^{\prime }}\right) \geq 2^{s\left(
1+\delta \right) }\ell \left( M^{\prime }\right) }}}\frac{\left\Vert \mathsf
Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega
\right) }^{\spadesuit 2}}{\left\vert c_{M}-c_{M^{\prime }}\right\vert
^{2-\alpha }}\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{B\left(
M,M^{\prime }\right) }\sigma \right) }{\left\vert M\right\vert } \\
&\lesssim &2^{-\delta s}\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right)
^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt{A_{2}^{\alpha }}\ .
\end{eqnarray*}
We writ
\begin{eqnarray}
S_{s}\left( M^{\prime }\right) &\approx &\sum_{k\geq s\left( 1+\delta
\right) }\frac{1}{\left( 2^{k}\left\vert M^{\prime }\right\vert \right)
^{2-\alpha }}\sum_{F}\overset{\ast }{\sum_{M:\ d\left( c_{M},c_{M^{\prime
}}\right) \approx 2^{k}\ell \left( M^{\prime }\right) }}\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1}_{B\left(
M,M^{\prime }\right) }\sigma \right) }{\left\vert M\right\vert }
\label{def Sks} \\
&=&\sum_{k\geq s\left( 1+\delta \right) }\frac{1}{\left( 2^{k}\left\vert
M^{\prime }\right\vert \right) ^{2-\alpha }}S_{s}^{k}\left( M^{\prime
}\right) \ ; \notag \\
S_{s}^{k}\left( M^{\prime }\right) &\equiv &\sum_{F}\overset{\ast }
\sum_{M:\ d\left( c_{M},c_{M^{\prime }}\right) \approx 2^{k}\ell \left(
M^{\prime }\right) }}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast
}}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}\frac{\mathrm{P
^{\alpha }\left( M,\mathbf{1}_{B\left( M,M^{\prime }\right) }\sigma \right)
}{\left\vert M\right\vert }, \notag
\end{eqnarray
where by $d\left( c_{M},c_{M^{\prime }}\right) \approx 2^{k}\ell \left(
M^{\prime }\right) $ we mean $2^{k}\ell \left( M^{\prime }\right) \leq
d\left( c_{M},c_{M^{\prime }}\right) \leq 2^{k+1}\ell \left( M^{\prime
}\right) $. Moreover, if $d\left( c_{M},c_{M^{\prime }}\right) \approx
2^{k}\ell \left( M^{\prime }\right) $, then from the fact that the radius of
$B\left( M,M^{\prime }\right) $ is $\frac{1}{2}d\left( c_{M},c_{M^{\prime
}}\right) $, we obtain
\begin{equation*}
B\left( M,M^{\prime }\right) \subset C_{0}2^{k}M^{\prime },
\end{equation*
where $C_{0}$ is a positive constant ($C_{0}=6$ works).
For fixed $k\geq s\left( 1+\delta \right) $, we invoke yet again the \emph
`prepare to puncture'} argument. Choose an augmented interval $\widetilde
M^{\prime }}\in \mathcal{AD}$ such that $C_{0}2^{k}M\subset \widetilde
M^{\prime }}$ and $\ell \left( \widetilde{M^{\prime }}\right) \leq
C2^{k}\ell \left( M^{\prime }\right) $. Define $\widetilde{\omega }=\omega
-\omega \left( \left\{ p\right\} \right) \delta _{p}$ where $p$ is an atomic
point in $\widetilde{M^{\prime }}$ for which
\begin{equation*}
\omega \left( \left\{ p\right\} \right) =\sup_{q\in \mathfrak{P}_{\left(
\sigma ,\omega \right) }:\ q\in \widetilde{M^{\prime }}}\omega \left(
\left\{ q\right\} \right) .
\end{equation*
(If $\omega $ has no atomic point in common with $\sigma $ in $\widetilde
M^{\prime }}$, set $\widetilde{\omega }=\omega $.) Then we have $\left\vert
\widetilde{M^{\prime }}\right\vert _{\widetilde{\omega }}=\omega \left(
\widetilde{M^{\prime }},\mathfrak{P}_{\left( \sigma ,\omega \right) }\right)
$ and so from (\ref{key obs}) and (\ref{key fact})
\begin{equation*}
\frac{\left\vert \widetilde{M^{\prime }}\right\vert _{\widetilde{\omega }}}
\left\vert \widetilde{M^{\prime }}\right\vert ^{1-\alpha }}\frac{\left\vert
\widetilde{M^{\prime }}\right\vert _{\sigma }}{\left\vert \widetilde
M^{\prime }}\right\vert ^{1-\alpha }}\leq A_{2}^{\alpha ,\limfunc{punct}
\text{ and }\sum_{F\in \mathcal{F}}\left\Vert \mathsf{Q}_{F,A}^{\omega
\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit
2}\lesssim \ell \left( A\right) ^{2}\left\vert A\right\vert _{\widetilde
\omega }}\ \text{for any interval }A.
\end{equation*}
Now we are ready to apply Cauchy-Schwarz and the augmented local estimate
\ref{shifted local}) in Lemma \ref{shifted} with $M=\widetilde{M^{\prime }}\
$to the second line below, and to apply the argument in (\ref{stan pig'}) to
the first line below, to get the following estimate for $S_{s}^{k}\left(
M^{\prime }\right) $ defined in (\ref{def Sks}) above
\begin{eqnarray*}
S_{s}^{k}\left( M^{\prime }\right) &\leq &\left( \sum_{F}\sum_{M:\ d\left(
c_{M},c_{M^{\prime }}\right) \approx 2^{k}\ell \left( M^{\prime }\right)
}\left\Vert \mathsf{Q}_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}\right) ^{\frac{1}{2}} \\
&&\times \left( \sum_{F}\sum_{M:\ d\left( c_{M},c_{M^{\prime }}\right)
\approx 2^{k}\ell \left( M^{\prime }\right) }\left\Vert \mathsf{Q
_{F,M}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert _{L^{2}\left( \omega \right)
}^{\spadesuit 2}\left( \frac{\mathrm{P}^{\alpha }\left( M,\mathbf{1
_{B\left( M,M^{\prime }\right) }\sigma \right) }{\left\vert M\right\vert
\right) ^{2}\right) ^{\frac{1}{2}} \\
&\lesssim &\left( 2^{2s}\left\vert \widetilde{M^{\prime }}\right\vert
^{2}\left\vert \widetilde{M^{\prime }}\right\vert _{\widetilde{\omega
}\right) ^{\frac{1}{2}}\left( \left[ \left( \mathfrak{E}_{2}^{\alpha
}\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy}}\right] \left\vert \widetilde
M^{\prime }}\right\vert _{\sigma }\right) ^{\frac{1}{2}} \\
&\lesssim &\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
,\limfunc{energy}}}2^{s}\left\vert \widetilde{M^{\prime }}\right\vert \sqrt
\left\vert \widetilde{M^{\prime }}\right\vert _{\widetilde{\omega }}}\sqrt
\left\vert \widetilde{M^{\prime }}\right\vert _{\sigma }} \\
&\lesssim &\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
,\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}2^{s}\left\vert
\widetilde{M^{\prime }}\right\vert \left\vert \widetilde{M^{\prime }
\right\vert ^{1-\alpha } \\
&\approx &\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}2^{s}2^{k\left(
1-\alpha \right) }\left\vert M^{\prime }\right\vert ^{2-\alpha },\ \ \ \ \ \
\text{since }\ell \left( \widetilde{M^{\prime }}\right) \approx 2^{k}\ell
\left( M^{\prime }\right) .
\end{eqnarray*}
Altogether then we hav
\begin{eqnarray*}
S_{s}\left( M^{\prime }\right) &=&\sum_{k\geq \left( 1+\delta \right) s
\frac{1}{\left( 2^{k}\left\vert M^{\prime }\right\vert \right) ^{2-\alpha }
S_{s}^{k}\left( M^{\prime }\right) \\
&\lesssim &\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
,\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\sum_{k\geq \left(
1+\delta \right) s}\frac{1}{\left( 2^{k}\left\vert M^{\prime }\right\vert
\right) ^{2-\alpha }}2^{s}2^{k\left( 1-\alpha \right) }\left\vert M^{\prime
}\right\vert ^{2-\alpha } \\
&=&\sqrt{\left( \mathfrak{E}_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha
\limfunc{energy}}}\sqrt{A_{2}^{\alpha ,\limfunc{punct}}}\sum_{k\geq \left(
1+\delta \right) s}2^{s-k}\lesssim 2^{-\delta s}\sqrt{\left( \mathfrak{E
_{2}^{\alpha }\right) ^{2}+A_{2}^{\alpha ,\limfunc{energy}}}\sqrt
A_{2}^{\alpha ,\limfunc{punct}}},
\end{eqnarray*
which is summable in $s$. This completes the proof of (\ref{Us bound}), and
hence of the estimate for $\mathbf{Back}\left( \widehat{I}\right) $ in (\re
{e.t2 n'}).
The proof of Proposition \ref{func ener control} is now complete.
\section{Appendix C: Errata for the Revista paper}
The current paper adapts the arguments of our 2016 Revista paper \cit
{SaShUr7} whenever possible (which in turn adapted arguments from many
earlier papers of various authors). To aid the reader in consulting \cit
{SaShUr7}, we include here a list of up-to-date \emph{errata} for the
Revista paper \cite{SaShUr7}.
\bigskip
{\Large \#1:} Lemma 3.1 on page 90 and its proof should be replaced with the
following taken from \cite[\texttt{arXiv:1505.07816}v3]{SaShUr6}.
\textbf{Lemma 3.1} Given $\mathbf{r}\geq 3$, $\mathbf{\tau }\geq 1$ and
\frac{1}{\mathbf{r}}<\varepsilon <1-\frac{1}{\mathbf{r}}$, we have
\begin{equation*}
\mathcal{D}_{\left( \mathbf{r}-1,\delta \right) -\limfunc{good}}\subset
\mathcal{D}_{\left( \mathbf{r},\varepsilon \right) -\limfunc{good}}^{\mathbf
\tau }}\ ,
\end{equation*
provide
\begin{equation*}
0<\delta \leq \frac{\mathbf{r}\varepsilon -1}{\mathbf{r}+\mathbf{\tau }}.
\end{equation*}
\begin{proof}
Suppose that $I\in \mathcal{D}_{\left( \mathbf{r}-1,\delta \right) -\limfunc
good}}$ where $\delta $ is as above. If $J$ is a child of $I$, then $J\in
\mathcal{D}_{\left( \mathbf{r},\delta \right) -\limfunc{good}}$, and since
\delta <\varepsilon $, we also have $\mathcal{D}_{\left( \mathbf{r},\delta
\right) -\limfunc{good}}\subset \mathcal{D}_{\left( \mathbf{r},\varepsilon
\right) -\limfunc{good}}$. It remains to show that $\pi _{\mathcal{D
}^{\left( m\right) }I\in \mathcal{D}_{\left( \mathbf{r},\varepsilon \right)
\limfunc{good}}$ for $0\leq m\leq \mathbf{\tau }$. For this it suffices to
show that if $K\in \mathcal{D}$ satisfies\thinspace $\pi _{\mathcal{D
}^{\left( m\right) }I\subset K$ and $\ell \left( \pi _{\mathcal{D}}^{\left(
m\right) }I\right) \leq 2^{-\mathbf{r}}\ell \left( K\right) $ , the
\begin{equation*}
\frac{1}{2}\left( \frac{\ell \left( \pi _{\mathcal{D}}^{\left( m\right)
}I\right) }{\ell \left( K\right) }\right) ^{\varepsilon }\ell \left(
K\right) \leq \limfunc{dist}\left( \pi _{\mathcal{D}}^{\left( m\right)
}I,K^{c}\right) .
\end{equation*
Now $\ell \left( I\right) =2^{-m}\ell \left( \pi _{\mathcal{D}}^{\left(
m\right) }I\right) \leq 2^{-\left( m+\mathbf{r}\right) }\ell \left( K\right)
\leq 2^{-\left( \mathbf{r}-1\right) }\ell \left( K\right) $ and $I\in
\mathcal{D}_{\left( \mathbf{r}-1,\delta \right) -\limfunc{good}}$ imply tha
\begin{equation*}
\frac{1}{2}\left( \frac{\ell \left( I\right) }{\ell \left( K\right) }\right)
^{\delta }\ell \left( K\right) \leq \limfunc{dist}\left( I,K^{c}\right) ,
\end{equation*
and since the triangle inequality give
\begin{equation*}
\limfunc{dist}\left( I,K^{c}\right) \leq \limfunc{dist}\left( \pi _{\mathcal
D}}^{\left( m\right) }I,K^{c}\right) +2^{m}\ell \left( I\right) ,
\end{equation*
we see that it suffices to sho
\begin{equation*}
\frac{1}{2}\left( \frac{\ell \left( \pi _{\mathcal{D}}^{\left( m\right)
}I\right) }{\ell \left( K\right) }\right) ^{\varepsilon }\ell \left(
K\right) +2^{m}\ell \left( I\right) \leq \frac{1}{2}\left( \frac{\ell \left(
I\right) }{\ell \left( K\right) }\right) ^{\delta }\ell \left( K\right) ,\ \
\ \ \ 0\leq m\leq \mathbf{\tau }.
\end{equation*
This is equivalent to successively
\begin{eqnarray*}
\frac{1}{2}\left( \frac{2^{m}\ell \left( I\right) }{\ell \left( K\right)
\right) ^{\varepsilon }\ell \left( K\right) +2^{m}\ell \left( I\right) &\leq
&\frac{1}{2}\left( \frac{\ell \left( I\right) }{\ell \left( K\right)
\right) ^{\delta }\ell \left( K\right) ; \\
\left( \frac{2^{m}\ell \left( I\right) }{\ell \left( K\right) }\right)
^{\varepsilon }+2^{m+1}\frac{\ell \left( I\right) }{\ell \left( K\right) }
&\leq &\left( \frac{\ell \left( I\right) }{\ell \left( K\right) }\right)
^{\delta }; \\
2^{m\varepsilon }\left( \frac{\ell \left( I\right) }{\ell \left( K\right)
\right) ^{\varepsilon -\delta }+2^{m+1}\left( \frac{\ell \left( I\right) }
\ell \left( K\right) }\right) ^{1-\delta } &\leq &1,\ \ \ \ \ 0\leq m\leq
\mathbf{\tau }.
\end{eqnarray*
Since $0<\delta <\varepsilon <1$ by our restriction on $\varepsilon $ and
our choice of $\delta $, and since $\frac{\ell \left( I\right) }{\ell \left(
K\right) }\leq 2^{-\left( m+\mathbf{r}\right) }$, it thus suffices to show
tha
\begin{eqnarray*}
2^{m\varepsilon }\left( 2^{-\left( m+\mathbf{r}\right) }\right)
^{\varepsilon -\delta }+2^{m+1}\left( 2^{-\left( m+\mathbf{r}\right)
}\right) ^{1-\delta } &\leq &1, \\
\text{i.e. }2^{m\varepsilon -\left( m+\mathbf{r}\right) \left( \varepsilon
-\delta \right) }+2^{m+1-\left( m+\mathbf{r}\right) \left( 1-\delta \right)
} &\leq &1,
\end{eqnarray*
for $0\leq m\leq \mathbf{\tau }$. In particular then it suffices to show bot
\begin{eqnarray*}
m\varepsilon -\left( m+\mathbf{r}\right) \left( \varepsilon -\delta \right)
&\leq &-1, \\
m+1-\left( m+\mathbf{r}\right) \left( 1-\delta \right) &\leq &-1,
\end{eqnarray*
equivalently bot
\begin{eqnarray*}
\left( \mathbf{r}+m\right) \delta &\leq &\mathbf{r}\varepsilon -1, \\
\left( \mathbf{r}+m\right) \delta &\leq &\mathbf{r}-2,
\end{eqnarray*
for $0\leq m\leq \mathbf{\tau }$. Finally then it suffices to show bot
\begin{equation*}
\delta \leq \frac{\mathbf{r}\varepsilon -1}{\mathbf{r}+\mathbf{\tau }}\text{
and }\delta \leq \frac{\mathbf{r}-2}{\mathbf{r}+\mathbf{\tau }}.
\end{equation*
Because of the restriction that $\frac{1}{\mathbf{r}}<\varepsilon <1-\frac{
}{\mathbf{r}}$, we see that $0<\mathbf{r}\varepsilon -1<\mathbf{r}-2$, and
it is now clear that the above display holds for our choice of $\delta $.
\end{proof}
\bigskip
{\Large \#2:} In the integral in line $3$ of page 93, the factor
s_{I}\left( y\right) $ should be raised to the power $n-\alpha $.
\bigskip
{\large \#3:} The definition of $\varphi _{F}$ in line $-2$ of page 124
should rea
\begin{equation*}
\varphi _{F}\equiv \sum_{k,\theta :\ \theta \left( I_{k}\right) \in \mathcal
F}}\mathbf{1}_{\theta \left( I_{k}\right) }\left( \mathbb{E}_{\theta \left(
I_{k}\right) }^{\sigma }f-\mathbb{E}_{I_{K}}^{\sigma }f\right) ,
\end{equation*
and the display in line $3$ of page 125 should rea
\begin{equation*}
\left\vert \sum_{F\in \mathcal{F}}\left\langle T_{\sigma }^{\alpha }\varphi
_{F},g_{F}\right\rangle \right\vert \lesssim \sqrt{A_{2}^{\alpha }
\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }\ ,
\end{equation*
and finally in line $-3$ of page 125, the constant $\mathcal{NTV}_{\alpha }$
should be replaced by $\mathfrak{T}_{T^{\alpha }}$ in both of its
appearances.
\bigskip
{\Large \#4:} The\ display beginning with the term $S$ in line $4$ of page
134 should be replaced with
\begin{eqnarray*}
S &=&\sum_{F\in \mathcal{F}_{I}}\sum_{J\in \mathcal{M}_{\mathbf{r}-\limfunc
deep}}(F)}\left( \sum_{F^{\prime }\in \mathcal{F}:\ F\subset F^{\prime
}\subsetneqq I}\frac{d\left( F^{\prime }\right) }{d\left( F^{\prime }\right)
}\frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\pi _{\mathcal{F
_{I}}F^{\prime }\setminus F^{\prime }}\sigma \right) }{\left\vert
J\right\vert ^{1/n}}\right) ^{2}\left\Vert \mathsf{P}_{F,J}^{\omega }\mathbf
x}\right\Vert _{L^{2}\left( \omega \right) }^{2} \\
&\leq &\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{F\in \mathcal{F}:\ F\subset F^{\prime }}\sum_{J\in \mathcal{M}_
\mathbf{r}-\limfunc{deep}}(F)}\left( \sum_{F^{\prime }\in \mathcal{F}:\
F\subset F^{\prime }\subsetneqq I}\frac{1}{d\left( F^{\prime }\right) ^{2}
\right) \left( \frac{\mathrm{P}^{\alpha }\left( J,\mathbf{1}_{\pi _{\mathcal
F}_{I}}F^{\prime }\setminus F^{\prime }}\sigma \right) }{\left\vert
J\right\vert ^{1/n}}\right) ^{2}\left\Vert \mathsf{P}_{F,J}^{\omega }\mathbf
x}\right\Vert _{L^{2}\left( \omega \right) }^{2} \\
&\leq &C\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left( F^{\prime
}\right) }\sum_{F\in \mathcal{F}:\ F\subset F^{\prime }}\sum_{J\in \mathcal{
}_{\mathbf{r}-\limfunc{deep}}(F):\ J\subset K}\left( \frac{\mathrm{P
^{\alpha }\left( J,\mathbf{1}_{\pi _{\mathcal{F}_{I}}F^{\prime }\setminus
F^{\prime }}\sigma \right) }{\left\vert J\right\vert ^{1/n}}\right)
^{2}\left\Vert \mathsf{P}_{F,J}^{\omega }\mathbf{x}\right\Vert _{L^{2}\left(
\omega \right) }^{2} \\
&\lesssim &\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left( F^{\prime
}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{\pi _
\mathcal{F}_{I}}F^{\prime }\setminus F^{\prime }}\sigma \right) }{\left\vert
K\right\vert ^{1/n}}\right) ^{2}\sum_{F\in \mathcal{F}:\ F\subset F^{\prime
}}\sum_{J\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}(F):\ J\subset
K}\left\Vert \mathsf{P}_{F,J}^{\omega }\mathbf{x}\right\Vert _{L^{2}\left(
\omega \right) }^{2},
\end{eqnarray*
and then the display at the bottom of page 134 should be replaced wit
\begin{eqnarray*}
S &\lesssim &\sum_{F^{\prime }\in \mathcal{F}_{I}}d\left( F^{\prime }\right)
^{2}\sum_{K\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left( F^{\prime
}\right) }\left( \frac{\mathrm{P}^{\alpha }\left( K,\mathbf{1}_{\pi _
\mathcal{F}_{I}}F^{\prime }\setminus F^{\prime }}\sigma \right) }{\left\vert
K\right\vert ^{1/n}}\right) ^{2}\left\Vert \widehat{\mathsf{P}}_{F^{\prime
},K}^{\omega }\mathbf{x}\right\Vert _{L^{2}\left( \omega \right) }^{2} \\
&=&\sum_{k=0}^{\infty }k^{2}\sum_{F^{\prime }\in \mathcal{F}_{I}:\ d\left(
F^{\prime }\right) =k}\sum_{K\in \mathcal{M}_{\mathbf{r}-\limfunc{deep
}\left( F^{\prime }\right) }\left( \frac{\mathrm{P}^{\alpha }\left( K
\mathbf{1}_{\pi _{\mathcal{F}_{I}}F^{\prime }\setminus F^{\prime }}\sigma
\right) }{\left\vert K\right\vert ^{1/n}}\right) ^{2}\left\Vert \widehat
\mathsf{P}}_{F^{\prime },K}^{\omega ,\mathbf{b}^{\ast }}\mathbf{x
\right\Vert _{L^{2}\left( \omega \right) }^{2}\equiv \sum_{k=0}^{\infty
}A_{k}.
\end{eqnarray*}
\bigskip
{\Large \#5:} Beginning two lines above the display at the bottom of page
144, and ending after the display at the top of page 145, replace with the
following
\begin{equation*}
\sum_{F^{\prime }\in \mathcal{F}}\left\Vert \mathsf{Q}_{F^{\prime
},A}^{\omega }x\right\Vert _{L^{2}\left( \omega \right) }^{2}\lesssim
\mathbf{\tau }\left\vert A\right\vert ^{\frac{2}{n}}\left\vert A\right\vert
_{\omega }\ ,\ \ \ \ \ \text{for any cube }A.
\end{equation*
Note that if $c_{J^{\prime }}\in 2J$ and $\ell \left( J^{\prime }\right)
<\ell \left( J\right) $, then $J^{\prime }\subset \frac{5}{2}J$. Then with
\begin{equation*}
\mathcal{W}_{J}^{s}\equiv \dbigcup\limits_{F^{\prime }\in \mathcal{F
}\left\{ J^{\prime }\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left(
F^{\prime }\right) :J^{\prime }\subset \frac{5}{2}J\text{ and }\ell \left(
J^{\prime }\right) =2^{-s}\ell \left( J\right) \right\} ,
\end{equation*
we hav
\begin{eqnarray*}
\mathcal{A}_{s,far}^{0}\left( J\right) &\leq &\sum_{F^{\prime }\in \mathcal{
}}\overset{\ast }{\sum_{c_{J^{\prime }}\in 2J}}\int_{I\setminus \left(
3J\right) }S_{\left( J^{\prime },J\right) }^{F^{\prime }}\left( y\right)
d\sigma \left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{J}^{s}}\sum_{F^{\prime }\in \mathcal{F}:\
A\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left( F^{\prime }\right)
}\int_{I\setminus \left( 3J\right) }\frac{\left\Vert \mathsf{Q}_{F^{\prime
},A}^{\omega }x\right\Vert _{L^{2}\left( \omega \right) }^{\spadesuit 2}}
\left( \left\vert J\right\vert ^{\frac{2}{n}}+\left\vert y-c_{J}\right\vert
\right) ^{2\left( 2-\alpha \right) }}d\sigma \left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{J}^{s}}\int_{I\setminus \left( 3J\right)
\frac{\left\vert A\right\vert ^{\frac{2}{n}}\left\vert A\right\vert _{\omega
}}{\left( \left\vert J\right\vert ^{\frac{2}{n}}+\left\vert
y-c_{M}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma \left(
y\right) \\
&=&\left( \sum_{A\in \mathcal{W}_{J}^{s}}\left\vert A\right\vert ^{\frac{2}{
}}\left\vert A\right\vert _{\omega }\right) \int_{I\setminus \left(
3J\right) }\frac{1}{\left( \left\vert J\right\vert ^{\frac{2}{n}}+\left\vert
y-c_{J}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma \left(
y\right) .
\end{eqnarray*
Now we use the standard pigeonholing of side length of $A$ to conclude that
\begin{eqnarray*}
\sum_{A\in \mathcal{W}_{J}^{s}}\left\vert A\right\vert ^{\frac{2}{n
}\left\vert A\right\vert _{\omega } &=&\sum_{k=s}^{\infty }\sum_{A\in
\mathcal{W}_{J}^{s}:\ \ell \left( A\right) =2^{-k}\ell \left( J\right)
}\left\vert A\right\vert ^{\frac{2}{n}}\left\vert A\right\vert _{\omega
}\leq \sum_{k=s}^{\infty }2^{-2k}\left\vert J\right\vert ^{\frac{2}{n
}\sum_{A\in \mathcal{W}_{J}^{s}:\ \ell \left( A\right) =2^{-k}\ell \left(
J\right) }\left\vert A\right\vert _{\omega } \\
&\leq &\sum_{k=s}^{\infty }2^{-2k}\left\vert J\right\vert ^{2}\left\vert
\frac{5}{2}J\right\vert _{\omega }\lesssim 2^{-2s}\left\vert J\right\vert
^{2}\left\vert \frac{5}{2}J\right\vert _{\omega },
\end{eqnarray*
so that combining the previous two displays we hav
\begin{eqnarray*}
\mathcal{A}_{s,far}^{0}\left( J\right) &\lesssim &2^{-2s}\left\vert
J\right\vert ^{2}\left\vert \frac{5}{2}J\right\vert _{\omega
}\int_{I\setminus \left( 3J\right) }\frac{1}{\left( \left\vert J\right\vert
^{\frac{2}{n}}+\left\vert y-c_{J}\right\vert \right) ^{2\left( 2-\alpha
\right) }}d\sigma \left( y\right) \\
&\leq &2^{-2s}\left\vert \frac{5}{2}J\right\vert _{\omega }\int_{I\setminus
\left( 3J\right) }\frac{1}{\left( \left\vert J\right\vert ^{\frac{2}{n
}+\left\vert y-c_{J}\right\vert \right) ^{2\left( 1-\alpha \right) }}d\sigma
\left( y\right) \\
&\approx &2^{-2s}\frac{\left\vert \frac{5}{2}J\right\vert _{\omega }}
\left\vert \frac{5}{2}J\right\vert ^{1-\alpha }}\int_{I\setminus \left(
3J\right) }\left( \frac{\left\vert J\right\vert }{\left( \left\vert
J\right\vert ^{\frac{2}{n}}+\left\vert y-c_{J}\right\vert \right) ^{2}
\right) ^{1-\alpha }d\sigma \left( y\right) \\
&\lesssim &2^{-2s}\frac{\left\vert \frac{5}{2}J\right\vert _{\omega }}
\left\vert \frac{5}{2}J\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left(
\frac{5}{2}J,\mathbf{1}_{I\setminus \left( 3J\right) }\sigma \right)
\lesssim 2^{-2s}\mathcal{A}_{2}^{\alpha }\ .
\end{eqnarray*}
\bigskip
{\Large \#6:} Lines $3$ through $11$ on page 146 should be replaced with
this:
\bigskip
Similarly, for $\ell \geq 1$, we can estimate the far term $\mathcal{A
_{s,far}^{\ell }\left( J\right) $ by the argument used for $\mathcal{A
_{s,far}^{0}\left( J\right) $ but applied to $2^{\ell }J$ in place of $J$.
For this need the following variant of $\mathcal{W}_{J}^{s}$ given b
\begin{equation*}
\mathcal{W}_{J}^{s,\ell }\equiv \dbigcup\limits_{F^{\prime }\in \mathcal{F
}\left\{ J^{\prime }\in \mathcal{W}\left( F^{\prime }\right) :J^{\prime
}\subset 3\left( 2^{\ell }J\right) \text{ and }\ell \left( J^{\prime
}\right) =2^{-s-\ell }\ell \left( 2^{\ell }J\right) \right\} .
\end{equation*
Then we have
\begin{eqnarray*}
\mathcal{A}_{s,far}^{\ell }\left( J\right) &\leq &\sum_{F^{\prime }\in
\mathcal{F}}\overset{\ast }{\sum_{c_{J^{\prime }}\in \left( 2^{\ell
+1}J\right) \setminus \left( 2^{\ell }J\right) }}\int_{I\setminus 2^{\ell
+2}J}S_{\left( J^{\prime },J\right) }^{F^{\prime }}\left( y\right) d\sigma
\left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{J}^{s,\ell }}\sum_{F^{\prime }\in
\mathcal{F}:\ A\in \mathcal{M}_{\mathbf{r}-\limfunc{deep}}\left( F^{\prime
}\right) }\int_{I\setminus 4\left( 2^{\ell }J\right) }\frac{\left\Vert
\mathsf{Q}_{F^{\prime },A}^{\omega ,\mathbf{b}^{\ast }}x\right\Vert
_{L^{2}\left( \omega \right) }^{\spadesuit 2}}{\left( \left\vert
J\right\vert ^{\frac{2}{n}}+\left\vert y-c_{J}\right\vert \right) ^{2\left(
2-\alpha \right) }}d\sigma \left( y\right) \\
&\lesssim &\sum_{A\in \mathcal{W}_{J}^{s,\ell }}\int_{I\setminus 4\left(
2^{\ell }J\right) }\frac{\left\vert A\right\vert ^{\frac{2}{n}}\left\vert
A\right\vert _{\omega }}{\left( \left\vert J\right\vert ^{\frac{2}{n
}+\left\vert y-c_{J}\right\vert \right) ^{2\left( 2-\alpha \right) }}d\sigma
\left( y\right) \\
&=&\left( \sum_{A\in \mathcal{W}_{J}^{s,\ell }}\left\vert A\right\vert ^
\frac{2}{n}}\left\vert A\right\vert _{\omega }\right) \int_{I\setminus
4\left( 2^{\ell }J\right) }\frac{1}{\left( \left\vert J\right\vert ^{\frac{
}{n}}+\left\vert y-c_{J}\right\vert \right) ^{2\left( 2-\alpha \right) }
d\sigma \left( y\right) ,
\end{eqnarray*
where, just as for the sum over $A\in \mathcal{W}_{J}^{s,0}$
\begin{eqnarray*}
\sum_{A\in \mathcal{W}_{J}^{s,\ell }}\left\vert A\right\vert ^{\frac{2}{n
}\left\vert A\right\vert _{\omega } &=&\sum_{k=s}^{\infty }\sum_{A\in
\mathcal{W}_{J}^{s,\ell }:\ \ell \left( A\right) =2^{-k-\ell }\ell \left(
2^{\ell }J\right) }\left\vert A\right\vert ^{\frac{2}{n}}\left\vert
A\right\vert _{\omega }\leq \sum_{k=s}^{\infty }2^{-2k-2\ell }\left\vert
J\right\vert ^{\frac{2}{n}}\sum_{A\in \mathcal{W}_{M}^{s,\ell }:\ \ell
\left( A\right) =2^{-k-\ell }\ell \left( 2^{\ell }J\right) }\left\vert
A\right\vert _{\omega } \\
&\leq &\sum_{k=s}^{\infty }2^{-2k-2\ell }\left\vert 2^{\ell }J\right\vert ^
\frac{2}{n}}\left\vert 3\left( 2^{\ell }J\right) \right\vert _{\omega
}\lesssim 2^{-2s-2\ell }\left\vert 2^{\ell }J\right\vert ^{\frac{2}{n
}\left\vert 3\left( 2^{\ell }J\right) \right\vert _{\omega }\ .
\end{eqnarray*
Now we can continue wit
\begin{eqnarray*}
&&\mathcal{A}_{s,far}^{\ell }\left( J\right) \lesssim 2^{-2s}2^{-2\ell
}\left\vert 3\left( 2^{\ell }J\right) \right\vert _{\omega }\int_{I\setminus
4\left( 2^{\ell }J\right) }\frac{1}{\left( \left\vert 2^{\ell }J\right\vert
^{\frac{2}{n}}+\left\vert y-c_{2^{\ell }J}\right\vert \right) ^{2\left(
1-\alpha \right) }}d\sigma \left( y\right) \\
&\approx &2^{-2s}2^{-2\ell }\frac{\left\vert 3\left( 2^{\ell }J\right)
\right\vert _{\omega }}{\left\vert 3\left( 2^{\ell }J\right) \right\vert
^{1-\alpha }}\int_{I\setminus 4\left( 2^{\ell }J\right) }\left( \frac
\left\vert 2^{\ell }J\right\vert ^{\frac{2}{n}}}{\left( \left\vert 2^{\ell
}J\right\vert ^{\frac{2}{n}}+\left\vert y-c_{2^{\ell }J}\right\vert \right)
^{2}}\right) ^{1-\alpha }d\sigma \left( y\right) \\
&\lesssim &2^{-2s}2^{-2\ell }\left\{ \frac{\left\vert 3\left( 2^{\ell
}J\right) \right\vert _{\omega }}{\left\vert 3\left( 2^{\ell }J\right)
\right\vert ^{1-\alpha }}\mathcal{P}^{\alpha }\left( 3\left( 2^{\ell
}J\right) ,\mathbf{1}_{I\setminus 4\left( 2^{\ell }J\right) }\sigma \right)
\right\} \lesssim 2^{-2s}2^{-2\ell }\mathcal{A}_{2}^{\alpha }\ .
\end{eqnarray*}
\bigskip
{\Large \#7:} The three lines above Section 11 should be replaced with these
three lines in quotes\footnote
A detailed argument is given in Subsection 9.3 of \cite[v3]{SaShUr6}, and in
Subsection \ref{Subsec back test} of Appendix B above for the one
dimensional case adapted to $Tb$.}: "and then expanding the square and
calculating as in the proof of the local part given earlier to obtain the
bound $\mathcal{A}_{2}^{\alpha }+\left( \mathcal{E}_{\alpha }^{\func{plug}}
\sqrt{A_{2}^{\alpha ,\func{energy}}}\right) \sqrt{A_{2}^{\alpha ,\limfunc
punct}}}$. The details are similar and they are left to the reader."
\bigskip
{\Large \#8:} In line -3 of page 164 the final factor on the right hand side
should instead be $\omega _{\mathcal{P}_{L,0}^{\limfunc{small}}}\left(
\mathbf{T}^{\mathbf{\tau }-\limfunc{deep}}\left( K\right) \right) $.
\bigskip
{\Large \#9:} There is a gap in the proof of the Orthogonality Lemma at the
top of page 170, where the restricted norm of the sublinear form used there
doesn't match the definition on page 161. The required change in definition
of the restricted norm, then forces an additional argument - due to Michael
Lacey in \cite{Lac} - that uses another Calder\'{o}n-Zygmund decomposition
in the proof of the Straddling Lemma on page 166. Here are the changes
required (see also Section 7 above, which adapts Lacey's additional argument
to the $Tb$ setting, in which the dual martingale differences are no longer
orthogonal projections).
First, the display on page 161 right before Proposition 11.8 should be
replaced wit
\begin{equation*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\mathcal{P}}\left( f,g\right) \leq \mathfrak{N}_{\limfunc{stop
,1,\bigtriangleup ^{\omega }}^{A,\mathcal{P}}\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\left\Vert g\right\Vert _{L^{2}\left( \omega
\right) }\ ,
\end{equation*
in which the term $\alpha _{\mathcal{A}}\left( A\right) \sqrt{\left\vert
A\right\vert _{\sigma }}$ no longer appears on the right hand side.
Second, the sentence on page 169 right before Subsubsection 11.4.2 should be
replaced with the following material:
\bigskip
Now we sum these bounds in $s$ and $\ast $ and use $\sup_{S\in \mathcal{S}
\mathcal{S}_{\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \leq
\mathcal{S}_{\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q}\right) $ to obtai
\begin{equation*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) \leq \mathcal{S}_{\limfunc{size
}^{\alpha ,A}\left( \mathcal{Q}\right) \left\{ \alpha _{\mathcal{A}}\left(
A\right) \sqrt{\left\vert A\right\vert _{\sigma }}+\left\Vert f\right\Vert
_{L^{2}\left( \sigma \right) }\right\} \left\Vert g\right\Vert _{L^{2}\left(
\omega \right) }\ .
\end{equation*
However, this inequality has the unwanted term $\alpha _{\mathcal{A}}\left(
A\right) \sqrt{\left\vert A\right\vert _{\sigma }}$ included on the right
hand side, and we must apply an argument of M. Lacey \cite[see the proof of
Lemma 3.19]{Lac} to eliminate this term. We begin wit
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &=&\left\vert \mathsf{B}\right\vert _
\limfunc{stop},1,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}}\left( h,g\right)
, \\
\text{where }h &\equiv &\mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right)
}^{\sigma }f=\sum_{I\in \Pi _{1}\mathcal{Q}}\bigtriangleup _{\pi I}^{\sigma
}f,
\end{eqnarray*
which follows from the formula $\varphi _{J}^{\mathcal{Q}}\equiv \sum_{I\in
\mathcal{C}_{A}^{\prime }:\ \left( I,J\right) \in \mathcal{Q}}\mathbb{E
_{I}^{\sigma }\left( \bigtriangleup _{\pi I}^{\sigma }f\right) \ \mathbf{1
_{A\setminus I}$ since $\bigtriangleup _{\pi I}^{\sigma }f=\bigtriangleup
_{\pi I}^{\sigma }h$ for $I\in \Pi _{1}\mathcal{Q}$, which holds in turn
because the Haar projections are orthogonal. Now define Calder\'{o}n-Zygmund
stopping times $\mathcal{H}$ for the function $h=\mathsf{P}_{\pi \left( \Pi
_{1}\mathcal{Q}\right) }^{\sigma }f\in L^{2}\left( \sigma \right) $, so that
the quasiorthogonal inequalit
\begin{equation*}
\sum_{H\in \mathcal{H}}\alpha _{\mathcal{H}}\left( H\right) ^{2}\left\vert
H\right\vert _{\sigma }\lesssim \left\Vert h\right\Vert _{L^{2}\left( \sigma
\right) }^{2}=\left\Vert \mathsf{P}_{\pi \left( \Pi _{1}\mathcal{Q}\right)
}^{\sigma }f\right\Vert _{L^{2}\left( \sigma \right) }^{2}
\end{equation*
holds with $\alpha _{\mathcal{H}}\left( H\right) =\mathbb{E}_{H}^{\sigma
}\left\vert h\right\vert $. We also define in the usual way coronas
\mathcal{C}_{H}^{\mathcal{H}}$ and $\mathcal{C}_{H}^{\mathcal{H},\mathbf
\tau }-\func{shift}}$.
Now we return to the previous inequalities we obtained for $\left\vert
\mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega }}^{A
\mathcal{Q}_{s}}\left( f,g\right) $ and $\left\vert \mathsf{B}\right\vert _
\limfunc{stop},1,\bigtriangleup ^{\omega }}^{A,\mathcal{Q}_{\ast }}\left(
f,g\right) $, and replace the collection $\mathcal{Q}$ with the $A
-admissible collection $\mathcal{Q}_{H}\equiv \left\{ \left( I,J\right) \in
\mathcal{Q}:J\in \mathcal{C}_{H}^{\mathcal{H},\mathbf{\tau }-\func{shift
}\right\} $, so that the arguments given there can be adapted to yiel
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\left( \mathcal{Q}_{H}\right) _{s}}\left( f,g\right) &\leq &\sup_{S\in
\mathcal{S}}\mathcal{S}_{\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q
_{H}\right) \alpha _{\mathcal{H}}\left( H\right) \sqrt{\left\vert
H\right\vert _{\sigma }}\left\Vert \mathsf{P}_{\mathcal{C}_{H}^{\mathcal{H}
\mathbf{\tau }-\func{shift}}}^{\omega }g\right\Vert _{L^{2}\left( \omega
\right) }\ ,\ \ \ \ \ \mathbf{\tau }\leq s\leq \mathbf{\rho }-1, \\
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\left( \mathcal{Q}_{H}\right) _{\ast }}\left( f,g\right) &\leq
&\sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{size}}^{\alpha ,A;S}\left(
\mathcal{Q}_{H}\right) \alpha _{\mathcal{H}}\left( H\right) \sqrt{\left\vert
H\right\vert _{\sigma }}\left\Vert \mathsf{P}_{\mathcal{C}_{H}^{\mathcal{H}
\mathbf{\tau }-\func{shift}}}^{\omega }g\right\Vert _{L^{2}\left( \omega
\right) }\ .
\end{eqnarray*
We then sum these improved bounds in $s$ and $\ast $ to obtai
\begin{eqnarray*}
\left\vert \mathsf{B}\right\vert _{\limfunc{stop},1,\bigtriangleup ^{\omega
}}^{A,\mathcal{Q}}\left( f,g\right) &\leq &\sup_{S\in \mathcal{S}}\mathcal{S
_{\limfunc{size}}^{\alpha ,A;S}\left( \mathcal{Q}\right) \sum_{H\in \mathcal
H}}\alpha _{\mathcal{H}}\left( H\right) \sqrt{\left\vert H\right\vert
_{\sigma }}\left\Vert \mathsf{P}_{\mathcal{C}_{H}^{\mathcal{H},\mathbf{\tau
-\func{shift}}}^{\omega }g\right\Vert _{L^{2}\left( \omega \right) } \\
&\leq &\sup_{S\in \mathcal{S}}\mathcal{S}_{\limfunc{size}}^{\alpha
,A;S}\left( \mathcal{Q}\right) \sqrt{\sum_{H\in \mathcal{H}}\alpha _
\mathcal{H}}\left( H\right) ^{2}\left\vert H\right\vert _{\sigma }}\sqrt
\sum_{H\in \mathcal{H}}\left\Vert \mathsf{P}_{\mathcal{C}_{H}^{\mathcal{H}
\mathbf{\tau }-\func{shift}}}^{\omega }g\right\Vert _{L^{2}\left( \omega
\right) }^{2}} \\
&\leq &\mathcal{S}_{\limfunc{size}}^{\alpha ,A}\left( \mathcal{Q}\right)
\left\Vert f\right\Vert _{L^{2}\left( \sigma \right) }\left\Vert
g\right\Vert _{L^{2}\left( \omega \right) }\ ,
\end{eqnarray*
where we have used the quasiorthogonal inequality and $\left\Vert \mathsf{P
_{\pi \left( \Pi _{1}\mathcal{Q}\right) }^{\sigma }f\right\Vert
_{L^{2}\left( \sigma \right) }^{2}\leq \left\Vert f\right\Vert _{L^{2}\left(
\sigma \right) }^{2}$ in the last line.
\section{Appendix D: Glossary}
\subsubsection{Section 1}
\begin{enumerate}
\item $C_{CZ}$; (\ref{sizeandsmoothness'})
\item $\mathfrak{N}_{T_{\sigma }^{\alpha }}$; (\ref{two weight'})
\item $T_{\sigma ,\delta ,R}^{\alpha }f\left( x\right) $; (\ref{def
truncation})
\item $p$\emph{-weakly }$\mu $\emph{-accretive} family $\mathbf{b}=\left\{
b_{Q}\right\} _{Q\in \mathcal{P}}$ of functions on $\mathbb{R}$; (\ref{local
accretive})
\item $\mathfrak{T}_{T^{\alpha }}^{\mathbf{b}},\mathfrak{T}_{T^{\alpha ,\ast
}}^{\mathbf{b}^{\ast }}$; (\ref{b testing cond})
\item $\mathrm{P}^{\alpha }\left( Q,\mu \right) ,\mathcal{P}^{\alpha }\left(
Q,\mu \right) $; (\ref{def Poisson})
\item $\mathcal{A}_{2}^{\alpha },\mathcal{A}_{2}^{\alpha ,\ast }$; (\ref{def
call A2})
\item $\mathfrak{P}_{\left( \sigma ,\omega \right) }$; (\ref{def common
point mass})
\item $A_{2}^{\alpha ,\limfunc{punct}},A_{2}^{\alpha ,\ast ,\limfunc{punct}}
; (\ref{def punct})
\item $\mathfrak{A}_{2}^{\alpha }$; (\ref{def A2})
\item $\mathcal{E}_{2}^{\alpha },\mathcal{E}_{2}^{\alpha ,\ast },\mathfrak{E
_{2}^{\alpha },\mathcal{NTV}_{\alpha }$;$\ $(\ref{strong b* energy}), (\re
{strong b energy}), (\ref{def frak energy}), (\ref{def NTV})
\end{enumerate}
\subsubsection{Section 2}
\begin{enumerate}
\item \emph{gradient elliptic} kernel $K^{\alpha }\left( x,y\right) $; (\re
{def grad elliptic})
\item $PLBP,$pointwise lower bound property; (\ref{plb})
\item reverse H\"{o}lder control of children (\ref{rev Hol con})
\item Calder\'{o}n-Zygmund stopping intervals; Definition \ref{CZ stopping
times}
\item $\mathbf{b}$-accretive/weak testing stopping intervals; Definition \re
{accretive stopping times gen}
\item $\sigma $-energy stopping intervals; Definition \ref{def energy corona
3}
\item $\mathbf{X}_{\alpha }\left( \mathcal{C}_{S}\right) $ energy stopping
times; (\ref{def stopping energy 3})
\item $\mathcal{D}_{\beta }$: (\ref{def dyadic grid})
\end{enumerate}
\subsubsection{Section 3}
\begin{enumerate}
\item $\limfunc{skel}K$,$\ \mathbb{S}_{x}^{\limfunc{dy}},\limfunc{body}K$
skeleton, spray, body of an interval; (\ref{skel spray body})
\item $\varepsilon -\limfunc{good}$ \emph{with respect to} an interval; (\re
{eps far})
\item $\mathcal{G}_{\left( k,\varepsilon \right) -\limfunc{good}}^{\mathcal{
}}$; Definition \ref{good two grids}
\item $k$-$\limfunc{bad}$ in a grid; (\ref{bad in grid})
\item $R^{\maltese },\kappa \left( R\right) $: \ref{def sharp cross}
\item $\Theta _{2}^{\limfunc{bad}\natural }\left( f,g\right) $; (\re
{Theta_2^bad sharp})
\item $\mathcal{G}_{k-\limfunc{bad}}^{\mathcal{D}}=\mathcal{G}_{\left(
k,\varepsilon \right) -\limfunc{bad}}^{\mathcal{D}}$; Definition \ref{def
Gbad}
\item $\Theta _{2}^{\limfunc{good}}\left( f,g\right) $; (\ref{def Theta 2
good})
\item $\mathfrak{3T}_{T^{\alpha }}^{\mathbf{b}}$; (\ref{triple b testing
cond})
\item $\mathfrak{FT}_{T^{\alpha }}^{\mathbf{b}}$; (\ref{full b testing})
\item $\mathcal{E}_{2}^{\alpha ,\limfunc{triple}}$; Definition \ref{def
triple energy}
\end{enumerate}
\subsubsection{Section 4}
\begin{enumerate}
\item $\Theta _{1}^{\limfunc{long}}\left( f,g\right) ,\Theta _{1}^{\limfunc
short}}\left( f,g\right) $; (\ref{decomp long short})
\end{enumerate}
\subsubsection{Section 5}
\begin{enumerate}
\item $\partial _{\eta }L$; (\ref{def halo})
\item $\mathcal{K}\left( I^{\prime },J^{\prime }\right) $; (\ref{def
K(I',J')})
\item $\left\{ E,F\right\} $; (\ref{def E,F})
\item $\mathrm{P}_{\delta }^{\alpha }\mathsf{Q}^{\omega }\left( J,\upsilon
\right) $; (\ref{def compact})
\item $\left\{ K_{\limfunc{out}},K_{\limfunc{in}}\right\} ^{\limfunc{orig}}
; (\ref{def orig})
\end{enumerate}
\subsubsection{Section 6}
\begin{enumerate}
\item $\mathcal{C}_{B}^{\mathcal{G},\limfunc{shift}}$; Definition \re
{shifted corona}
\item $\left\langle T_{\sigma }^{\alpha }\left( \mathsf{P}_{\mathcal{C}_{A}^
\mathcal{D}}}^{\sigma ,\mathbf{b}}f\right) ,\mathsf{P}_{\mathcal{C}_{B}^
\mathcal{G},\limfunc{shift}}}^{\omega ,\mathbf{b}^{\ast }}g\right\rangle
_{\omega }^{\Subset _{\mathbf{r},\varepsilon }}$; (\ref{def shorthand})
\item $\mathsf{B}_{\Subset _{\mathbf{r},\varepsilon }}^{A}\left( f,g\right)
; (\ref{def local})
\item \emph{Whitney} subintervals $\mathcal{W}\left( F\right) $; (\ref{def
Whitney})
\item $\mathfrak{F}_{\alpha }=\mathfrak{F}_{\alpha }^{\mathbf{b}^{\ast
}}\left( \mathcal{D},\mathcal{G}\right) $; (\ref{functional energy n})
\item $\mathsf{B}_{\limfunc{stop}}^{A}\left( f,g\right) $; (\ref{bounded
stopping form}), (\ref{def stop}), (\ref{dummy})
\item $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$; Definition \ref{def
shift}
\item $\mathsf{Q}_{\mathcal{H}}^{\omega ,\mathbf{b}^{\ast }}$; (\ref{def
localization})
\item $\mathsf{B}_{\limfunc{broken}}^{A}\left( f,g\right) $; (\ref{broken
vanish})
\item $\mathsf{B}_{\limfunc{neighbour}}^{A}\left( f,g\right) $; (\ref{def
neighbour})
\item $\mathsf{B}_{\limfunc{stop}}^{A,\mathcal{P}}\left( f,g\right) $; (\re
{def stop P})
\end{enumerate}
\subsubsection{Section 7}
\begin{enumerate}
\item $\varphi _{J}^{\mathcal{P}}$; (\ref{def phi P})
\item $\widehat{\mathfrak{N}}_{\limfunc{stop},\bigtriangleup ^{\omega }}^{A
\mathcal{P}}$; (\ref{Norm hat})
\item $\Pi _{2}^{K}\mathcal{P}$; (\ref{rest K})
\item $\Pi _{1}^{\limfunc{below}}\mathcal{P}$; (\ref{Pi below})
\item $\mathcal{S}_{\limfunc{init}\limfunc{size}}^{\alpha ,A}\left( \mathcal
P}\right) ^{2}$; (\ref{def ext size})
\item $\Pi _{2}^{K,\limfunc{aug}}\mathcal{P}$; Definition \ref{augs}
\item $\mathcal{S}_{\limfunc{aug}\limfunc{size}}^{\alpha ,A}\left( \mathcal{
}\right) $; Definition \ref{augs}, (\ref{def P stop energy' 3})
\item $\mathcal{P}_{\func{cor}}^{A}$; (\ref{def cor})
\item $J^{\flat }=J_{\searrow J}^{\maltese }$; (\ref{def aug})
\item $\mathcal{Q}$ $\flat $\textbf{straddles} $\mathcal{S}$; Definition \re
{flat straddles}
\item $\mathcal{W}^{\ast }\left( S\right) $; (\ref{def Whit})
\item $\mathcal{S}_{\limfunc{loc}\limfunc{size}}^{\alpha ,A;S}\left(
\mathcal{Q}\right) ^{2}$; (\ref{localized size ref})
\item $\mathcal{Q}$ \textbf{substraddles} $L$; (\ref{def substraddles})
\item $\omega _{\mathcal{P}}$ and $\omega _{\flat \mathcal{P}}$; (\ref{def
atomic})
\item $\mathcal{G}_{d}$; (\ref{geom depth})
\item $\mathcal{P}_{L,t}^{\mathcal{H}}$; (\ref{def PHLt})
\item $\mathcal{P}^{big}$, $\mathcal{P}^{small}$; (\ref{def big small})
\item $\mathcal{P}^{\flat big}$, $\mathcal{P}^{\flat small}$; (\ref{def big
small flat})
\end{enumerate}
\subsubsection{Section 9}
\begin{enumerate}
\item $\mathbb{E}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) $; (\ref{def
expectation})
\item $\widehat{\mathbb{F}}_{Q}^{\mu ,\mathbf{b}}f\left( x\right) $; (\ref{F
hat})
\item $\bigtriangleup _{Q}^{\mu ,\mathbf{b}}f\left( x\right) $, $\square
_{Q}^{\mu ,\mathbf{b}}f\left( x\right) $; (\ref{def diff})
\item $\bigtriangledown _{Q}^{\mu }$, $\widehat{\bigtriangledown }_{Q}^{\mu
} $; (\ref{Carleson avg op})
\item $\mathbb{E}_{Q}^{\mu ,\pi ,\mathbf{b}}f$, $\mathbb{F}_{Q}^{\mu ,\pi
\mathbf{b}}f$; (\ref{def pi exp})
\item $\bigtriangleup _{Q}^{\mu ,\pi ,\mathbf{b}}f$, $\square _{Q}^{\mu ,\pi
,\mathbf{b}}f$; (\ref{def pi box})
\item $\square _{I}^{\sigma ,\flat ,\mathbf{b}}f$; (\ref{flat box})
\item $\widehat{\square }_{I}^{\sigma ,\flat ,\mathbf{b}}f$; (\ref{flat box
hat})
\item $\bigtriangleup _{I,\limfunc{broken}}^{\mu ,\flat ,\mathbf{b}}f$,
\square _{I,\limfunc{broken}}^{\mu ,\flat ,\mathbf{b}}f$; (\ref{def flat
broken})
\item $\Psi _{\mathcal{B},\mathbf{\lambda }}^{\mu ,\mathbf{b}}f$: Definition
\ref{Psi op}
\item $\mathsf{P}_{\mathcal{B}}^{\mu }f$; (\ref{Haar proj})
\end{enumerate}
\subsubsection{Section 10}
\begin{enumerate}
\item $\mathsf{Q}_{\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};M}^{\omega
\mathbf{b}^{\ast }}$; (\ref{def pseudo rest})
\item $\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}}$, $\mathcal{C}_{F}^
\mathcal{G},\limfunc{shift}};K$; (\ref{def shift cor rest})
\item $J\Subset _{\mathbf{\rho },\varepsilon }K$; (\ref{def deep embed})
\item $\mathcal{M}_{\left( \mathbf{\rho },\varepsilon \right) -\limfunc{deep
,\mathcal{G}}\left( K\right) $, $\mathcal{M}_{\left( \mathbf{\rho
,\varepsilon \right) -\limfunc{deep},\mathcal{D}}\left( K\right) $,
\mathcal{W}\left( K\right) $; (\ref{def M_r-deep})
\item augmented dyadic grid $\mathcal{AD}$; Definition \ref{def dyadic}
\item $\mathsf{Q}_{K}^{\omega ,\mathbf{b}^{\ast }}$; (\ref{large pseudo})
\item $\mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{partial}}$; (\re
{plug})
\item $A_{2}^{\alpha ,\limfunc{energy}}$, $A_{2}^{\alpha ,\ast ,\limfunc
energy}}$; (\ref{def energy A2})
\item $\mathcal{E}_{2}^{\alpha ,\func{Whitney}\limfunc{plug}}$; (\ref{def
deep plug})
\item $\mu $; (\ref{def mu n})
\item $\mathsf{P}_{F,K}^{\omega ,\mathbf{b}^{\ast }}\equiv \mathsf{P}_
\mathcal{C}_{F}^{\mathcal{G},\limfunc{shift}};K}^{\omega ,\mathbf{b}^{\ast
}} $; (\ref{def F,K})
\item $\mathbf{Local}\left( I\right) $, $\overline{\mu }$; (\ref{def local
forward})
\item $\mathcal{J}^{\ast }$; (\ref{def J*})
\item $\mathbf{Back}\left( \widehat{I}\right) $; (\ref{e.t2 n'})
\item $U_{s}$; (\ref{def Us})
\item $B\left( M,M^{\prime }\right) $; (\ref{def BMM'})
\item $T_{s}^{\limfunc{intersection}}$; (\ref{def Tints})
\item $T_{s}^{\limfunc{proximal}}$; (\ref{def Tproxs})
\item $T_{s}^{\limfunc{difference}}$; (\ref{def Tdiffs})
\item $\overset{\ast }{\sum }$; Notation \ref{Sum *}
\item $\mathcal{W}_{M}^{s}$; (\ref{def WMs})
\item $\mathcal{W}_{M}^{s,\ell }$; (\ref{def WMsell})
\end{enumerate}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,673 |
{"url":"http:\/\/basilisk.fr\/sandbox\/jmf\/tutorial","text":"# sandbox\/jmf\/tutorial\n\nTutorial Basilisk\n\n# Introduction (basic) Basilisk\n\nThis tutorial is a very very basic introduction to Basilisk. We propose a slow journey into Basilisk starting from the study of the diffusion equation, doing a comparison between a classical C code and the equivalent in Basilisk.\n\n## Example : solving a diffusion equation\n\nSo we want to solve the diffusion equation\n\n\\displaystyle {\\partial A \\over \\partial t} = \\nabla^2 A\n\nusing a simple scheme in C (we have supposed that the diffusion coefficient is 1) over a rectangular squared grid of NxN points and physical side equal to 1.\n\nIn terms of an algorithmic approach we have to\n\n\u2022 declare and initialize the variables\n\u2022 set the initial and boundary conditions\n\u2022 compute the discrete problem\n\u2022 write the solution\n\nIn a sequential C code we have then\n\n### C code\n\n\n#include <stdio.h>\n#include <math.h>\n\nint main ()\n{\nint i,j,k;\nint N = 256 ;\ndouble L = 1. ;\ndouble x,y;\ndouble dx = L\/N ;\ndouble dt = 0.00001;\ndouble A[N][N];\ndouble dA[N][N];\n\n\/\/ boundary conditions\nfor (i = 0 ; i < N ; i++) A[i][0] = A[i][N-1] = 0. ;\nfor (j = 0 ; j < N ; j++) A[0][j] = A[N-1][j] = 0. ;\n\n\/\/ initial conditions\n\nfor (i = 0 ; i < N ; i++)\n{\nfor (j = 0 ; j < N ; j++)\n{\nx = i*dx - 0.5 ;\ny = j*dx - 0.5 ;\nA[i][j] = 1.\/0.1*((fabs(x*x+y*y) < 0.05)) ;\n}\n}\n\nfor (j = 0 ; j < N ; j++)\n{\nprintf(\"%f \\n\",A[(int)N\/2][j]);\n}\nprintf(\"\\n\\n\");\n\n\/\/ time integration\n\nfor (k = 0 ; k < 10 ; k++)\n{\n\nfor (i = 1 ; i < N-1 ; i++)\n{\nfor (j = 1 ; j < N-1 ; j++)\n{\ndA[i][j] = (A[i+1][j] + A[i-1][j] - 2. * A[i][j])\/dx\/dx +\n(A[i][j+1] + A[i][j-1] - 2. * A[i][j])\/dx\/dx ;\n}\n}\n\n\/\/ update\n\nfor (i = 0 ; i < N ; i++)\n{\nfor (j = 0 ; j < N ; j++)\n{\nA[i][j] = A[i][j] + dt* dA[i][j] ;\n}\n}\n\n}\n\n\/\/ print solution (centerline)\n\nfor (j = 0 ; j < N ; j++)\n{\nprintf(\"%f \\n\",A[(int)N\/2][j]);\n}\n\n}\n\n### Basilisk Code\n\nSame code using Basilisk appears more compact in a first sight\n\n#include \"grid\/cartesian.h\"\n#include \"run.h\"\n\nscalar A[];\nscalar dA[];\n\ndouble dt;\n\nint main() {\nL0 = 1.;\nN = 256;\n\nrun();\n}\n\n\/\/ boundary conditions\n\nA[left] = 0.0 ;\nA[top] = 0.0 ;\nA[right] = 0.0 ;\nA[bottom] = 0.0 ;\n\n\/\/ initial conditions\nevent init (t = 0) {\nforeach()\nA[] = 1.\/0.1*(fabs(x*x+y*y)<0.05);\nboundary ({A});\n}\n\n\/\/ integration\n\nevent integration (i++) {\nforeach()\ndA[] = (A[1,0] + A[-1,0] - 2. * A[])\/Delta\/Delta +\n(A[0,1] + A[0,-1] - 2. * A[])\/Delta\/Delta ;\n\nforeach()\nA[] = A[] + dt*dA[];\nboundary ({A});\n\n}\n\n\/\/ print\nevent print (i=10) {\n\nfor (double y = 0 ; y <= L0; y += 0.01){\nprintf(\"%g %g \\n\",\ny, interpolate (A, 0, y));\n}\n\n}\n\n### First observations\n\nObserving both piece of code we recognize some differences, about\n\n\u2022 Some reserved words (N, L0, \u2026 )\n\u2022 Automatic grid setting\n\u2022 New types (scalar, ) with automatic memory allocation\n\u2022 Function run()\n\u2022 Boundary conditions\n\u2022 Position in the grid : A[1,0], A[-1,0], A[]\n\u2022 New \u201citerators\u201d like foreach() (replacing \u201cfor ( \u2026\u201d)\n\u2022 New method \u201cevents\u201d managing code actions\n\n## Exploring the differences\n\nWe list now the difference to introduce the Basilisk syntax.\n\n### Reserved word\n\nSome variables (and consequently their names) are global and reserved in Basilisk, some of them are\n\nSome reserved words\nWords meaning\nu v w velocities\np pressure\nt time\nN grid size\nL0 physical size\n\nyou have to learn them, in particular you could take a look to the solvers to understand what variable is global and reserved.\n\n### Automatic grid\n\nBasilisk computed equation over a cartesian grid, when you declare in the code\n\n N = 256\n\nyou are setting the grid size. Attention N must be multiple of 2. Another way is using the following function\n\n init_grid (128);\n\n### New types\n\nBasilisc adds some new types to the classical C types (double, float, int, \u2026). The first we have seen is scalar\n\n scalar A[]; \n\nwhich do an implicit allocation. An explicit allocation and deallocation can be done like this:\n\n scalar A = new scalar;\n...\ndelete ({A});\n\nIn both cases Basilisk allocates the memory for a NxN grid (or N if in 1D or NxNxN if in 3D) for the variable A. In a finite volume approach the elementary cell has several positions to define the variables : the center, the sides and the corners.\n\nThe scalar A is defined at the center of the cell.\n\nThere exist two other types of fields defined in Basilisk : vector and tensor.\n\n### Function run()\n\nThe run() function implements a generic time loop which executes events until termination. The time t and time step dt can be accessed as global variables.\n\nThis function appears in all Basilisk programs, and basically\n\n1. Set the grid if the variable N is done\n2. List all events in order until termination\n\nA typical usage is\n\nint main() {\nN = 128\n\ninit_grid (N);\nrun();\n}\n}\n\n### Boundary conditions\n\nBasilisk creates stencil values outside the domain (called ghost cell values) which need to be initialised. These values can be set in order to provide the discrete equivalents of different boundary conditions. In our case we use the reserved words left, right, top or bottom to impose such values. Doing\n\n A[left] = dirichlet(0.);\n\nwe impose the value zero to the left column in the matrix A doesn\u2019t matter the gird size. This is equivalent to the C code\n\n for (j = 0 ; j < N ; j++) A[0][j] = 0.0 ;\n\nWe can also use a given function of spatial and temporal variables as\n\n A[left] = dirichlet(y * cos(2 Pi t);\n\nGhost values usually depend on the values inside the domain (for example when using symmetry conditions). It is necessary to update them when values inside the domain are modified. This can be done by calling\n\n boundary ({A});\n\nwhich sets all boundary conditions defined in the code. Normally we must to update the boundary conditions after each change in the stencil.\n\n### Field values over a stencil\n\nStencils are used to access field values and their local neighbours. By default Basilisk guarantees consistent field values in a 3x3 neighbourhood (in 2D). This can be represented like this\n\nWhen you are inside of the loop foreach() you are every time at [0,0] (the center of the stencil) and you can access to all values over the stencil only calling them by their local position. The neighboring values, necessary to define integration schema, are accessed directly using the indexing scheme of the Figure. Note that A[] is a shortcut of A[0,0]. As an example we can compute a centered spatial 1st derivative\n\n (A[-1,0]+A[1,0])\/Delta\n\nas well as the 2nd one\n\n (A[-1,0]+A[1,0]-2. * A[])\/Delta\/Delta\n\n### Iterators\n\nAs observed before foreach() iterates over the whole grid, in 2D the double loop over i and j in a C code is\n\n for (i = 1 ; i < N-1 ; i++)\n{\nfor (j = 1 ; j < N-1 ; j++)\n{\n\u2026\n}\n}\n\nbecomes now in Basilisk\n\n foreach() {\n\u2026\n}\n\nNote that inside iterators some variables are implicitly defined :\n\n double x, y; \/\/ coordinates of the center of the stencil\ndouble \u0394; \/\/ size of the stencil cell\n\n\nOthers iterators are also defined : foreach_dimension(), foreach_face() and foreach_vertex().\n\n### Events\n\nNumerical simulations need to perform actions (inputs or outputs for example) at given time intervals. Basilisk C provides events to manage all actions.\n\nThe overall syntax of events is\n\n event name (t = 1; t <= 5; t += 1) {\n...\n}\n\nwhere name is the user-defined name of the event, is this case t = 1 specifies the starting time, t <= 5 is the condition which must be verified for the event to carry on and t += 1 is the iteration operator. We can use both the specified times t or a specified number of time steps i using a C syntax, like\n\n event othername (i++) {\n...\n}\n\nwhich means do it at every iteration\n\n# Basilisc C (a bit more)\n\nNow we go inside Basilisk syntax a bit more deep\n\n## Types and stencils\n\nVector and tensor fields are used in a similar way. Vector fields are a collection of D scalar fields and tensor fields are a collection of D vector fields.\n\naccess\nEach of the components of the vector or tensor fields are accessed using the x, y or z field of the corresponding structure.\n\nvector v[];\ntensor t = new tensor;\n...\nforeach() {\nv.x[] = 1.;\nt.x.x[] = (v.x[1,0] - v.x[-1,0])\/\u0394;\nt.y.x[] = (v.y[1,0] - v.y[-1,0])\/\u0394;\n}\n\nWhen we write numerical scheme we need often a special arrangements of discretisation variables relative to grid (this is sometimes called variable staggering). Basilisk provides support for the three most common types of staggering:\n\n1. centered staggering (default case),\n2. face staggering\n3. vertex staggering\n\nThe following Figure shows the three staggering\n\nIn this case we have defined\n\nscalar p[];\nface vector u[];\nvertex scalar \u03c9[];\n\nImportant : some operations performed by Basilisk (such as interpolation and boundary conditions) need to know that these fields are staggered, you need then to know the kind of variable you are using.\n\nlist\nA new concept in Basilisk is list which can combine elements of different types (e.g.\u00a0scalar fields and vector fields) is a single row.\n\nBy the way an automatic list of scalars can be declared and allocated like this:\n\n scalar * list = {a,b,c,d};\n\nLists are used to do a repetitive things, for example to iterate over all the elements of a list use\n\nscalar * list = {a,b,c,d};\n...\nfor (scalar s in list)\ndosomething (s);\n\nor setting boundary conditions\n\nboundary({a,b,c,d})\n\nwhich update all defined boundary conditions for scalars a,b,c,d.\n\n## Boundary conditions\n\nThe default boundary condition is symmetry for all the fields : scalars, vectors or tensors.\n\nThere exists somme reserved conditions for the boundary condition as the classical neumann or dirichlet\n\n### Scalars\n\nBoundary conditions can be changed for scalar fields using the following syntax:\n\n A[top] = a[];\n\nwhere a[top] is the ghost value of the scalar field a immediately outside the top (respectively bottom, right, left as stated above) boundary. This corresponds to a Neumann condition (i.e.\u00a0a condition on the normal derivative of field a) which can be written as\n\n A[top] = neumann(0.0);\n\n### Vectors and tensor\n\nFor vector fields, boundary conditions are defined in a coordinate system local to the boundary where the x and y components are replaced by the normal n and tangential t components i.e.\u00a0imposing no-flux of a vector v through the top and left boundary, together with a no-slip boundary condition would be written\n\n v.n[top] = dirichlet(0);\nv.t[top] = dirichlet(0);\nv.n[left] = dirichlet(0);\nv.t[left] = dirichlet(0); \n\n### Periodic\n\nPeriodic boundary conditions can be imposed on the right\/left and top\/bottom boundaries using for example\n\nint main()\n{\n...\nperiodic (right);\n\u2026\n}\n\nWhere all existing fields and all the fields allocated after the call will be periodic in the right\/left direction. Boundary conditions on specific fields can still be set to something else. For example, one could use\n\nint main()\n{\n...\nperiodic (right);\np[left] = dirichlet(0);\np[right] = dirichlet(1);\n\u2026\n}\n\nto impose a pressure gradient onto an otherwise periodic domain.\n\n### Boundary Internal Domain (bid)\n\nIn Basilsik, the simulation domain is by default a square box with right, left, top and bottom associated boundary conditions. It is possible to define domains of arbitrary shape, with an arbitrary number of associated boundary conditions, using the mask() function. This function associates a given boundary condition to each cell of the grid.\n\nFor example, to turn the domain into a rectangle with the variable y between 0 and 0.5\n\n mask (y > 0.5 ? top : none);\n\n1. The argument of the function is the value of the boundary condition to assign to each cell. In this example, all grid points of our new domain will be assigned the (pre-defined) top boundary condition.\n2. the boundary condition of all other grid points will be unchanged (the none value is just ignored).\n\nMore complex boundary conditions can be done using the Boundary Internal Domain (or bid) by defining\n\n bid circle;\n\nwhere circle is a user-defined identifier. For example for a no-slip boundary condition for a vector field u could be defined using\n\n u.t[circle] = dirichlet(0);\n\nmask (sq(x - 0.5) + sq(y - 0.5) < sq(0.5) ? circle : none);\n\n# Outputs functions\n\nLater when you manage correctly the Basilisk solvers you will need only to know the output functions. You can write yourself your own output function using the standard C as we have done in the 1st Basilsik code\n\n event print (i=10) {\n\nfor (double y = 0 ; y <= L0; y += 0.01){\nprintf(\"%g %g \\n\",\ny, interpolate (A, 0, y));\n}\n}\n\nThe function interpolate()is very useful to do slices over data, it does a bilinear interpolation over the grid and the syntax is interpolate(A,x,y).\n\nThe following output code write every 10 time units the x,y position of the grid together with the x and y components of the velocity field\n\n event print (t += 10) {\n\nforeach(){\nprintf(\"%f %f %f %f\\n\",x,y,u.x[],u.y[]);\n}\nprintf(\"\\n\\n\");\n}\n\nthe double blank line printf(\u201c\\n\\n\u201d); is useful for using the block notion in Gnuplot to graphic vectors, like\n\n gnuplot> plot \"field\" index 1:10 using 1:2:3:4 with vector\n\nBasilisk includes several output function, we present some of them\n\n## output_field(): regular grid in a text format\n\nDoes interpolation over multiple fields on a regular grid in a text format.\n\n\u2022 This function interpolates a list of fields on a n x n regular grid.\n\u2022 The resulting data are written in text format in the file pointed to by fp.\n\u2022 The correspondance between column numbers and variables is summarised in the first line of the file.\n\u2022 The data are written row-by-row and each row is separated from the next by a blank line.\n\nThis format is compatible with the splot command of gnuplot i.e.\u00a0one could use something like\n\n gnuplot> set pm3d map\ngnuplot> splot 'fields' u 1:2:4 \n\nThe arguments and their default values are:\n\n\u2022 list : is a list of fields to output. Default is all.\n\u2022 fp : is the file pointer. Default is stdout.\n\u2022 n : is the number of points along each dimension. Default is N.\n\u2022 linear : use first-order (default) or bilinear interpolation.\nevent output (t = 5) {\nchar name[80];\nsprintf (name, \"pressure.dat\", nf);\nFILE * fp = fopen (name, \"w\");\noutput_field ({p}, fp, linear = true);\nfclose (fp);\n\n## output_ppm(): Portable PixMap (PPM) image output\n\nGiven a field, this function outputs a colormaped representation as a Portable PixMap image. If ImageMagick is installed on the system, this image can optionally be converted to any image format supported by ImageMagick.\n\nThe arguments and their default values are:\n\n\u2022 f : is a scalar field (compulsory).\n\u2022 fp : is a file pointer. Default is stdout.\n\u2022 n : is number of pixels. Default is N.\n\u2022 file : sets the name of the file used as output for ImageMagick.\n\nFor example, one could use C output_ppm (f, file = \u201cf.png\u201d); to get a PNG image. You can use output_ppm()to generate movies, like\n\nevent movie (t += 0.2; t <= 30) {\nstatic FILE * fp = popen (\"ppm2mpeg > vort.mpg\", \"w\");\nscalar \u03c9[];\nvorticity (u, \u03c9);\noutput_ppm (\u03c9, fp, linear = true);\n}\n\nwhere we supposed that exist a C function vorticiy()which computers the vorticity from the velocity field.\n\n## output_vtk - Write data in a VTK format\n\nVTK format are used in softwares as paraviewor visit\n\nThe arguments and their default values are:\n\n\u2022 list : is a list of fields to output. Default is all.\n\u2022 fp : is a file pointer. Default is stdout.\n\u2022 n : is number of pixels. Default is N.\n\u2022 linear : a boolean for linear or bilinear interpolation\n\nThe syntax is\n\n output_vtk (scalar * list, int n, FILE * fp, bool linear)\n\n# Examples using solvers\n\nBasilisk provides an ensemble of solvers (Saint Venant, Navier-Stokes, diffusion, etc) that could be used to solve simple and more complex systems by adding them. Now what\u2019s a solver and how do you use it?\n\n1. A solver is a C file which contains variable definitions and functions for solving a specific general problem.\n2. When you include a solver file some variables as well as functions are reserved\n3. You need then first to read the solver file to know the reserved variables and functions.\n4. And you need to know the inputs for the solver!!\n\n## (still) diffusion equation\n\nWe come back to the diffusion equation \\displaystyle \\partial_t A = \\nabla^2 A\n\nwhich is a particular case of a reaction\u2013diffusion equation\n\n\\displaystyle \\theta\\partial_tf = \\nabla\\cdot(D\\nabla f) + \\beta f + r where \\beta f + r is a reactive term, D is the diffusion coefficient and \\theta could be a kind of density term. Including the diffusion solver into the program as\n\n #include \"diffusion.h\"\n\nyou include an implicit solver fro the reaction\u2013diffusion equation, for a scalar field f, scalar fields r and \\beta defining the reactive term, the time step dt and a face vector field containing the diffusion coefficient D. By the way a complete calling of the solver is\n\n diffusion (C, dt, D, r, \u03b2);\n\nwhich solves the diffusion-reaction problem for a scalar C, with a diffusion coefficient D, r and \\beta as defined. In particular\n\n\u2022 If D or \\theta are omitted they are set to one.\n\u2022 If \\beta is omitted it is set to zero.\n\u2022 Both D and \\beta may be constant fields.\n\nThen for\n\n\\displaystyle \\partial_t A = \\nabla^2 A\n\nthe syntax for the diffusion() function is\n\ndiffusion (A, dt);\n\nFor information using a time-implicit backward Euler discretisation, our equation can be written as\n\n\\displaystyle \\frac{A^{n+1} - A^{n}}{dt} = \\nabla^2 A^{n+1}\n\nRearranging the terms we get\n\n\\displaystyle \\nabla^2 A^{n+1} + \\frac{1}{dt} A^{n+1} = - \\frac{1}{dt}A^{n}\n\nThis is a Poisson\u2013Helmholtz problem which can be solved with a multigrid solver.\n\nWe can now re-write the Basilsik program\n\n#include \"grid\/cartesian.h\"\n#include \"run.h\"\n#include \"diffusion.h\"\n\nscalar A[];\nscalar dA[];\n\ndouble dt;\n\nint main() {\nL0 = 1.;\nN = 256;\n\nrun();\n}\n\nevent init (t = 0) {\nforeach()\nA[] = 1.\/0.1*(fabs(x*x+y*y)<0.05);\nboundary ({A});\n}\n\nevent integration (i++) {\n\ndiffusion(A,dt);\nboundary ({A});\n\n}\n\nevent print (i=10) {\n\nfor (double y = 0 ; y <= L0; y += 0.01){\nprintf(\"%g %g \\n\",\ny, interpolate (A, 0, y));\n}\n\n}\n\n## Shallow water equation\n\nFor conservation of mass and momentum in the shallow-water context we solve\n\n\\displaystyle \\partial_t \\mathbf{q} + \\nabla \\mathbf{f} = 0\n\nfor the conserved vector \\mathbf{q} and flux function \\mathbf{f}(\\mathbf{q}), explicitly\n\n\\displaystyle \\mathbf{q} = \\left(\\begin{array}{c} h\\\\ hu_x\\\\ hu_y \\end{array}\\right), \\;\\;\\;\\;\\;\\; \\mathbf{f} (\\mathbf{q}) = \\left(\\begin{array}{cc} hu_x & hu_y\\\\ hu_x^2 + \\frac{1}{2} gh^2 & hu_xu_y\\\\ hu_xu_y & hu_y^2 + \\frac{1}{2} gh^2 \\end{array}\\right) where \\mathbf{u} is the velocity vector, h the water depth and z_b the height of the topography.\n\nThe primary fields are the water depth h, the bathymetry z_b and the flow speed \\mathbf{u}. \\eta is the water level i.e. z_b + h. Note that the order of the declarations is important as z_b needs to be refined before h and h before \\eta.\n\nscalar zb[], h[], eta[];\nvector u[];\n\nThe only physical parameter is the acceleration due to gravity G. Cells are considered dry when the water depth is less than the dry parameter (very small number).\n\ndouble G = 1.;\ndouble dry = 1e-10;\n\n~c #include \u201csaint-venant.h\u201d\n\nint LEVEL = 9;\n\nWe define a new boundary for the cylinder.\n\nbid cylinder;\n\nint main() {\nsize (5.);\nG = 9.81;\norigin (-L0\/2., -L0\/2.);\ninit_grid (1 << LEVEL);\nrun();\n}\n\nWe impose height and velocity on the left boundary.\n\n#define H0 3.505271526\n#define U0 6.29033769408481\n\nh[left] = H0;\neta[left] = H0;\nu.n[left] = U0;\n\nevent init (i = 0) {\n\nThe geometry is defined by masking and the initial step function is imposed.\n\n mask (sq(x - 1.5) + sq(y) < sq(0.5) ? cylinder : none);\nmask (y > 2.-x*0.2 ? top : y < -2.+x*0.2 ? bottom : none);\nforeach() {\nh[] = (x <= -1 ? H0 : 1.);\nu.x[] = (x <= -1 ? U0 : 0.);\n}\n}\n\nevent logfile (i++) {\nstats s = statsf (h);\nfprintf (ferr, \"%g %d %g %g %.8f\\n\", t, i, s.min, s.max, s.sum);\n}\n\nWe generate movies of depth and level of refinement.\n\nc event movie (t += 0.005; t < 0.4) { static FILE * fp = popen (\u201cppm2mpeg > depth.mpg\u201d, \u201cw\u201d); output_ppm (h, fp, min = 0.1, max = 6, map = cool_warm, n = 400, linear = true); } event adapt (i++) { astats s = adapt_wavelet ({h}, (double[]){1e-2}, LEVEL); fprintf (ferr, \u201c# refined %d cells, coarsened %d cells\u201d, s.nf, s.nc); } The movie of the depth of water ## Navier-Stokes equations We simulate the lid-driven cavity problem using the centered solver We wish to approximate numerically the incompressible, variable-density Navier-Stokes equations \\displaystyle \\partial_t\\mathbf{u}+\\nabla\\cdot(\\mathbf{u}\\otimes\\mathbf{u}) = \\frac{1}{\\rho}\\left[-\\nabla p + \\nabla\\cdot(\\mu\\nabla\\mathbf{u})\\right] + \\mathbf{a} \\displaystyle \\nabla\\cdot\\mathbf{u} = 0 When we analyze the solver (file centered.h) we learn that 1. reserved words scalar p[]; vector u[]; vector g[]; are reserved variables (all centered, the pressure p and the vectors \\mathbf{u} an \\mathbf{g}) and also scalar pf[]; face vector uf[]; the auxiliary face velocity field \\mathbf{u}_f and the associated centered pressure field p_f. 2. parameters a. In the case of variable density, the user will need to define both the face and centered specific volume fields (\\alpha and \\alpha_c respectively) i.e. 1\/\\rho. If not specified by the user, these fields are set to one i.e.\u00a0the density is unity. b. Viscosity is set by defining the face dynamic viscosity \\mu; default is zero. c. The face field \\mathbf{a} defines the acceleration term; default is zero. d. If stokes (a boolean variable) is set to true, the velocity advection term \\nabla\\cdot(\\mathbf{u}\\otimes\\mathbf{u}) is omitted. The code is C #include \u201cgrid\/multigrid.h\u201d #include \u201cnavier-stokes\/centered.h\u201d int main() { \/\/ coordinates of lower-left corner origin (-0.5, -0.5); \/\/ number of grid points init_grid (64); \/\/ viscosity const face vector muc[] = {1e-3,1e-3}; \u03bc = muc; \/\/ maximum timestep DT = 0.1; \/\/ CFL number CFL = 0.8; run(); } \/\/ boundary condition u.t[top] = dirichlet(1); u.t[bottom] = dirichlet(0); u.t[left] = dirichlet(0); u.t[right] = dirichlet(0); event outputfile (i += 100) { output_matrix (u.x, stdout, N, linear = true); } event movie (i += 4; t <= 15.) { static FILE * fp = popen (\u201cppm2mpeg > norm.mpg\u201d, \u201cw\u201d); scalar norme[]; foreach() norme[] = norm(u); boundary ({norme}); output_ppm (norme, fp, linear = true); } ~ We generate a mpeg file of the norm of the velocity field ## Navier Stokes : Flow over a cylinder An example of 2D viscous flow around a simple solid boundary. Fluid is injected to the left of a channel bounded by solid walls with a slip boundary condition. The Reynolds number is set to 160. C #include \u201cnavier-stokes\/centered.h\u201d\n\nThe domain is eight units long, centered vertically.\n\nint main() {\nL0 = 8.;\norigin (-0.5, -L0\/2.);\nN = 512;\nrun();\n}\n\nThe fluid is injected on the left boundary with a unit velocity. The tracer is injected in the lower-half of the left boundary. An outflow condition is used on the right boundary.\n\nu.n[left] = dirichlet(1.);\np[left] = neumann(0.);\npf[left] = neumann(0.);\n\nu.n[right] = neumann(0.);\np[right] = dirichlet(0.);\npf[right] = dirichlet(0.);\n\nWe add a new boundary condition for the cylinder. The tangential velocity on the cylinder is set to zero.\n\nbid cylinder;\nu.t[cylinder] = dirichlet(0.);\n\nevent init (t = 0) {\n\nTo make a long channel, we set the top boundary for y > 0.5 and the bottom boundary for y < -0.5. The cylinder has a radius of 0.0625.\n\n mask (y > 0.5 ? top :\ny < -0.5 ? bottom :\nsq(x) + sq(y) < sq(0.0625) ? cylinder :\nnone);\n\nWe set a constant viscosity corresponding to a Reynolds number of 160, based on the cylinder diameter (0.125) and the inflow velocity (1). We also set the initial velocity field and tracer concentration.\n\n const face vector muc[] = {0.00078125,0.00078125};\nmu = muc;\nforeach() {\nu.x[] = 1.;\n}\n}\n\nWe check the number of iterations of the Poisson and viscous problems.\n\n~c event logfile (i++) fprintf (stderr, \u201c%d %g %d %d\u201d, i, t, mgp.i, mgu.i); event movies (i += 4; t <= 15.) { static FILE * fp = popen (\u201cppm2mpeg > vort.mpg\u201d, \u201cw\u201d); scalar vorticity[]; foreach() vorticity[] = (u.x[0,1] - u.x[0,-1] - u.y[1,0] + u.y[-1,0])\/(2.*Delta); boundary ({vorticity}); output_ppm (vorticity, fp, box = {{-0.5,-0.5},{7.5,0.5}}, min = -10, max = 10, linear = true); } ~ We generate a mpeg file of the vorticity","date":"2022-08-12 22:04:28","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5598247647285461, \"perplexity\": 4326.265378824593}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571758.42\/warc\/CC-MAIN-20220812200804-20220812230804-00322.warc.gz\"}"} | null | null |
Product prices and availability are accurate as of 2019-04-18 10:50:16 UTC and are subject to change. Any price and availability information displayed on http://www.amazon.com/ at the time of purchase will apply to the purchase of this product.
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2,Color:If the color you need is not listed, please sent us color picture and we can custom make the dress color according to your party theme.
*If these sizes do not belong to one standard size,choose the size which fits the largest part of your body.The dress can then be altered to fit your figure correctly.
*If you are not sure your size,please contact us for suggestion or go with custom make with your personal measurement.
*Custom make dress return is unaccepted.
Gorgeous short one shoulder design,handmade,made of chiffon,soft material,good feeling,popular color,elegant,fashion,practical and perfect gift for classy and elegant ladies.Please check the measurement in product description carefully,make sure the seller is ThaliaDress to avoid buying fake items. ThaliaDress,we always try our best to provide the best products to you with the lowest price. We provide 100% customer satisfication,please contact us if any problem you met. Material:Chiffon ,Back:Lace up; Custom make service is acceptable,please contact us by Amazon message freely. Occasion:Prom,Party,Homecoming,Cocktail,Ball,Wedding,or other formal outdoor activities etc. Delivery:Generally speaking,If you choose the expedited delivery way,you will receive the item in 2-5 days after we ship out; please use the physical address for expedited shipping;if you choose the standard delivery way,you will receive the item in 7-15 days after shipping. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,478 |
package com.fangxuele.tool.push.ui.form;
import com.fangxuele.tool.push.App;
import com.fangxuele.tool.push.util.UIUtil;
import com.fangxuele.tool.push.util.UndoUtil;
import com.intellij.uiDesigner.core.GridConstraints;
import com.intellij.uiDesigner.core.GridLayoutManager;
import com.intellij.uiDesigner.core.Spacer;
import lombok.Getter;
import javax.swing.*;
import javax.swing.plaf.FontUIResource;
import javax.swing.text.StyleContext;
import java.awt.*;
import java.util.Locale;
/**
* <pre>
* PushForm
* </pre>
*
* @author <a href="https://github.com/rememberber">RememBerBer</a>
* @since 2019/5/6.
*/
@Getter
public class PushForm {
private JPanel pushPanel;
private JPanel pushUpPanel;
private JLabel pushSuccessCount;
private JLabel pushFailCount;
private JLabel pushTotalProgressLabel;
private JProgressBar pushTotalProgressBar;
private JLabel pushLastTimeLabel;
private JLabel pushLeftTimeLabel;
private JLabel jvmMemoryLabel;
private JLabel availableProcessorLabel;
private JLabel pushTotalCountLabel;
private JLabel pushMsgName;
private JLabel scheduleDetailLabel;
private JPanel pushDownPanel;
private JPanel pushControlPanel;
private JTextField threadCountTextField;
private JButton ScheduleRunButton;
private JButton pushStopButton;
private JButton pushStartButton;
private JCheckBox dryRunCheckBox;
private JPanel pushCenterPanel;
private JTextArea pushConsoleTextArea;
private JTable pushThreadTable;
private JLabel countPerThread;
private JSlider threadCountSlider;
private JLabel threadTipsLabel;
private JLabel dryRunHelpLabel;
private JCheckBox saveResponseBodyCheckBox;
private JLabel tpsLabel;
private static PushForm pushForm;
private PushForm() {
UndoUtil.register(this);
}
public static PushForm getInstance() {
if (pushForm == null) {
pushForm = new PushForm();
}
return pushForm;
}
/**
* 初始化推送tab
*/
public static void init() {
pushForm = getInstance();
pushForm.getPushMsgName().setText(App.config.getMsgName());
pushForm.getThreadCountTextField().setText(String.valueOf(App.config.getThreadCount()));
initSlider();
pushForm.getDryRunCheckBox().setSelected(App.config.isDryRun());
if (UIUtil.isDarkLaf()) {
Color bgColor = new Color(43, 43, 43);
pushForm.getPushConsoleTextArea().setBackground(bgColor);
Color foreColor = new Color(187, 187, 187);
pushForm.getPushConsoleTextArea().setForeground(foreColor);
}
}
public static void initSlider() {
pushForm = getInstance();
Integer maxThreads = App.config.getMaxThreads();
pushForm.getThreadCountSlider().setMaximum(maxThreads);
int threadCount = App.config.getThreadCount();
if (threadCount > maxThreads) {
threadCount = maxThreads;
}
pushForm.getThreadCountSlider().setValue(threadCount);
}
{
// GUI initializer generated by IntelliJ IDEA GUI Designer
// >>> IMPORTANT!! <<<
// DO NOT EDIT OR ADD ANY CODE HERE!
$$$setupUI$$$();
}
/**
* Method generated by IntelliJ IDEA GUI Designer
* >>> IMPORTANT!! <<<
* DO NOT edit this method OR call it in your code!
*
* @noinspection ALL
*/
private void $$$setupUI$$$() {
final JPanel panel1 = new JPanel();
panel1.setLayout(new GridLayoutManager(1, 1, new Insets(0, 0, 0, 0), -1, -1));
pushPanel = new JPanel();
pushPanel.setLayout(new GridLayoutManager(3, 1, new Insets(0, 2, 0, 2), -1, -1));
panel1.add(pushPanel, new GridConstraints(0, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, null, null, null, 0, false));
pushUpPanel = new JPanel();
pushUpPanel.setLayout(new GridLayoutManager(7, 10, new Insets(0, 0, 0, 0), -1, -1));
pushPanel.add(pushUpPanel, new GridConstraints(0, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, null, null, null, 0, false));
pushSuccessCount = new JLabel();
Font pushSuccessCountFont = this.$$$getFont$$$(null, -1, 72, pushSuccessCount.getFont());
if (pushSuccessCountFont != null) pushSuccessCount.setFont(pushSuccessCountFont);
pushSuccessCount.setForeground(new Color(-13587376));
pushSuccessCount.setText("0");
pushUpPanel.add(pushSuccessCount, new GridConstraints(0, 0, 7, 1, GridConstraints.ANCHOR_EAST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushFailCount = new JLabel();
Font pushFailCountFont = this.$$$getFont$$$(null, -1, 72, pushFailCount.getFont());
if (pushFailCountFont != null) pushFailCount.setFont(pushFailCountFont);
pushFailCount.setForeground(new Color(-2200483));
pushFailCount.setText("0");
pushUpPanel.add(pushFailCount, new GridConstraints(0, 2, 7, 1, GridConstraints.ANCHOR_EAST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushTotalProgressLabel = new JLabel();
pushTotalProgressLabel.setText("总进度");
pushUpPanel.add(pushTotalProgressLabel, new GridConstraints(6, 8, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushTotalProgressBar = new JProgressBar();
pushTotalProgressBar.setStringPainted(true);
pushUpPanel.add(pushTotalProgressBar, new GridConstraints(6, 9, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JLabel label1 = new JLabel();
label1.setText("成功");
pushUpPanel.add(label1, new GridConstraints(2, 1, 3, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JLabel label2 = new JLabel();
label2.setText("失败");
pushUpPanel.add(label2, new GridConstraints(2, 3, 3, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JSeparator separator1 = new JSeparator();
separator1.setOrientation(1);
pushUpPanel.add(separator1, new GridConstraints(0, 4, 7, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_WANT_GROW, null, null, null, 0, false));
pushLastTimeLabel = new JLabel();
pushLastTimeLabel.setEnabled(true);
Font pushLastTimeLabelFont = this.$$$getFont$$$("Microsoft YaHei UI Light", -1, 36, pushLastTimeLabel.getFont());
if (pushLastTimeLabelFont != null) pushLastTimeLabel.setFont(pushLastTimeLabelFont);
pushLastTimeLabel.setForeground(new Color(-6710887));
pushLastTimeLabel.setText("0s");
pushUpPanel.add(pushLastTimeLabel, new GridConstraints(0, 6, 3, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JLabel label3 = new JLabel();
label3.setHorizontalAlignment(0);
label3.setHorizontalTextPosition(0);
label3.setText("耗时");
pushUpPanel.add(label3, new GridConstraints(0, 5, 3, 1, GridConstraints.ANCHOR_EAST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JSeparator separator2 = new JSeparator();
separator2.setOrientation(1);
pushUpPanel.add(separator2, new GridConstraints(0, 7, 7, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_WANT_GROW, null, null, null, 0, false));
jvmMemoryLabel = new JLabel();
jvmMemoryLabel.setText("JVM内存占用:--");
pushUpPanel.add(jvmMemoryLabel, new GridConstraints(4, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
availableProcessorLabel = new JLabel();
availableProcessorLabel.setText("可用处理器核心:--");
pushUpPanel.add(availableProcessorLabel, new GridConstraints(3, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushTotalCountLabel = new JLabel();
pushTotalCountLabel.setText("消息总数:0");
pushUpPanel.add(pushTotalCountLabel, new GridConstraints(1, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushMsgName = new JLabel();
Font pushMsgNameFont = this.$$$getFont$$$(null, -1, 24, pushMsgName.getFont());
if (pushMsgNameFont != null) pushMsgName.setFont(pushMsgNameFont);
pushMsgName.setForeground(new Color(-276358));
pushMsgName.setText("消息标题");
pushUpPanel.add(pushMsgName, new GridConstraints(0, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
scheduleDetailLabel = new JLabel();
scheduleDetailLabel.setForeground(new Color(-276358));
scheduleDetailLabel.setText("");
pushUpPanel.add(scheduleDetailLabel, new GridConstraints(5, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
countPerThread = new JLabel();
countPerThread.setText("平均每个线程分配:0");
pushUpPanel.add(countPerThread, new GridConstraints(2, 8, 1, 2, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JLabel label4 = new JLabel();
label4.setText("预计剩余");
pushUpPanel.add(label4, new GridConstraints(3, 5, 3, 1, GridConstraints.ANCHOR_EAST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushLeftTimeLabel = new JLabel();
Font pushLeftTimeLabelFont = this.$$$getFont$$$("Microsoft YaHei UI Light", -1, 36, pushLeftTimeLabel.getFont());
if (pushLeftTimeLabelFont != null) pushLeftTimeLabel.setFont(pushLeftTimeLabelFont);
pushLeftTimeLabel.setForeground(new Color(-6710887));
pushLeftTimeLabel.setText("0s");
pushUpPanel.add(pushLeftTimeLabel, new GridConstraints(3, 6, 3, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_VERTICAL, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final JLabel label5 = new JLabel();
label5.setText("TPS");
pushUpPanel.add(label5, new GridConstraints(6, 5, 1, 1, GridConstraints.ANCHOR_EAST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
tpsLabel = new JLabel();
tpsLabel.setText("0");
pushUpPanel.add(tpsLabel, new GridConstraints(6, 6, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushDownPanel = new JPanel();
pushDownPanel.setLayout(new GridLayoutManager(1, 1, new Insets(0, 0, 0, 0), -1, -1));
pushPanel.add(pushDownPanel, new GridConstraints(2, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, null, null, null, 0, false));
pushControlPanel = new JPanel();
pushControlPanel.setLayout(new GridLayoutManager(1, 11, new Insets(0, 0, 0, 0), -1, -1));
pushDownPanel.add(pushControlPanel, new GridConstraints(0, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, null, null, null, 0, false));
final JLabel label6 = new JLabel();
label6.setText("线程数");
label6.setToolTipText("当前版本受http连接池限制建议不要设置过多线程,推荐100以内");
pushControlPanel.add(label6, new GridConstraints(0, 1, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
threadCountTextField = new JTextField();
threadCountTextField.setEditable(false);
threadCountTextField.setFocusable(false);
threadCountTextField.setRequestFocusEnabled(false);
threadCountTextField.setToolTipText("当前版本受http连接池限制建议不要设置过多线程,推荐100以内");
pushControlPanel.add(threadCountTextField, new GridConstraints(0, 2, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, new Dimension(60, -1), null, 0, false));
ScheduleRunButton = new JButton();
ScheduleRunButton.setIcon(new ImageIcon(getClass().getResource("/icon/clock.png")));
ScheduleRunButton.setText("按计划执行");
pushControlPanel.add(ScheduleRunButton, new GridConstraints(0, 8, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushStopButton = new JButton();
pushStopButton.setEnabled(false);
pushStopButton.setIcon(new ImageIcon(getClass().getResource("/icon/suspend.png")));
pushStopButton.setText("停止");
pushControlPanel.add(pushStopButton, new GridConstraints(0, 9, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushStartButton = new JButton();
pushStartButton.setIcon(new ImageIcon(getClass().getResource("/icon/run@2x.png")));
pushStartButton.setText("开始");
pushControlPanel.add(pushStartButton, new GridConstraints(0, 10, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
dryRunCheckBox = new JCheckBox();
dryRunCheckBox.setText("空跑");
dryRunCheckBox.setToolTipText("空跑勾选时不会真实发送消息");
pushControlPanel.add(dryRunCheckBox, new GridConstraints(0, 6, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
final Spacer spacer1 = new Spacer();
pushControlPanel.add(spacer1, new GridConstraints(0, 4, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_WANT_GROW, 1, null, null, null, 0, false));
threadCountSlider = new JSlider();
threadCountSlider.setDoubleBuffered(true);
threadCountSlider.setExtent(0);
threadCountSlider.setFocusCycleRoot(false);
threadCountSlider.setFocusTraversalPolicyProvider(false);
threadCountSlider.setFocusable(false);
threadCountSlider.setInverted(false);
threadCountSlider.setMajorTickSpacing(10);
threadCountSlider.setMinimum(1);
threadCountSlider.setMinorTickSpacing(5);
threadCountSlider.setOpaque(false);
threadCountSlider.setOrientation(0);
threadCountSlider.setPaintLabels(false);
threadCountSlider.setPaintTicks(true);
threadCountSlider.setPaintTrack(true);
threadCountSlider.setRequestFocusEnabled(false);
threadCountSlider.setSnapToTicks(false);
threadCountSlider.setValueIsAdjusting(false);
pushControlPanel.add(threadCountSlider, new GridConstraints(0, 3, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_HORIZONTAL, GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
threadTipsLabel = new JLabel();
threadTipsLabel.setIcon(new ImageIcon(getClass().getResource("/icon/helpButton.png")));
threadTipsLabel.setText("");
pushControlPanel.add(threadTipsLabel, new GridConstraints(0, 0, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
dryRunHelpLabel = new JLabel();
dryRunHelpLabel.setIcon(new ImageIcon(getClass().getResource("/icon/helpButton.png")));
dryRunHelpLabel.setText("");
pushControlPanel.add(dryRunHelpLabel, new GridConstraints(0, 7, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_FIXED, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
saveResponseBodyCheckBox = new JCheckBox();
saveResponseBodyCheckBox.setText("保存请求返回的Body");
pushControlPanel.add(saveResponseBodyCheckBox, new GridConstraints(0, 5, 1, 1, GridConstraints.ANCHOR_WEST, GridConstraints.FILL_NONE, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_CAN_GROW, GridConstraints.SIZEPOLICY_FIXED, null, null, null, 0, false));
pushCenterPanel = new JPanel();
pushCenterPanel.setLayout(new GridLayoutManager(2, 1, new Insets(0, 0, 0, 0), -1, -1));
pushPanel.add(pushCenterPanel, new GridConstraints(1, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, null, null, null, 0, false));
final JScrollPane scrollPane1 = new JScrollPane();
pushCenterPanel.add(scrollPane1, new GridConstraints(0, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, null, null, null, 0, false));
pushConsoleTextArea = new JTextArea();
scrollPane1.setViewportView(pushConsoleTextArea);
final JScrollPane scrollPane2 = new JScrollPane();
pushCenterPanel.add(scrollPane2, new GridConstraints(1, 0, 1, 1, GridConstraints.ANCHOR_CENTER, GridConstraints.FILL_BOTH, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, GridConstraints.SIZEPOLICY_CAN_SHRINK | GridConstraints.SIZEPOLICY_WANT_GROW, null, null, null, 0, false));
pushThreadTable = new JTable();
pushThreadTable.setGridColor(new Color(-12236470));
pushThreadTable.setRowHeight(36);
pushThreadTable.setShowVerticalLines(false);
scrollPane2.setViewportView(pushThreadTable);
}
/**
* @noinspection ALL
*/
private Font $$$getFont$$$(String fontName, int style, int size, Font currentFont) {
if (currentFont == null) return null;
String resultName;
if (fontName == null) {
resultName = currentFont.getName();
} else {
Font testFont = new Font(fontName, Font.PLAIN, 10);
if (testFont.canDisplay('a') && testFont.canDisplay('1')) {
resultName = fontName;
} else {
resultName = currentFont.getName();
}
}
Font font = new Font(resultName, style >= 0 ? style : currentFont.getStyle(), size >= 0 ? size : currentFont.getSize());
boolean isMac = System.getProperty("os.name", "").toLowerCase(Locale.ENGLISH).startsWith("mac");
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Biden Nominates Samantha Power To Run U.S. Agency For International Development
by Laurel Wamsley
Power is "a world-renowned voice of conscience and moral clarity," President-elect Joe Biden said of the former U.N. ambassador. The post will be elevated to become a National Security Council member.
Laurel Wamsley
President-elect Joe Biden has nominated former U.N. Ambassador Samantha Power to lead the U.S. Agency for International Development.
Biden also said he was elevating that role — USAID Administrator — to be a member of the White House National Security Council.
"Samantha Power is a world-renowned voice of conscience and moral clarity — challenging and rallying the international community to stand up for the dignity and humanity of all people," Biden said in a statement on Wednesday.
"I know firsthand the unparalleled knowledge and tireless commitment to principled American engagement she brings to the table, and her expertise and perspective will be essential as our country reasserts its role as a leader on the world stage," he said. "As USAID Administrator, Ambassador Power will be a powerful force for lifting up the vulnerable, ushering in a new era of human progress and development, and advancing American interests globally."
Power, 50, served in the Obama administration as U.S. ambassador to the United Nations from 2013 to 2017. From 2009 to 2013, she served on the National Security Council staff as special assistant to the president and as senior director for multilateral affairs and human rights.
Before her diplomacy career, Power was a journalist who reported from Bosnia, East Timor, Kosovo, Rwanda, Sudan and Zimbabwe. She was the founding executive director of the Carr Center for Human Rights Policy at the Harvard Kennedy School, and is currently a professor at Harvard. She was born in Dublin in 1970 and immigrated to the U.S. from Ireland with her family at the age of 9.
Power won a Pulitzer Prize in 2003 for her book A Problem From Hell: America and the Age of Genocide.
"One of the most pressing challenges facing our nation is restoring and strengthening America's global leadership as a champion of democracy, human rights, and the dignity of all people," Vice President-elect Kamala Harris said in a statement. "Few Americans are better equipped to help lead that work than Ambassador Samantha Power."
"Clear-eyed, resolute, and guided by a true moral compass, Ambassador Power is a seasoned leader and innovative thinker," Harris added. "And she will not only help lift up the world's most vulnerable and advance our nation's interests around the world, she will be a powerful voice for the values and ideals we cherish as Americans."
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Do soutěže mužské čtyřhry na tenisovém turnaji Argentina Open 2015 nastoupilo šestnáct dvojic. Obhájcem titulu byl španělský pár Marcel Granollers a Marc López, jehož členové zvolili start na paralelně probíhajícím acapulské události Abierto Mexicano Telcel.
Vítězem čtyřhry se stal nenasazený finsko-brazilský pár Jarkko Nieminen a André Sá, který ve finále zdolal španělsko-rakouskou dvojici Pablo Andújar a Oliver Marach výsledkem 4–6, 6–4 a [10–7]. Oba šampioni si do žebříčku ATP připsali 250 bodů.
Všechny nasazené páry vypadly v úvodním kole.
Nasazení párů
Pablo Cuevas / David Marrero (1. kolo)
Máximo González / Horacio Zeballos (1. kolo)
Johan Brunström / Nicholas Monroe (1. kolo)
František Čermák / Jiří Veselý (1. kolo)
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Reference
Externí odkazy
ATP Buenos Aires
Tenis v Argentině v roce 2015
ATP World Tour 2015 | {
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\section{Introduction}
Degeneracies in energy-momentum relations, so-called band structures, play an important role in many fields of physics from classical mechanics to condensed matter physics and optics.
Recently, the study on degenerate points in band structures opened a new path to observe many exotic topological behaviors.
For instance, Dirac points, point degeneracies with a linear dispersion relation in two-dimensional momentum space \cite{Rechtsman_Nature_2013,Yang2015}, have been used to show chiral one-way edge/surface states in a band gap \cite{Li2018,Gong_ACSPhotonics_2020,Wong_PRR_2020,Kim_2020_LSA}.
Weyl points, which are also point degeneracies with a linear dispersion but in three-dimensional momentum space \cite{Lu2013,Lu_2015_Science,Yang_Science_2018}, have been used to generate Fermi arc-like surface states \cite{Yang_OptExp_2017,Noh_NatPhys_2017,Guo_PRL_2019,Yang2019,He_NatComm_2020_4}.
In particular, Weyl points are very interesting and important in topological classification because they are stable meaning that the Weyl points are robust to perturbations when only one of the inversion ($\mathcal{P}$) symmetry and time-reversal ($\mathcal{T}$) symmetry is broken in three-dimensional space \cite{Lu2013}.
By recovering the broken symmetry in a structure with Dirac/Weyl points, i.e., making it $\mathcal{P}$ and $\mathcal{T}$ symmetric, a nodal line
\cite{Kennedy_PRB_2016,Ahn_2018_PRL,Wu2019,Park_2021a_ACSPhotonics}, a one-dimensional degeneracy, can be created.
Nodal lines have drawn attention because they can feature two-dimensional surface states bounded by the projected nodal lines, called drumhead surface states \cite{Rui_Yu_PRL_2015,Kim_PRL_2015,Wang_SciRep_2021}, and exhibit non-Abelian band topology \cite{Wu2019}.
Moreover, as nodal lines have higher dimensions than Dirac/Weyl points, they show various shapes, for instance, a simple nodal line, a nodal ring \cite{Gao2018,Lingbo_PRL_2019,Yang_2021_NatComm}, nodal knots \cite{Kedia_PRL_2013,Bi_2017_PRB,Lee2020} for a single nodal line or a Hopf link \cite{Xie_2019_PRB,Lee2020,Tiwari_2020_PRB,Unal_PRL_2020,Yang2020,Park_2021a_ACSPhotonics}, a nodal chain \cite{Bzdusek2016,Yan_2018_NatPhys,YangZhesen_2020_PRL_ZonesPolynomial,Park_2021a_ACSPhotonics} for multiple nodal lines.
To study the topological characteristics of the nodal lines, the starting point is to define topological invariants which are the conserved quantities when any topological phase transitions do not occur, i.e., the topological phase do not change under certain perturbations \cite{Lu_NatPhoton_Review_2014}.
Topological invariants of Dirac/Weyl points and nodal lines are often called \emph{topological charges} similarly to electric/magnetic charges of electric/magnetic monopoles.
While the topological charge of a simple nodal line, that is the degeneracy between two bands, can be described by a Berry phase \cite{Berry_1984}, the definition of topological charge for multiple nodal lines becomes more complex because three or more bands are involved.
Recently, it was shown that the topological charges of the nodal lines in a three-band system can be described by quaternion numbers, forming a non-Abelian group \cite{Wu2019} and the experimental observation of the quaternion charges has been reported \cite{Guo2021}.
Along with the progress in understanding band topology, recent years saw the rapid development of topological photonics which studies and demonstrates the topological states of photons in metamaterials (including metallic photonic crystals) \cite{Khanikaev_NatMat_2013,Gao_PRL_2015,Gao_NatComm_2016,Guo_PRL_2017,Gao2018,Yan_2018_NatPhys,Yang_2021_NatComm} and dielectric photonic crystals \cite{Lu2013,Slobozhanyuk2017,Park_2020_ACSPhotonics,Jo_2021_AppliedMaterialsToday,Park_2021a_ACSPhotonics}.
In contrast to electronic systems these photonic materials have great advantages because one can design a structure and manipulate the propagation properties of photons with more freedom and in a wide range of frequency spectra.
For this reason, many exciting advances have been made, for example, Weyl fermions have been demonstrated using a dielectric photonic crystal called double gyroid \cite{Lu2013,Park_2020_ACSPhotonics,Jo_2021_AppliedMaterialsToday} and unidirectional edge modes have been demonstrated using a magnetic photonic crystal \cite{Wang2009c,Yang_OptExp_2017}.
On the other hand, however, the progress on nodal lines in photonics has been rather slow compared to the study on Dirac/Weyl photonic crystals.
The possible reasons can be summarized as follows. First, investigating nodal lines in three-dimensional momentum space requires more amount of computations than Dirac or Weyl points. Second, it was very recent that nodal chains \cite{Yan_2018_NatPhys} or nodal links \cite{Yang2020,Wang_LSA_2021,Park_2021a_ACSPhotonics} started to be realized although nodal ring by a dielectric photonic crystal was already reported in Ref.~\cite{Lu2013}. Lastly, discussions on new topological invariants of nodal lines such as non-Abelian topological charges have been made very recently \cite{Wu2019}.
Therefore, reviewing the recent work on nodal lines is very timely and summarising important concepts is essential for more exciting outcomes that will be generated in the field of topological photonics in the near future.
In this review, we will aim to cover the basic theory of topological physics with a focus on topological nodal lines and introduce important examples of topological nodal lines demonstrated in various artificial material systems.
In Section~\ref{section:theory}, we describe the degeneracies in band structures including Weyl points and nodal lines.
In Section~\ref{section:Topological_invariants}, we explain Abelian and non-Abelian topological invariants using two examples, the Berry phase and the Wilczek-Zee connection. In Section~\ref{section:PhotonicExample}, we highlight examples of topological nodal lines in metamaterials, photonic crystals and photonic systems with synthetic dimensions. In Section~\ref{section:example}, we discuss examples in electronic crystals, phononic crystals and electrical circuits.
\begin{figure}
\includegraphics{Fig_WeylPoint_NodalLine.pdf
\caption{
\label{fig:WeylPoint_and_NodalLine}
(a),(b) Schematics of a Weyl point and a nodal line, respectively.
}
\end{figure}
\section{Band degeneracies}
\label{section:theory}
In an electronic band structure, two adjacent bands may touch each other in one or more $\mathbf k$-point(s) meaning that the two bands have the same energy but different eigenstates.
This is `degeneracy'.
The concept of degeneracy can also be applied to photonic/phononic band structures which are the frequency-wavevector relations for waves in a periodic array of photonic/phononic atoms.
In three-dimensional momentum space, the degeneracies can be classified as zero-, one-, and two-dimensional depending on their dimensionality.
In this section, we will explain zero- and one-dimensional degeneracies which can be Weyl points and nodal lines, respectively, and introduce diverse categories of nodal lines that are classified by their shapes.
\subsection{Zero-dimensional degeneracies: Weyl points}
\label{section:WeylPoints}
A representative example of a zero-dimensional degeneracy is a Weyl point \cite{Lu2013,Lu_2015_Science,Yang_Science_2018}.
The Weyl point was named thus because the dispersion around the degenerate point is governed by the Weyl Hamiltonian $H(\mathbf k) = v_1 k_1 \sigma_1 + v_2 k_2 \sigma_2 + v_3 k_3 \sigma_3$ where $\sigma_i$ are the Pauli matrices.
In three-dimensional momentum space, the Weyl point acts as a monopole that emits or soaks the Berry flux similar to a magnetic monopole where the magnetic flux departs or terminates.
The mathematical definition of the Berry flux will be introduced in Section~\ref{section:BerryPhase}.
In the Weyl Hamiltonian, the $\sigma_2$ term can exist only when one of the inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries is broken \cite{Lu2013}. This is a necessary condition for the existence of Weyl points.
If we set $v_1 = v_2 = v_3 = 1$, the eigenenergies of the Weyl Hamiltonian are expressed as $E = \pm \left| \mathbf k \right|$.
Thus, the band structure shows a point degeneracy at $\mathbf k = \mathbf 0$
and a linear dispersion around the degeneracy, as shown in Figure~\ref{fig:WeylPoint_and_NodalLine}a.
The eigenstates of the Weyl Hamiltonian at $\mathbf k \neq \mathbf 0$ can be expressed as $\psi_1 = \left[ \cos \left( {\theta / 2} \right), e^{i \phi} \sin \left( {\theta / 2} \right) \right]$ and $\psi_2 = \left[ e^{-i \phi} \sin \left( {\theta / 2} \right) , -\cos \left( {\theta / 2} \right) \right]$ using the spherical coordinate system $\left( r, \theta, \phi \right)$.
Then, the Berry curvature is expressed as $\pm 1/\left(2 k^2 \right) \hat{\mathbf r}$ where $\hat{\mathbf r}$ is the unit vector in the radial direction in momentum space \cite{Griffiths_QM_2005,Lu_2015_Science} implying that the Weyl point becomes a sink or source of the Berry flux.
In the early 2010s, significant efforts have been made to realize Weyl points \cite{Wan2011,Burkov_2011_PRL,Xu_PRL_2011,Yang2011a,Halasz_PRB_2012,Hosur_PRL_046602,Aji_PRB_2011},
and a double gyroid structure was theoretically proposed as a Weyl photonic crystal in 2013 \cite{Lu2013}.
Two years after this theoretical work, Weyl points were experimentally observed in the microwave frequency range \cite{Lu_2015_Science}.
Following this first experimental demonstration, the realization of Weyl points has been achieved using photonic crystals \cite{Yang_OptExp_2017,Luyang_PRA_2016,Goi_LaserPhotonRev_2018,Lu_NatComm_2018,Yang_Science_2018,Jia_Science_2019,Fruchart_PNAS_2018,Park_2020_ACSPhotonics,Yang_PRL_2020,Jo_2021_AppliedMaterialsToday},
phononic crystals \cite{Xiao_NatPhys_2015,Li2018,He_NatComm_2020_4,He_Nature_2018,Peri_NatPhys_2019,Takahashi_PRB_2019,He_NatComm_2020_5,Wang_NatComm_2021}, metals \cite{Burkov_AnnRevCondMattPhys_2018,Galitski_PRL_2018,Mizobata_PRB_2020,Sorn_PRB_2021} and semimetals \cite{Soluyanov2015,Sie_Nature_2019,Heidari_PRB_2020,Ilan_NatRevPhys_2020,Kim_PRL_2017,Wang_PRL_2020,Yuan_NatComm_2018}.
\begin{figure*}[!ht]
\centering
\includegraphics{Fig_Classification_of_NodalLine.pdf
\caption{
\label{fig:Classification_of_NodalLine}
Classification of nodal lines.
}
\end{figure*}
\subsection{One-dimensional degeneracies: nodal lines}
\label{section:NodalLines}
One-dimensional degeneracies (co-dimension $N-1$ in a $N$-dimensional space), called `nodal lines', can be found when two bands touch each other on a line in momentum space \cite{Burkov2011,Kennedy_PRB_2016,Ahn_2018_PRL,Wu2019,Park_2021a_ACSPhotonics}, as illustrated in Figure~\ref{fig:WeylPoint_and_NodalLine}b.
The nodal lines can be classified by their shapes and the connectivities between other nodal lines, as shown in Figure~\ref{fig:Classification_of_NodalLine}.
As shown in Figure~\ref{fig:Classification_of_NodalLine}a, a nodal line can have a loop shape to form a nodal ring \cite{Lu2013,Fang_PRB_2015,Weng_PRB_2015,Chen_NatComm_2015,Xie_APLMat_2015,Ezawa_PRL_2016,Zhao_PRB_2016,Chan_PRB_2016,Li_PRL_2016,Bian_NatComm_2016,Gao2018,Nomura_PRMat_2018,Deng2019,Lingbo_PRL_2019,Tiwari_2020_PRB,Yang_2021_NatComm,Li_2021_PRB}
or knot, such as a trefoil knot \cite{Kedia_PRL_2013,Bi_2017_PRB,Lee2020}, a double trefoil knot \cite{Bi_2017_PRB,Ezawa_PRB_2017}, a cinquefoil knot \cite{Bi_2017_PRB,Kedia_PRL_2013}, or a figure 8-knot \cite{Lee2020}.
Such knots cannot be transformed into a nodal ring without cutting or intersecting them.
If two rings intersect each other, they form a nodal chain \cite{Yan_PRB_2017,Sun_PRL_2018,Gong_PRL_2018,Zhou_PRB_2018,Belopolski2019,Merkel_CommPhys_2019,Chang_2017_PRL}.
If two rings are tied without touching so that they cannot be separated by cutting or passing them, they form a nodal link \cite{Wilczek_PRL_1983,Moore_PRL_2008,Neupert_PRB_2012,Kedia_PRL_2013,Deng_PRB_2013,Deng_PRB_2014,Yan_PRB_2017,Chang_2017_PRL,Ezawa_PRB_2017,Liu_PRB_2017,Lian_PRB_2017,Zhou_PRB_2018,Xie_2019_PRB,Lee2020,Tiwari_2020_PRB,Unal_PRL_2020} or Solomon's knots \cite{Ezawa_PRB_2017,Bi_2017_PRB} (Figure~\ref{fig:Classification_of_NodalLine}b). Each ring of the nodal chain/link can be formed by the same or a different pair of bands. Especially, if the two rings of the nodal chain originate from different set of bands in a three-band system, the three bands meet at a single point where the nodal rings touch \cite{Wu2019,Lenggenhager_2021_PRB,SPark_2021_arXiv,Lange_2021_arXiv}. This is called a triple point.
Multiple nodal rings may form an infinite nodal chain \cite{Bzdusek2016,Wang_NatComm_2017,Yan_2018_NatPhys,Gong_PRL_2018,YangZhesen_2020_PRL_ZonesPolynomial,Park_2021a_ACSPhotonics} or a link \cite{Xie_2019_PRB,Lenggenhager_2021_PRB,Park_2021a_ACSPhotonics} due to the periodicity of momentum space (Figure~\ref{fig:Classification_of_NodalLine}c). Although Figure~\ref{fig:Classification_of_NodalLine}c shows only a one-dimensional infinite chain and a link, they can also form a two- or three-dimensional infinite chain \cite{Yan_2018_NatPhys} or a link \cite{Xie_2019_PRB}.
In some cases, multiple nodes of different types can appear mixed.
First, the mixed shape of the nodal rings appear as earring nodal links \cite{Wu2019,Tiwari_2020_PRB}, multiple Hopf links \cite{Kedia_PRL_2013,Ezawa_PRB_2017,Zhou_PRB_2018,He_2020_PRA}, mixed nodal links \cite{Tiwari_2020_PRB}, and the linked nodal ring and a chain (Figure~\ref{fig:Classification_of_NodalLine}d) \cite{Yang2020,Wang_2021_arXiv}.
Second, the nodal lines and a nodal ring/chain can be linked to show the non-touching between nodal lines and rings \cite{Tiwari_2020_PRB,Yang2020,SPark_2021_arXiv}.
Inversely, the nodal lines and nodal ring/link can be chained to show the touching between nodal lines and ring \cite{Bzdusek2016,Gong_PRL_2018,Lu_PRA_2020,Wang_2021_arXiv} or the touching between nodal lines and link \cite{Wu2019,Wang_2021_arXiv}, as shown in Figure~\ref{fig:Classification_of_NodalLine}e.
Finally, if the linked nodal ring and chain \cite{Wang_LSA_2021} and the nodal lines touch, the linked nodal ring, chain, and lines \cite{Yang2020} are generated (Figure~\ref{fig:Classification_of_NodalLine}f).
For the above classification, we make one assumption for brevity. As we will look at a periodic system, the momentum space is periodic. When a nodal line crosses a Brillouin zone boundary, it intersects at the same point at the opposite Brillouin zone boundary.
Therefore, a nodal line and a double helix (Figure~\ref{fig:Classification_of_NodalLine}b) \cite{YangZhesen_2020_PRL_ZonesPolynomial,Kennedy_PRB_2016,Sun_PRL_2017,Chang_PRB_2017,Tan_APL_2018,Chen_PRB_2017,Unal_PRRes_2019,Wang_Nature_2021} can be considered as a nodal ring and a nodal link, respectively.
In the above, however, we assumed that the first Brillouin zone is distinct from the neighboring Brillouin zones, which means that a nodal line that crosses the boundary extends from negative infinity to positive infinity.
\section{Topological invariants}
\label{section:Topological_invariants}
The physical behaviors of bands in momentum space are interpreted using several kinds of topological invariants.
The topological invariant is a quantized number that characterizes the topological status of a given system, and the Chern number or Berry phase are examples of the topological invariants.
The topological invariants are related to the various phenomena (e.g., surface states) of the topological insulators, which act as an insulators in their bulk and permit the electronic/photonic/phononic waves on their boundaries.
For example, in $\mathcal{T}$-symmetry broken topological insulators, one-way surface states are formed at the interface of two band-gapped materials due to the difference in Chern numbers between two bands \cite{Raghu2008, Wang2009c,Kruthoff2017,Lu_NatPhoton_Review_2014}.
In pseudo-$\mathcal{T}$-symmetry broken or $\mathcal{P}$-symmetry broken topological insulators, surface states exist due to the difference in the spin-Chern numbers or valley-Chern numbers \cite{Wu2015a,Slager2013,Saba2020}.
In addition to the topological invariants of these gapped phases, several efforts also have been carried out to understand relations between surface states and the topological invariants of gapless phases (band degeneracies), such as Dirac points \cite{Jiang2021}, Weyl points \cite{Lu2013,Chen_NatComm_2016} or nodal lines \cite{Wu2019}.
Thus, describing the topological invariants of the band degeneracy is an important step to understanding the degeneracy and finding appropriate applications.
In the following, we review Abelian and non-Abelian topological invariants. We introduce the Berry phase \cite{Berry_1984} as an example of the Abelian topological invariants. We also explain the Wilczek-Zee phase \cite{Wilczek_PRL_1984}, which is the starting point toward the non-Abelian topological quaternion charges \cite{Wu2019}.
Simple examples of the quaternion charges are shown, and the use of correlation vectors in full-vector field systems is explained.
Finally, the patch Euler patch class is explained with an example of non-Abelian topology.
For more information, refer to Ref.~\cite{Berry_1984} for the Berry phase, Ref.~\cite{Wilczek_PRL_1984} for the Wilczek-Zee connection, Ref.~\cite{Wu2019} for the non-Abelian quaternion charges, and Ref.~\cite{Bouhon2020} for the Euler class derived using the Wilson loop.
\subsection{Abelian and non-Abelian topological invariants}
\label{section:Abelian_and_non_Abelian}
As noted in Section~\ref{section:theory}, a Weyl point and a nodal line are zero- and one-dimensional degeneracies, respectively.
In fact, a nodal line in three-dimensional space has the similar features to a Dirac point \cite{Rechtsman_Nature_2013,Yang2015,Lu_NatPhys_2016,Slobozhanyuk2017,Jin_PRL_2017,Abbaszadeh_PRL_2017,Brendel_PNAS_2017,Yang_Nature_2019,Wen_NatPhys_2019,Gong_ACSPhotonics_2020,Liu_NatComm_2020,Shao_NatNanotech_2020,Wong_PRR_2020} which is a point degeneracy in two-dimensional momentum space.
The nodal line and the Dirac point have the same co-dimension $N-1$
and commonly correspond to $H(\mathbf k) = v_1 k_1 \sigma_1 + v_3 k_3 \sigma_3$ that does not have the $\sigma_2$ term compared to the Weyl Hamiltonian mentioned in Section~\ref{section:WeylPoints}.
In addition, calculating the topological invariant of a nodal line and a Dirac point starts from considering a closed loop around the degeneracies, whereas calculating the topological charge of a Weyl point is associated with the Berry flux on the surface enclosing the Weyl point.
In case of the Abelian charge of nodal lines, to describe the topological nature of multiple degeneracies between the same pair of bands, e.g., in a two-band system, the topological invariants are obtained by simply summing up the invariants of all the degeneracies. However, such an invariant cannot express the full topological nature of the multi-band systems. For example, when the Abelian charges are used, the nodal lines between the first and second bands and between the second and third bands commonly exhibit a topological charge of $\pm \pi$. Thus, this invariant cannot distinguish which bands make the degeneracy, and the relation of the charges between a different pair of bands cannot be described.
The non-Abelian band topology gives a solution for the multi-band systems \cite{Wu2019}. Degeneracies by a different pair of bands have different topological charges, the quaternion numbers. The mutual interaction between the different pairs of bands can be written clearly, and the topological charges satisfy the anticommutative relation.
\subsection{Berry phase}
\label{section:BerryPhase}
Berry phase is a geometrical phase that is obtained by a system when it moves along a closed path in a parameter space \cite{Berry_1984}.
The Berry phase is path-dependent and is useful in studying the topology of the parameter space by providing a way to calculate topological invariants.
The mathematical description of the Berry phase starts with a Hamiltonian that depends on time-varying parameters $\mathbf k = \left[ k_1 , k_2 ,\cdots \right]$, i.e., $H = H \left( \mathbf k \left( t \right) \right)$.
We denote the orthonormal bases of $H \left( \mathbf k \left( t \right) \right)$ as $\left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle$:
\begin{equation}
H \left( \mathbf k \left( t \right) \right)
\left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle
=
E_n \left( \mathbf k \left( t \right) \right)
\left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle
\label{eqn:eigenvalue} ,
\end{equation}
\begin{equation}
\left\langle u_m \left( \mathbf k \left( t \right) \right)
\middle|
u_n \left( \mathbf k \left( t \right) \right) \right\rangle
=
\delta_{mn} .
\label{eqn:orthonormality}
\end{equation}
Here, we assume that the eigenstates in Eqs.~(\ref{eqn:eigenvalue})-(\ref{eqn:orthonormality}) are given by not only $\left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle$ but also $e^{i \gamma_n \left( t \right)} \left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle$.
A state $\left| \psi \left( t \right) \right\rangle$ satisfying the time-dependent Schr\"{o}dinger equation is considered. If the parameter $\mathbf k$ changes adiabatically (that is, $\mathbf k$ varies slowly with time), the state $\left| \psi \left( t \right) \right\rangle$ that was in $n$-th state at $t = 0$ remains in the same state at $t = T$ where $T$ is the period of the cycle \cite{Kato_JPhysSocJpn_1950,Messiah_QM_1962}. The state $\left| \psi \left( t \right) \right\rangle$ is given by \cite{Berry_1984}
\begin{equation}
\left| \psi \left( t \right) \right\rangle
=
e^{
- \frac i \hbar
\int _0 ^t dt' E_n \left( \mathbf k \left( t' \right) \right)
}
e^{i \gamma_n \left( t \right)}
\left| u_n \left( \mathbf k \left( t \right) \right) \right\rangle
\end{equation}
where the first exponential term is the dynamical phase factor. By substituting $\left| \psi \left( t \right) \right\rangle$ into the time-dependent Schr\"{o}dinger equation, the geometric phase $\gamma_n$ can be obtained as an integral form. If a closed loop $\Gamma$ in $\mathbf k$-space is considered such that $\mathbf k \left( 0 \right) = \mathbf k \left( T \right)$, we have
\begin{equation}
\gamma_n
=
\oint_{\Gamma} {\mathbf A}_n \cdot d {\mathbf k}
\label{eqn:BerryPhase}
\end{equation}
where
\begin{equation}
{\mathbf A}_n \left( \mathbf k \right)
=
i
\left\langle
u_n \left( \mathbf k \right)
\middle|
\nabla_{\mathbf k}
u_n \left( \mathbf k \right)
\right\rangle .
\label{eqn:BerryConnection}
\end{equation}
As we consider a closed loop, the phase difference $\zeta \left( \mathbf k \left( T \right) \right) - \zeta \left( \mathbf k \left( 0 \right) \right)$ for the gauge transformation $\left| u_n \left( \mathbf k \right) \right\rangle \rightarrow e^{i \zeta \left( \mathbf k \right)} \left| u_n \left( \mathbf k \right) \right\rangle$ should be an integer multiple of $2 \pi$, so that $\gamma_n$ in Eq.~(\ref{eqn:BerryPhase}) becomes gauge-invariant. Here, $\gamma_n$ and $\mathbf A_n \left( \mathbf k \right)$ are called Berry phase and Berry connection, respectively \cite{Berry_1984,Xiao_RMP_2010,Griffiths_QM_2005}. The Berry flux (mentioned in Section~\ref{section:WeylPoints}) is given by $\iint {\mathbf F}_n \cdot d^2 \mathbf k$ where ${\mathbf F}_n \left( \mathbf k \right) = \nabla_{\mathbf k} \times {\mathbf A}_n \left( \mathbf k \right)$ is called the Berry curvature \cite{Lu_NatPhoton_Review_2014}. Regarding the closed loop $\Gamma$, a more practical form of Eqs.~(\ref{eqn:BerryPhase}) and (\ref{eqn:BerryConnection}) is
\begin{equation}
\gamma_n
=
\oint_{\Gamma} i
\left\langle
u_n \left( \mathbf k \right)
\middle|
\frac \partial {\partial k}
u_n \left( \mathbf k \right)
\right\rangle
d k .
\label{eqn:BerryPhase_Practical}
\end{equation}
From the orthonormality relation $\left\langle u_m \middle| u_n \right\rangle = \delta_{mn}$, we have $\left\langle \partial u_m / \partial k \middle| u_n \right\rangle + \left\langle u_m \middle| \partial u_n / \partial k \right\rangle = 2 \operatorname{Re} \left( \left\langle u_m \middle| \partial u_n / \partial k \right\rangle \right ) = 0$. Thus, the integrand in Eq.~(\ref{eqn:BerryPhase_Practical}) is purely imaginary, and $\gamma_n$ is real. If $\left| u_n \right\rangle$ consists of only real numbers, $\gamma_n$ becomes zero. In other words, to get a non-zero Berry phase, $\left| u_n \right\rangle$ should consist of complex numbers having one or more non-zero imaginary components. One representative example for this case is the Weyl Hamiltonian with $v_1 = v_2 = v_3 = 1$, mentioned in Section~\ref{section:WeylPoints}. The Weyl Hamiltonian has $\sigma_2$ so that its eigenstates consist of complex numbers. Possessing this $\sigma_2$ term corresponds to the $\mathcal{P} \mathcal{T}$-symmetry breaking in three-dimensional space and is a necessary condition for the existence of Weyl points \cite{Lu2013}.
\subsection{Wilczek-Zee connection}
Nodal lines are generated when both $\mathcal{P}$ and $\mathcal{T}$ symmetries are conserved. Such a situation corresponds to the lack of $\sigma_2$ in the Weyl Hamiltonian. In this case, the Berry phase in Eq.~(\ref{eqn:BerryPhase_Practical}) becomes zero because one can choose a gauge that keeps the Hamiltonian and its eigenvectors real.
Therefore, it is useful to define a non-vanishing topological invariant, for example,
the Wilczek-Zee connection \cite{Wilczek_PRL_1984} which will be explained in the following.
A state $\left| \eta_m \left( t \right) \right\rangle$ of the $m$-th band satisfying the Schr\"{o}dinger equation can be expressed as a linear combination of the basis $\left| u_n \left( \mathbf k \left ( t \right ) \right) \right\rangle$ in Eqs.~(\ref{eqn:eigenvalue})~(\ref{eqn:orthonormality}):
\begin{equation}
\left| \eta_m \left( t \right) \right\rangle
=
\sum _n W_{mn} \left| u_n \right\rangle
\end{equation}
If we assume that $\left| \eta_m \left( t \right) \right\rangle$ remain normalized, we have
\begin{equation}
0 = \left\langle \eta_l \middle| \frac {\partial \eta_m} {\partial t} \right\rangle
= \left\langle \eta_l \right|
\sum_n \left[
\frac {\partial W_{mn}} {\partial t} \left| u_n \right\rangle + W_{mn} \frac {\partial \left| u_n \right\rangle} {\partial t}
\right] .
\end{equation}
This equation can be arranged as follows:
\begin{equation}
\frac {\partial \mathbf W} {\partial t} \left| \mathbf u \right\rangle
=
- \mathbf W \frac {\partial \left| \mathbf u \right\rangle} {\partial t}
\end{equation}
or
\begin{equation}
\left[
{\mathbf W}^{-1} \frac {\partial \mathbf W} {\partial t}
\right]
\left| \mathbf u \right\rangle
=
- \frac {\partial \left| \mathbf u \right\rangle} {\partial t}
\end{equation}
where $\left| \mathbf u \right\rangle = \left[ \left| u_1 \right\rangle, \left| u_1 \right\rangle, \cdots \right]^T$ and $\left[ \mathbf W \right]_{mn} = W_{mn}$. Using the orthonormality condition in Eq.~(\ref{eqn:orthonormality}), we have the following skew-symmetric matrix:
\begin{equation}
\left[
{\mathbf W}^{-1} \frac {\partial \mathbf W} {\partial t}
\right]_{mn}
= - \left\langle u_n \middle| \frac {\partial u_m} {\partial t} \right\rangle
= \left\langle u_m \middle| \frac {\partial u_n} {\partial t} \right\rangle
= A_{mn} .
\label{eqn:skewsymmWZ_1}
\end{equation}
Then the path-ordered integrals of this can be expressed as
\begin{equation}
\mathbf W = \operatorname{exp} \int _0 ^T \mathbf A \left( t' \right) d t' ,
\end{equation}
or if we consider a closed loop $\Gamma$, and if we recall that $\partial u_n / \partial t = \left( \nabla_{\mathbf k} u_n \right) \cdot \left( \partial {\mathbf k} / \partial t \right)$, we can get the Wilson loop
\begin{equation}
\mathbf W
= \operatorname{exp} \left\{ \oint_{\Gamma} \mathbf A \left( \mathbf k \right) \cdot d {\mathbf k} \right\}
= \operatorname{exp} \left\{ \oint_{\Gamma} \mathbf A \left( k \right) d k \right\} .
\label{eqn:WilsonLoop}
\end{equation}
Here, the component of $\mathbf A \left( \mathbf k \right)$ is given by
\begin{equation}
A_{mn}
= \left\langle u_m \middle| \nabla_{\mathbf k} u_n \right\rangle
= \left\langle u_m \middle| \frac {\partial u_n} {\partial k} \right\rangle
\label{eqn:WZConnection}
\end{equation}
and is called the Wilczek-Zee connection \cite{Wilczek_PRL_1984}.
\begin{figure*}
\centering
\includegraphics{Fig_Schematic_Charge_IJK.pdf
\caption{
\label{fig:Schematic_Charge_IJK}
Schematics of the non-Abelian quaternion charges. (a) A nodal line and a closed loop encircling the nodal line. (b) An example of the eigenstates along the closed loop. (c) Quaternion multiplication table. (d),(e),(f) Eigenstates gathered at the origin indicating the topological charges ${\boldsymbol k}$, ${\boldsymbol i}$, and $+{\boldsymbol j}$, respectively.
}
\end{figure*}
\subsection{Three-band system}
In this section, the expressions derived in the previous section are applied to a three-band system.
Following formalism will be the basis of the rigorous description of quaternion charges \cite{Wu2019}.
To calculate the topological charges of an arbitrary nodal line, first, a closed loop $\Gamma \left( \alpha \right)$ ($\alpha \in \left(0, 2\pi \right]$) around the nodal line is considered (see Figure~\ref{fig:Schematic_Charge_IJK}a).
From the eigenstates of the $m$- and $n$-th bands ($m,n=1,2,3$), the Wilczek-Zee connection (Eq.~(\ref{eqn:WZConnection})) is rewritten as follows:
\begin{equation}
\mathbf A \left( k \right)
=
\left[
\begin{array}{ccc}
A_{11} & A_{12} & A_{13} \\
A_{21} & A_{22} & A_{23} \\
A_{31} & A_{32} & A_{33} \\
\end{array}
\right].
\end{equation}
Eq.~(\ref{eqn:skewsymmWZ_1}) leads to the skew-symmetric $\mathbf A \left( \mathbf k \right)$:
\begin{equation}
\mathbf A \left( k \right)
=
\left[
\begin{array}{ccc}
0 & A_{12} & -A_{31} \\
-A_{12} & 0 & A_{23} \\
A_{31} & -A_{23} & 0 \\
\end{array}
\right]
=
{\boldsymbol \beta} \left( k \right) \cdot {\mathbf L}
\label{eq_A_BWZ_skewsymm}
\end{equation}
where ${\boldsymbol \beta} = \left[ -A_{23}, -A_{31}, -A_{12} \right]$ and
$\left( L_i \right)_{jk} = -\epsilon_{ijk}$ \cite{Wu2019,Yang2020}.
If the closed loop encircles the nodal line formed by the bands $m=1$ and $n=2$, $\mathbf A \left( \mathbf k \right)$ is rewritten as
\begin{eqnarray}
\mathbf A \left( k \right)
&=&
\left[
\begin{array}{ccc}
0 & A_{12} & 0 \\
-A_{12} & 0 & 0 \\
0 & 0 & 0 \\
\end{array}
\right]
\nonumber\\
&=&
{\boldsymbol \beta}_{12} \left( k \right) \cdot {\mathbf L} = -A_{12} L_3 ,
\label{eq_A_BWZ_Mat_12}
\end{eqnarray}
where ${\boldsymbol \beta}_{12} \left( k \right) = \left[ 0,0, -A_{12} \right]$.
Substituting Eq.~(\ref{eq_A_BWZ_Mat_12}) into Eq.~(\ref{eqn:WilsonLoop}) gives
\begin{eqnarray}
W
&=&
\operatorname{exp}
\left\{
\oint_{ \Gamma(\alpha) } {\boldsymbol \beta}_{12} \left( k \right) \cdot {\mathbf L} d k
\right\}
\nonumber\\
&=&
\operatorname{exp}
\left\{
\left(
\oint_{ \Gamma(\alpha) } -A_{12} d k
\right)
L_3
\right\}.
\label{eq_A_Wilson12}
\end{eqnarray}
The spin Wilczek-Zee connection which is a $\mathfrak{spin} \left( N \right)$-valued 1-form is then written as \cite{Wu2019}
\begin{equation}
\bar{A} \left( k \right)
= {\boldsymbol \beta}_{12} \left( k \right) \cdot {\mathbf t}
= {\boldsymbol \beta}_{12} \left( k \right) \cdot \left( - \frac i 2 {\boldsymbol \sigma} \right),
\label{eq_A_BWZ_2x2_12}
\end{equation}
where the components of ${\boldsymbol \sigma} = \left[ \sigma_1, \sigma_2, \sigma_3 \right]$ are the Pauli matrices
and ${\mathbf t} = \left(- i / 2 \right) {\boldsymbol \sigma}$.
From Eqs.~(\ref{eq_A_Wilson12}) and (\ref{eq_A_BWZ_2x2_12}), the topological charge \cite{Wu2019,Yang2020} is then expressed as
\begin{eqnarray}
n_\Gamma
&=&
\operatorname{exp}
\left\{
\oint_{ \Gamma(\alpha) } \bar{A} \left( k \right) d k
\right\}
\nonumber\\
&=&
\operatorname{exp}
\left\{
\oint_{ \Gamma(\alpha) } {\boldsymbol \beta}_{12} \left( k \right) \cdot \left( - \frac i 2 {\boldsymbol \sigma} \right) d k
\right\}
\nonumber\\
&=&
\operatorname{exp}
\left\{
- \frac i 2 \sigma_3 \oint_{ \Gamma(\alpha) } -A_{12} d k.
\right\}.
\label{eq_A_n_Gamma_12}
\end{eqnarray}
If the integral $\oint_{ \Gamma(\alpha) } -A_{12} d k$ is $\pm \pi$, the charge $n_\Gamma$ becomes $\mp i\sigma_3$.
Now, if the closed loop encircles the nodal line formed by the bands $m=2$ and $n=3$, $\mathbf A \left( \mathbf k \right)$ in Eq.~(\ref{eq_A_BWZ_skewsymm}) is rewritten as $\mathbf A \left( k \right)={\boldsymbol \beta}_{23} \left( k \right) \cdot {\mathbf L} = -A_{23} L_1$ where ${\boldsymbol \beta}_{23} \left( k \right) = \left[ -A_{23}, 0,0 \right]$.
Substituting this into Eq.~(\ref{eqn:WilsonLoop}) gives
\begin{eqnarray}
W
&=&
\operatorname{exp}
\left\{
\left(
\oint_{ \Gamma(\alpha) } -A_{23} d k
\right)
L_1
\right\}.
\label{eq_A_Wilson23}
\end{eqnarray}
The spin Wilczek-Zee connection is $\bar{A} \left( k \right)= {\boldsymbol \beta}_{23} \left( k \right) \cdot {\mathbf t} = {\boldsymbol \beta}_{23} \left( k \right) \cdot \left( - \frac i 2 {\boldsymbol \sigma} \right)$, and the charge in Eq.~(\ref{eq_A_n_Gamma_12}) is then rewritten as
\begin{eqnarray}
n_\Gamma
&=&
\operatorname{exp}
\left\{
- \frac i 2 \sigma_1 \oint_{ \Gamma(\alpha) }^{ } -\left[ A \left( k \right) \right]_{23} d k.
\right\}.
\label{eq_A_n_Gamma_23}
\end{eqnarray}
In the same way as the first case, we obtain $n_\Gamma = \mp i \sigma_1$ when $\oint_{ \Gamma(\alpha) }^{ } -A_{23} d k = \pm \pi$.
\subsection{Non-Abelian quaternion charges}
\label{section:non-Abelian}
To describe the non-Abelian band topology, Q. Wu et al. \cite{Wu2019} employed the quaternion numbers, ${\mathbb Q} = \left\{ \pm \boldsymbol i, \pm \boldsymbol j, \pm \boldsymbol k, \pm 1 \right\}$ (first written by the Irish mathematician, William Rowan Hamilton in 1843).
The basis elements ${\boldsymbol i}$, ${\boldsymbol j}$, and ${\boldsymbol k}$ are defined such that ${\boldsymbol i}^2 = {\boldsymbol j}^2 = {\boldsymbol k}^2 = -1$.
Their multiplication relations are ${\boldsymbol i}{\boldsymbol j} = {\boldsymbol k}$, ${\boldsymbol j}{\boldsymbol k} = {\boldsymbol i}$, and ${\boldsymbol k}{\boldsymbol i} = {\boldsymbol j}$.
They all anticommute, that is, ${\boldsymbol i}{\boldsymbol j} = -{\boldsymbol j}{\boldsymbol i}$, ${\boldsymbol j}{\boldsymbol k} = -{\boldsymbol k}{\boldsymbol j}$, and ${\boldsymbol k}{\boldsymbol i} = -{\boldsymbol i}{\boldsymbol k}$.
All these are summarized in Figure~\ref{fig:Schematic_Charge_IJK}c.
Interestingly, the Pauli matrices $\left\{ \mp i \sigma_1, \mp i \sigma_2, \mp i \sigma_3, \pm I \right\}$ exhibit the same properties as the quaternions and are isomorphic so that we can map ${\boldsymbol i} \mapsto -i \sigma_1$, ${\boldsymbol j} \mapsto -i \sigma_2$, ${\boldsymbol k} \mapsto -i \sigma_3$, and $1 \mapsto I$.
Due to the anticommutative properties of ${\boldsymbol i}$, ${\boldsymbol j}$, ${\boldsymbol k}$, or $-i \sigma_1$, $-i \sigma_2$, $-i \sigma_3$, their relations are regarded as non-Abelian.
Thus, the topological charges calculated in Eqs.~(\ref{eq_A_n_Gamma_12}) and (\ref{eq_A_n_Gamma_23}) can be regarded as the quaternions $\pm {\boldsymbol k}$ and $\pm {\boldsymbol i}$, respectively.
\subsection{Behavior of eigenstates of a 3$\times$3 Hamiltonian}
We apply the above formalism to a system expressed by a $3\times3$ Hamiltonian.
First, the rightmost side of Eq.~(\ref{eq_A_Wilson12}) is rewritten in rotation matrix form:
\begin{equation}
W
=
\left[
\begin{array}{ccc}
\operatorname{cos} \left( \phi_{12} \right) & -\operatorname{sin} \left( \phi_{12} \right) & 0 \\
\operatorname{sin} \left( \phi_{12} \right) & \operatorname{cos} \left( \phi_{12} \right) & 0 \\
0 & 0 & 1
\end{array}
\right],
\label{eq_A_WRot_12}
\end{equation}
where
\begin{equation}
\phi_{12} = \oint_{ \Gamma(\alpha) } -A_{12} d k.
\label{eq_A_phi_12}
\end{equation}
Let us suppose that $\left| u_{\mathbf k}^n \right\rangle$ ($n=1,2,3$) are the eigenstates of the Hamiltonian of the given system.
To satisfy Eqs.~(\ref{eq_A_BWZ_Mat_12}) and (\ref{eq_A_WRot_12}) we fix $\left| u_{\mathbf k}^3 \right\rangle$ as $\left[ 0,0,1 \right]$.
We also assume
$\left| u_{{\mathbf k} \left( \alpha \right)}^1 \right\rangle
=
\left[
\operatorname{cos} \left( g \left( \alpha \right) \right),
\operatorname{sin} \left( g \left( \alpha \right) \right),
0
\right]$
and
$\left| u_{{\mathbf k} \left( \alpha \right)}^2 \right\rangle
=
\left[
-\operatorname{sin} \left( g \left( \alpha \right) \right),
\operatorname{cos} \left( g \left( \alpha \right) \right),
0
\right]$
where $g \left( \alpha \right)$ is a real-valued arbitrary function that depends on the position of the closed-loop $\Gamma \left( \alpha \right)$ parametrized by $\alpha$.
For convenience, we set $g \left( 0 \right) = 0$.
The integral in Eq.~(\ref{eq_A_phi_12}) is written as
\begin{equation}
\phi
=
\int_{ \alpha = 0 }^{ \alpha = 2\pi }
- \left\langle u_{{\mathbf k} \left( \alpha \right)}^1
\middle| \frac {\partial u_{{\mathbf k} \left( \alpha \right)}^2} {\partial \alpha}
\right\rangle
d \alpha
= g \left( 2\pi \right)
\label{eq_A_phi_12_2pi}
\end{equation}
In the above, we mentioned that the quaternion $\mp i \sigma_3$ is obtained if this integral is $\pm \pi$.
Thus, we can deduce $g \left( 2\pi \right) = \pm \pi$, and one can define $g \left( \alpha \right) = \pm \alpha / 2$ \cite{Wu2019,Yang2020}.
This means that we can plot the eigenstates $\left| u_{\mathbf k}^1 \right\rangle$ and $\left| u_{\mathbf k}^2 \right\rangle$ rotating around the fixed $\left| u_{\mathbf k}^3 \right\rangle$ by $\pm \pi$.
In the same manner, the rightmost side of Eq.~(\ref{eq_A_Wilson23}) becomes
\begin{equation}
W
=
\left[
\begin{array}{ccc}
1 & 0 & 0 \\
0 & \operatorname{cos} \left( \phi_{23} \right) & -\operatorname{sin} \left( \phi_{23} \right) \\
0 & \operatorname{sin} \left( \phi_{23} \right) & \operatorname{cos} \left( \phi_{23} \right) \\
\end{array}
\right],
\label{eq_A_WRot_23}
\end{equation}
where
\begin{equation}
\phi_{23} = \oint_{ \Gamma(\alpha) }^{ } -A_{23} d k.
\label{eq_A_phi_23}
\end{equation}
We assume $\left| u_{\mathbf k}^1 \right\rangle = \left[ 1,0,0 \right]$,
$\left| u_{{\mathbf k} \left( \alpha \right)}^2 \right\rangle
=
\left[
0,
\operatorname{cos} \left( g \left( \alpha \right) \right),
\operatorname{sin} \left( g \left( \alpha \right) \right)
\right]$
, and
$\left| u_{{\mathbf k} \left( \alpha \right)}^3 \right\rangle
=
\left[
0,
-\operatorname{sin} \left( g \left( \alpha \right) \right),
\operatorname{cos} \left( g \left( \alpha \right) \right)
\right]$
with the same $g \left( \alpha \right)$. The integral in Eq.~(\ref{eq_A_phi_23}) becomes
\begin{equation}
\phi
=
\int_{ \alpha = 0 }^{ \alpha = 2\pi }
- \left\langle u_{{\mathbf k} \left( \alpha \right)}^2
\middle| \frac {\partial u_{{\mathbf k} \left( \alpha \right)}^3} {\partial \alpha}
\right\rangle
d \alpha
= g \left( 2\pi \right)
\label{eq_A_phi_23_2pi}
\end{equation}
and we get similar results; the eigenstates $\left| u_{\mathbf k}^2 \right\rangle$ and $\left| u_{\mathbf k}^3 \right\rangle$ rotate by $\pm \pi$ around $\left| u_{\mathbf k}^1 \right\rangle$ that corresponds to the quaternion $\mp i \sigma_1$.
Thus, if the nodal line system is described by a 3$\times$3 Hamiltonian, the eigenstates ${\mathbf u}_{\mathbf k}^1$, ${\mathbf u}_{\mathbf k}^2$, ${\mathbf u}_{\mathbf k}^3$ along the closed loop can be calculated and plotted along an arbitrary coordinate system (see Figure~\ref{fig:Schematic_Charge_IJK}b). After collecting the eigenstates at the origin, the rotation behavior of the eigenstates indicates the corresponding topological charge. For example, in Figure~\ref{fig:Schematic_Charge_IJK}d, ${\mathbf u}_{\mathbf k}^3$ is fixed while ${\mathbf u}_{\mathbf k}^1$ and ${\mathbf u}_{\mathbf k}^2$ show the $+\pi$-rotation. Then, this is considered as the quaternion charge ${\boldsymbol k}$. Figure~\ref{fig:Schematic_Charge_IJK}e and f also show similar behaviors, thereby their charges are ${\boldsymbol i}$ and ${\boldsymbol j}$, respectively.
\subsection{Correlations for full-vector field problems}
Let us think more about the rotation behaviors of the eigenstates $\left| u_{\mathbf k}^1 \right\rangle$, $\left| u_{\mathbf k}^2 \right\rangle$, and $\left| u_{\mathbf k}^3 \right\rangle$.
If we denote the starting point of the closed loop as ${\mathbf k}_0$ and choose the orthonormal coordinates placed along $\left| u_{{\mathbf k}_0}^1 \right\rangle$, $\left| u_{{\mathbf k}_0}^2 \right\rangle$, and $\left| u_{{\mathbf k}_0}^3 \right\rangle$, these eigenstates $\left| u_{{\mathbf k}_0}^n \right\rangle$ are the same as the unit vectors of the coordinate system. For an arbitrary orthonormal coordinate system $\left| e^n \right\rangle$, the eigenstates $\left| u_{\mathbf k}^n \right\rangle$ can be mapped to $\left| {\left( u' \right)}_{\mathbf k}^n \right\rangle$ by the following rotation matrix
\begin{equation}
{\mathbf R} = \sum_{n=1}^3 \left| e^n \right\rangle \left\langle u_{{\mathbf k}_0}^n \right|.
\end{equation}
The resulting new eigenstates $\left| {\left( u' \right)}_{\mathbf k}^n \right\rangle$ are explicitly written as
\begin{equation}
\left| {\left( u' \right)}_{\mathbf k}^n \right\rangle
=
\left[
\left\langle
u_{{\mathbf k}_0}^1
|
u_{\mathbf k}^n
\right\rangle
,
\left\langle
u_{{\mathbf k}_0}^2
|
u_{\mathbf k}^n
\right\rangle
,
\left\langle
u_{{\mathbf k}_0}^3
|
u_{\mathbf k}^n
\right\rangle
\right] ^T .
\label{eq_A_3x3_eigenstate_in_newcoord}
\end{equation}
Now, we want to render a plot similar to Figure~\ref{fig:Schematic_Charge_IJK}d-f when the system is described not only by a 3$\times$3 matrix but also a geometry-dependent Hamiltonian, e.g., as in photonic crystals. In this case, an eigenstate $\left| \psi_{\mathbf k}^n \right\rangle$ is a function of the three-dimensional position vector \cite{beekman_2017}. Similar to Eq.~(\ref{eq_A_3x3_eigenstate_in_newcoord}) the following correlations can be defined \cite{Park_2021a_ACSPhotonics}:
\begin{equation}
{\mathbf C}_{\mathbf k}^n
=
\left[
\left\langle
\psi_{{\mathbf k}_0}^1
|
\psi_{\mathbf k}^n
\right\rangle
,
\left\langle
\psi_{{\mathbf k}_0}^2
|
\psi_{\mathbf k}^n
\right\rangle
,
\left\langle
\psi_{{\mathbf k}_0}^3
|
\psi_{\mathbf k}^n
\right\rangle
\right].
\label{eq_A_Correlation}
\end{equation}
In the same manner, if the correlations ${\mathbf C}_{\mathbf k}^1$ and ${\mathbf C}_{\mathbf k}^2$ rotate by $\pm \pi$ around ${\mathbf C}_{\mathbf k}^3$, it corresponds to the quaternion $\mp i \sigma_3$.
And if the correlations ${\mathbf C}_{\mathbf k}^2$ and ${\mathbf C}_{\mathbf k}^3$ rotate by $\pm \pi$ around ${\mathbf C}_{\mathbf k}^1$, it corresponds to the quaternion $\mp i \sigma_1$.
\begin{figure}
\centering
\includegraphics{Fig_PatchEulerClass.pdf
\caption{
\label{fig:PatchEulerClass}
Schematic of a three-band system exhibiting the non-Abelian band nodes in two-dimensional momentum space. (a) Band structure with oppositely charges nodes. The blue triangles and red circles indicate the nodes whose charges are $\pm \boldsymbol i$ and $\pm \boldsymbol j$, respectively. The open and filled symbols correspond to the plus and minus signs, respectively. (b) Braiding of the open triangle node around the open circle node. (c),(d) Band structure with the nodes whose charges changed by a factor of $-1$ after the tuning of (b). The patch Euler class is now $1$. Reproduced with permission from Ref.~\cite{Jiang2021}. Copyright 2021, Nature Portfolio (a-d).
}
\end{figure}
\subsection{Evolution of degeneracies and the patch Euler class}
\label{section:patchEulerClass}
In the previous sections, the non-Abelian topological charges in three-band systems were introduced. On top of discovering the nodal line systems with non-Abelian charges, it has been studied how the non-Abelian charged degeneracies evolve with the tuning of the Hamiltonian. Such evolutions include the annihilation or creation of the degeneracies \cite{Bzdusek2017,Bouhon2019,Sun_PRL_2018,Bouhon_NatPhys_2020,Peng2021}.
To understand whether the degeneracies are annihilated or not, the patch Euler class was introduced in Ref. \cite{Bouhon2020,Bouhon_NatPhys_2020}.
In a recent experimental work \cite{Jiang2021}, B. Jiang et al. demonstrated the evolution of degeneracies with non-Abelian charges and patch Euler class. The experimentally observed evolution of degeneracies clearly shows the creation and annihilation of degeneracies although they considered Dirac nodes in two-dimensional momentum space instead of the nodal lines in three-dimensional momentum space. For the three-band system in two-dimensional momentum space illustrated in Figure~\ref{fig:PatchEulerClass}a, the nodes between the first and second (second and third) bands have the charges $\pm \boldsymbol i$ ($\pm \boldsymbol j$). Here, the two charges $\pm \boldsymbol i$ ($\pm \boldsymbol j$) have opposite signs. Thus, the oppositely charged $\pm \boldsymbol i$ (or $\pm \boldsymbol j$) can be annihilated pairwise and the patch Euler class is zero. However, braiding the open triangle around the open circle (see Figure~\ref{fig:PatchEulerClass}b) makes these two flip their charges to positive as shown in Figure~\ref{fig:PatchEulerClass}(c). As a result, the patch Euler class becomes one (see Figure~\ref{fig:PatchEulerClass}d). A patch Euler class of $\pm 1$ means that when the two nodes merge together, they form a stable quadratic node with frame charge $q=-1$ rather than undergoing pairwise annihilation.
\section{Topological nodal lines in photonic systems}
\label{section:PhotonicExample}
\begin{figure*}[ht]
\centering
\includegraphics{Fig_Wenlong.pdf
\caption{
\label{fig:NodalLine_Metamaterials}
Examples of nodal line topological metamaterials. (a) Schematic of the metallic mesh structure for generating a nodal chain. (b) The illustration of the nodal chain consists of two types of nodal lines (blue and red) in the cubic Brillouin zone. (c) Equifrequency contour and band structure of the non-Abelian nodal line metamaterial reminiscent of a biaxial crystal at low frequencies (d) Schematic of the unit cell and the nodal lines in the Brillouin zone. (e) The transitions between orthogonal nodal chain, in-plane nodal chain, and separated nodal lines are observed in the illustrated bianisotropic metamaterial. Reproduced with permission from Ref.~\cite{Yan_2018_NatPhys}. Copyright 2018, Nature Portfolio (a-b); Reproduced with permission from Ref.~\cite{Yang2020}. Copyright 2020, American Physical Society (c-d); Reproduced with permission from Ref.~\cite{Wang_LSA_2021}. Copyright 2021, Nature Portfolio (e).}
\end{figure*}
As we described in Section~\ref{section:non-Abelian}, nodal line degeneracies can be described by the Weyl Hamiltonian without $\sigma_2$ term. Therefore, the goal of designing a structure with nodal lines is to find geometrical or coupling parameters for a structure that has such a dispersion.
Although there is no general recipe that can be applied to different systems, the spatial symmetry consideration can be a good guide to find nodal lines because a structure with nodal lines respects $\mathcal{P}$ and $\mathcal{T}$ symmetry as mentioned earlier. For instance, if we know a structure with Weyl degeneracy, that respects $\mathcal{P}$ or $\mathcal{T}$ symmetry only, we can start from the structure to recover both symmetries. Alternatively, if we start with a structure with excessive symmetries including $\mathcal{P}$ and $\mathcal{T}$ symmetries, which is the case for most of the Bravais lattices, we need to introduce perturbations in the direction of reducing the number of symmetries.
Numerical simulations are often used to see how the dimension of degeneracies changes during the symmetry reduction process.
In this section, we will introduce examples of nodal lines in photonic systems using metamaterials, metallic photonic crystals and dielectric photonic crystals.
\subsection{Metamaterials and metallic photonic crystals}
\label{section:metamaterial}
Photonic metamaterials emerge as a prominent light matter interaction platform and have attracted enormous research interest within the past two decades \cite{simovski_tretyakov_2020,Soukoulis_NatPhoton_2011,Kadic_NatRevPhys_2019,Jahani_NatNanotech_2016}. Utilizing the quasi-homogeneity and by manipulating the constituent deep-subwavelength units, so-called meta-atoms, they can create collective responses to photons far beyond the scope of natural materials, for instance, negative refraction \cite{Pendry_PRL_2000,Hoffman_NatMat_2007}, strong anisotropy \cite{Luo_PRL_2014,Fang_PRB_2009}, hyperbolicity \cite{Yao_Science_2008,Poddubny_NatPhoton_2013}, strong optical activity \cite{Pendry_Science_2004,SZhang_PRL_2009}, etc.
Metamaterials have provided a reliable and convenient guideline towards artificial photonic materials, assisted by the rich design experience accumulated in the past two decades. Providing new degrees of freedom in photonic material design, researchers have found rich fundamentally new physics and applications with metamaterials.
Topological photonic metamaterials have emerged in recent years as a salient topic within the grand regime of topological photonics.
For instance, bianisotropic metamaterials were used to realize a photonic topological insulator \cite{Khanikaev_NatMat_2013}. In fact, topological phenomena in continuous photonic medium have a long-standing history including the renowned Pancharatnam-Berry phase in polarization space \cite{Pancharatnam_PIAS_1956}, and the conical diffraction in biaxial crystals that is a direct consequence of the quantized Berry phase of the Dirac point \cite{Turpin_lpor_2016}.
The first topological metamaterial was designed in 2015 \cite{Gao_PRL_2015}, in which a composite response from hyperbolicity and chirality introduces Fermi surfaces with distinct Chern numbers that uni-directional surface state connects.
Various topological semimetals have also been discovered in metamaterials, including Weyl nodes \cite{Gao_NatComm_2016} and Dirac nodes \cite{Guo_PRL_2017}.
Naturally, metamaterials, or equivalently effective medium methods, play an important role in the construction of topological nodal lines.
An ideal photonic nodal line was discovered in a type-I hyperbolic metamaterial \cite{Gao2018}. Here `ideal' refers to that the nodal line is free from coexisting trivial modes in the bulk. The band crossing happens between effective longitudinal and transverse modes in which interactions are eliminated due to their mismatched field polarizations. Despite being an accidental degeneracy, the band crossing is imposed by the engineered nonlocal response in the metamaterial exerted by the glide reflection symmetry.
In terms of crystallographic symmetry, these nodal lines are protected by mirror symmetry and by introducing mirror symmetry breaking terms, for instance bianisotropy, these nodal lines are instantly gapped and give rise to vortex-like distributed Berry curvatures \cite{Yang_2021_NatComm}.
An ideal type-II nodal line has been discovered recently in Bragg reflection mirror type layered photonic crystals as the phase transition point between trivial and non-trivial Zak phase regimes \cite{Deng_2021_arXiv}.
Definition of the 'type-II' follows the classification of Weyl points \cite{Soluyanov2015}, meaning the highly tilted contact between bands. It exhibits a ring-like contact between electron and hole pocket, distinguished from the donut-like Fermi surface in type-I nodal line semimetals \cite{Gao2018}.
The nodal chain was experimentally introduced using a three-dimensional metallic-mesh structure (Figure~\ref{fig:NodalLine_Metamaterials}a) in microwave scale \cite{Yan_2018_NatPhys}, which was the original design of a metallic metamaterial with extremely low plasma frequency \cite{Pendry_PRL_1996}, though the nodal chain was found far above the plasma frequency and cannot be explained by effective medium theory.
The nodal chain in Figure~\ref{fig:NodalLine_Metamaterials}b, although it consists of two colored nodal rings, is formed by the same adjacent two bands.
This study also examined the drumhead surface states, a sheet of surface dispersion enclosed by the projected nodal line bulk states on the surface Brillouin zone.
Another example of crystallographic symmetry mediated optical response of meta-atoms was demonstrated in the discovery of hourglass nodal lines in a photonic metamaterial \cite{Lingbo_PRL_2019}.
Although it may seem that band topologies can be solely determined by global crystallographic symmetries, the interplay between them and local optical responses is surprisingly rich in new physics, for instance, the hidden symmetries that are unforeseen by crystallographic group theory \cite{Xiong_LSA_2020}.
It is, however, worth noting that without exquisite design, the touching point between equi-frequency contours in natural biaxial crystal forms a 3-dimensional nodal chain if the bands at higher momentum are considered cut off and flattened by the Brillouin zone boundary (Figure~\ref{fig:NodalLine_Metamaterials}c). Utilizing this property and the extreme anisotropy provided by metamaterials, researchers have constructed and measured nodal-link metamaterials in the microwave regime \cite{Yang2020}.
The nodal link in this metamaterial is formed by the lowest three bands (see Figure~\ref{fig:NodalLine_Metamaterials}c, d). The surface bound states in the continuum are another achievement of this study.
Moreover, in a metamaterial with explicitly broken inversion symmetry through bianisotropic optical activities (Figure~\ref{fig:NodalLine_Metamaterials}e), transition between different types of nodal chains is observed by engineering the optical resonances of meta-atoms \cite{Wang_LSA_2021}. One remarkable aspect of these studies is that the topological nature of the nodal line was well described by the non-Abelian band topology \cite{Wu2019}, and can be conveniently derived from the effective Hamiltonian model stemming from the effective medium theory without referring to microscopic electromagnetic fields within the structures.
\begin{figure*}[ht]
\centering
\includegraphics{Fig_Dielectric_NodalLine.pdf
\caption{
\label{fig:Dielectric_NodalLine}
Examples of nodal lines in dielectric photonic crystals. (a) Inversion symmetric double gyroid structure that has air spheres. (b) Band structure of (a) that exhibits a nodal ring in the momentum space. (c) Inversion symmetric and anisotropic double diamond structure. (d) Nodal link, nodal chain, and nodal lines in momentum space of the structure in (c). Reproduced with permission from Ref.~\cite{Lu2013}. Copyright 2013, Nature Portfolio (b); Reproduced with permission from Ref.~\cite{Park_2021a_ACSPhotonics}. Copyright 2021, American Chemical Society (c-d).
}
\end{figure*}
\subsection{Dielectric photonic crystals}
\label{section:photonic}
In this section, we explain how to realize nodal lines in dielectric materials with two examples: double gyroid and double diamond structures. Although fabricating dielectric photonic crystals is challenging, these crystals have advantages such as scalability and convenience in theoretical descriptions.
The nodal ring was theoretically realized using an $\mathcal{P}$-symmetric double gyroid \cite{Lu2013}.
The well-known single gyroid in $O$-symmetry is defined by a set $\mathbf x = \left[ x_1 , x_2 , x_3 \right]$ such that
\begin{eqnarray}
g \left( \mathbf x \right)
&=&
\operatorname{sin} \left( X_1 \right) \operatorname{cos} \left( X_2 \right)
+ \operatorname{sin} \left( X_2 \right) \operatorname{cos} \left( X_3 \right)
\nonumber\\
&+& \operatorname{sin} \left( X_3 \right) \operatorname{cos} \left( X_1 \right) > g_c > 0 ,
\end{eqnarray}
and its space group is $I 4_1 32$ (No. 214) \cite{Fruchart_PNAS_2018,Park_2020_ACSPhotonics}. Here, $X_i = \left( 2\pi / a \right)x_i$ is a local coordinate where $a$ is a lattice constant.
In the study in Ref.~\cite{Lu2013}, the reduced-symmetric single gyroid is created by introducing an air sphere of radius $0.13a$ located at $\left[ 1/4, -1/8, 1/2 \right]a$ in the single gyroid. Then, a double gyroid was created by combining this single gyroid and its counterpart while satisfying inversion symmetry, as shown in Figure~\ref{fig:Dielectric_NodalLine}a. The photonic band structure reveals that the set of degeneracies between the forth and fifth bands form the nodal ring, as shown in Figure~\ref{fig:Dielectric_NodalLine}b.
Very recently, a double diamond photonic crystal exhibiting nodal link, nodal chain, and nodal lines all at once was reported \cite{Park_2021a_ACSPhotonics}. When the lattice vectors are given by ${\mathbf a}_1 = a/2 \left[ 0,1,1 \right]$, ${\mathbf a}_2 = a/2 \left[ 1,0,1 \right]$, and ${\mathbf a}_1 = a/2 \left[ 1,1,0 \right]$ with the lattice constant $a$, the double diamond is defined by a set $\mathbf x$ satisfying $f \left( \pm \mathbf x \right) > f_c > 0 $. Each inequality with plus or minus sign corresponds to each single diamond, so that the two single diamonds are inversion symmetric. Here, the function $f \left( \mathbf x \right)$ is given by
\begin{eqnarray}
f \left( \mathbf x \right)
&=&
A_0 \operatorname{sin} \left( X_1 + X_2 + X_3 \right)
\nonumber\\
&+& \sum_{i=1} ^3 A_i \operatorname{sin} \left( X_1 + X_2 + X_3 - 2 X_i \right)
\end{eqnarray}
where $\mathbf X = \left[ X_1 , X_2 , X_3 \right] = \left( 2 \pi / a \right) \left( \mathbf x -\gamma /2 \sum_{i=1} ^3 {\mathbf a}_i \right)$ is a local coordinate that expresses the translation of the single diamonds along the $\pm \left[ 1,1,1 \right]$-directions, adjusted by the coefficient $\gamma$. Selecting $A_0 = A_1 = A_2 = A_3$ and $\gamma = 0$ generates the conventional diamond structure \cite{Wohlgemuth_Macromelecules_2001,Angelova_JInorgOrgPolMat_2015,Barriga_SoftMatter_2015,La_NatComm_2018,Sheng_NanoRes_2020}. However, this study selected different coefficients $A_i$ and non-zero $\gamma$ (see Figure~\ref{fig:Dielectric_NodalLine}c) to destroy as many symmetries as possible. The resulting structure is anisotropic so that the double diamond has only inversion and translational symmetries.
In momentum space the degeneracies between the first and second bands form the nodal chain, the degeneracies between the third and fourth bands form the simple nodal lines, and the degeneracies between the third, fourth, and fifth bands form the nodal link (see Figure~\ref{fig:Dielectric_NodalLine}d).
The nodal chain is centered at the $\Gamma$-point. Two pink nodal lines (between the first and second bands) depart the $\Gamma$-point and arrive on a single boundary. Another two pink nodal lines are centrosymmetric to the first two pink nodal lines. The two boundaries that these nodal lines touch are parallel, and their normal vectors are commonly along the ${\mathbf b}_2$-direction. Thus, the nodal chain is infinitely connected along the ${\mathbf b}_2$-direction.
Meanwhile, the nodal link consists of the non-touching orange (between the third and fourth bands) and cyan rings (between the fourth and fifth bands), as shown in Figure~\ref{fig:Dielectric_NodalLine}d. The cyan ring is centered at the $\Gamma$-point, and the center of the orange ring is located on a boundary. Due to the periodicity of the first Brillouin zone, this link is infinitely connected.
One more significance of this research \cite{Park_2021a_ACSPhotonics} is that the correlation vectors in Eq.~(\ref{eq_A_Correlation}) were first introduced. By employing the correlation vectors, the non-Abelian topological nature of the nodal link could be directly calculated from the full-vector field eigenstates, instead of using the $3\times3$ effective Hamiltonian, thus the topological charges ${\mathbb Q} = \left\{ \pm \boldsymbol i, \pm \boldsymbol j, \pm \boldsymbol k, \pm 1 \right\}$ were completely deduced from the numerically calculated nodal link.
A simple post-processing calibration was also proposed in this photonic study \cite{Park_2021a_ACSPhotonics}. To get the topological charge $\pm \boldsymbol j$, a closed loop that encloses two nodal lines exhibiting the topological charges $\pm \boldsymbol k$ and $\pm \boldsymbol i$ should be considered. Although we illustrate the quaternion charge $\pm \boldsymbol j$ in Figure~\ref{fig:Schematic_Charge_IJK}f, the eigenstates or correlations calculated by Figure~\ref{fig:Schematic_Charge_IJK}a and b do not generally show results as in Figure~\ref{fig:Schematic_Charge_IJK}f. In many cases, they exhibit the $\pi$-disclinations of ${\mathbf u}_{\mathbf k} ^3$ and ${\mathbf u}_{\mathbf k} ^1$ at ${\mathbf k}_0$ of the loop, implying the topological charge $\pm \boldsymbol j$. Here, the $\pi$-disclinations mean that, before and after winding along the loop, the directions of ${\mathbf u}_{\mathbf k} ^3$ and ${\mathbf u}_{\mathbf k} ^1$ are rotated by $\pi$. To observe the rotation behaviors of the eigenstates or correlations more clearly, this study \cite{Park_2021a_ACSPhotonics} introduces a post-processing calibration method. First, at a point $\mathbf k$, a rotation matrix $\mathbf R \left( \mathbf k \right)$ is defined so that it maps ${\mathbf u}_{\mathbf k} ^2$ to ${\mathbf u}_{{\mathbf k}_0} ^2$ around $\mathbf r \left( \mathbf k \right) = {\mathbf u}_{\mathbf k}^2 \times {\mathbf u}_{{\mathbf k}_0} ^2$. Then, ${\mathbf u}_{\mathbf k} ^1$ and ${\mathbf u}_{\mathbf k} ^3$ are also rotated by $\mathbf R \left( \mathbf k \right)$. This process is done for all $\mathbf k$ on the closed loop, to get a result like Figure~\ref{fig:Schematic_Charge_IJK}f.
\subsection{Optical frequency synthetic dimension}
\label{section:synthetic}
As described in the previous sections, diverse topologies of nodal lines can be created using metamaterials and photonic crystals because one can design a periodic structure and tune geometric and optical parameters. However, the freedom in photonic designs is not simply limited to a spatially periodic system but one can extend the design freedom into the frequency domain based on a new concept called `synthetic dimension'.
Indeed, this can provide another possibility to create more complex topologies.
Very recently, K. Wang et al. \cite{Wang2021} showed that topologies such as unknots, Hopf links and trefoils can be created using two coupled ring resonators and a dielectric waveguide by modulating the phase and amplitude in one of the ring resonators.
In their work, the frequency synthetic dimension is created by multiple resonance frequencies of the unperturbed ring resonators which are periodic in frequency space and the complex energy spectra are obtained by measuring the detuning of the resonance frequencies.
By doing so, the wavevector-energy space $(k, \mathrm{Re}(E), \mathrm{Im}(E))$ becomes the parameter space where the optical bands with different topologies can exist.
It is worth noting that the topology of two bands is considered instead of nodal lines originating from two different bands.
\section{Topological nodal lines in other systems}
\label{section:example}
In this section, we review recent achievement in finding nodal lines in electronic crystals, phononic crystals and electrical circuits.
Historically, the study on nodal lines in electronic crystals started earlier than all other systems.
Several review papers have been published on nodal line semimetals \cite{Fang2016, Gao2019}.
Therefore, in our review, we highlight a few important achievements in electronic crystals.
\begin{figure}
\centering
\includegraphics{Fig_ElectronicCrystals.pdf
\caption{
\label{fig:ElectronicCrystals}
Nodal lines and nodal chains in electronic crystals.
(a) Drumhead surface states (green) bounded by the nodal line in Ca$_3$P$_2$.
(b) Surface states in Mg$_3$Bi$_3$ measured by ARPES.
(c) Nodal chains.
(d) nodal lines in IrF$_4$.
Reproduced with permission from Ref.~\cite{Chan_PRB_2016}. Copyright 2021, American Physical Society (a); Reproduced with permission from Ref.~\cite{Chang_AdvSci_2019}. Copyright 2019, WILEY-VCH Verlag GmbH \& Co. KGaA, Weinheim (b); Reproduced with permission from Ref.~\cite{Bzdusek2016}. Copyright 2016, Nature Portfolio (c); Reproduced with permission from Ref.~\cite{Wu2019}. Copyright 2019, American Association for the Advancement of Science (d).
}
\end{figure}
\begin{figure*}[ht]
\centering
\includegraphics{Fig_PhononNodalRing.pdf
\caption{
\label{fig:PnononNodalRing}
Nodal lines in phononic crystals. (a) Schematic of a granular metamaterial toward phononic nodal chain. Four types of interactions between two beads in the nearest-neighbors are also illustrated. (b). Simulation results showing the nodal chain in momentum space. (c) An illustration of the tight-binding Hamiltonian in Eq.~(\ref{eq_Phonon_Hamiltonian}) with different values of $t$ and $\delta t$. (d) Photograph (left) and schematics (right) of a phononic crystal exhibiting nodal rings. (e-f) Simulation and experimental results showing nodal rings, respectively. Reproduced with permission from Ref.~\cite{Merkel_CommPhys_2019}. Copyright 2019, Nature Portfolio (c-f); Reproduced with permission from Ref.~\cite{Deng2019}. Copyright 2019, Nature Portfolio (c-f).
}
\end{figure*}
\subsection{Electronic crystals}
\label{section:electronic}
The discovery of nodal lines starts from the prediction of the cubic
antiperovskite material Cu$_{3}$NX, where X=\{Ni; Cu; Pd; Ag; Cd\}, as ${\mathbb Z}_2$ protected topological semimetals when ignoring spin-orbit interaction \cite{Rui_Yu_PRL_2015,Kim_PRL_2015}.
This material holds one-dimensional Dirac line nodes and two-dimensional nearly-flat surface states, protected by $\mathcal{P}$ and $\mathcal{T}$ symmetries.
In particular, the 2D surface states are bounded by the projected Dirac line nodes and because of this, they were called drumhead states (Figure \ref{fig:ElectronicCrystals}a) in the field afterwards.
In the same work by Y. Kim et al. \cite{Kim_PRL_2015}, they showed that nearly flat surface states exist in Cu$_{3}$NX.
Additionally, Y. H. Chan et al. showed that the drumhead surface states of Ca$_3$P$_2$ exist due to a quantized Berry phase and the ${\mathbb Z}_2$ topological invariants were defined similarly as in strong topological insulators \cite{Chan_PRB_2016}.
It is worth noting that, before the surge of the search for the drumhead edge states, a topologically protected flat band has drawn attention because it can promote surface superconductivity with an infinite density of states \cite{Kopnin2011}.
Recently, the drumhead surface states were also shown in phononic crystals \cite{Deng2019}. Y. Wang et al. analyzed the flatness and boundedness of photonic drumhead surface states using a simple cubic lattice of metals \cite{Wang_SciRep_2021}.
Soon after, other nodal line materials in the absence of spin-orbital interaction were predicted, such as alkaline-earth compounds AX$_{2}$ (A = Ca, Sr, Ba; X = Si, Ge, Sn) \cite{Huang_PRB_2016}, the CaP$_3$ family of materials \cite{Xu_PRB_2017,Takane_PRB_2018}, some of which are experimentally demonstrated, such as CaCdSn \cite{Laha_PRB_2020} and Mg$_3$Bi$_3$ \cite{Chang_AdvSci_2019} (Figure \ref{fig:ElectronicCrystals}b).
Furthermore, topological semimetals can be classified into two types according to the tilting degree of the fermion cone. Type-II nodal lines in CaPd \cite{Liu_PRB_2018} and Mg$_3$Bi$_3$ \cite{Chang_AdvSci_2019} were proposed. Finally, topological spinful nodal lines were found to exist in TlTaSe$_2$ \cite{Bian_PRB_2016}, ZrSiSe and ZrSiTe \cite{Jin_Hu_PRL_2016} by including spin-orbit interaction as well.
More recently, diverse topologies have been reported in electronic crystals. For example, nodal chains were predicted in IrF$_4$ (Figure \ref{fig:ElectronicCrystals}c) \cite{Bzdusek2017} and nodal links were shown in Sc \cite{Wu2019}. Interestingly, non-Abelian topological invariants (see Subsection~\ref{section:non-Abelian}) were shown to exist in those multiple nodal line structures and this work has inspired much additional research on non-Abelian topology including the one in photonic crystals \cite{Park_2020_ACSPhotonics}.
In addition, it was recently reported that braiding happens in electrons with strain and phase transitions \cite{Viktor_arXiv_2021,Chen_arXiv_2021}.
\subsection{Phononic crystals}
\label{section:phononic}
Phononic crystals, also known as acoustic crystals, are also a good platform to demonstrate the topological physics because the couplings between meta-atoms can be easily controlled and the displacement field of the modes can be easily measured giving the full profile of the modes and band structure.
A nodal chain by the phononic wave was theoretically proposed using a granular metamaterial \cite{Merkel_CommPhys_2019}. For the beads consisting a face-centered cubic arrangement, one can assume that tension, shear, bending, and torsion exist on all contacts between any two nearest-neighbors grains, denotes as $K_N$, $K_S$, $G_B$, and $G_T$, respectively, in Figure~\ref{fig:PnononNodalRing}a. By applying these assumptions to the linear equations of motion for each bead, the result in Figure~\ref{fig:PnononNodalRing}b clearly shows the nodal chain in momentum space.
W. Deng et al. proposed a phononic crystal that exhibits nodal rings in momentum space \cite{Deng2019}. The effective Hamiltonian for this realization is written as
\begin{equation}
H = d_x \sigma_1 + d_y \sigma_2
\label{eq_Phonon_Hamiltonian}
\end{equation}
where
\begin{equation}
d_x = -2 \operatorname{cos} k_x -2 \operatorname{cos} k_y -2 t \operatorname{cos} k_z
\label{eq_Phonon_dx}
\end{equation}
and
\begin{equation}
d_y = -2 \delta t \operatorname{sin} k_z
\label{eq_Phonon_dy}
\end{equation}
are the sublattice pseudospins. For $t>0$ and $\delta t > 0$, the hopping amplitude in the $z$-direction is $-t \pm \delta t$ as illustrated in Figure~\ref{fig:PnononNodalRing}c by thin and thick vertical rods. The hopping in the $x$- and $y$-directions are $-1$ as represented in Figure~\ref{fig:PnononNodalRing}c.
The eigenvalue of Eq.~(\ref{eq_Phonon_Hamiltonian}) is $E = \pm \sqrt{d_x ^2 + d_y ^2}$, and the degeneracies are formed when $d_x = d_y = 0$. This indicates that the pseudospins are arranged as that corresponds to nodal rings formed on the $k_z = n \pi$ plane ($n$ is an integer).
For experimental observation of the nodal rings, they prepared a layer-stacked-phononic crystal made of plastic stereolithography material. In the three-dimensional structures shown in Figure~\ref{fig:PnononNodalRing}d, there are several types of holes, the propagation paths of sound waves. Along the $z$-direction, two types of holes exist. The smaller and larger holes correspond to the thinner and thicker rods in Figure~\ref{fig:PnononNodalRing}c.
Along the horizontal directions, the sound waves meet the same types of rectangular holes, and this corresponds to the same size of rods in Figure~\ref{fig:PnononNodalRing}c.
With this phononic crystal design, the nodal rings could be observed in momentum space (see Figure~\ref{fig:PnononNodalRing}e and f), as predicted by Eq.~(\ref{eq_Phonon_Hamiltonian}).
\begin{figure}
\centering
\includegraphics{Fig_PhononPatchEulerClass.pdf
\caption{
\label{fig:PhononPatchEulerClass}
Experimental realization of Euler class using a phononic crystal. (a) Schematics of a kagome tight-binding model with tunning parameters $t$ and $t'$. (b) Phononic band structure of (a). (c) Experimental design of (a). (d) Simulation (white plots) and experimental band structure of (c). Reproduced with permission from Ref.~\cite{Jiang2021}. Copyright 2021, Nature Portfolio (a-d).
}
\end{figure}
Very recently, B. Jiang et al. used a two-dimensional phononic crystal to observe non-Abelian topological charges and topological phase transitions \cite{Jiang2021}. Although this work is not directly related to nodal lines in three-dimensional momentum space, this work has its significance in experimental demonstration of the non-Abelian phononic nodes (degenerate point in two-dimensional momentum space) and the patch Euler class mentioned in Section \ref{section:patchEulerClass}.
They employed a tight-binding model of a kagome lattice as shown in Figure~\ref{fig:PhononPatchEulerClass}a.
The hopping between lattice points A, B, and C can be adjusted by $t$ and $t'$. Band structures by tuning these two variables exhibit several types of non-Abelian charges and Euler classes, as shown in Figure~\ref{fig:PhononPatchEulerClass}b.
Then, the tight-binding model is realized using cylindrical acoustic resonators as shown in Figure~\ref{fig:PhononPatchEulerClass}c to observe topological phase transitions and the new topological invariant ($-1$ of the Euler class) (see Figure~\ref{fig:PhononPatchEulerClass}d) .
\begin{figure*}[ht]
\centering
\includegraphics{Fig_Xiao_Topolectricity.pdf
\caption{
\label{fig:Topolectricity}
(a) Mathematical models construct a nodal knot/link from a braid. A braid closure can be embedded onto the three-dimensional Brillouin zone torus differently through different choices of $F \left( \mathbf k \right)$ functions. Depending on its topological charge density distribution, it produces different numbers of nodal knots in the Brillouin zone, i.e. either a single copy ($F_1$) or two copies ($F_2$) related by mirror symmetry. (b) Experimental setup for impedance measurement of the Hopf-link circuit. Reproduced with permission from Ref.~\cite{Lee2020}. Copyright 2020, Nature Portfolio (a-b).
}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics{Fig_Xiao_Non_Hermitian.pdf
\caption{
\label{fig:Non_Hermitian}
Illustration of different constituents of the non-Hermitian trefoil knot circuit. (a) On-site hopping and nearest-neighbor hopping along the $x$ direction, (b) next-nearest-neighbor hopping along the $x$ direction, (c) nearest-neighbor hopping along the $y$ direction, (d) nearest-neighbor hopping along the $z$ direction, (e) next-nearest-neighbor hopping along the $y$ direction, and (f) diagonal hopping within the $x-y$ plane. Reproduced with permission from Ref.~\cite{Zhang_CommPhys_2021}. Copyright 2021, Nature Portfolio (a-f).
}
\end{figure*}
\subsection{Electrical circuits}
\label{section:electrical}
Knots, such as everyday-life ropes are intricate nodal lines. They are difficult to construct because they require finely tuned long-ranged hoppings. To realize models with those long-ranged hoppings, it is naturally suggested to use artificial structures, which allow for unprecedented control over individual couplings \cite{Lee2020}. Most importantly, electrical circuits, whose connections transcend locality and dimensionality constraints, put the implementation of couplings between distant sites of a high-dimensional system and nearest-neighbor connections on equally accessible footing. This advantage is found to be crucial to the realization of nodal knots, which contains many strong long-ranged hoppings.
Having explained the necessity to implement nodal knots using electrical circuits, we now describe how they can be concretely implemented and detected. An electrical $RLC$ circuit is an undirected network with nodes $\alpha=1,\cdots,N$ connected by resistors, inductors, capacitors or combinations of them. Its behavior can be fully characterized by Kirchhoff's law as $I_\alpha=J_{\alpha\beta}V_\beta$, where $I_\alpha$ is the external current at junction $\alpha$ and $V_\beta$ is the voltage at junction $\beta$. Each entry $J_{\alpha\beta}$ of the Laplacian $J$ contributes $r_{ab}$ to the Laplacian, where $r_{ab}=R,i\omega L, (i \omega C)^{-1}$ for a single resistor, inductor and capacitor and whose combinations follows. The strictly reciprocal (symmetric) form of $J_{\alpha\beta}$ constrains the possible Laplacian, which prevents nodal knot circuits from being developed using mathematical models of nodal knots proposed before, since those imply broken reciprocity. Thus in order to construct nodal knot circuits, new models which preserve reciprocity need to be invented \cite{Lee2020}.
A very recent work discovered a method to overcome this obstacle, which goes beyond existing approaches that requires broken reciprocity, by pairing nodal knots with their mirror images to realize pairs of nodal knots in a fully reciprocal setting \cite{Lee2020}. The key insight is that pairs of nodal knots can preserve reciprocity while a single knot cannot, such that they can be realized in a circuit as shown in Figure~\ref{fig:Topolectricity}a. A highlight of this work is the experimental verification of surface drumhead states in a design of the nodal Hopf link circuit shown in Figure~\ref{fig:Topolectricity}b. This experimental setting is used to extract the admittance band structure through linearly independent measurements, whereas the number of measurements needed is the same as the number of inequivalent nodes $N$. Each step consists of a local excitation in this circuit and a global measurement of the voltage response from which one can extract all components of the Laplacian in reciprocal space. By diagonalizing the Laplacian $J(k)$, the admittance band structure can be found, reflecting that the Laplacian plays the same role as the Hamiltonian in an electronic material.
There are other advantages that we have not mentioned so far in using electronic circuits to construct nodal knot as well as other nodal lines. Take the non-Hermitian nodal knots as an example, the positive, negative and non-reciprocal couplings needed for non-Hermitian nodal knots can be realized through carefully chosen combinations of $RLC$ components and operation amplifiers, which can introduce non-reciprocal feedback needed for the skin effect \cite{Zhang_CommPhys_2021} that $RLC$ components can not. By active elements in those circuits we can realize non-Hermitian models straightforwardly in electrical circuits, given the non-Hermitian trefoil circuit as an example in Figure~\ref{fig:Non_Hermitian}, which reflects the flexibility of topolectrical circuits. In those circuits, topological zero modes manifest themselves through a divergent impedance which we call topolectrical resonances. Finally, but not least, electronic circuits provide possibilities to simulate nodal lines and drumhead states beyond three dimensions, in an analogous way as simulating topological insulators in Class AI with electronic circuits in four dimensions \cite{Price_PRB_2020,Wang2020}.
\section{Conclusions and outlook}
To summarize, we have explained the concept of nodal lines in band structures and showed that a wide range of types from a simple nodal line, a nodal link and a chain to mixed nodal lines can exist. Indeed, the extension from zero-dimensional degeneracy allows for a wider range of topology for the degeneracies in band structures of particles/waves including electrons, photons and phonons.
Then, we have reviewed the theoretical description of the topological invariants of nodal lines with the Berry phase and the Wilczek-Zee connection based on the two seminal papers by M. V. Berry \cite{Berry_1984} and Wilczek and Zee \cite{Wilczek_PRL_1983}. This provides us with an essential toolset to describe the topology of nodal lines. Using the Wilczek-Zee connection, we explained how non-Abelian topology can be considered in a three-band system, following the recent work on non-Abelian topology by Q. Wu et al. \cite{Wu2019}.
Finally, we have reviewed recent advances in implementing nodal lines using metamaterials and photonic crystals. These two photonic systems have successfully demonstrated those exotic behaviors and they remain very promising in finding new topological states related to nodal lines. However, they are not the only systems and they also benefit from the earlier work in other fields. To supplement, we have introduced the examples of nodal lines in electronic crystals, phononic crystals, and electrical circuits. Moreover, non-Hermitian systems, which have complex energy eigenvalues due to the exchange of energy with the environment, can be used to extend the dimension of parameters space. Indeed, new non-Hermitian systems with optical ring resonators and RLC circuits are being introduced to implement more complex topology of nodal lines such as knots and we expect many undiscovered topological nodal lines will be implemented soon with these new approaches.
However, there are still challenges in understanding the physical consequence of topological phase of nodal lines and also implementing the proposed ideas in artificial materials.
First, no bulk-edge/surface correspondence for the non-Abelian charges has been mathematically proven \cite{Jiang2021} although one can naturally think that there could be a relation similar to the bulk-edge/surface correspondence in Chern insulators, which is Abelian.
To answer this question, more in-depth theoretical investigation on multiple nodal lines systems is required.
Second, the refractive index required to have nodal links and other topology using dielectric photonic crystals is too high, normally requiring a refractive index higher than 3.5 \cite{Park_2020_ACSPhotonics}.
This limits the choices of materials in the microwave range, adding another challenge in fabrication. For example, L. Lu et al. used a ceramic material with a high refractive index to observe Weyl points experimentally, and they drilled the material in different directions to prepare a Weyl photonic crystal \cite{Lu_2015_Science}.
Third, the fabrication of nanostructures for dielectric photonic crystals working at optical wavelengths can be challenging. As the nodal lines are normally implemented in three-dimensional momentum space that requires a three-dimensional array of high index dielectric materials. A self-assembly method using block copolymers \cite{Jo_2021_AppliedMaterialsToday} can be used but it requires the additional step of inserting high-refractive index materials and it is hard to control the local geometrical perturbations as we want. A direct-laser writing or other polymerization methods can be used but they still require an additional step to add high-refractive index material.
However, these challenges could be overcome in a few years considering the rapid progress in nanofabrication techniques.
There are also unexplored areas in relation to nodal line physics in photonic systems.
First, 3D nanoplasmonic systems can be used to implement and observe nodal lines of surface plasmon polaritons. In 2D topological photonics, there is already a theoretical study that shows unidirectional propagation and corner states in 2D metallic arrays \cite{Proctor2021}. However, due to the difficulty in fabricating 3D plasmonic structures, no nodal lines in 3D plasmonic band structures have been observed yet. Second, quantum optic systems can be used to implement the nodal line. An array of coupled quantum emitters has been studied in 2D topological photonics structures \cite{Perczel2020}. If we extend the system to 3D structures, 3D exciton polaritonic systems could be a good platform to study the topology of nodal lines. Recently, also an array of dipolar molecules in 3D optical lattices has been proposed to implement Hopf insulators \cite{Schuster2021}.
The remaining question is what applications could be enabled using nodal lines. There are very few reports or proposals regarding practical applications and it is hard to discuss a general way of applying nodal lines.
For electronic crystals with nodal lines, high-temperature surface superconductivity using drumhead surface states \cite{Kopnin2011} has been one major motivation of study of nodal lines. Additionally, applications for surface ferromagnetism \cite{Chen2019PRL} and high harmonic generation \cite{Lee2020_HHG} have been theoretically proposed.
For photonic applications, high density of states of surface states in nodal lines systems \cite{Wang_SciRep_2021, Gao2018} are expected to enhance spontaneous emission, resonant scattering, nonlinearities and blackbody radiation.
The bound states in the continuum related to nodal lines \cite{Yang2020} may find applications in lasing \cite{Hwang_NatComm_2021} and sensing \cite{Yanik_PNAS_2011}.
Although direct applications seem limited for the moment, a better understanding of topological phases of nodal lines would give us rich knowledge and interesting physics of artificially designed materials system as well as electronic materials. We believe this would open a new avenue to exciting applications as well as expand our human knowledge.
\begin{acknowledgments}
The authors thank Robert-Jan Slager, Andreas Pusch and Stephan Wong for critical reading and comments. The work is part-funded by the European Regional Development Fund through
the Welsh Government (80762- CU145 (East)). This research
is also supported by the National Natural Science Foundation of China (Grant No.11874431), the National Key R\&D Program of China (Grant No. 2018YFA0306800),
and the Guangdong Science and Technology Innovation Youth Talent Program (Grant No.2016TQ03X688).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,891 |
Q: ArrayIndexOutOfBoundsException: in my array I have an array that runs on recursion it finds the minimum number on the array. I run the program and I get an ArrayIndexOutOfBoundsException error on (Assignment9.java:36)
if (previousMin > numbers[endIndex]) and (Assignment9.java:20)
double min = findMin(numbers, 0, numbers.length); I know why this problem usually occurs but I cant find the fix for my code. I dont know if my actual code works since I cant run the program. Any suggestions..
import java.io.*;
import java.text.*;
public class Assignment9
{
public static void main(String[] args) throws IOException
{
int [] numbers = new int[100];
InputStreamReader streamR = new InputStreamReader(System.in);
BufferedReader inFile = new BufferedReader(streamR);
String reader = inFile.readLine();
double min = findMin(numbers, 0, numbers.length);
System.out.print ("The minimum number is " + min + ('\n'));
}
public static int findMin (int [] numbers, int startIndex, int endIndex)
{
if (startIndex == endIndex)
{
return numbers[startIndex];
}
else
{
double previousMin = findMin (numbers, startIndex, endIndex - 1);
if (previousMin > numbers[endIndex])
return numbers[endIndex];
else
return numbers[endIndex];
}
}
A: You are accessing numbers[endIndex] where endIndex = numbers.length. This is not possible in java since array indexing starts at 0, the last element is at index length-1, hence the exception.
A: numbers.length will return the length of your array, starting from 1. You want to use numbers.length - 1 in your method call
A: An array of 100 elements has them numbered from 0 to 99. But you are using 100 as endIndex in your initial call to findMin. So when you reference numbers[endIndex], you've gone past the end of the array - numbers[100] does not exist.
A: You should use numbers.length-1 for your end index because java is zero-indexed, and numbers.length will be outside the bounds of the array. Returning the value numbers[startIndex] will be out of bounds when startIndex==endIndex.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,190 |
is a 1969 Japanese yakuza film directed by Akinori Matsuo.
Plot
The brothers of Tetsujirō and Tetsugorō, who lost their parents and home after the Great Kanto Earthquake, were picked up and grew up by a Yakuza Goi clan. One day, Tetsugorō fights the confronted yakuza Honma clan, and is expelled from Goi clan's Boss. Later, boss Goi is attacked by Honma clan's assassin and seriously injured. The assassin is Tetsutaōr, Tetsugorō's eldest brother, who had been missing after the Great Kanto Earthquake.
Cast
Hideki Takahashi as Nonaka Tetsugorō
Shigeru Tsuyuguchi as Nonaka Tetsujirō
Yumiko Nogawa as Ochō
Yoshirō Aoki as Nonaka Tetsutarō
Ryōhei Uchida as Obuse Keita
Hei Enoki as Tamura Gunji
Shouki Fukae as Honma Ginzō
Tomiko Ishii as Shibata Chiyo
Yoko Machida as Nonaka Shima
Ichirō Sugai as Goi Kiichirō
Toru Abe as Omori Giichirō
References
External links
Nikkatsu films
Yakuza films
Japanese crime films
1960s Japanese-language films
1970s Japanese films
1960s Japanese films | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,788 |
\section{Introduction}
Charge colloidal particles do not usually conform to the simple and popular idea that they can be characterized
either as insulators with fixed surface charges or conductors with constant surface potential~\cite{Adamson}.
In fact, when two colloidal particles with ionizable surface groups (immersed in an aqueous electrolyte solution) are brought together,
both their surface charge-density and surface electrostatic-potential change with the particle (surface) inter-distance~\cite{Borkovec1,Borkovec2}.
This ubiquitous phenomenon stems from the dissociation/association of surface ionizable groups and is referred to as {\it charge regulation} (CR).
It was elegantly formalized within the Poisson-Boltzmann (PB) theory of electrostatic interactions by Ninham and Parsegian in the 1970's~\cite{NP-regulation}.
The CR formalism can be implemented either through a chemical dissociation equilibrium of surface binding sites
(law of mass action)~\cite{Regulation2,Regulation3,Regulation4,Regulation5}, or through a surface-site partition function (free energy)
\cite{Olvera,Olvera2,Natasa1,Natasa2,epl,diamant,maarten}. In both cases, it yields the same self-consistent boundary condition
for an effective surface-charge density that differs from the boundary condition of
constant charge (CC) for charged insulators or constant potential (CP) for conducting surfaces.
The concept of charge regulation has been widely applied in different situations:
analysis of the stability of the electrostatic double-layer and its relation to inter-surface forces \cite{stab,instab},
dissociation of amino acids and protein interactions~\cite{Leckband,Lund,Fernando},
charge regulation in protein aggregates such as viral shells \cite{Nap}
and polyelectrolytes and polyelectrolyte brushes~\cite{Netz-CR,Borukhov,Kilbey,Zhulina},
as well as for charged membranes~\cite{membranes1,membranes2,membranes3}.
Although the theory of charge-regulated electrostatic interactions has been previously used in numerous situations,
some conceptually important issues have not been addressed with sufficient generality.
Usually, the CR disjoining pressure, $\Pi_{_{\rm CR}}$, is bounded
by those stemming from the CC and CP boundary conditions~\cite{always}
(for some exceptions, see {\it e.g.}, Refs.~\cite{notalways1,notalways2}).
However, this {\it does not} imply that, in general, the expression of $\Pi_{_{\rm CR}}(d)$ as function of the inter-surface separation, $d$,
will properly reduce to the two implied limits.
In this Letter, we show on general grounds that the disjoining pressure, $\Pi_{_{\rm CR}}(d)$,
based on the CR boundary condition
has scaling properties in the limit of small inter-plate separations,
which differ from the scaling behavior of the CC or CP boundary conditions.
\section{Model}
Consider an ionic solution that contains monovalent
symmetric (1:1) salt of charge $\pm e$ of bulk concentration $n_b$,
immersed in aqueous solvent between two symmetric planes separated by distance $d$,
and of infinite lateral extent, as depicted in Fig~1.
We consider three types of boundary conditions: constant charge (CC), constant potential (CP) and charge regulation (CR).
The water solvent is assumed to be a continuum dielectric medium characterized by the water dielectric constant, $\varepsilon_w$.
We choose for convenience to locate the two planes at $z=\pm d/2$ such that $z=0$ is a symmetry plane.
Thus, the electrostatic potential is symmetric about the mid-plane, yielding a zero electric field, $E_m \propto \psi_m' = 0$ at $z = 0$.
\begin{figure
\centering
\includegraphics[scale=0.4]{fig1.eps}
\caption{ \textsf{Schematic drawing of two symmetric surfaces at $z=\pm d/2$ with dissociable ionic groups.
The charge regulation boundary-condition is described with a surface interaction parameter, $\alpha$.
The ions are dissolved in an aqueous solution of dielectric constant, $\varepsilon_w$.
}
}
\label{figure1}
\end{figure}
The equation that governs the distribution of mobile ions in solution at finite temperature is the
well-known Poisson-Boltzmann (PB) equation (for details see Ref. \cite{Safinya}).
For 1:1 monovalent salts it has the simple form:
\begin{eqnarray}
\label{e1}
\frac{\mathrm{d}^2\psi}{\mathrm{d} z^2} = \kappa_\mathrm{D}^2 \sinh\psi(z) \, ,
\end{eqnarray}
where $\psi$ is redefined as a dimensionless electrostatic potential ($e \psi/k_{\mathrm{B}}T \rightarrow \psi$)
and $\lambda_\mathrm{D} = \kappa_\mathrm{D}^{-1} = (8\pi e^2 n_b / \varepsilon_w k_{\mathrm{B}}T)^{-1/2}$
is the Debye length, with $k_B$ the Boltzmann constant and $T$ the temperature.
This one-dimensional PB equation is obtained by taking into account the translation symmetry in the $x-y$ plane.
The solution of the one-dimensional PB equation
can be expressed in terms of elliptic functions.
Exploiting the symmetry of the system, it is then sufficient to consider the interval $[0,d/2]$,
with $\psi_m' = 0$ at the $z=0$ mid-plane.
The general solution in such a symmetric setup
can be written in terms of the Jacobi elliptic function~\cite{abramowitz,Safinya}, ${\rm cd}(u|a^2)$, as
\begin{eqnarray}
\label{e2}
\psi = \psi_m + 2\ln\left[ {\rm cd} \left( \frac{z}{2\lambda_\mathrm{D}\sqrt{m}} \Big| m^2 \right) \right] \, .
\end{eqnarray}
with $m \equiv \exp\left(\psi_m\right)$ and $\psi_m \equiv \psi(z=0)$.
The additional boundary condition at $z=d/2$ will determine a different $\psi_m$ for the three different cases: CC, CP, and CR at finite $d$.
Evaluating the above equation and considering the boundary condition at $z=d/2$
result in an explicit relation $\sigma = \sigma(\psi_s;d)$ between the surface charge density, $\sigma$,
the surface potential $\psi_s$ and $d$.
In order to understand how the three boundary conditions differ
and when they indeed merge, we use the general expression for the disjoining pressure,
which is valid in all three cases (CC, CP and CR) {as explained in Ref.~\cite{Harries}},
\begin{eqnarray}
\label{e2a}
\Pi(d) = 4k_{\mathrm{B}}T n_b\sinh^2(\psi_m/2) > 0 \, .
\end{eqnarray}
This pressure is a macroscopic measurable quantity that strongly depends on the inter-plane separation, $d$.
The difference in the disjoining pressure for the three boundary conditions becomes substantial only for
relatively small separations, $d \lesssim \lambda_\mathrm{D}$,
while in the large separation limit, $d\ \gg \lambda_\mathrm{D}$, the three pressure expressions coincide.
To gain further insight into the different behavior of the disjoining pressure,
we focus on the small separation limit, $d \ll \lambda_\mathrm{D}$ and $d \ll \ell_\mathrm{GC}$,
{where $\ell_\mathrm{GC} \equiv e/(2\pi\ell_\mathrm{B}|\sigma|)$ is the Gouy-Chapman length and $\ell_\mathrm{B} = e^2/\varepsilon_wk_{\mathrm{B}}T$ is the Bjerrum length}.
Our analytical results give the scaling of $\Pi$ with $d$ and clearly distinguish between the three boundary conditions.
Let us start with the most common CC boundary condition,
\begin{eqnarray}
\label{e3}
\psi^{\, \prime}\Big|_{z=d/2} \equiv \psi_s^{\, \prime} = 4\pi\ell_\mathrm{B} \frac{\sigma}{e} \, .
\end{eqnarray}
Equations~(\ref{e2}) and (\ref{e3}) give a relation between the surface charge density $\sigma$,
the mid-plane potential $\psi_m$ and $d$, in terms of the Jacobi elliptic functions~\cite{abramowitz,Safinya},
\begin{eqnarray}
\label{e4}
\frac{\sigma}{e} = \frac{\kappa_\mathrm{D}}{4\pi\ell_\mathrm{B}}\frac{m^2-1}{\sqrt{m}}\frac{{\rm sn}(u_s|m^2)}{{\rm cn}(u_s|m^2){\rm dn}(u_s|m^2)} \, ,
\end{eqnarray}
with $u_s \equiv d/(4\lambda_\mathrm{D}\sqrt{m})$ and $m \equiv \exp\left(\psi_m\right)$ as defined above.
For fixed surface charge, this relation gives the mid-plane potential, $\psi_m$.
Then, the disjoining pressure can be calculated from eq.~(\ref{e2a}).
When $d$ is the smallest length scale in the system, it can be shown
that the disjoining pressure in the so-called {\it ideal-gas} regime~\cite{Safinya,andelman2005,andelman1995} scales as,
\begin{eqnarray}
\label{e5}
\Pi_{_{\rm CC}} \simeq \frac{k_{\mathrm{B}}T}{\pi\ell_\mathrm{B}\ell_\mathrm{GC}}\,\frac{1}{d} \sim d^{-1} \, .
\end{eqnarray}
The density (per unit volume) of the counter-ions is almost
constant between the two charged plates and is equal to $2|\sigma|/(ed)$.
This counter-ion density neutralizes the surface charge density, $\sigma$,
and the main contribution to the pressure comes from the entropy of an ideal-gas behavior of the counter-ion cloud.
In the second case of a CP boundary-condition $\psi_s$ is fixed, and unlike
the CC case, here the counter-ion concentration remains constant near
each of the planes, as it uniquely depends on the value of the surface potential, $\psi_s$, through the Boltzmann factor~\cite{banyaakov2010}.
The corresponding surface charge density, $\sigma$, is proportional to $d$ at small separation, $\sigma \sim d$.
Therefore, it vanishes for $d\to 0$.
Using the Taylor expansion of elliptic functions~\cite{abramowitz},
one can evaluate the leading terms in the surface potential for small $d$ as,
\begin{eqnarray}
\label{e6}
\psi_s \simeq \psi_m - \left(1 - m^2\right)u_s^2 \, .
\end{eqnarray}
Substituting the above equation into the disjoining pressure expression, eq.~(\ref{e2a}), yields to second order in $d$,
\begin{eqnarray}
\label{e7}
\nonumber \Pi_{_{\rm CP}} &\simeq& k_{\mathrm{B}}T n_b \left( 4\sinh^2(\psi_s/2) - \sinh^2(\psi_s)\frac{(\kappa_\mathrm{D} d)^2}{8} \right) \\
&\simeq& {const.} + {\cal O}\left(d^2\right) \, .
\label{CP1}
\end{eqnarray}
The above equation shows that the disjoining pressure for the CP boundary-condition
goes to a constant value, $\Pi_0$, for vanishing inter-plate separation, $d\to 0$,
with a leading correction term proportional to $d^2$.
\section{Single-site process}
As an example of a CR boundary-condition, we consider a surface that is composed of ionizable groups ({\it e.g.}, charged phospholipids).
Each group can release a counter-ion into the solution in a {\it single-site} dissociation process.
We first focus on such single-site CR dissociation process and refer to it as ${\rm CR_1}$. It is the simplest and most common CR process, and it will be extended below to multi-site processes.
The surface dissociation/association can be described by the reaction:
\begin{eqnarray}
\label{e8}
{\rm A}^+ + {\rm B}^- \rightleftarrows {\rm AB} \, ,
\end{eqnarray}
where A denotes a surface site that can be either ionized $({\rm A}^+)$
or neutral (AB).
The process of dissociation/association is characterized by an equilibrium constant $K_{\rm d}$
through the law of mass action
\begin{eqnarray}
\label{e9}
K_{\rm d} = \frac{[{\rm A}^+][{\rm B}^-]_{\rm s}}{[{\rm AB}]} \, ,
\end{eqnarray}
where $[{\rm A}^+]$, $[{\rm B}^-]_{\rm s}$ and [AB] denote the three corresponding surface concentrations.
The equilibrium condition of eq.~(\ref{e9}) can be written in terms of $\phi_s \equiv \sigma(\psi_s) a^2/e \sim [{\rm A}^+]$,
\begin{eqnarray}
\label{e10}
\nonumber\phi_s &=& \frac{1}{1 + \phi_b\mathrm{e}^{ - \alpha + \psi_s }} \\
&=& \frac{1}{2} - \frac{1}{2}\tanh\left[(\ln{\phi_b} - \alpha + \psi_s )/2\right] \, ,
\end{eqnarray}
where $a^3$ is the ion volume, $\phi_b = a^3n_b$ is the ionic volume fraction
and we have introduced a surface interaction parameter $\alpha = \ln(a^3 K_{\rm d})$.
From $\sigma(\psi_s)$, eq.~(\ref{e4}) and eq.~(\ref{e6})
one obtains explicitly $\psi_m$.
By using the Taylor expansion of elliptic functions~\cite{abramowitz}
in eqs.~(\ref{e4}) and (\ref{e10}), it is clear that as $d \to 0$, $m$ diverges, but this divergency is weaker than $d^{-1}$.
It yields a diverging ${\rm CR_1}$ disjoining pressure for small $d$,
\begin{eqnarray}
\label{e11}
\Pi^{(1)}_{_{\rm CR}} \simeq \sqrt{2}k_{\mathrm{B}}T\mathrm{e}^{\alpha/2}a^{-5/2}d^{-1/2} \sim d^{-1/2}\, ,
\label{res1}
\end{eqnarray}
where the superscript in $\Pi^{(1)}_{_{\rm CR}}$ indicates that it corresponds to a CR$_1$ process.
Note that just like $\Pi_{_{\rm CC}}$, $\Pi^{(1)}_{_{\rm CR}}$ does not depend on the bulk salt concentration, $n_b$.
\begin{figure
\centering
\includegraphics[scale=0.7]{fig2.eps}
\caption{ \textsf{ The dimensionless disjoining pressure, $\Pi$ [in units of $k_{\mathrm{B}}T/(4\pi\ell_\mathrm{B}\lambda_\mathrm{D}^2)$],
for the three boundary conditions: constant charge (CC - dashed red line), constant potential (CP - dotted-dashed blue line)
and charge regulation for {\it single-site} dissociation process (${\rm CR_1}$ - solid black line).
The pressure inequality, seen in the figure, $\Pi_{_{\rm CC}} \geq \Pi_{_{\rm CR}} \geq \Pi_{_{\rm CP}}$,
is an inequality that holds in general for the case of charge regulation consisting of a single-site process.
The parameters used are: $a = 5 \, {\rm \AA}$, $n_b = 0.1\, {\rm M}$ and $\alpha = -6$ (${\rm pK} \simeq 1.48$).
In the inset we present the same disjoining pressure on a log-log plot, demonstrating its scaling with the inter-plate distance, $d/\lambda_\mathrm{D}$.
}}
\label{figure2}
\end{figure}
Another possible single-site process is the process of charging a neutral surface,
\begin{eqnarray}
\label{a1}
{\rm A} + {\rm B}^{+} \rightleftarrows {\rm AB^{+}} \, .
\end{eqnarray}
The equilibrium condition for this ${\rm CR_1}$ process can be written as,
\begin{eqnarray}
\label{a2}
\nonumber \phi_s &=& \frac{\phi_b}{\phi_b + \mathrm{e}^{\alpha + \psi_s}} \\
&=& \frac{1}{2} - \frac{1}{2}\tanh\left[(-\ln\phi_b + \alpha + \psi_s )/2\right] \, .
\end{eqnarray}
Note the eq.~(\ref{a2}) reduces to eq.~(\ref{e10}) for the mapping:
$\phi_s\to 1-\phi_s$ and $\psi_s\to -\psi_s$.
Repeating the same procedure as above, we obtain a somewhat different disjoining pressure,
\begin{eqnarray}
\label{a3}
\Pi^{(1)}_{_{\rm CR}} \simeq \sqrt{2}k_{\mathrm{B}}T n_b \mathrm{e}^{-\alpha/2}a^{1/2}d^{-1/2} \sim d^{-1/2} \, ,
\label{res2}
\end{eqnarray}
which also diverges as $d^{-1/2}$ but with a different prefactor that is linear
in the bulk concentration.
Note that these results do not depend on the sign of the surface site.
{
A typical pressure isotherm, $\Pi(d)$, is computed numerically for the ${\rm CR_1}$ process and is shown in Fig.~\ref{figure2}.
The mid-plane potential, $\psi_m$, is obtained as a function of inter-membrane separation, $d/\lambda_\mathrm{D}$,
using eqs.~(\ref{e2}), (\ref{e4}) and (\ref{e10}).
The pressure is calculated via eq.~(\ref{e2a}) and the surface potential from eq.~(\ref{e2}).
From the extrapolation of the surface-potential value at large
inter-plate separations,
we obtain the surface potential and the surface charge for CP and CC, respectively.}
The different pressure isotherms obey the inequality: $\Pi_{_{\rm CC}} \geq \Pi_{_{\rm CR}} \geq \Pi_{_{\rm CP}}$.
This is a general inequality that holds for charge regulations consisting of single-site dissociation process.
The log-log plot in Fig.~\ref{figure2} clearly shows the distinct $d^{-1/2}$ scaling for ${\rm CR_1}$,
confirming our analytical results, eqs.~(\ref{e11}) and (\ref{a3}).
An additional interesting observation can be made for the vanishing inter-plate separation, $d\to 0$.
The results for $\Pi^{(1)}_{_{\rm CR}}(d\to0)$ can be obtained from $\Pi_{_{\rm CC}}(d\to0)$ of eq.~(\ref{e5})
by substituting the surface charge, $\sigma(d)$, for the single-site process into the Gouy-Chapman length.
This shows a resemblance of the ${\rm CR}_1$ and CC processes, and gives some insight to the understanding of the different $\Pi^{(1)}_{_{\rm CR}}$ scalings.
The $\Pi^{(1)}_{_{\rm CR}}$ divergence is due to counter-ions that are bounded between the planes and neutralize the surface charge, as in the CC case.
However, the surface charge itself is not constant but decreases with $d$ as explained above. Namely,
some of the counter-ions adsorb onto the surface in order to neutralize it.
Therefore, less counter-ions are bounded between the planes and the entropic penalty is reduced, as compare to the CC case.
The CR scaling results differ substantially from the disjoining pressure
of the CC $(\Pi_{_{\rm CC}} \sim 1/d)$ as well as CP $(\Pi_{_{\rm CP}} \sim {\rm const})$ boundary conditions.
There is a fundamental difference between the three boundary conditions,
making it clear that the disjoining-pressure scaling for small separations for the CC and CP boundary conditions
{\it cannot} be obtained by any limiting behavior of the CR boundary condition.
In previous works on charge regulation{~\cite{Borkovec1,Borkovec2,notalways1,Chan,Chan1}},
based on the same dissociation model,
additional approximations were used, including linearization of the CR boundary condition or linearization of the PB equation.
In these works, it has been shown that CR can reduce to CC or CP in different limits {of the differential capacitance}.
In contrast, our results show that the three disjoining pressures, $\Pi_{_{\rm CR}}$, $\Pi_{_{\rm CC}}$ and $\Pi_{_{\rm CP}}$ scale differently
in the small $d$ limit (as in eqs.~(\ref{res1}) and~(\ref{res2})), and point out that the CR case
does not generally reduce to the CC or CP ones.
It is not their generalization but rather a third distinct case of its own merit.
\section{Multi-site process}
{The inadequacy of the presumed limiting nature of CC and CP boundary conditions is equally apparent when one considers
more complicated surface dissociation/association processes that involve several ionic species.
In fact, although the disjoining pressure in this case is also bound between $\Pi_{\rm CC}$ and $\Pi_{\rm CP}$, it has a different scaling law than in the single-site CR process.} This
shows that CR has a rich behavior that depends on the number of surface dissociation/assciation processes.
As an illustrative example, we consider a dissociable surface with two independent dissociation/association processes
referred to as ${\rm CR_2}$ and described by:
\begin{eqnarray}
\label{e12}
\nonumber& &{\rm A}_1 + {\rm B}_1^+ \rightleftarrows {\rm A_1B_1^+} \, , \\
& &{\rm A}_2 + {\rm B}_2^- \rightleftarrows {\rm A_2B_2^-} \, ,
\end{eqnarray}
where ${\rm A_{1,2}}$ are two different surface binding sites.
A specific example for two dissociation processes can be~\cite{healy}:
\begin{eqnarray}
\label{e13a}
\nonumber{\rm A_1 H} + {\rm H}^+ &\rightleftarrows& {\rm A_1H_2^+} \, , \\
{\rm A_2H} + {\rm OH}^- &\rightleftarrows& {\rm A_2^- + H_2O} \, .
\end{eqnarray}
The equilibrium condition yields
\begin{eqnarray}
\label{e13}
\phi_s = \frac{p \phi_b}{\phi_b + \mathrm{e}^{\,\alpha_1 + \psi_s}} - \frac{(1-p)\phi_b}{\phi_b + \mathrm{e}^{\,\alpha_2 - \psi_s}} \, ,
\end{eqnarray}
where $\alpha_{1,2}$ are two surface interaction parameters for dissociation/association of ${\rm B}_{1,2}$,
and $p = N_{1}/N$ is the surface fraction of A$_1$ sites with $N$ being the total number of sites and $N_{1}$ the number of the ${\rm A}_{1}$ sites.
For $p=1$ (or $p=0$), eq.~(\ref{e13}) reduces to eq.~(\ref{a2}) (or a similar equation for negative binding ions)
and the single-site process case (${\rm CR_1}$) is recovered.
Note that bulk electro-neutrality dictates the equality of the bulk concentration of ${\rm B}_{1}^+$ and ${\rm B}_{2}^-$.
Without loss of generality we can focus on the situation in which $\alpha_2 \to -\infty$, {\it i.e.}, strong
adsorption of the ${\rm B}_{2}^-$ ions, giving the approximate form of eq.~(\ref{e13}) (similar to eq.~(6) in Ref. \cite{Borkovec1}),
\begin{eqnarray}
\label{e14}
\phi_s \simeq \frac{p \phi_b}{\phi_b + \mathrm{e}^{\,\alpha_1 + \psi_s}} - (1-p) \, ,
\end{eqnarray}
such that the adsorption of ${\rm B}_{2}^-$ ion is similar to a constant surface charge that remains fixed.
We repeat the same steps as done above for the ${\rm CR_1}$ process in order
to derive the limiting form of the disjoining pressure for $d \to 0$, and obtain, to first order in $d$,
\begin{eqnarray}
\label{e15}
\nonumber \Pi^{(2)}_{_{\rm CR}} \!\!\!\! &\simeq& \!\!\!\!k_{\mathrm{B}}T n_b \frac{(m_0 - 1)^2 }{m_0} \Bigg[ 1 - d \cdot \frac{n_b a^2}{2}
\frac{ p(m_0 + 1)^2}{(1-p)(2p-1)m_0} \Bigg] \\
&\simeq& \Pi_0 - \Pi_1 d \, ,
\end{eqnarray}
where $m = m_0 + m_1d + \ldots$, and
\begin{eqnarray}
\label{e15a}
m_0 = \frac{2p - 1 }{1 - p}n_ba^3\mathrm{e}^{-\alpha} \, ,
\end{eqnarray}
is the first term in the expansion of $m=\exp(\psi_m)$.
This result is similar to the CP result as the disjoining pressure goes to a constant value for $d \to 0$,
but the first correction in ${\rm CR_2}$ is linear in $d$, unlike the first CP correction that scales as $d^2$.
This pressure expression is valid for $0.5 < p < 1$, while for $p \to 1/2$,
$\Pi^{(2)}_{_{\rm CR}} \sim d^{-1/2}$ as for the ${\rm CR_1}$ case.
For smaller $0< p < 0.5$, there are always some fixed surface charges and the pressure expression for $d\to0$
reduces to the one of CC, eq.~(\ref{e5}), with $|\sigma|/e = 1 - 2p$.
Note that these expressions hold in the limit $\alpha_2 \to -\infty$,
while for any finite $\alpha_2$, the pressure expression always has the same limiting behavior as eq.~(\ref{e15}),
$\Pi^{(2)}_{_{\rm CR}} \simeq C_0 - C_1 d$, but with different coefficients, $C_0$ and $C_1$.
\begin{figure
\centering
\includegraphics[scale=0.7]{fig3.eps}
\caption{ \textsf{ The dimensionless disjoining pressure, $\Pi$ [in units of $k_{\mathrm{B}}T/(4\pi\ell_\mathrm{B}\lambda_\mathrm{D}^2)$],
for the three different boundary conditions: constant charge (CC - dashed red line), constant potential (CP - dotted-dashed blue line)
and charge regulation for {\it two-site} dissociation process (${\rm CR_2}$ - solid black line).
The dotted-dashed thin (black) line indicates the non-zero slope of the ${\rm CR_2}$ at $d=0$,
and the parameters used are as in Fig.~\ref{figure2} with $p=0.7$ ($m_0 > 1$).
In the inset, we present the same disjoining pressures on a log-log plot, demonstrating their
scaling with the inter-plate distance, $d/\lambda_\mathrm{D}$.
}}
\label{figure3}
\end{figure}
{A typical pressure isotherm, $\Pi(d)$, is computed numerically for the ${\rm CR_2}$ process and is shown in Fig.~\ref{figure3}.
The calculation is exactly the same as for Fig.~\ref{figure2}, but instead of eq.~(\ref{e10}), we use eq.~(\ref{e14}),
which corresponds to the CR$_2$ boundary condition.
The log-log plot (inset) clearly shows that $\Pi(d)$ tends towards a constant value, $\Pi_0$, as $d\to 0$,
with a constant negative slope, $\Pi_1$ (dotted-dashed thin black line), as derived in eq.~(\ref{e15}).}
The same calculation can be performed for any {\it multi-site} dissociation processes ${\rm CR}_{n\geq 2}$.
It can be shown that
$\Pi^{(n\geq2)}_{_{\rm CR}}(d\to 0) \simeq C_0^{(n)} - C_1^{(n)} d$,
where $C_{0,1}^{(n)}$ are the two coefficients in the small $d$ expansion, whose value depends on $n\geq 2$.
The value of $m_0$ also depends on $n\geq 2$ and is determined by examining the vanishing $\phi_s$ limit
in the equilibrium condition [eq.~(\ref{e13}) for the ${\rm CR_2}$ dissociation process].
This pressure scaling is a result of the competition between the two (or more) dissociation/association processes of anions and cations.
Unlike the CC and ${\rm CR_1}$ cases, where counter-ions have to stay bounded between the planes to neutralize the surface charge,
in the multi-site process the planes are neutralized by the two (ot more) competing processes.
Therefore, no counter-ions remain between the plane and there is no entropic penalty.
\section{Conclusions}
In this Letter, we have shown that the CR boundary condition implies a much richer behavior than just an interpolation between
the limiting forms of the CC and CP boundary conditions.
Our conclusions are based on the full non-linear PB equation, as well as the non-linear form of
the charge regulation conditions.
They differ from previous claims that are based on linearization schemes{~\cite{Borkovec1,Borkovec2,notalways1,Chan,Chan1}}.
{We have shown that for both single-site (${\rm CR_1}$) and multi-site (${\rm CR_{n\geq2}}$) surface dissociation/association processes
the disjoining pressure is indeed bounded by the CC and CP limits,
while its scaling for small separations depends on the process type and, generally, is at odds with both the CC and CP limiting cases.}
This is clear from the different scaling of the single-site process (${\rm CR_1}$),
$\Pi^{(1)}_{_{\rm CR}}$ that scales as $d^{-1/2}$, while $\Pi_{_{\rm CC}}$ scales as $d^{-1}$ and $\Pi_{_{\rm CP}}$ tends to a constant.
We note that all considered boundary conditions lead to an identical (universal) separation scaling for large $d$.
The single-site case is more similar to the CC boundary condition as it diverges for small separations.
As was explained above, the pressure isotherm, $\Pi^{(1)}_{\rm CR}$,
can be obtained by substituting the surface charge density, $\sigma(d)$,
into the Gouy-Chapman length of the disjoining pressure expression, $\Pi_{_{\rm CC}}$ of eq.~(\ref{e5}),
for small $d$.
Furthermore, for multi-site dissociation processes, we have shown that $\Pi^{(n)}_{_{\rm CR}}$ ($n\geq 2$)
is similar to the CP case, as it tends towards a constant value for small separations, $\Pi^{(n)}_{_{\rm CR}} \simeq C_0^{(n)} - C_1^{(n)} d$.
Nevertheless, as is apparent from its negative slope at $d=0$, it differs from the CP case whose slope is zero.
In summary, the CR process is shown to be a distinct type of boundary condition with particular scaling
behavior,
and cannot be considered as a generalization of the CC and CP cases.
One should also keep in mind the fundamental difference between the CR single-site process and the multi-site ones.
\acknowledgments
{\bf Acknowledgements.~~~}
We thank F. Mugele for discussions that push-started this research
and to M. Biesheuvel, N. Boon, M. Borkovec, A. Cohen, Y. Nakayama, R. Netz, S. Safran, and especially R. van Roij
for useful comments and numerous suggestions.
This work was supported in part by the Israel
Science Foundation (ISF) under Grant No. 438/12,
the US-Israel Binational Science Foundation (BSF) under
Grant No. 2012/060.
R.P. would like to thank the Slovene research agency ARRS for support through Grant No. P1-0055.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,011 |
//
// Copyright (c) 20015-2017 Jingping Yu.
//
// Permission is hereby granted, free of charge, to any person obtaining a copy
// of this software and associated documentation files (the "Software"), to deal
// in the Software without restriction, including without limitation the rights
// to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
// copies of the Software, and to permit persons to whom the Software is
// furnished to do so, subject to the following conditions:
//
// The above copyright notice and this permission notice shall be included in
// all copies or substantial portions of the Software.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
// IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
// FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
// AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
// LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
// OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
// THE SOFTWARE.
using UnityEngine;
using System;
using System.IO;
using System.Collections;
using System.Collections.Generic;
using SQLite4Unity3d;
namespace Mistral.UniDialogue
{
/// <summary>
/// The types of the entries.
/// </summary>
public enum EntryType
{
Conversation,
Execution,
Condition,
Content
}
///Why are all the variables in capital? Because of SQLite4Unity3d.
/// <summary>
/// The Base Abstract Class for all DialogueEntries.
/// </summary>
public abstract class DialogueDBEntry
{
/// <summary>
/// The Primary ID Key for the Entry. If set to 0, it means End.
/// </summary>
/// <value>The ID.</value>
[PrimaryKey]
public int ID { get; set; }
/// <summary>
/// Gets the type of the entry.
/// </summary>
/// <returns>The entry type.</returns>
public abstract EntryType GetEntryType();
public DialogueDBEntry ()
{
ID = -1;
}
}
public class ConversationEntry : DialogueDBEntry
{
/// <summary>
/// The name of the Conversation.
/// </summary>
public string ConversationName { get; set; }
/// <summary>
/// The ID of the first Entry. Could be ExecutionEntry, ConditionEntry or ContentEntry.
/// </summary>
public int FirstEntryID { get; set; }
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ConversationEntry"/> class.
/// </summary>
/// <param name="cn">Cn.</param>
/// <param name="nid">Nid.</param>
public ConversationEntry (string cn, int nid) : base ()
{
ConversationName = cn;
FirstEntryID = nid;
}
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ConversationEntry"/> class.
/// </summary>
/// <param name="id">Identifier.</param>
/// <param name="cn">Cn.</param>
/// <param name="nid">Nid.</param>
public ConversationEntry (int id, string cn, int nid)
{
ID = id;
ConversationName = cn;
FirstEntryID = nid;
}
/// <summary>
/// Default Constructor
/// </summary>
public ConversationEntry () : base()
{
}
public override EntryType GetEntryType ()
{
return EntryType.Conversation;
}
}
public class ExecutionEntry : DialogueDBEntry
{
/// <summary>
/// The YCode to be Executed.
/// </summary>
/// <value>The execution code.</value>
public string ExecutionCode { get; set; }
/// <summary>
/// The ID of the Next Entry. Could be ExecutionEntry, ConditionEntry of ContentEntry.
/// </summary>
/// <value>The next entry I.</value>
public int NextEntryID { get; set; }
public ExecutionEntry (string ec, int nid) : base ()
{
ExecutionCode = ec;
NextEntryID = nid;
}
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ExecutionEntry"/> class.
/// </summary>
/// <param name="id">Identifier.</param>
/// <param name="ec">Ec.</param>
/// <param name="nid">Nid.</param>
public ExecutionEntry (int id, string ec, int nid)
{
ID = id;
ExecutionCode = ec;
NextEntryID = nid;
}
/// <summary>
/// Default Constructor
/// </summary>
public ExecutionEntry () : base()
{
}
public override EntryType GetEntryType ()
{
return EntryType.Execution;
}
}
public class ConditionEntry : DialogueDBEntry
{
/// <summary>
/// The Condition of Execution this Entry.
/// </summary>
/// <value>The condition code.</value>
public string ConditionCode { get; set; }
/// <summary>
/// The ID of the entry to enter when the condition is met.
/// </summary>
public int SuccessID { get; set; }
/// <summary>
/// If the condition is failed, move to the next ConditionEntry.
/// </summary>
public int NextConditionID { get; set; }
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ConditionEntry"/> class.
/// </summary>
/// <param name="cc">Cc.</param>
/// <param name="sid">Sid.</param>
/// <param name="nid">Nid.</param>
public ConditionEntry (string cc, int sid, int nid) : base ()
{
ConditionCode = cc;
SuccessID = sid;
NextConditionID = nid;
}
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ConditionEntry"/> class.
/// </summary>
/// <param name="id">Identifier.</param>
/// <param name="cc">Cc.</param>
/// <param name="sid">Sid.</param>
/// <param name="nid">Nid.</param>
public ConditionEntry (int id, string cc, int sid, int nid)
{
ID = id;
ConditionCode = cc;
SuccessID = sid;
NextConditionID = nid;
}
/// <summary>
/// Default Constructor
/// </summary>
public ConditionEntry () : base()
{
}
public override EntryType GetEntryType ()
{
return EntryType.Condition;
}
}
public class ContentEntry : DialogueDBEntry
{
/// <summary>
/// The Actors who say the content. In the case of multiple actors, split the actors using '#'.
/// </summary>
public string Actor { get; set; }
/// <summary>
/// The content being told.
/// </summary>
public string Content { get; set; }
/// <summary>
/// The ID of the next Entry. Could be ExecutionEntry, COnditionEntry or ContentEntry.
/// </summary>
public int NextEntryID { get; set; }
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ContentEntry"/> class.
/// </summary>
/// <param name="a">The alpha component.</param>
/// <param name="c">C.</param>
/// <param name="nid">Nid.</param>
public ContentEntry (string a, string c, int nid) : base ()
{
Actor = a;
Content = c;
NextEntryID = nid;
}
/// <summary>
/// Initializes a new instance of the <see cref="Mistral.UniDialogue.ContentEntry"/> class.
/// </summary>
/// <param name="id">Identifier.</param>
/// <param name="a">The alpha component.</param>
/// <param name="c">C.</param>
/// <param name="nid">Nid.</param>
public ContentEntry (int id, string a, string c, int nid)
{
ID = id;
Actor = a;
Content = c;
NextEntryID = nid;
}
/// <summary>
/// Default Constrcutor
/// </summary>
public ContentEntry () : base()
{
}
public override EntryType GetEntryType ()
{
return EntryType.Content;
}
}
/// <summary>
/// The Administrator of UniDialogue. In charge of creating and deleting database.
/// Singleton!
/// </summary>
public static class DialogueDBAdmin
{
#region Public Static Variables
public static string streamingPath
{
///DEBUG: Compiler Options are now toggled. With these options exist the MonoDevelop can't function appropriately ...
get
{
//#if UNITY_EDITOR
return @"Assets/StreamingAssets/";
//#elif UNITY_ANDROID
// return "jar:file://" + Application.dataPath + "!/assets/";
//#elif UNITY_IOS
// return Application.dataPath + "/Raw/";
//#else
// return Application.dataPath + "/StreamingAssets/";
//#endif
}
}
#endregion
#region Public Static Methods
/// <summary>
/// Delete the indicated database file.
/// </summary>
/// <returns><c>true</c>, if database was deleted, <c>false</c> otherwise.</returns>
/// <param name="_dbName">Db name.</param>
public static bool DeleteDatabase (string _dbName)
{
if (File.Exists(streamingPath + _dbName))
{
File.Delete(streamingPath + _dbName);
Debug.Log("The Indicated Database has been successfully Removed! ");
return true;
}
else
{
Debug.Log("Deleting Database Failed! File not Found! ");
return false;
}
}
/// <summary>
/// Creates a new empty database. Do not use the database immediately after this function!
/// </summary>
/// <param name="_dbName">Db name.</param>
public static bool CreateDatabase (string _dbName)
{
if (File.Exists(streamingPath + _dbName))
{
Debug.Log("Unable to Create the Database! A file with the same name has already been created! ");
return false;
}
else
{
SQLiteConnection _connection = new SQLiteConnection(streamingPath + _dbName, SQLiteOpenFlags.ReadWrite | SQLiteOpenFlags.Create);
return true;
}
}
/// <summary>
/// Create a new UniDilaougeDatabase.
/// </summary>
/// <param name="_dbName">Db name.</param>
public static bool CreateUniDialogueDB (string _dbName)
{
if (!CreateDatabase(_dbName))
return false;
InitializeUniDialogueDB(_dbName);
return true;
}
/// <summary>
/// Initialize a database into a UniDialogue Database.
/// Warning: Won't Drop tables other than the ones used by UniDialogue. And meanwhile all the tables that UniDialogue uses
/// will be dropped. This operation is not revorable.
/// </summary>
/// <param name="_dbName">Db name.</param>
public static bool InitializeUniDialogueDB (string _dbName)
{
if (File.Exists(streamingPath + _dbName))
{
SQLiteConnection _connection = new SQLiteConnection(streamingPath + _dbName, SQLiteOpenFlags.ReadWrite);
///Drop Old Tables.
_connection.DropTable<ConversationEntry>();
_connection.DropTable<ContentEntry>();
_connection.DropTable<ConditionEntry>();
_connection.DropTable<ExecutionEntry>();
///Create New Tables
_connection.CreateTable<ConversationEntry>();
_connection.CreateTable<ContentEntry>();
_connection.CreateTable<ConditionEntry>();
_connection.CreateTable<ExecutionEntry>();
return true;
}
else
{
Debug.Log("Unable to Initialized the database! The indicated file is not found! ");
return false;
}
}
/// <summary>
/// Copy the Database to the indicated path.
/// </summary>
/// <returns><c>true</c>, if up database was backed, <c>false</c> otherwise.</returns>
/// <param name="_originDB">Origin D.</param>
/// <param name="_targetPath">Target path.</param>
public static bool BackUpDatabase (string _originDB, string _targetPath)
{
if (!File.Exists(streamingPath + _originDB))
{
Debug.Log("The dabatase to backup does not exist. ");
return false;
}
if (File.Exists(_targetPath + _originDB))
{
File.Delete(_targetPath + _originDB);
}
File.Copy(streamingPath + _originDB, _targetPath + _originDB);
return true;
}
#endregion
}
/// <summary>
/// The Dialogue Database Manager. Used Based on a Connection.
/// </summary>
public class DialogueDBManager
{
#region Public Variables
/// <summary>
/// The Connection to the UniDialogueDB.
/// </summary>
public DialogueDBConnection _connection;
#endregion
#region Public Attributes
public int NextContentID
{
get
{
return nextContentID;
}
private set
{
nextContentID = value;
}
}
public int NextExecutionID
{
get
{
return nextExecutionID;
}
private set
{
nextExecutionID = value;
}
}
public int NextConditionID
{
get
{
return nextConditionID;
}
private set
{
nextConditionID = value;
}
}
public int NextConversationID
{
get
{
return nextConversationID;
}
private set
{
nextConversationID = value;
}
}
#endregion
#region Private Variables
private int nextContentID;
private int nextExecutionID;
private int nextConditionID;
private int nextConversationID;
#endregion
#region Constructors
public DialogueDBManager ()
{
}
public DialogueDBManager (DialogueDBConnection con)
{
_connection = con;
///Get the Rows with maximum ID.
ConversationEntry _maxConversationEntry = _connection.Query<ConversationEntry>("SELECT *, MAX(ID) FROM ConversationEntry")[0];
ContentEntry _maxContentEntry = _connection.Query<ContentEntry>("SELECT *, MAX(ID) FROM ContentEntry")[0];
ExecutionEntry _maxExecutionEntry = _connection.Query<ExecutionEntry>("SELECT *, MAX(ID) FROM ExecutionEntry")[0];
ConditionEntry _maxConditionEntry = _connection.Query<ConditionEntry>("SELECT *, MAX(ID) FROM ConditionEntry")[0];
///And then set the IDs to the Manager. If a table is empty, then set the start ID.
if (_maxConversationEntry.ID != 0)
NextConversationID = _maxConversationEntry.ID + 1;
else
NextConversationID = 1;
if (_maxContentEntry.ID != 0)
NextContentID = _maxContentEntry.ID + 10;
else
NextContentID = 1;
if (_maxExecutionEntry.ID != 0)
NextExecutionID = _maxExecutionEntry.ID + 10;
else
NextExecutionID = 2;
if (_maxConditionEntry.ID != 0)
NextConditionID = _maxConditionEntry.ID + 10;
else
NextConditionID = 3;
}
#endregion
#region Insert Methods
/// <summary>
/// Inserts an entry into the database.
/// </summary>
/// <param name="entry">Entry.</param>
public void InsertEntry (DialogueDBEntry entry)
{
EntryType eType = entry.GetEntryType();
int newID;
switch (eType)
{
case EntryType.Conversation:
newID = NextConversationID;
break;
case EntryType.Condition:
newID = NextConditionID;
break;
case EntryType.Content:
newID = NextContentID;
break;
case EntryType.Execution:
newID = NextExecutionID;
break;
default:
newID = 0;
break;
}
entry.ID = newID;
try
{
_connection.Insert(entry);
}
catch (SQLiteException sex)
{
Debug.Log("Entry is not inserted. An error has occured. Check the constraints of the database scheme. "
+ "The Result is: " + sex.Result
+ "The Message is: " + sex.Message);
throw;
}
switch (eType)
{
case EntryType.Conversation:
NextConversationID++;
break;
case EntryType.Condition:
NextConditionID += 10;
break;
case EntryType.Content:
NextContentID += 10;
break;
case EntryType.Execution:
NextExecutionID += 10;
break;
default:
break;
}
}
//These Methods are not welcomed to use ... However in test mode or some situations they could be pretty handy ...
public void InsertConversationEntry (string cname, int startID)
{
ConversationEntry toInsert = new ConversationEntry(NextConversationID, cname, startID);
try
{
_connection.Insert(toInsert);
}
catch (SQLiteException sex)
{
Debug.Log("Entry is not inserted. An error has occured. Check the constraints of the database scheme. "
+ "The Result is: " + sex.Result
+ "The Message is: " + sex.Message);
throw;
}
NextConversationID++;
}
public void InsertContentEntry (string aname, string cname, int nextID)
{
ContentEntry toInsert = new ContentEntry(NextContentID, aname, cname, nextID);
try
{
_connection.Insert(toInsert);
}
catch (SQLiteException sex)
{
Debug.Log("Entry is not inserted. An error has occured. Check the constraints of the database scheme. "
+ "The Result is: " + sex.Result
+ "The Message is: " + sex.Message);
throw;
}
NextContentID += 10;
}
public void InsertExecutionEntry (string ycode, int nextID)
{
ExecutionEntry toInsert = new ExecutionEntry(NextExecutionID, ycode, nextID);
try
{
_connection.Insert(toInsert);
}
catch (SQLiteException sex)
{
Debug.Log("Entry is not inserted. An error has occured. Check the constraints of the database scheme. "
+ "The Result is: " + sex.Result
+ "The Message is: " + sex.Message);
throw;
}
NextExecutionID += 10;
}
public void InsertConditionEntry (string ycode, int sucID, int nextID)
{
ConditionEntry toInsert = new ConditionEntry(NextConditionID, ycode, sucID, nextID);
try
{
_connection.Insert(toInsert);
}
catch (SQLiteException sex)
{
Debug.Log("Entry is not inserted. An error has occured. Check the constraints of the database scheme. "
+ "The Result is: " + sex.Result
+ "The Message is: " + sex.Message);
throw;
}
NextConditionID += 10;
}
//These Methods are not welcomed to use ... However in test mode or some situations they could be pretty handy ...
#endregion
}
/// <summary>
/// The Connection to a UniDialogueDatabase.
/// Wraps the SQLiteConnection to make sure that the low-level API is not exposed.
/// Always be ready to change the model level.
/// </summary>
public class DialogueDBConnection
{
#region Public Variables
#endregion
#region Private Variables
/// <summary>
/// The Connection to the SQLite DB.
/// </summary>
private static SQLiteConnection _connection = null;
#endregion
#region Constructors
public DialogueDBConnection ()
{
_connection = null;
}
public DialogueDBConnection (string _dbName, SQLiteOpenFlags authentication)
{
EstablishConnection(_dbName, authentication);
}
#endregion
#region Connection Methods
/// <summary>
/// Establishes a new connection to the indicated database.
/// </summary>
/// <param name="_dbName">Db name.</param>
/// <param name="authentication">Authentication.</param>
public bool EstablishConnection (string _dbName, SQLiteOpenFlags authentication)
{
_connection = null;
if (!File.Exists(DialogueDBAdmin.streamingPath + _dbName))
{
Debug.Log("Failed to Establish the Connection! The indicated Database does not exist! ");
return false;
}
_connection = new SQLiteConnection(DialogueDBAdmin.streamingPath + _dbName, authentication);
Debug.Log("The Connection has been established successfully! ");
return true;
}
/// <summary>
/// Disconnects from the current database.
/// </summary>
public void Disconnect ()
{
_connection = null;
Debug.Log("Disconnected! ");
}
#endregion
#region SQL Methods
public List<T> Query<T> (string query, params object[] args) where T : new()
{
return _connection.Query<T> (query, args);
}
/// <summary>
/// Safe Insert will never cause an interruption. Mostly used during runtime.
/// </summary>
/// <returns>The insert.</returns>
/// <param name="obj">Object.</param>
/// <typeparam name="T">The 1st type parameter.</typeparam>
public int SafeInsert (object obj)
{
try
{
int x = _connection.Insert(obj);
return x;
}
catch (SQLiteException sex)
{
Debug.Log("An Error is Occured While Trying to Insert Data. However in Safe Mode the Error is Ignored. "
+ "The result is: " + sex.Result + "The Message is: " + sex.Message
);
return 0;
}
}
/// <summary>
/// The same with SQLite4Unity Insert Function.
/// Possibly Trigger an error and stops a MonoBehaviour.
/// </summary>
/// <param name="obj">Object.</param>
public int Insert (object obj)
{
return _connection.Insert(obj);
}
#endregion
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,880 |
\section{Introduction}
\label{sec:intro}
The authors of~\cite{C} allege to have shown that the conclusions
of~\cite{HARDY} regarding the inconsistency of Time Asymmetric Quantum
Theory (TAQT) with quantum mechanics are false. In this reply, we will
show that the arguments of~\cite{C} are missing essential aspects
of~\cite{HARDY}, and that therefore the conclusions of~\cite{HARDY}
still stand.
The most important claims of~\cite{C} are the following:
\begin{enumerate}
\item[{\bf 1}.] There are many examples of TAQT, and the present author
has inadvertently constructed another one.
\item[{\bf 2}.] The flaws of the Quantum Arrow of Time (QAT)
pointed out in~\cite{HARDY} are actually not flaws, because the original
derivation of the QAT was misquoted from its source~\cite{JMP95}.
\item[{\bf 3}.] The crucial argument of~\cite{HARDY} regarding the
exponential blowup of the test functions $\widehat{\varphi}^{\pm}(z)$
does not prevent $\widehat{\varphi}^{\pm}(z)$ from being of Hardy class.
\end{enumerate}
As we shall see, all these claims do not stand close scrutiny. In order to
show why, in Sec.~\ref{sec:stamet} we will outline the method to
construct rigged Hilbert spaces in quantum mechanics based on the
theory of distributions~\cite{GELFAND}. We shall refer to this method
as the ``standard method'' and show that the resulting rigged Hilbert spaces
are not of Hardy class. We shall also explain the
meaning of the exponential blowup of $\widehat{\varphi}^{\pm}(z)$ and why
it implies that the spaces of test functions are not of Hardy class. In
Sec.~\ref{sec:TAQTvsSQM}, we briefly outline the method to introduce
rigged Hilbert spaces of Hardy class in TAQT and compare such method with
the ``standard method.'' It will then be apparent that using the
method of TAQT, one can introduce any arbitrary rigged Hilbert space
for the Gamow states. In order to address claim~{\bf 2}, we show (again)
in Sec.~\ref{seec:QAT} that no matter how one introduces it, the Quantum
Arrow of Time has little to do with the actual time evolution of a quantum
system. To address claim~{\bf 3}, in Sec.~\ref{sec:clasins} we use classic
results of Paley and Wiener and of Gelfand and Shilov to show that the
``standard method'' of dealing with the Lippmann-Schwinger equation leads to
rigged Hilbert spaces that are {\it not} of Hardy class. Section~\ref{sec:con}
concludes that the arguments of~\cite{HARDY} still stand.
\section{The ``standard method''}
\label{sec:stamet}
In this section, we illustrate the main features of the ``standard method''
to construct rigged Hilbert spaces in quantum
mechanics~\cite{RELBO}. Such ``standard
method'' is based on the theory of distributions~\cite{GELFAND}. For the sake
of clarity, we shall use the spherical shell potential of height $V_0$,
\begin{equation}
V(\vec{x})=V(r)=\left\{ \begin{array}{ll}
0 &0<r<a \\
V_0 &a<r<b \\
0 &b<r<\infty \, .
\end{array}
\right.
\label{potential}
\end{equation}
For $l=0$, the Hamiltonian acts as (we take $\hbar ^2/2m=1$)
\begin{equation}
H = -\frac{\rmd ^2}{\rmd r^2} + V(r) \, .
\label{doh}
\end{equation}
The regular solution is
\begin{equation}
\chi (r;E)=\left\{ \begin{array}{lll}
\sin (\sqrt{E \,} r) \quad &0<r<a \\
{\cal J}_1(E)\rme ^{\rmi \sqrt{E-V_0 \,} r}
+{\cal J}_2(E)\rme ^{-\rmi \sqrt{E-V_0 \,} r}
\quad &a<r<b \\
{\cal J}_3(E) \rme ^{\rmi \sqrt{E \,} r}
+{\cal J}_4(E)\rme ^{-\rmi \sqrt{E \,} r}
\quad &b<r<\infty \, .
\end{array}
\right.
\label{chi}
\end{equation}
The Jost functions and the $S$ matrix are given by
\begin{equation}
{\cal J}_+(E)=-2\rmi {\cal J}_4(E) \, , \quad
{\cal J}_-(E)=2\rmi {\cal J}_3(E) \, ,
\label{josfuc1}
\end{equation}
\begin{equation}
S(E) =\frac{{\cal J}_-(E)}{{\cal J}_+(E)} \, .
\label{smatrix1}
\end{equation}
The solutions of the Lippmann-Schwinger equation can be written as
\begin{equation}
\langle r|E^{\pm}\rangle \equiv
\chi ^{\pm}(r;E)= \sqrt{\frac{1}{\pi}
\frac{1}{\sqrt{E \,}\,}\,} \,
\frac{\chi (r;E)}{{\cal J}_{\pm}(E)} \, .
\label{LSdnesb}
\end{equation}
When $V$ tends to zero, these eigensolutions tend to the ``free''
eigensolution:
\begin{equation}
\langle r|E\rangle \equiv
\chi _0(r;E)= \sqrt{\frac{1}{\pi}
\frac{1}{\sqrt{E \,}\,}\,} \, \sin (\sqrt{E\,}r) \, .
\label{LSdnesb0}
\end{equation}
These eigenfunctions are delta-normalized and therefore their associated
unitary operators,
\begin{equation}
(U_{\pm}f)(E)=\int_0^{\infty}\rmd r \,
\overline{\chi ^{\pm}(r;E)}\, f(r) \equiv
\widehat{f}_{\pm}(E) \, , \quad E\geq 0 \, ,
\label{Upm}
\end{equation}
\begin{equation}
(U_0f)(E)=\int_0^{\infty}\rmd r \,
\overline{\chi _0(r;E)}\, f(r) \equiv
\widehat{f}_0(E) \, , \quad E\geq 0 \, ,
\label{Upm0}
\end{equation}
transform from $L^2([0,\infty),\rmd r)$ onto $L^2([0,\infty),\rmd E)$.
The Lippmann-Schwinger and the ``free'' eigenfunctions can be analytically
continued from the scattering spectrum into the whole complex plane. We shall
denote such analytically continued eigenfunctions by $\chi ^{\pm}(r;z)$
and $\chi _0(r;z)$. Whenever they exist, the analytic continuations
of~(\ref{Upm}) and (\ref{Upm0}) are denoted by
\begin{equation}
\widehat{f}_{\pm}(z)=\int_0^{\infty}\rmd r \,
\overline{\chi ^{\pm}(r;\overline{z})}\, f(r) \, ,
\label{Upmac}
\end{equation}
\begin{equation}
\widehat{f}_{0}(z)=\int_0^{\infty}\rmd r \,
\overline{\chi _0(r;\overline{z})}\, f(r) \, ,
\label{Upm0ac}
\end{equation}
where here and in the following $z$ belongs to a two-sheeted Riemann surface.
The resonant energies are given by the poles $z_n$ of the $S$ matrix, and their
associated Gamow states are
\begin{equation}
u(r;z_n) = N_n\left\{ \begin{array}{ll}
\frac{1}{{\mathcal J}_3(z_n)}\sin(\sqrt{z_n\,}r) &0<r<a \\ [1ex]
\frac{{\mathcal J}_1(z_n)}{{\mathcal J}_3(z_n)}
\rme ^{\rmi \sqrt{z_n-V_0\,}r}
+\frac{{\mathcal J}_2(z_n)}{{\mathcal J}_3(z_n)}
\rme ^{-\rmi \sqrt{z_n-V_0\,}r}
&a<r<b
\\ [1ex]
\rme ^{\rmi \sqrt{z_n\,}r} &b<r<\infty \, ,
\end{array}
\right.
\label{dgv0p}
\end{equation}
where $N_n$ is a normalization factor.
The theory of distributions~\cite{GELFAND} says that a test function
$\varphi (r)$ on which
a distribution $d(r)$ acts is such that the following
integral is finite:\footnote{In quantum mechanics, we need to impose a few
more requirements, but we will not need to go into such details here.}
\begin{equation}
\langle \varphi|d\rangle \equiv
\int_0^{\infty}\rmd r \, \overline{\varphi (r)} d(r) <\infty \, ,
\label{basic}
\end{equation}
where $\langle \varphi |d\rangle$ represents the action of the
functional $|d\rangle$ on the test function $\varphi$. With some variations,
this is the ``standard method'' followed
by~\cite{SUDARSHAN,BOLLINI,FP02,JPA02,IJTP03,JPA04,EJP05,LS1,LS2} to introduce
spaces of test functions in quantum mechanics. Thus, contrary to what the
authors of~\cite{C} assert, the method followed by the present author runs
(somewhat) parallel to~\cite{BOLLINI}, not to~TAQT.
In order to use~(\ref{basic}) to construct the rigged Hilbert
spaces for the analytically continued Lippmann-Schwinger eigenfunctions and
for the Gamow states, we need to obtain the growth of $\chi ^{\pm}(r;z)$,
$\chi _0(r;z)$ and $u(r;z_n)$. Because the regular solution blows up
exponentially~\cite{TAYLOR},
\begin{equation}
\left| \chi (r; z)\right| \leq C \,
\frac{\left|z\right|^{1/2}r}{1+\left|z\right|^{1/2}r} \,
\rme ^{|{\rm Im}\sqrt{z\,}|r} \, ,
\label{boundrs}
\end{equation}
the growth of the eigenfunctions~(\ref{LSdnesb}), (\ref{LSdnesb0}) and
(\ref{dgv0p}) blows up exponentially:
\begin{equation}
|\chi ^{\pm}(r;z)| \leq C \, \frac{1}{{\cal J}_{\pm}(z)} \,
\frac{\left|z\right|^{1/4}r}{1+\left|z\right|^{1/2}r} \,
\rme ^{|{\rm Im}\sqrt{z\,}|r} \, ,
\label{boundpms}
\end{equation}
\begin{equation}
|\chi _0(r;z)| \leq C \,
\frac{\left|z\right|^{1/4}r}{1+\left|z\right|^{1/2}r} \,
\rme ^{|{\rm Im}\sqrt{z\,}|r} \, ,
\end{equation}
\begin{equation}
|u(r;z_n)| \leq C_n \,
\frac{\left|z _n\right|^{1/2}r}{1+\left|z_n \right|^{1/2}r}
\rme ^{|{\rm Im}\sqrt{z_n\,}|r} \, .
\end{equation}
When we plug this exponential blowup into the basic requirement~(\ref{basic})
of the ``standard method,'' we see that the test functions on which those
distributions act must fall off at least exponentially.
By using the Gelfand-Shilov theory of $M$ an $\Omega$
functions~\cite{GELFAND}, it was shown in~\cite{LS2} that when $a$ and $b$
are positive real numbers satisfying
\begin{equation}
\frac{1}{a}+\frac{1}{b} = 1 \, ,
\label{ab}
\end{equation}
and when $\varphi ^+(r)$ is an infinitely differentiable function
whose tails fall off like $\rme ^{-r^a/a}$, then $\varphi ^+(z)$ grows
like $\rme ^{|{\rm Im}(\sqrt{z})|^b/b}$ in the infinite arc of the lower
half-plane of the Riemann surface:
\begin{equation}
\hskip-1cm {\rm If} \ |\varphi ^+(r)| < C \rme ^{-\frac{\, r^a}{a}}
\ {\rm as} \ r \to \infty , \ {\rm then} \
|\widehat{\varphi}^+(z)| \leq
C \rme ^{\frac{\, |{\rm Im}(\sqrt{z})|^b}{b}} \
{\rm as} \ |z| \to \infty \,.
\label{blowupfex}
\end{equation}
It was shown in~\cite{HARDY} that when $\varphi ^+(r) \in C_0^{\infty}$,
$\widehat{\varphi}^+(z)$ blows up exponentially in the infinite arc of
the lower half-plane of the Riemann surface:
\begin{equation}
{\rm If} \ |\varphi ^+(r)| = 0 \ {\rm when} \ r>A , \
{\rm then} \ |\widehat{\varphi}^+(z)| \leq C
\rme ^{A|{\rm Im}\sqrt{z\,}|}
\ {\rm as} \ |z| \to \infty \, .
\label{grofainf}
\end{equation}
From the above estimates, we concluded in~\cite{HARDY}
that the $\varphi ^+$'s obtained from the ``standard method'' cannot be
Hardy functions, since $\widehat{\varphi}^+(z)$ does not tend to zero as
$|z|$ tends to infinity.
The authors of~\cite{C} argue that one cannot draw any conclusion
on the limit $|z|\to \infty$ from estimates such as~(\ref{blowupfex}) or
(\ref{grofainf}), and therefore they conclude that nothing
prevents $\widehat{\varphi}^+(z)$ from tending to zero and therefore from
being Hardy functions. Their conclusion is not true, because their argument
does not take the nature of~(\ref{blowupfex}) and
(\ref{grofainf}) into account. After we explain the meaning of those estimates,
it will be clear why they prevent $\widehat{\varphi}^{\pm}(z)$ from tending
to zero in any infinite arc of the Riemann surface.
In order to understand what~(\ref{blowupfex}) and
(\ref{grofainf}) mean, we start with the simple sine function
$\sin (\sqrt{E\, }r)$. When $E\geq 0$, the sine function oscillates between
$+1$ and $-1$:
\begin{equation}
|\sin (\sqrt{E\, }r)| \leq 1 \, , \quad E\geq 0 \, .
\end{equation}
As $E$ tends to infinity, such oscillatory behavior remains, and in such limit
the sine function does not tend to zero. When we analytically
continue the sine function,
\begin{equation}
\sin (\sqrt{z\, }r) \, ,
\label{sineac}
\end{equation}
the oscillations are bounded by
\begin{equation}
|\sin (\sqrt{z\, }r)| \leq C \,
\frac{\left|z\right|^{1/2}r}{1+\left|z\right|^{1/2}r} \,
\rme ^{|{\rm Im}\sqrt{z\,}|r} \, .
\label{sineabv}
\end{equation}
Thus, as $|z|$ tends to infinity, $\sin (\sqrt{z\,}r)$ oscillates wildly, and
the magnitude of its oscillation is tightly bounded by the exponential
function. It is certain that as $|z|$ tends to infinity, $\sin (\sqrt{z\,}r)$
does not tend to zero, even though the function vanishes when
$\sqrt{z\, }r = \pm n \pi$, $n=0,1,\ldots$
It just happens that the solutions of the Lippmann-Schwinger equation follow
the same pattern. When $E$ is positive, the eigensolutions are oscillatory
and bounded by
\begin{equation}
|\chi ^{\pm}(r;E)| \leq C \, \frac{1}{{\cal J}_{\pm}(E)} \,
\frac{\left|E\right|^{1/4}r}{1+\left|E\right|^{1/2}r} \, .
\end{equation}
When the energy is complex, their oscillations get wild and are bounded
by Eq.~(\ref{boundpms}).\footnote{The points at which
${\cal J}_{\pm}(z)=0$ do not affect the essence of the argument.} Thus, the
analytic continuations of the Lippmann-Schwinger eigenfunctions
oscillate wildly, and the magnitude of their oscillation is tightly bounded
by an exponential function (multiplied by factors that don't cancel the
exponential blowup when $|z|\to \infty$).
Because in Eqs.~(\ref{Upmac}) and (\ref{Upm0ac}) we are integrating
over $r$, the exponentially-bounded oscillations of
$\chi ^{\pm}(r;z)$ get transmitted into $\widehat{\varphi}^{\pm}(z)$. The
estimates~(\ref{blowupfex}) and (\ref{grofainf}) bound the oscillation
of the test functions of the ``standard method,'' except for
factors that don't cancel the exponential blowup. It is the
exponentially-bounded oscillations of $\widehat{\varphi}^{\pm}(z)$ what
prevent $\widehat{\varphi}^{\pm}(z)$ from tending to
zero in any infinite arc of the Riemann surface and therefore from being
of Hardy class.
A somewhat simpler way to understand the above estimates is by looking at
the ``free'' incoming and outgoing wave functions
${\varphi}^{\rm in}$ and $\varphi ^{\rm out}$. Because
in the energy representation such wave functions are the same as the
``in'' and ``out'' wave functions,
\begin{equation}
\widehat{\varphi}^{\rm in}(E) = \langle E|{\varphi}^{\rm in}\rangle
=\langle ^+E|{\varphi}^{+}\rangle = \widehat{\varphi}^{+}(E)\, ,
\label{asstrco1}
\end{equation}
\begin{equation}
\widehat{\varphi}^{\rm out}(E) = \langle E|{\varphi}^{\rm out}\rangle
=\langle ^-E|{\varphi}^{-}\rangle =\widehat{\varphi}^{-}(E) \, ,
\label{asstrco2}
\end{equation}
in TAQT the analytic continuation of
$\widehat{\varphi}^{\rm in}(E)$ and $\widehat{\varphi}^{\rm out}(E)$ are also
of Hardy class. Since
\begin{equation}
\widehat{\varphi}^{\rm in, out}(z) =\int_0^{\infty}\rmd r \,
\frac{1}{\sqrt{\pi}} \frac{1}{z^{1/4}} \,
\sin (\sqrt{z}\,r) \,\varphi ^{\rm in, out}(r) \, ,
\label{Upm0in}
\end{equation}
it is evident that the exponential blowup~(\ref{sineabv}) of
$\sin (\sqrt{z\,}r)$ will prevent $\widehat{\varphi}^{\rm in, out}(z)$ from
tending to zero as $|z| \to \infty$ in any half-plane of the Riemann
surface. Thus, $\widehat{\varphi}^{\rm in, out}(z)$ are not of Hardy class,
contrary to TAQT.
Strictly speaking, the bounds~(\ref{blowupfex}) and (\ref{grofainf}) are
not the tightest ones. We should include polynomial corrections, see
Eq.~(B.15) in~\cite{LS2}, and the effect of
$\frac{\left|z\right|^{1/4}r}{1+\left|z\right|^{1/2}r}$ and
$\frac{1}{{\cal J}_{\pm}(z)}$ to obtain the tightest bounds. We shall not
obtain those corrections here, because they do not cancel
the exponential blowup at infinity, and because in this reply we shall
use instead other classic bounds, see Sec.~\ref{sec:clasins}.
Let us summarize this section. In standard quantum mechanics, once the
Lippmann-Schwinger equation is solved, the properties of
$\widehat{\varphi}^{\pm}(z)$ are already determined by
Eqs.~(\ref{Upmac}) and (\ref{Upm0ac}), and there
is no room for any extra assumption on their properties. This means, in
particular, that the Hardy axiom cannot be
simply assumed. Rather, the Hardy axiom must be proved using
Eqs.~(\ref{Upmac}) and (\ref{Upm0ac}).\footnote{This is what in~\cite{HARDY}
it was meant by the assertion that the Hardy axiom is not a matter of
assumption but a matter of proof.} It simply happens that the
``standard method'' yields $\widehat{\varphi}^{\pm}(z)$
and $\widehat{\varphi}^{\rm in,out}(z)$ that oscillate wildly. Because these
oscillations are bounded by exponential functions, $\widehat{\varphi}^{\pm}(z)$
and $\widehat{\varphi}^{\rm in,out}(z)$ do not tend to zero as $|z|$ tends
to infinity in any half-plane of the Riemann surface---hence they are not of
Hardy class.
\section{TAQT vs.~the ``standard method''}
\label{sec:TAQTvsSQM}
In TAQT, one doesn't solve the Lippmann-Schwinger equation in order to
afterward obtain the properties of $\widehat{\varphi}^{\pm}(z)$
using Eq.~(\ref{Upmac}). Instead, one transforms into the
energy representation (using $U_{\pm}$ in our example) and then imposes
the Hardy axiom. If ${\cal H}_{\pm}^2$ denotes the
spaces of Hardy functions from above ($+$) and below ($-$), $\cal S$
denotes the Schwartz space, and $\tilde{\Phi}_{\pm}$ denote their intersection
restricted to the positive real line,
\begin{equation}
\tilde{\Phi}_{\pm} = {\cal H}_{\pm}^2\cap {\cal S}|_{{\mathbb R}^+} \, ,
\end{equation}
then the Hardy axiom states that the functions $\widehat{\varphi}^{\pm}(z)$
belong to $\tilde{\Phi}_{\mp}$:
\begin{equation}
\widehat{\varphi}^{\pm}(z) \in \tilde{\Phi}_{\mp} \, .
\label{sssusm}
\end{equation}
This means that in the position representation, the Gamow
states and the analytic continuation of the Lippmann-Schwinger eigenfunctions
act on the following spaces:
\begin{equation}
\Phi _{{\rm BG}\mp}= U_{\pm}^{-1} \tilde{\Phi}_{\mp} \, .
\label{BGchoice}
\end{equation}
It is obvious that the choices~(\ref{sssusm})-(\ref{BGchoice}) are
arbitrary. One may as well choose another dense subset of
$L^2([0,\infty ),\rmd E)$ with different properties
and obtain a different space of test functions for the Gamow states. What is
more, $\Phi _{{\rm BG}\pm}$ are different from the spaces of test
functions obtained through the ``standard method,'' because the
functions $\widehat{\varphi}^{\pm}(z)$ of the ``standard method'' are not
of Hardy class.
The authors of~\cite{C} claim that the present author has inadvertently
constructed an example of TAQT. That such is not the case can be seen not
only from the differences between the ``standard method'' and the method used
in TAQT to introduce rigged Hilbert spaces, but also from the
outcomes. For example, whereas in the position representation the
``standard method'' calls for just {\it one} rigged Hilbert space for the Gamow
states and for the analytically continued Lippmann-Schwinger
eigenfunctions~\cite{LS2}, TAQT uses {\it two} rigged Hilbert spaces
\begin{equation}
\Phi _{{\rm BG}\pm}
\subset L^2([0,\infty ),\rmd r) \subset
\Phi _{{\rm BG}\pm}^{\times} \,.
\label{BGchoicetwo}
\end{equation}
One of the rigged Hilbert spaces is used for the ``in'' solutions and for
the anti-resonant states, whereas the other one is used for the ``out''
solutions and for the resonant states. Another difference is that in TAQT, the
solutions of the
Lippmann-Schwinger equation for scattering energies have a time asymmetric
evolution~\cite{BLUNDER}, whereas the ``standard method'' yields that
such time evolution runs from $t=-\infty$ to $t=+\infty$,
see~\cite{LS1}. Incidentally, this is an instance where TAQT differs not only
mathematically but also physically from standard quantum mechanics, because in
standard scattering theory, the time evolution of a scattering process goes
from the asymptotically remote past ($t \to -\infty$) to the asymptotically far
future ($t \to +\infty$). This is not so in TAQT~\cite{BLUNDER}.
It seems hardly necessary to clarify what the present author means by
``standard quantum mechanics.'' Standard quantum mechanics means the
Schr\"odinger
equation, and standard scattering theory means the Lippmann-Schwinger
equation. In standard quantum mechanics, one assumes that these equations
describe the physics and then solves them. Because of the scattering and
resonant spectra, their solutions lie within rigged Hilbert spaces. The
construction of such rigged Hilbert spaces follows by application of the
``standard method.'' By contrast, TAQT simply assumes that the solutions of
the Schr\"odinger and the Lippmann-Schwinger equations comply with the Hardy
axiom, without ever showing that the actual solutions of those equations
comply with such axiom.
It was claimed in~\cite{HARDY} that there is no example of TAQT. The
authors of~\cite{C} dispute such claim and assert that there are
many examples. The present author disagrees with their assertion, because
{\it assuming} that for a large class of potentials the solutions of the
Lippmann-Schwinger equation comply with the Hardy axiom is not the same as
having an example where it is shown that the {\it actual} solutions of the
Lippmann-Schwinger equation comply with the Hardy axiom. In fact,
to the best of the present author's knowledge, no advocate of TAQT has ever
used Eq.~(\ref{Upmac}) to discuss the analytic properties of
$\widehat{\varphi}^{\pm}(E) = \langle ^{\pm}E|\varphi ^{\pm}\rangle$ in
terms of the actual solutions $\chi^{\pm}(r;E)$ of the Lippmann-Schwinger
equation.
The authors of~\cite{C} inadvertently acknowledge that there is no example
of TAQT when they say that they still need ``{\it to identify the
form and properties}'' of the functions of~(\ref{BGchoice}), see the last
paragraph in section~2 of~\cite{C}. By saying so, they are
acknowledging that they don't know whether the standard Gamow states defined
in the position representation are well defined as functionals acting
on $\Phi _{{\rm BG}\pm}$. If TAQT had an example, it would be known.
\section{The Quantum Arrow of Time (QAT)}
\label{seec:QAT}
Advocates of TAQT argue that their choice~(\ref{BGchoice}) is not
arbitrary but rather is rooted on a causality principle. Such causality
principle is the ``preparation-registration arrow of time,'' sometimes referred
to as the ``Quantum Arrow of Time'' (QAT). For the ``in'' states
$\varphi ^+$, the causal statement of the QAT is written as
\begin{equation}
\widetilde{\varphi}^+(t) \equiv \int_{-\infty}^{+\infty}\rmd E \,
\rme ^{-\rmi Et} \widehat{\varphi}^{+}(E) =0
\, , \quad \mbox{for} \ t>0 \, .
\label{ferFtvph+}
\end{equation}
By one of the Paley-Wiener theorems, Eq.~(\ref{ferFtvph+}) is equivalent to
assuming that $\widehat{\varphi}^+(E)$ is of Hardy class from below. The
corresponding causal statement for the ``out'' wave functions $\varphi ^-$
implies that $\varphi ^-$ is of Hardy class from above. Hence, in TAQT, the
choice~(\ref{BGchoice}) is not arbitrary but a consequence of causality.
It was pointed out in~\cite{HARDY} that the QAT is flawed. The argument was
twofold. First, it was pointed out that the original derivation~\cite{JMP95}
of Eq.~(\ref{ferFtvph+}) made use of the following flawed assumption:
\begin{equation}
0=\langle E|\varphi ^{\rm in}(t)\rangle =
\langle ^+E|\varphi ^{+}(t)\rangle =
\rme ^{-\rmi Et} \widehat{\varphi}^{+}(E)
\, , \quad {\rm for \ all \ energies,}
\label{flawassum1}
\end{equation}
which can happen only when $\varphi ^+$ and $\varphi ^{\rm in}$ are
identically 0. It was then pointed out that even though one may simply assume
the causal statement~(\ref{ferFtvph+}) and forget about how it was derived,
such causal statement says little about the actual time evolution of a quantum
system, because the quantum mechanical time evolution of $\varphi ^+$ is not
given by Eq.~(\ref{ferFtvph+}):
\begin{equation}
\varphi ^+(t)= \rme ^{-\rmi Ht} \varphi ^+ \ \neq \
\widetilde{\varphi}^+(t) \, .
\label{neoe}
\end{equation}
To counter this argument, the authors of~\cite{C} claim that the derivation
of the QAT was misquoted from the original source~\cite{JMP95}, and that the
flawed assumption~(\ref{flawassum1}) was never used to derive
the QAT~(\ref{ferFtvph+}). It seems therefore necessary to quote the original
derivation (see~\cite{JMP95}, page~2597):\footnote{In this quote,
$\phi ^{\rm in}$, $\phi ^+$, ${\cal F}(t)$ and Eq.~(\ref{Ftdkd}) correspond,
respectively, to $\varphi ^{\rm in}$, $\varphi ^+$, $\widetilde{\varphi}^+(t)$
and Eq.~(\ref{ferFtvph+}).}
\begin{quote}
{\small \it ``We are now in the position to give a mathematical formulation of
the QAT: we choose $t=0$ to be the time before which all preparations of
$\phi ^{\rm in}(t)$ are completed and after which the registration of
$\psi ^{\rm out}(t)$ begins. This means that for $t>0$ the energy distribution
of the preparation apparatus must vanish:
$\langle E,\eta |\phi ^{\rm in}(t) \rangle =0$ for all values of the quantum
numbers $E$ and $\eta$ ($\eta$ are the additional quantum numbers which we
usually suppress). As the mathematical statement for `no preparations for
$t>0$' we therefore write (the slightly weaker condition)
\begin{equation}
\hskip-0.5cm 0= \int \rmd E \, \langle E|\phi ^{\rm in}(t)\rangle =
\int \rmd E \, \langle ^+E|\phi ^{+}(t)\rangle =
\int \rmd E \, \langle ^+E|\rme ^{-\rmi Ht}|\phi ^{+}\rangle
\end{equation}
or
\begin{equation}
\hskip-0.5cm
0= \int_{-\infty}^{+\infty} \rmd E \,
\langle ^+E|\phi ^{+}\rangle \rme ^{-\rmi Et}
\equiv {\cal F}(t) \quad {\rm for \ } t>0 \, .
\label{Ftdkd}
\end{equation}
}
\end{quote}
The readers can decide whether or not the flawed
hypothesis~(\ref{flawassum1}) was used to derive the QAT~(\ref{Ftdkd}).
Nevertheless, it is actually not very relevant whether the authors
of~\cite{JMP95} used~(\ref{flawassum1}) to derive~(\ref{ferFtvph+}). As
pointed out in~\cite{HARDY}, and as mentioned
above, even though one can forget~(\ref{flawassum1}) and simply
assume~(\ref{ferFtvph+}) as the causal condition to be satisfied by
$\varphi ^+$, such causal condition has little to do with the time evolution
of a quantum system, see again Eq.~(\ref{neoe}). In particular, as even the
author of~\cite{BAUMGARTEL} has asserted, the $t$ that appears in
Eq.~(\ref{ferFtvph+}) is not the same as the parametric time $t$
that labels the evolution of a quantum system.\footnote{All this shows that
the new term TAQT is a misnomer. A better name is Bohm-Gadella theory, because
it was these two authors who proposed the theory and summarized it
in~\cite{BG}.} Thus, as far as standard quantum mechanics is concerned, the
causal content of the QAT is physically vacuous, and therefore, regardless of
how one motivates it, there is no physical justification for the
choice~(\ref{BGchoice}).
\section{TAQT vs.~the ``classic results''}
\label{sec:clasins}
In this section, we are going to compare the Hardy axiom of TAQT with
some classic results of Paley and Wiener, of Gelfand and Shilov and of
the theory of ultradistributions, which we shall collectively refer to
as the ``classic results.'' More precisely, we will see that the spaces of
test functions $\widehat{\varphi}^{\pm}$ obtained by the ``standard method''
would be of Hardy class only if the ``classic results'' were wrong.
The direct comparison with the ``classic results'' is more easily done
in one dimension, and therefore we shall use the example of the one-dimensional
rectangular barrier potential:
\begin{equation}
V(x)=\left\{ \begin{array}{ll}
0 &-\infty <x<a \\
V_0 &a<x<b \\
0 &b<x<\infty \, .
\end{array}
\right.
\label{sbpotential1}
\end{equation}
For this potential, the ``in'' and ``out'' eigensolutions are well known
and can be found for example in~\cite{JPA04}. We shall denote them by
$\chi _{\rm l,r}^{\pm}(x;E)$, where the labels l,r denote left and right
incidence. When we analytically continue these eigenfunctions, or when
we consider the Gamow states for this potential, the ``standard method''
calls for test functions $\varphi _{\rm l,r}^{\pm}(x)$ for which the
following integrals are finite:
\begin{equation}
\widehat{\varphi}_{\rm l,r}^{\pm}(z)=\int_{-\infty}^{\infty}\rmd x \,
\overline{\chi _{\rm l,r}^{\pm}(x;\overline{z})}\, {\varphi}(x)
\, .
\label{Upmac1D}
\end{equation}
Just as in the example discussed in Sec.~\ref{sec:stamet}, the
test functions $\varphi (x)$ must at least fall off
faster than exponentials.
To further simplify the discussion, we need to recall that, because
of Eqs.~(\ref{asstrco1}) and (\ref{asstrco2}), the Hardy axiom assumes
that the ``free'' wave functions $\widehat{\varphi}_{\rm l,r}^{\rm in}(E)$ and
$\widehat{\varphi}_{\rm l,r}^{\rm out}(E)$
are also of Hardy class. These ``free'' functions are given by (hereafter, we
just consider $\varphi _{\rm l,r}^{\rm in}$, since the analysis
for $\varphi _{\rm l,r}^{\rm out}$ is the same)
\begin{equation}
\widehat{\varphi}_{\rm l}^{\rm in}(E)=
\frac{1}{\sqrt{4\pi k \,}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{-\rmi kx} \, {\varphi}^{\rm in}(x)
\, ,
\label{Upmac1D0in}
\end{equation}
\begin{equation}
\widehat{\varphi}_{\rm r}^{\rm in}(E)=
\frac{1}{\sqrt{4\pi k \,}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{\rmi kx} \, {\varphi}^{\rm in}(x)
\, ,
\label{Upmac1D0inr}
\end{equation}
where $k=\sqrt{E}$ is the wave number. The total wave function is given
by the sum of left and right components:
\begin{equation}
\widehat{\varphi}^{\rm in}(E)= \widehat{\varphi}_{\rm l}^{\rm in}(E) +
\widehat{\varphi}_{\rm r}^{\rm in}(E) \, .
\label{lrimoc}
\end{equation}
It is simpler to work with $k$ rather than with $E$ and define
\begin{equation}
\widehat{\varphi}_{\rm l,r}^{\rm in}(k) \equiv
\sqrt{2k \,} \,
\widehat{\varphi}_{\rm l,r}^{\rm in}(E)
\, ;
\label{Upmac1D0in2}
\end{equation}
that is,
\begin{equation}
\widehat{\varphi}_{\rm l}^{\rm in}(k)= \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{-\rmi kx} \, {\varphi}^{\rm in}(x)
\, , \quad k\geq 0 \, ,
\label{Upmac1D0ink}
\end{equation}
\begin{equation}
\widehat{\varphi}_{\rm r}^{\rm in}(k)= \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{\rmi kx} \, {\varphi}^{\rm in}(x)
\, , \quad k\geq 0 \, .
\label{Upmac1D0irdk}
\end{equation}
The ``total'' wave function in the wave-number representation,
$\widehat{\varphi}^{\rm in}(k)= \widehat{\varphi}_{\rm l}^{\rm in}(k) +
\widehat{\varphi}_{\rm r}^{\rm in}(k)$, is
thus the Fourier transform of $\varphi (x)$,
\begin{equation}
\widehat{\varphi}^{\rm in}(k)= \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{-\rmi kx} \, {\varphi}^{\rm in}(x)
\, , \quad k \in {\mathbb R} \, .
\label{lrimockks}
\end{equation}
Its analytic continuation will be denoted as
\begin{equation}
\widehat{\varphi}^{\rm in}(q)= \frac{1}{\sqrt{2\pi}}
\int_{-\infty}^{\infty}\rmd x \,
\rme ^{-\rmi qx} \, {\varphi}^{\rm in}(x)
\, , \quad q \in {\mathbb C} \, .
\label{Upmac1D0inlrqFT}
\end{equation}
At this point, we are ready to introduce two classic theorems. The first one
is due to Paley and Wiener (see Theorem IX.11 in~\cite{SIMON}):
\vskip0.5cm
\theoremstyle{plain}
\newtheorem*{Th1}{Theorem~1 (Paley-Wiener)}
\begin{Th1}
An entire analytic function $\widehat{\varphi}(q)$ is the Fourier transform
of a $C_0^{\infty}({\mathbb R})$ function $\varphi (x)$ with support in
the segment $\{ x \, | \ |x|<A \}$
if, and only if, for each $N$ there is a $C_N$ so that
\begin{equation}
|\widehat{\varphi}(q)|\leq
\frac{C_N \, \rme ^{A|{\rm Im}(q)|}}{ (1+|q|)^N}
\label{PWbound}
\end{equation}
for all $q \in {\mathbb C}$.
\end{Th1}
\vskip0.5cm
This theorem says that the Fourier transform of a $C_0^{\infty}$ function
is an analytic function that grows exponentially,
and that such exponential growth is mildly corrected (but not canceled)
by a polynomial falloff.
The second theorem we shall use is due to Gelfand and
Shilov~\cite{GELFAND}. Before stating it, we need some definitions. Let
$a$ and $b$ denote two positive real numbers satisfying~(\ref{ab}). Let
us define $\Phi _{a,b}$ as the set of all differentiable
functions $\varphi (x)$ ($-\infty <x <\infty$) satisfying the inequalities
\begin{equation}
\left| \frac{\rmd ^n \varphi (x)}{\rmd x ^n} \right|
\leq C_n \rme ^{-\alpha \frac{|x|^a}{a}}
\end{equation}
with constants $C_n$ and $\alpha >0$ which may depend on the function
$\varphi$. Let us define the space $\widehat{\Phi}_{a,b}$ as the set of entire
analytic functions $\widehat{\varphi}(q)$, $q={\rm Re}(q)+\rmi \,{\rm Im}(q)$,
which satisfy the inequalities
\begin{equation}
|q^n\widehat{\varphi}(q)|\leq C_n
\rme ^{+\beta \frac{|{\rm Im}(q)|^b}{b}}
\, ,
\label{GSbound}
\end{equation}
where the constants $C_n$ and $\beta >0$ depend on the function $\varphi$. It
is obvious that the elements of $\Phi _{a,b}$ are functions that, together with
their derivatives, decrease at infinity faster than
$\rme ^{-\frac{|x|^a}{a}}$, whereas the elements of
$\widehat{\Phi}_{a,b}$ are analytic
functions that grow exponentially at infinity as
$\rme ^{+\frac{|{\rm Im}(q)|^b}{b}}$, except for a polynomial correction that
doesn't cancel the exponential blowup.
\vskip0.5cm
\theoremstyle{plain}
\newtheorem*{Th2}{Theorem~2 (Gelfand-Shilov)}
\begin{Th2}
The space $\widehat{\Phi}_{a,b}$ is the Fourier transform of
$\Phi _{a,b}$.
\end{Th2}
\vskip0.5cm
This theorem means that the smooth functions that fall off at infinity
faster than $\rme ^{-|x|^a/a}$ are, in Fourier space, analytic
functions that grow exponentially like $\rme ^{+|{\rm Im}(q)|^b/b}$.
The bounds~(\ref{PWbound}) and (\ref{GSbound}) are to be understood in
the same way as the bounds~(\ref{blowupfex}) and (\ref{grofainf}). That is,
the bounds~(\ref{PWbound}) and (\ref{GSbound}) mean that
$\widehat{\varphi}(q)$ is an oscillatory function
that grows exponentially in the infinite arc of the $q$-plane, the oscillation
being tightly bounded by Eqs.~(\ref{PWbound}) and (\ref{GSbound}) when
$\varphi (x)$ belongs to $C_0^{\infty}$ and $\Phi _{a,b}$, respectively. Note
that after the addition of the corresponding polynomial corrections, the
bounds~(\ref{blowupfex}) and (\ref{grofainf}) are entirely analogous to the
bounds~(\ref{PWbound}) and (\ref{GSbound})---the operators $U_{\pm}$ are
after all Fourier-like transforms~\cite{JPA04}.
Let us now apply the above theorems to the functions $\varphi ^{\rm in}(x)$
obtained by the ``standard method.'' In order for Eq.~(\ref{Upmac1D0inlrqFT})
to make sense, $\varphi ^{\rm in}(x)$ must fall
off faster than exponentials. If we choose $\varphi ^{\rm in}(x)$ to fall off
like $\rme ^{-|x|^a/a}$, then the Gelfand-Shilov theorem tells us that
$\widehat{\varphi}^{\rm in}(q)$ grows like $\rme ^{+|{\rm Im}(q)|^b/b}$. Even
when we impose that $\varphi ^{\rm in}(x)$ is $C_0^{\infty}$, which is already
a very strict requirement, the Paley-Wiener theorem says that
$\widehat{\varphi}^{\rm in}(q)$ grows
exponentially. This means, in particular, that the
$\widehat{\varphi}^{\rm in}(q)$
do in general {\it not} tend to zero in the infinite arc of the $q$-plane,
because if they did, the Paley-Wiener and the Gelfand-Shilov theorems would be
wrong. Because of Eq.~(\ref{Upmac1D0in2}), $\widehat{\varphi}^{\rm in}(z)$
does in general not tend to zero as $|z|$ tends to infinity in the lower
half-plane of the second sheet. Hence
the space of $\widehat{\varphi}^{\rm in}$'s is not of Hardy class from below.
The space of $\widehat{\varphi}^{+}$'s cannot be of Hardy class from below
either, because if it were, then
\begin{equation}
\lim _{|z|\to \infty} \widehat{\varphi}^{+}(z) =0 \, ,
\label{sskdks}
\end{equation}
where the limit is taken in the lower half plane of the second sheet. By
Eq.~(\ref{asstrco1}), this implies that also the space of
$\widehat{\varphi}^{\rm in}$'s would be of Hardy class and comply with this
limit, which we know is not possible due to the ``classic results.'' Thus,
the ``standard method'' yields spaces of test functions that
do {\it not} comply with the Hardy axiom. This is precisely what it was
meant in~\cite{HARDY} by the assertion that TAQT is inconsistent with standard
quantum mechanics.
To finish this section, we note that if we chose the test functions as
in~\cite{BOLLINI}, then we would be dealing with ultradistributions. In
Fourier space, the test functions for ultradistributions grow faster
than any exponential as we follow the imaginary axis, see~\cite{BOLLINI}
and references therein. Thus, if the
``standard method'' yielded spaces of Hardy functions, that property of
ultradistributions would be false.
\section{Further remarks}
\label{sec:further remarks}
The authors of~\cite{C} claim that it is inaccurate to state that the
proponents of TAQT dispense with asymptotic completeness. This statement should
be compared with the first quote in section~6 of~\cite{HARDY}.
The authors of~\cite{C} also claim that TAQT obtains the resonant states
by solving the Schr\"odinger equation subject to purely outgoing boundary
conditions. This claim should be compared with the second quote in
section~6 of~\cite{HARDY}.
The authors of~\cite{C} also dispute the assertion of~\cite{HARDY} that TAQT
sometimes uses the whole real line as though it coincided with the scattering
spectrum of the Hamiltonian. A glance at, for example, the
QAT~(\ref{ferFtvph+}) seems to support such assertion.
\section{Conclusions}
\label{sec:con}
In standard scattering theory, one assumes that the physics is
described by the Lippmann-Schwinger equation. When one solves such equation,
one finds that its solutions must be accommodated by a rigged
Hilbert space, and that its time evolution runs from $t=-\infty$ till
$t=+\infty$~\cite{LS1}. When one analytically continues the solutions of
the Lippmann-Schwinger equation, one finds that they must be accommodated
by {\it one} rigged Hilbert space, which also accommodates the resonant (Gamow)
states. The construction of such rigged Hilbert space is determined by
standard distribution theory.
By contrast, TAQT assumes that the solutions of the Lippmann-Schwinger
equations belong to {\it two} rigged Hilbert spaces of Hardy
class. In TAQT, one never explicitly solves the Lippmann-Schwinger
equation for specific potentials in the position representation. Instead, one
assumes that its solutions satisfy the Hardy axiom. Unlike
in standard scattering theory, in TAQT the time evolution of the solutions
of the Lippmann-Schwinger equation does not run from $t=-\infty$ till
$t=+\infty$.
By comparing the properties of the actual solutions of the Lippmann-Schwinger
equation with the Hardy axiom, we have seen that such actual solutions
would comply with the Hardy axiom only if classic results of
Paley and Wiener, of Gelfand and Shilov, and of the theory of
ultradistributions were wrong. We have (again) stressed the fact that the
Quantum Arrow of Time, which is the justification for using the rigged
Hilbert spaces of Hardy class, has little to do with the time evolution of
a quantum system. We have stressed that using the method of TAQT to
introduce rigged Hilbert spaces, we could accommodate the Gamow states in
a landscape of arbitrary rigged Hilbert spaces, see also~\cite{RELBO}.
Our claim of inconsistency should not be taken as a claim that TAQT is
mathematically inconsistent or that TAQT doesn't have a beautiful mathematical
structure. What the present author claims
is that TAQT is not applicable in quantum mechanics and is in fact a different
theory.
To finish, we would like to mention that the ``classic theorems'' are not
in conflict with using Hardy functions in quantum mechanics. They are in
conflict only with the Hardy axiom. Thus, our results do not apply to other
works that use Hardy functions in a different way~\cite{YOSHI}.
\ackn
This research was supported by MEC and DOE.
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,732 |
<?PHP
class Photo
{
public $Title, $Description, $Type, $AlbumID, $AlbumTitle;
protected $_SQL, $_Template, $id;
private $_CreatedBy, $_CreatedOn, $_EditedBy, $_EditedOn;
function __construct( $p_id, $sql, $template )
{
if(!is_numeric($p_id))
return 0;
$this->_SQL = $sql;
$this->_Template = $template;
$this->id = $p_id;
$this->_getData();
}
private function _getData()
{
$q_str = "SELECT * FROM `gallery_photos` WHERE `id` = {$this->id}";
$query = $this->_SQL->query($q_str);
if($query->num_rows > 0)
{
$result = $query->fetch_assoc();
$this->Title = $result['photo_title'];
$this->Description = $result['photo_description'];
$this->AlbumID = $result['album_id'];
$this->AlbumTitle = Album::getTitle($result['album_id'], $this->_SQL);
$this->Type = $result['type'];
$this->_CreatedBy = $result['created_by'];
$this->_CreatedOn = $result['date_created'];
$this->_EditedBy = $result['edited_by'];
$this->_EditedOn = $result['date_last_edited'];
}
}
public function getImageSrc()
{
$src = CMS_Core::getInstance()->parentDirectory() . CMS_Core::getInstance()->Settings['uploadDir'] . strtolower($this->_getRaw($this->AlbumTitle)) . DIR_SPACER . $this->_getRaw($this->Title) . "." . $this->Type;
return $src;
}
public function getImageSmallSrc()
{
$src = CMS_Core::getInstance()->parentDirectory() . CMS_Core::getInstance()->Settings['uploadDir'] . strtolower($this->_getRaw($this->AlbumTitle)) . DIR_SPACER . $this->_getRaw($this->Title) . SMALL_IMAGE_EXT . "." . $this->Type;
return $src;
}
public function photoSrc()
{
return $this->_getRaw($this->Title);
}
private function _getRaw($name)
{
return implode("_", explode(" ", $name));
}
}
?> | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,438 |
Christian Elias, right, a 17-year veteran of the inner workings of the Green Monster, manned the left-field scoreboard with the help of Nate Moulter in May 2007. Players and scoreboard operators have left their mark on the walls for decades.
© 2012 by the Boston Globe
Published by Running Press,
A Member of the Perseus Books Group
All rights reserved under the Pan-American and International
Copyright Conventions
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ISBN 978-0-7624-4204-1
Library of Congress Control Number: 2011925148
E-book ISBN 978-0-7624-4490-8
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Digit on the right indicates the number of this printing
Cover and Interior Design: Joshua McDonnell
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"The game opens in other stadiums in the country, in those giant modern saucers, and the day is a joyous, modern, klieg-light event. The game opens here, and it is a continuation. It is a pleasant click on the calendar. It is a celebration of the past, the present and everything in between. It is newly painted history."
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DEDICATION
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—John Powers
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—Ron Driscoll
ACKNOWLEDGMENTS
The histories of Fenway Park and the Boston Red Sox have been intertwined with the _Boston Globe_ from the outset, and also with the Taylor family, which owned the _Globe_ for much of the newspaper's first 125 years, and which played a key role with the team and its ballpark at various times. Thus we would like to especially thank the Red Sox, present and former staff members of the _Globe_ , and the Taylor family for their involvement in helping to create 100 years of Fenway Park history, and in making it come alive for readers and sports fans in New England, and increasingly, around the world.
A huge thank you as well to Janice Page, the _Globe_ 's book development editor, who masterfully guided the project from start to fruition; to _Globe_ editor Martin Baron, publisher Christopher Mayer, deputy managing editor Mark Morrow, and the entire Sports staff, especially columnists Dan Shaughnessy and Bob Ryan and editor Joe Sullivan. Our appreciation also goes to the book's keen-eyed photo director, Susan Vermazen, as well as Jim Wilson, Leanne Burden, David Ryan, Jim Davis, Stan Grossfeld, and all members of the photo department, along with graphics staffers Daigo Fujiwara, Javier Zarracina, and David Schutz. Thanks as well to the indefatigable Lisa Tuite and the library staff for their research efforts, and Stephanie Schorow (research and fact checking), Alan Wirzbicki (fact checking), Richard Kassirer, Paul Colton, William Herzog, Jim Matte (proofreading), and Ray Marsden and John Ioven (imaging).
At Running Press, special thanks to editor Greg Jones, designer Joshua McDonnell, and every exacting copy editor who had a hand in these pages.
Cheers to Ben Taylor, for sharing both memories and memorabilia, and to Jim Lonborg, who's just as classy off the field as he was on it.
As always, we are grateful for the support of Lane Zachary and Todd Shuster at Zachary, Shuster, Harmsworth Literary Agency. We also thank the good people at the Boston Public Library (Jane Winton, Tom Blake, Catherine Wood), Tim Wiles of the National Baseball Hall of Fame, and our friends at Dorian Color Lab in Arlington, Massachusetts. And we especially appreciate the generosity of Dan Rea, Susan Goodenow, David Friedman, and everyone in the Fenway Park front office.
"The ballpark is the star. In the age of Tris Speaker and Babe Ruth, the era of Jimmie Foxx and Ted Williams, through the empty-seats epoch of Don Buddin and Willie Tasby and unto the decades of Carl Yastrzemski and Jim Rice, the ballpark is the star. A crazy-quilt violation of city planning principles, an irregular pile of architecture, a menace to marketing consultants, Fenway Park works. It works as a symbol of New England's pride, as a repository of evergreen hopes, as a tabernacle of lost innocence. It works as a place to watch baseball."
**—Martin F. Nolan, former Boston _Globe_ editorial page editor**
Foreword | CONTENTS
---|---
Special Introduction
Introduction
1910s
1920s
1930s
1940s
1950s
1960s
1970s
1980s
1990s
2000s
Photography Credits
Index
FOREWORD
BY
JIM LONBORG
On the final day of the 1967 season, teammates and fans rushed Jim Lonborg after he pitched the Red Sox to victory over the Minnesota Twins to gain at least a tie for the American League pennant.
I first saw Fenway Park in 1965 when I was a rookie pitcher for the Red Sox. We had played a couple of exhibitions on the way home from spring training in Scottsdale, Arizona, and had flown into Boston on Saturday night. We were staying at the Kenmore Hotel and we walked over for a workout on Sunday. I was used to ballparks like Candlestick Park and Dodger Stadium in California, so it was unique to be in a city setting and enter through a beautiful brick facade.
I remember coming through the tunnel on the first-base side and the first thing I saw was the Wall. I thought, this is where I have to work? I stepped off the distance from home plate to the Wall to see whether the posted distance of 315 feet was an accurate reading and it wasn't. But our coaching staff did a really good job of preparing us mentally to pitch in Fenway. The Wall can help you as much as it hurts you because a lot of line drives are knocked down by it. Since I was a sinkerball pitcher, if balls were up in the sky I wasn't making very good pitches.
My first major-league victory came in Fenway against the Yankees on May 10. It was a thrill to pitch against Mickey Mantle, who'd been one of my boyhood heroes, and I struck him out in his first at-bat. After that, though, he had a single and a homer, and when Mickey hit a double with two out in the ninth Billy Herman, our manager, came out and said, "I think it's time to bring in the Big Guy." So Dick Radatz came in for the save and we won 3-2.
There weren't many fans at our games during my first two years and sounds travel very well at Fenway so you could hear what players and fans were yelling. But in 1967 after we won 10 in a row and returned from that road trip, we came back to packed seats. People still tell me that it was the greatest summer of their lives. None of us ever had been in a pennant race before. That year we had four teams involved—the Twins, White Sox, Tigers, and us—and you couldn't help but scoreboard-watch. At Fenway we had the best way of keeping score—the guy in the Wall. You would see that number disappear and wait for the next one to come up. It wasn't like it was being blurted out on a Jumbotron. You'd be in a situation where you wouldn't be expecting a cheer and you'd turn around and look at the scoreboard.
That final game against Minnesota was so tense all the way through. To make the comeback we did, to be on the field at the end and celebrate with all the players and then to turn around and see thousands of fans coming onto the field was the most exciting moment of my life. It was something a little kid would think about when he was a make-believe pitcher.
But as the fans were carrying me on their shoulders, the feeling went from jubilation to a little bit of fear. I was going places where I didn't want to go, out by Pesky's Pole. Finally the police got me to where I wanted to go, which was the clubhouse, where we still had to wait for the Angels-Tigers game to end in Detroit. For us to be sitting around a radio instead of a TV, it reminded me of an old-time movie where you were listening for news of some important event.
After the Angels beat the Tigers and we'd won the pennant, I went upstairs to see Mr. Yawkey to give him the game ball that my teammates had given to me. I knew that it was such a long time since he'd had anything good like that happen for him that I thought he should have it. It was almost like the owner's office in _The Natural_ , with the dark hallway and the dark-paneled room. Mr. Yawkey was there and I gave him the ball. He cried a lot that day. That was a special chapter of a fabled story. The beauty of the Red Sox is that every year is a different chapter—and there's still more to this book.
Whenever I come back to Fenway I try to go in through the same ramp that I did that first time in 1965, and it's the same feeling. It has so many great memories for me. The greenness, the majesty of the Wall. That image never goes away.
_Jim Lonborg was the first Red Sox pitcher to win the Cy Young Award (1967)_
SPECIAL INTRODUCTION
BY
BENJAMIN TAYLOR
After the death of my father, John I. Taylor, in June of 1987, I had to clean out his office at the _Boston Globe_ , where he had spent more than five decades as a newspaperman. Most of what I found was unremarkable, with one exception. In the dark recesses of a closet, I discovered a couple of blueprints. One was of Fenway Park and the other a detailed sketch of the hand-operated scoreboard still extant on the park's left-field wall. Upon closer inspection, I realized that the blueprints were of the renovation of the park in 1933 by then-owner Thomas A. Yawkey. I assumed that they were given to my father because his father, also named John I. Taylor, is credited with having built Fenway Park.
I never met my grandfather. He died in 1938, ten years before I was born. I was vaguely aware—no doubt from reading the great Peter Gammons—that he was the president of Boston's American League franchise from 1904 to 1911, that he changed the name of the team from the Boston Americans to the Boston Red Sox, and that he built Fenway Park. Other than this, I knew very little about him. Conversations with my older brothers confirm that our father rarely talked about his father. I cannot deny feeling a bit of family pride around his connection to Fenway Park, but I am also aware that my grandfather was thoroughly trashed as a meddlesome owner and dilettante by Glenn Stout and Richard A. Johnson in their book _Red Sox Century_.
I hung the blueprints in my office at the _Globe_ and then at home after I left the paper in late 1999. When I discovered they were fading because of exposure to the light, I took them off the wall and stored them in our attic where they now reside, gathering dust. (You can see them, restored to their original blue, at left and on the next page.)
As a fan, my experience is not unlike thousands of others in New England. In the first game I can remember attending at Fenway, the Red Sox played the great 1954 Cleveland Indians team that won 111 games during the 154-game regular season. I was seven years old. That same decade, I remember my parents telling me I couldn't go to a game with the rest of the family because of the polio scare. Like many Red Sox fans of a certain age, I was thrilled by the magical seasons of 1967, 1975, and 1986, when the Sox came within one game of winning the World Series and breaking the Curse of the Bambino. I became addicted to reading newspapers when I was young because of the Red Sox coverage. To this day, Sox stories in the _Globe_ remain one of the first staples I turn to each morning.
On my office wall now, I have a picture of Ted Williams and his beautiful swing during the All-Star Game in 1946. On the same wall, near the _Globe_ front page of August 9, 1974 announcing Nixon's resignation, is a framed reproduction of the _Globe_ 's front page of October 28, 2004 chronicling the Sox victory in the World Series for the first time since 1918. Like many Red Sox fans, I feel we are extremely fortunate that the current ownership chose to improve Fenway rather than build a new park. They have preserved a national treasure. That choice seems to have been a good business decision too, as the Sox continue to draw extraordinary crowds for home games.
My mother was a consummate Red Sox fan. She used to listen to games on the radio. Curt Gowdy's mellifluous tones and then Ned Martin's were familiar sounds in my house growing up. Born in 1915, which made her too young to remember the championship seasons of that year, 1916, and 1918, my mother used to say her one wish was that the Red Sox win a World Series before she died. She died in 1990, missing the championships of 2004 and 2007 that finally ended the club's long stretch of futility.
There is no way to predict the future, or how long Fenway will survive into the 21st century. It is important not to stay stuck in the past. Fenway, though vastly improved in recent years, is not without flaws. The right-field grandstand seats have lousy sightlines that apparently are very difficult to improve. Legroom is not the park's strongest suit. Many of the best seats have become financially prohibitive. Nevertheless, as it celebrates its 100th anniversary, the park still works remarkably well as a place where fans can enjoy professional baseball at the highest level. Those of us who love the game can only hope that Fenway Park remains an essential part of the soul of New England and of the national pastime for a long time to come.
_Benjamin Taylor is a former publisher of the_ Boston Globe
Blueprint of Fenway Park dated January 8, 1934. It was found in the Boston _Globe_ office of the late John I. Taylor, along with a blueprint of scoreboard renovations (page 16) dated October 25, 1933. Images restored and reproduced (see pullout poster) courtesy of the Taylor family.
INTRODUCTION
The storied home of the Red Sox for a century, "America's Most Beloved Ballpark" also is the oldest in the major leagues, and the most famous. From the classic brick entrance on Yawkey Way, to the unique left-field wall with its manual scoreboard, to Pesky's Pole in right field, its timeless features are recognized from the Bronx to the Dominican Republic to Japan.
John Updike's "lyric little bandbox," which he likened to "an old-fashioned peeping-type Easter egg," is so linked with Boston and baseball history that it is a destination in itself, equal to the Freedom Trail and the swan boats, with visitors taking guided ballpark tours even during winter. In _Cheers_ , the long-running situation comedy based in a Back Bay tavern, bartender Sam "Mayday" Malone was a former Red Sox relief pitcher. The fan film _Fever Pitch_ is based around Fenway. It is also where Kevin Costner took James Earl Jones for an inspirational outing in _Field of Dreams_.
Fenway's field is like no other. Because the park was jammed into a city lot bounded by narrow streets, its dimensions are a crazy confluence of oblique angles—like the three-sided oddity in center field that can turn the game into Pachinko, with the ball bouncing and rattling about. There is so little playable foul territory that dozens of balls end up in the stands, which are so close to the diamond that fans can hear the players' chatter.
Fenway is a charmingly auditory experience, from the scalpers on Brookline Avenue ("Who needs tickets?"), to the fans singing "Sweet Caroline" during the eighth inning, to the playing of "Dirty Water" over the public address system after victories.
Fenway's endearing quirkiness is the key to much of its allure. Except for some increased seating and creature comforts, the park has remained largely unchanged since it opened in 1912 in the same week that the Titanic sank.
"When I brought my kids to Fenway, they never complained about the inconveniences of the ancient ballpark," wrote _Boston Globe_ columnist Dan Shaughnessy, who confessed that he still took "some weird comfort in the knowledge that the poles that occasionally obscured our vision of the pitcher are the same green beams that blocked the vision of my dad and his dad when they would take the trolley from Cambridge to watch the Red Sox in the 1920s."
Babe Ruth threw his first pitch and Ted Williams hit his last home run at Fenway. From Christy Mathewson, to Ty Cobb, to Satchel Paige, to Joe DiMaggio, to Hank Aaron, most of baseball's greatest names have appeared on Fenway's stage, which also has accommodated an extraordinary variety of other athletes, politicians, and entertainers.
Three of Boston's professional football teams—the Redskins, the Yanks, and the Patriots—performed at Fenway. The Bruins and Flyers, two of hockey's fiercest rivals, played in the Winter Classic there on New Year's Day. Franklin Delano Roosevelt gave his final campaign address at Fenway. The Rolling Stones, Stevie Wonder, Paul McCartney, and Bruce Springsteen all sang there.
Through it all the _Boston Globe_ has been the consistent, respected chronicler—both in words and in pictures—of every important event in Fenway's history. The Taylor family, the newspaper's founders and longtime stewards, owned and named both the team and the park. So it's appropriate that the _Globe_ has produced the definitive book celebrating 100 years of Fenway Park, a collector's item featuring exceptional writing and unforgettable images from the _Globe_ 's incomparable archive of photographs, illustrations, and front pages.
Every significant moment from every year is here, and then some. The dramatic World Series victory over the Giants in 1912. The 1934 fire that scorched Tom Yawkey's renovated park. Ted Williams's "Great Expectoration" of 1956. Jim Lonborg's "hero's ride" after putting the Sox in position to secure the Impossible Dream pennant in 1967. Carlton Fisk's dramatic, "is-it-fair?" homer in the 12th inning of Game 6 of the 1975 World Series against the Reds. Bucky "Bleeping" Dent's heartbreaking screen shot in the 1978 divisional playoff game with New York. Roger Clemens's record 20 strikeouts against the Mariners in 1986. Dave Roberts' stolen base against the Yankees in 2004 that was the beginning of the end of 86 years of October frustration.
Fenway is all about lore. The Royal Rooters torturing visiting ballplayers with incessant renditions of "Tessie." Williams's monster bleacher shot knocking a hole in a fan's straw hat. Manny Ramirez's mystery disappearance inside the belly of the Monster. Jimmy Piersall oinking like a pig on the base paths. Luis Tiant's rhumba windup that the _New Yorker_ 's Roger Angell dubbed "Call the Osteopath." Pedro Martinez playing matador to former skipper Don Zimmer's enraged bull during a brawl with the Yankees. A midget coming out of the stands to cover third when the Indians used the "Williams Shift."
This is the story of 100 years of Fenway Park, in chapter and verse, by the people who lived it.
## 1910s
A member of the Royal Rooters, a group of passionate Red Sox fans, sounded the drumbeat during a 1903 World Series game with the Pittsburgh Pirates at Huntington Avenue Grounds. The Rooters continued their antics for several years after the Sox moved to Fenway.
"Now for the opening of Boston's magnificent new ballpark and a chance to see the Red Sox in action while leading the American League, a position gained while on the road."
**— _B OSTON GLOBE_, APRIL 20, 1912**
By the time Fenway Park debuted it was something of an anticlimax. The ballpark, which replaced the old Huntington Avenue Grounds, actually had been opened and used 11 days earlier when a handful of fans braved wintry weather to see the Red Sox shut out Harvard College in an exhibition. The game with the New York Highlanders (now Yankees) had been postponed by two days of rain. And most people were preoccupied with the _Titanic_ , which had sunk on April 15 with several dozen New Englanders among those aboard. The day the new ballpark opened it was packed with 24,000 spectators, yet attendance for the season would total only 597,000. _Sporting News_ predicted that the park would become more popular when people got accustomed to "journeying in the new direction." In its first season Fenway hosted the World Series, and two years later it did so again—for the crosstown Boston Braves. The park's colorful fandom featured saloonkeeper Michael McGreevy, society lady Isabella Stewart Gardner, and the raucous Royal Rooters, and they cheered the Sox to four world titles in seven seasons. But in Fenway's early days, it was much more than the home base for the Red Sox; it hosted football, lacrosse, hurling, parades, memorials, and political gatherings. Former President Teddy Roosevelt attended an outing in 1914 at Fenway, 30 years before his cousin, President Franklin D. Roosevelt, gave the final campaign speech of his life there. The decade was capped by a rally for Irish independence attended by 50,000, by a world title captured in a season abbreviated by the Great War, and by a trade that would become the 86-year symbol of Red Sox futility.
A rendering of the new ballpark from architect James E. McLaughlin was published in the _Boston Globe_ , which estimated the cost of the park at $1 million.
From the very beginning, the cherished and cursed home of the Boston Red Sox was the most misshapen and quirky collection of angles and corners in baseball. It was, John Updike wrote, "a compromise between Man's Euclidean determinations and Nature's beguiling irregularities." Even the much-beloved name started as a simple tribute to geography. "It's in the Fenway section, isn't it?" the team's owner said at the time.
The very genesis of Fenway Park was a matter of straightforward commerce. John I. Taylor was the owner of a ball club that played in a rented park. What he wanted was to own half of a club playing in a ballpark that he fully controlled, preferably in the embryonic neighborhood where his real estate company owned a large chunk of the reclaimed swampland that he and his partners hoped to develop into one of Boston's desirable districts.
So he bought more than 365,000 square feet from his company, had architect James McLaughlin draw up plans and sold half of the Red Sox to former Washington Senators manager Jimmy McAleer for $150,000. Taylor then set about building what the _Boston Globe_ promised would be a "magnificent baseball plant" between Lansdowne and Ipswich Streets. The new facility would be made of concrete and steel with a brick exterior that was a cross between a South End bowfront and a New England cotton mill and it would accommodate 28,000 spectators, twice as many as did the wooden Huntington Avenue Grounds in which the team had played since 1901.
Taylor, whose father Charles was publisher of the _Globe_ , opted for the obvious and commercially convenient name of Fenway Park. The constraints of the site, cost, calendar, and concern for squinting batsmen led to Fenway's endearing and infuriating dimensions.
Since Taylor didn't want hitters blinded by the setting sun in an era when games began at 3:30 p.m., he had the diamond oriented with home plate looking out toward Lansdowne.
He didn't want freeloaders sneaking into the standing areas in the outfield or peering down from nearby rooftops, so he erected a 25-foot wooden wall that he could cover with paid advertisements and that was buttressed by a 10-foot incline that made fielding fly balls something between art and accident.
Red Sox team president John I. Taylor, whose father was principal owner of the Globe, sold half of the Red Sox and built Fenway Park on land that he owned in the newly developed Fenway section of the city.
The first ball put into play in the first game at Fenway Park, with an inscription by umpire Tom Connolly.
The _Boston Post_ ran several photos of the new ballpark the day after the Red Sox defeated the New York Highlanders, 7-6, in 11 innings, in the park's inaugural game.
Hall of Fame outfielder Harry Hooper is the only man to play on four Red Sox world championship teams.
Although the foundation was designed to support an upper deck, Taylor wanted his $650,000 playpen finished in time for the 1912 season, which left only seven months from the September groundbreaking. So a grandstand was built to hold 15,000 ticket holders with additional seating along the lines. Quartets of box seats were offered at $250 for the season, with pavilion seats going for 50 cents a game and bleacher spaces for 25 cents.
Fenway Park was such a novelty and the April weather so inhospitable that what the _Globe_ called "the real down-to-the-book official dedication with the music stuff, the flowers and the flags" did not occur until May 17, when the Sox lost to Chicago, 5-2, after leading 2-1 until the ninth, before 17,000 fans. Only 3,000 witnesses had turned out for the dress rehearsal on April 9, a 2-0 exhibition game victory over Harvard that was played amid snow flurries. "It was no day for baseball," the _Globe_ concluded. Nor were April 18 and Patriots Day, when the scheduled league opener and rescheduled doubleheader were rained out.
When the sun finally appeared on April 20, more than 24,000 fans were on hand for the first official game and what would, in later decades, become a celebrated rarity—a victory over New York, this one achieved in 11 innings by a 7-6 count.
It took another four games for a ball to be knocked over the left-field wall, which then was more of a barrier than a monster. The man who did it, reserve first baseman Hugh Bradley, had hit only one other homer in his career and never managed another. But after sitting on one of southpaw Lefty Russell's corkscrew curves, Bradley lashed "a terrific smash" over the fence for a three-run shot that gave the Sox a 7-6 victory over the Athletics. "The scene that followed was indescribable," Tim Murnane wrote in the _Globe_. "Spectators jumped onto their seats and threw their hats in the air and howled like Indians until Bradley had ducked out of sight, with the Boston players offering congratulations." The Sox quickly proved themselves capable of running up dizzying numbers. "SWAT! SWAT! SWAT!" read the _Globe_ headline after the "Speed Boys" outgunned Washington, 33-19, in a May 29 home doubleheader, with the scorers for both clubs covering over three-and-a-half miles ("Hitting and Run-Getting on Wholesale Basis at Fenway Park"). On June 29, rain washed out the game with New York with Boston leading, 10-0, with two out in the first inning. But the hosts won, 8-0, the next day.
By July 20, when the club was in first place by seven games over the Senators, the _Globe_ essentially awarded it the pennant with this headline: "Keen Head Work Combined With Acutely Schemed Team Play; Excellent Pitching, Catching, Infield and Outfield Action; Timely Hitting and Shrewd Base Running Have Shined Gloriously Bright the Outlook for Winning the 1912 Championship."
Howard "Smoky Joe" Wood, the game's best "twirler," went an astounding 34-5 for the Sox in 1912, while his teammates Buck O'Brien and Hugh Bedient each won 20 games. Tris Speaker batted .383 and Duffy Lewis knocked in 109 runs. The Sox didn't even have to clinch the pennant on their own. When they were being rained out in Cleveland on September 18, the White Sox beat the Athletics to give Boston its first American League title since 1904. By then management had already begun expanding Fenway's capacity to 32,000, adding 900 box seats and new stands in left and right field.
"The game was full of interest, the crowd holding its seats to the end, figuring that the Red Sox would eventually nose out the Broadway swells."
—T. H. Murnane, _Boston Globe,_ April 21, 1912, in a
story headlined "Sox Open to Packed Park"
BACK IN THE DAY
BY JOHN POWERS
The Red Sox began the 1912 season with a new brick-and-steel ball yard ("the mammoth plant with the commodious fittings") and no ghosts. Their fifth-place finish in 1911 had been forgotten. By the time they hosted the New York Highlanders on April 20, a sunny Saturday afternoon, they were already sitting in first place in the eight-team American League, well on the way to the finest year in club history (105-47 and the world championship over the New York Giants).
J. Garland "Jake" Stahl, the new player-manager who worked as a Chicago banker in the off-season, had promised no less. The Red Sox, he said, would give the Boston public "only the best of baseball this season, barring accidents."
"It will be good to play some baseball," Stahl said. "We have come out of Hot Springs [Arkansas] ready to go. We started well and we do not need any more postponements."
The Red Sox were 4-1 after wins in New York and Philadelphia on the way north. The visiting Highlanders had not liked sitting around in their hotel—the Copley Plaza—for extra, wasteful days.
Opening Day was perhaps something of an anticlimax. Fenway Park, which replaced the old Huntington Avenue Grounds, actually had been opened and used 11 days earlier when 3,000 customers shivered amid snow flurries while the Red Sox shut out Harvard in an exhibition. The game with the Highlanders (now Yankees) had been postponed by two days of rain, from Thursday to Saturday. And most people were preoccupied with the _Titanic_ , which had sunk on Monday morning with several dozen New Englanders aboard.
The Boston-New York game was listed on the amusements page of the newspaper, merely one of several urban attractions that day. Billie Burke was playing in _The Runaway_ at the Hollis Street Theatre. There was a Taft rally at Faneuil Hall, the Textile and Power Show at the Mechanics Building (admission 25 cents), the BSO at Symphony Hall. And at the Park Theatre, _Hoopla! Father Doesn't Care!!_
If you wanted to attend Opening Day, you could buy a reserved seat in the grandstands for 75 cents or a bleacher seat for a quarter. Tickets were available at Wright and Ditson at 344 Washington Street or at the gate. The ballpark was packed with 24,000 spectators, with hundreds of them standing behind a rope in the outfield. Yet attendance for the season would total only 597,000. _The Sporting News_ predicted that the park would become more popular when people got accustomed to "journeying in the new direction."
If you owned a car in 1912, you could park it almost anywhere you pleased near the ballpark. The only two buildings near the park were a riding school and a garage out beyond right field. The West Fens was terra nova at the time. Artists, students, musicians, and assorted bohemians lived there, out beyond the water and marsh. Speculators owned the land, looking to sell it to developers who would erect apartment buildings near the trolley lines.
Most of the Opening Day crowd arrived by public transit, taking the Ipswich Street, Beacon Street or Commonwealth Avenue cars and walking past open lots to the park. A cadre of "big, fine-looking officers" from Division 16 preserved order. Inside the park customers drank Pureoxia ginger ale, Dr. Swett's Original Root Beer, and cold lager. If Fenway Franks were available, it was not recorded.
Mayor John "Honey Fitz" Fitzgerald (the future maternal grandfather of President John F. Kennedy) tossed out the first ball, with Governor Eugene Foss at his side, and Buck O'Brien threw the first pitch at 3:10 p.m. "There was no time wasted in childish parades," _Globe_ writer T.H. Murnane observed. The boisterous Royal Rooters, led by South End saloonkeeper Michael T. "Nuf Ced" McGreevy, serenaded the assemblage with their theme song, "Tessie."
The Red Sox, also called the Speed Boys, climbed out of a 5-1 hole and won, 7-6, in 11 innings, helped by five hits, including a pair of doubles, by second baseman Steve Yerkes. Murnane wrote, "Tristram Speaker, the Texas sharpshooter, with two down in the 11th inning and Yerkes on third, smashed the ball too fast for the shortstop to handle and the winning run came over the plate . . . the immense crowd leaving for home for a cold supper but wreathed in smiles." The size of the crowd may have hindered the Red Sox cause, said Murnane. "Before the game, the crowd broke into the outfield and remained behind the ropes, forcing the teams to make ground rules, all hits going for two bases. This ruling was a big disadvantage to the home team, for the Highland laddies never hit for more than a single, while three of Boston's hits went into the crowd, whereas with a clear field they would have gone for three-base drives and possibly home runs, and would have landed the home team a winner before the ninth inning."
The writers finished up their scorecards promptly at 6:20 and went back to Newspaper Row. Nobody bothered entering the clubhouse to talk to the players.
Boston Mayor John "Honey Fitz" Fitzgerald tipped his hat after throwing out the first pitch on April 20, 1912, the first official game in Fenway Park history.
A view from the stands in 1912.
Even if you didn't have a ticket, you could view the 1912 World Series through gaps in a fence at the new Fenway Park, where the Red Sox took on the New York Giants.
Mayor John Fitzgerald (standing) and star pitcher "Smoky Joe" Wood (in the back, with bow tie) took part in Boston's victory parade after the 1912 World Series.
"Smoky Joe" Wood dominated the American League in 1912 with a 34-5 record, including 10 shutouts. He also contributed three World Series victories as the Red Sox vanquished the New York Giants.
When the team returned from Detroit on September 23, a crowd of 200,000 jammed Summer Street for a parade as the players rode in cars from South Station to the Common. "No general and his army, returning victorious from war, were ever received with wilder or more enthusiastic acclaim," James O'Leary observed in the _Globe_. Pennant fever was contagious. Women wore scarlet hose and carried dolls dressed in Boston uniforms while men sported oversized neckties with large red stockings woven in.
After the Sox took the World Series opener from the Giants in New York, more than 1,000 fans were in line at dawn when bleacher seats went on sale for the first home date. "Many had attendants with them who did their bidding, such as running errands to procure cigars, eatables, and wraps when the night air was biting," the _Globe_ reported.
Not since the Sox defeated Pittsburgh for the 1903 championship had they played in the Series and it was the social event of the year. "Staid citizens of conservative Boston danced in their boxes," remarked the _Globe_. "They shouted, they hugged their neighbors and punched perfect strangers in the ribs, inquiring opinions they could not hear and didn't care about."
More than 6,000 supporters who couldn't acquire tickets stood 25 deep on Washington Street, watching the game unfold on a scoreboard in front of the _Globe_ offices downtown with "the stentorian tones of Frank J. Flynn announcing play after play."
The game was called for darkness after 11 innings with the score deadlocked at 6-6 and after the visitors won the replay a day later, the stage was set for a Series where winning at home was a challenge. After New York battered Wood with six first-inning runs and went on to win by an 11-4 count to knot the Series at three games each, the season came down to one game at Fenway, and one historic blunder—the "$30,000 Muff" of pinch hitter Clyde Engle's routine fly by New York outfielder Fred Snodgrass that put the tying run on second in the 10th inning.
HOLY SMOKY: WOOD'S SEASON FOR THE AGES
BY BOB RYAN
If you could be one Boston athlete for one year of the 20th century, who would it be? Bobby Orr in 1970? Larry Bird in 1986? Ted Williams in 1941? Doug Flutie in 1984? These are all worthy choices.
But my choice is a 22-year-old young man having the ultimate career year playing baseball in a baseball-mad town. There was an aura of freshness and spontaneity because the team had opened a new ballpark. Imagine being 34-5 and dominant enough to have two official nicknames. Imagine being able to help yourself continually with both the bat and the glove. Imagine staring down the immortal Walter Johnson in the most ballyhooed regular-season game ever played in Fenway Park. Imagine winning three games in the World Series. Imagine being that young, that intelligent, that handsome, that gracious, that talented, and that idolized. Imagine being Smoky Joe Wood in 1912. I can't think of anything better.
Joe Wood had been with the team since the late stages of 1908. He had come out of the West, the true "Wild West," in his own words—born in the southwestern Colorado town of Ouray on October 25, 1889, and raised in Ness City, Kansas.
He was a sturdy 5-11 and 180 pounds, and Joe Wood had such a fastball that sometimes batters only saw the vapors; hence the nickname "Smoky Joe."
"Can I throw harder than Joe Wood? Listen, my friend, there's no man alive who can throw harder than Smoky Joe Wood." So said Walter "Big Train" Johnson, who had a pretty good heater himself.
Smoky Joe, who was also known as the "Kansas Cyclone," had 35 complete games in 38 starts, and in the opening game in New York, he beat the Highlanders with a seven-hitter while driving in two runs and fielding his position like a circus acrobat. By the end of May, word was out that the great Johnson had a major pitching rival in this young Mr. Wood. A tremendous crowd turned out at Griffith Stadium to see the two compete on June 26, and Smoky Joe sent them home with newfound respect after dispatching the Washington Senators with a three-hit shutout in the second game of a doubleheader. That made him 15-3 and gave him at least two victories over every opposing club in the eight-team league.
Wood entered August with a 21-4 record. He came out 28-4, and now thoughts were turning to an upcoming visit by the Senators, for Johnson was not relinquishing his title as Mound King easily. He was en route to a record-tying 16-game winning streak. It was becoming clear that a showdown was in order, and it happened on September 6 at Fenway.
It was the custom in those days to allow overflow fans to spill onto the outfield. Ropes were put up, and balls bounding into the crowd were ground-rule doubles. But such was the interest in this game that spectators were also permitted to line the foul lines, which no one had ever seen and would never see again. For the era, it was a massive crowd, estimated at 29,000.
Johnson was 28-10 and had just had a 16-game winning streak snapped. Wood was 29-4, with a 13-game winning streak. "One of the greatest pitching duels that has been fought should result," said the _Globe_ in a front-page story.
With such hype, there was scant chance of the game living up to its billing, except that it did. On a glorious late-summer afternoon, the Red Sox scored the game's only run in the sixth when Tris Speaker doubled into the crowd in left and Duffy Lewis delivered him with a fly ball to right that ticked off Danny Moeller's glove for another double.
Wood gave up six hits and walked three, but in the ninth, with a man on second base with one away, he got two strikeouts to end the game and give himself victory No. 30.
The Red Sox clinched the pennant with two weeks to spare and were honored with a parade from South Station to the Common. And guess who rode in the spot of honor?
The Red Sox expected to win the Series for one very simple reason. Sure, John McGraw's Giants had the great Christy Mathewson, but in 1912, the premier pitcher in the land was Smoky Joe Wood. He beat the Giants, 4-3, in Game 1, finishing the game by striking out Otis Crandall to leave the tying run on second.
Wood pitched again in Game 4, with the Series tied at 1-1-1, Game 2 having ended at 6-6 when darkness prevailed. He mixed his pitches well and beat the Giants, 3-1, prompting Mathewson, in his ghostwritten newspaper column, to say, "His was the work of an artist." Wood also drove in the third run in the ninth inning.
But given a chance to conclude the Series with Boston ahead, 3-2-1 in Game 7, Joe Wood could not tie the ribbon on the package. He was removed after one horrible inning in which he gave up six hits and was reached for a double steal.
It looked as if Wood's storybook season would have a very inappropriate ending. The following day, Jake Stahl brought him out of the bullpen in the eighth inning of a 1-1 game. Wood threw two shutout innings before New York pushed across a run in the top of the 10th and would have scored another one if Joe hadn't made a tremendous barehanded stop of a Chief Meyers line drive to the box. At any rate, he was the losing pitcher of record until his team came to bat. But the Red Sox, aided by some storied Giants misplays (the Fred Snodgrass muff of a routine fly ball and a Speaker pop foul that fell among three Giants), scored twice in the bottom of the inning. Joe Wood had his third Series win and the Red Sox had the 1912 world championship.
It is all in the books. In 1912, Wood had a league-leading 10 shutouts. He won 16 straight games. He threw 344 innings. He hit .290 and slugged .435. He won games with his glove. He won three games in the World Series.
Mayor John Fitzgerald (the celebrated "Honey Fitz") orchestrated a massive civic celebration for the Red Sox. Joe Wood simply got up and said, "I did all I could, and I just want to thank you."
"When I brought my kids to Fenway, they never complained about the inconveniences of the ancient ballpark. . . . I still take some weird comfort in the knowledge that the poles that occasionally obscured our view of the pitcher are the same green beams that blocked the vision of my dad and his dad when they would take the trolley from Cambridge to watch the Red Sox in the 1920s."
—Dan Shaughnessy, Boston Globe sports columnist
Fans filled the third-base grandstand, as well as the temporary stands erected against the left-field wall, for the 1912 World Series against the New York Giants.
"I didn't seem to be able to hold the ball," he later told the _New York Times_ , saying that the error "froze him to the marrow." The ball "just dropped out of the glove and that's all there was to it." Snodgrass immediately made amends by snagging Harry Hooper's shot to deep center. The real killer miscue was a miscall by New York pitcher Christy Mathewson on Speaker's foul pop-up, which fell uncaught. Speaker then knocked in the tying run and Larry Gardner's sacrifice fly scored Boston's Steve Yerkes with the winner, setting off what the _Globe_ 's Murnane called an "outburst of insane enthusiasm."
There was another celebratory parade the next day, this one from Park Square to Faneuil Hall, where Mayor John F. "Honey Fitz" Fitzgerald proclaimed the Sox victory "an epoch in the history of this city."
The euphoria continued throughout the winter, with a reprise assumed all around. "The more I think it over, the more convinced I am that we will be stronger next season than we were last," McAleer said in February. So the 10-9 loss to Philadelphia in the home opener was shrugged off. "What's One Lost Game to a Team That Can Win 105 in a Year?" asked the _Globe_ headline.
But by April 20 the Sox found themselves in seventh place, and there was "shock over the Red Sox start." Injuries didn't help. Wood hurt his thumb while slipping as he was fielding a bunt and left fielder Duffy Lewis, shortstop Heinie Wagner, and the team's player-manager Jake Stahl were all sidelined. "Having had their spring vacation, it is to be hoped that the Red Sox are now going to work with renewed vigor," the _Globe_ observed on May 11, when the club was in sixth.
Babe Ruth in 1915, age 20, when he won 18 games for the Red Sox and helped them win the first of three world championships in four years, before his infamous trade to the Yankees.
"MRS. JACK" WAS AN EARLY FANATIC
As the _Globe's_ Jack Thomas wrote in 1988: "There was only one Isabella Stewart Gardner, which is too bad, for nobody was better at shocking Boston society in the late 19th and early 20th centuries, and Boston today could use another like her. She is remembered not only for the Gardner Museum that she built with her husband's money, but also for her impact on Boston's cultural and social history from her arrival in 1860 as Jack's 20-year-old bride until her burial in Mount Auburn Cemetery in July 1924.
"She was not even born in Boston, as her biographer pointed out. Jack met her in New York, married her two days before the Civil War began, brought her to Boston and moved into the Boylston Hotel, later the Touraine, until their house at 152 Beacon Street was built. She was aristocratic, eccentric, and scandalous. Her husband had money and patience and, being married to her, needed both. She demonstrated contempt for propriety by walking down Beacon Street with pet lions, and posed in a low-cut dress with pearls around her hips for a John Singer Sargent portrait considered so risqué that when it was displayed at the St. Botolph Club in the winter of 1888, Jack ordered it taken down, and it was never again shown in his lifetime."
Mrs. Jack, as she was known, once caused another huge commotion at Symphony Hall. In December 1912, two months after the Red Sox beat the New York Giants in the World Series, she appeared at a concert wearing "a white band around her head and on it the words, 'Oh you Red Sox' in red letters," as a Boston gossip columnist put it. "It looked as if the woman had gone crazy . . . almost causing a panic among those in the audience who discovered the ornamentation, and even for a moment upsetting [the musicians] so that their startled eyes wandered from their music stands."
Why the hubbub? "'Oh you Red Sox' was a song popular with the Royal Rooters, a group of Boston baseball fans known for rowdyism," Patrick McMahon, a Museum of Fine Arts curatorial project manager, told the _Globe_ in 2005. "Symphony-goers must have thought for a moment that one of those raucous drunks had slipped into the building."
Mrs. Jack wasn't just jumping on the bandwagon, insisted Gardner Museum archivist Kristin Parker. While perusing Gardner's scrapbooks in 2005, Parker found numerous Red Sox news items, photos, and notations of scores dating to the Boston Americans' triumph in the first World Series in 1903. In 1912, the 72-year-old Gardner purchased season tickets to the newly built Fenway Park, a stone's throw from her palazzo. "One of the only things that kept [humorist] Robert Benchley from going 'crazy with boredom' in Boston in the summer of 1912, I've read, was meeting Gardner," Parker said. "She took Benchley to Fenway, where she 'loudly encouraged all the Boston players by name.'"
She was once called Boston's most famous "insider outsider," and some wondered about the death of her infant son, her only child, and whether the magnificent palazzo in the Fenway was a memorial to him. In one of the museum's paintings, a Madonna and child by the Spanish artist Francisco de Zurbaran, she must have seen, as others have, that the child in the painting looked remarkably like one of the photographs of her little son. A _Globe_ story about a museum renovation in 1928 said: "Everything was put back in its place, so the museum looks exactly as it did when Mrs. Gardner died. Nothing has been added, nothing removed. That is as she wished—and willed—it should be."
More recently, in a nod to Mrs. Jack's baseball allegiance, anyone wearing a Red Sox-branded item receives $2 off admission to the Gardner. If you're named Isabella, you get in free—for life.
Portrait of Isabella Stewart Gardner by John Singer Sargent, 1888.
IT WAS HIS CLIFF
He was a member of the Red Sox outfield that many consider to be the best in baseball history. He saw the first home run Babe Ruth hit and the last, No. 714. He is also one of the few players to ever pinch-hit for the legendary slugger.
They even named a cliff after him.
He was the venerable George "Duffy" Lewis, a Red Sox outfielder in the early glory days of World Series victories and later a traveling secretary for the Boston Braves. He was one-third of the famed Lewis-Speaker-Hooper outfield that sparked the Red Sox to World Series triumphs in 1912 and 1915.
After an outstanding career as a player that started with the Red Sox in 1910, Lewis was also a coach, manager, owner, and, finally, traveling secretary for the Boston and Milwaukee Braves, before retiring in 1961.
Born April 18, 1888, in San Francisco, George Edward Lewis was one of three children. His mother's maiden name was Duffy and somehow that became his nickname. Along with his lifelong friend and fellow outfielder, Harry Hooper, Lewis attended St. Mary's College in Oakland, California. In 1908, the pair played in the so-called "Outlaw League" in California and then jumped to the state's City League.
Lewis came to the Red Sox from Oakland in 1910, joining Hooper, who had been brought up the year before. Lewis had been spotted and signed by John I. Taylor, owner of the Red Sox, for a $200 bonus. His first year's salary was $3,000 and his biggest contract as a Red Sox outfielder was for $5,000.
When Lewis came up, the Red Sox had Tris Speaker in center, Harry Niles in right, and Harry Hooper in left. The Sox lost the first three games. Lewis was put in left and Hooper moved to right. Lewis's hitting won the first game he played in for the Sox and he remained in left for the next 151 games. That established the Red Sox outfield for the next six seasons.
As the Red Sox left fielder, Lewis patrolled the most unusual parcel of real estate of any ballpark in America. Fenway's left field then included a precarious grassy slope rising about 10 feet to the base of the wall. A player had to be part mountain goat to scale it, catch an outfield fly, and then stride downhill to throw the ball into the infield.
"The first time I saw it," Lewis recalled in an April 21, 1978 _Globe_ story by Joe Dinneen, "I said to myself, 'Holy cow! What have we got here?' It looked pretty awesome."
Lewis would go out to the park early and have somebody hit the ball again and again out to the wall. He experimented with every angle of approach up the "cliff." He mastered the slope so well that it was referred to as "Duffy's Cliff."
Bulldozers knocked the cliff down to just about level when Tom Yawkey remodeled Fenway Park in 1934.
The Red Sox enjoyed their most successful seasons during the stewardship of their mighty outfield trio. In 1912, they beat the New York Giants in seven games in the World Series, but Lewis's real starring roles were to come in Boston's 1915 and 1916 Series wins.
In 1915, the Red Sox faced the Philadelphia Phillies, led by the great pitcher Grover Cleveland Alexander. But Boston clobbered the Phillies, winning four of five games, losing only the first to Alexander. Duffy led all the regulars with a .444 average.
"They taught me how to go up the hill, but they didn't teach me how to go down."
—A visiting left fielder in the days of "Duffy's Cliff"
T.H. Murnane of the _Globe_ wrote, "Duffy Lewis was the real hero of this Series, or any other. I have witnessed all of the contests for the game's highest honors in the last 30 years and I want to say that the all-around work of the modest Californian never has been equaled in a big series."
In 1916, the Red Sox had to defend their championship without Speaker, who was traded to Cleveland just before the season. After securing the pennant, they faced Brooklyn in the World Series, and again the Sox prevailed by a 4-1 margin as Lewis batted .353. Lewis played with the Red Sox through 1917, spent some time in the Navy during World War I, and then was with the Yankees for two seasons before winding up his playing career with one season in Washington.
Lewis was an established player when Babe Ruth came up to the Red Sox as a pitcher in 1914. Ruth had a terrible memory for names and always greeted everyone with, "Hi, kid." For some reason, Duffy recalled, Ruth remembered him and always saluted him with, "Hi, Duff."
The Babe, Lewis said, was a pretty fair hitter in his early days, but had a habit of striking out a lot. And that's how Lewis came to pinch-hit for the Sultan of Swat.
"I was on the bench nursing a bad ankle," Lewis recalled. "Bill Carrigan, our manager, asked me if I could hit. I said sure. So he sent me to pinch-hit for Ruth. I got a hit, too. I used to think it won the game, but someone told me a few years ago that it didn't."
In 1947, Lewis received a testimonial dinner at the Hotel Statler, with more than 1,000 friends and fans turning out. He was reunited for the evening with old outfield compatriots Speaker and Hooper.
Lewis threw out the first pitch to open the 1975 World Series at Fenway Park. Perhaps the only sad note in his long baseball career is that it did not end with a place in the Hall of Fame.
Speaker had been part of the Hall's second group in 1937. Hooper was inducted in 1971 and spent the remaining years of his life plugging for Duffy to make it.
Frank Frisch, a member of the Hall of Fame Veterans Committee, once told Lewis: "You, Speaker and Hooper all should have gone into the Hall of Fame at the same time. As the best outfield of your day, it would have been right."
Lewis shrugged it off. "A lot of people have said I should have gone in with the others. But I have no regrets. I'm doing all right."
Duffy Lewis died in Salem, New Hampshire, in 1979 at the age of 91. He was inducted into the Red Sox Hall of Fame in 2002.
The 10-foot-high incline in left field that abutted the wall (visible behind players) was named for Duffy Lewis (opposite page), the Red Sox left fielder who mastered its vagaries.
LENDING AN EAR TO DE VALERA
Eamon de Valera, president of the Irish Republic, came to America in 1919 to plead the cause of independence for the fledgling Irish state, and his visit to Boston brought a massive crowd of 50,000 to Fenway Park.
The _Globe_ 's A.J. Philpott wrote, "It was an inspiring assemblage—one in which the spirit of the Irish people rose above the spirit of faction, of group or party." Of the Irish president, he wrote, "The very mystery which attaches to this man, who was comparatively unheard of until recently, somehow fulfilled the dreams of the race—that some great figure would arise at the crucial moment and lead Ireland to freedom. In the thoughtful, militant, clean-cut face and gaunt personality of de Valera there is somehow also personified that new spirit which has come to Irishmen in which the demand has superseded the appeal for justice to Ireland. In that vast audience, you sensed this new dignity that has sunk into their consciousness."
A large audience was expected, according to the _Globe_. But instead of 25,000, some 50,000 descended on the grounds. They filled the grandstand first, and then the wings on the right and left, and then they poured into the field and filled the space between the platform and grandstand—jammed it—then flowed around and backward in all directions, and there were thousands on the streets outside.
It was an ideal day for a great outdoor meeting—clear, sunny, and not too warm—and the location could not have been much improved. When a series of resolutions demanding the recognition of the Irish Republic were read, the unanimous "Ayes" could be heard over in Dorchester, and the silence that followed when the call came for the "Nays" led to a shout of laughter.
Some excitement was caused when three mounted policemen forced a passageway through the crowd to enable the committee, with President de Valera, to reach the speaker's platform, which had been erected near home plate of the baseball diamond. On the whole, however, it was an orderly and patient audience. More than 20,000 members of the crowd stood from about 2 o'clock until after 5 o'clock, when "The Star-Spangled Banner" was sung and the audience dispersed quietly.
Two days later, as Eamon de Valera left New England for New York, he issued a message thanking the people of the region: "I did not need to come to Boston or to America to know that Americans would not lend themselves to an act of injustice against an ancient nation that clung to its traditions and maintained its spirit of independence through seven centuries of blood and tears. In the name of Ireland, I thank you."
By June 25, when the pennant finally was raised before a chilled crowd of only 6,500, the Sox were only in fifth. In mid-July, McAleer replaced Stahl, who'd played only two games due to a foot injury, with catcher Bill Carrigan. A player-manager who couldn't play wasn't much good, especially one who was rumored to want his job, McAleer reckoned. But the burly Carrigan, known as "Rough" for his rugged play behind the plate, couldn't pull his mates out of their hole and they finished fourth, more than 15 games behind Philadelphia. With no world championship to contend for, the Sox challenged the Braves to a best-of-seven series. But their crosstown neighbors begged off because a couple of their key players were hurt.
There would be another World Series at Fenway in 1914, but it would be the Braves acting as hosts after they staged the greatest comeback in the sport's young history. The threadbare neighbors hadn't had a winning season since 1902, when they were the Beaneaters, and had since changed their name three times. On July 18 they were in last place, 11 games behind the Giants. Then, guided by Manager George "Miracle Man" Stallings, the Braves went on a relentless hot streak, winning 59 of their final 75 to claim the pennant by 10½ games.
By early August, their revival had attracted so much attention that their home, South End Grounds, which seated only 5,000, was being overrun. So Sox owner Joseph Lannin, who'd bought the franchise during the preceding winter, offered the Braves the use of Fenway for the rest of the season. His own club was a distant second to the Athletics by then, but its renaissance already was underway in the form of pitcher George Herman Ruth, a 19-year-old son of a Baltimore saloon owner the _Globe_ described as "one of the most sensational moundsmen who ever toed a slab in the International League."
Lannin, who'd emigrated from Quebec and started as a hotel bellboy, had made his fortune in real estate and commodities. He saw rare potential in the raw and untutored Ruth, bought him from the minor-league Orioles with Ben Egan and Ernie Shore for a reported $25,000 (give or take several thousand, depending on your source), and promptly put him in uniform. But the Sox couldn't catch the Athletics, who went on to play the Braves in the World Series.
By then even the Royal Rooters had switched allegiances. The town was entranced—73,000 people had turned out for a separate-admission Labor Day doubleheader between the Braves and the Giants, with another 10,000 turned away. And nearly 70,000 showed up for the final two games of the Series as the Braves sealed the first sweep in World Series history with 5-4 and 3-1 victories. "There was joy last night in Boston," said the _Globe_ editorial, "the land of the free and home of the Braves."
The Braves' days as a glorified Twilight League team, playing in a sandlot at the corner of Walpole Street and Columbus Avenue, were over. Owner James Gaffney was building a capacious new park between Commonwealth Avenue and the Charles River that had foul lines of more than 400 feet and a distance of 440 to dead center and that seated better than 40,000.
PERFECT RELIEF FOR RUTH
Ernie Shore was acquired by the Red Sox in the same deal that brought them Babe Ruth in 1914. Shore became an outstanding pitcher for the Red Sox, going 19-8 with a 1.64 ERA in the 1915 world championship season, and winning 16 more games the following year when the Sox repeated as world champs. He even pitched a three-hitter in the title-clinching victory. However, Shore is forever linked with Ruth and best known for his performance in one of the oddest games in baseball history.
Shore was in the dugout on June 23, 1917, two days removed from a pitching outing when Ruth started on the mound at Fenway against the Washington Senators. Ruth walked the leadoff hitter, Ray Morgan, but his gripes with the strike zone were immediate and forceful. He complained to plate umpire Brick Owens repeatedly, and after the ball four call, Ruth and Owens met in front of the plate, where Ruth apparently threw a punch at Owens. Along with Sox catcher Chet Walker, Ruth was ejected and later received a 10-game suspension, and the Red Sox were now short a pitcher.
Shore came on in relief, getting only five warm-up throws according to the rules of the time. No matter, Shore thought, he would be replaced once another pitcher had sufficient time to warm up. Morgan was quickly caught attempting to steal second by new catcher Sam Agnew, and Shore went on to retire batter after batter after batter—26 in a row after the man was caught stealing, to complete a Red Sox no-hitter. In fact, Shore got credit for a perfect game for more than 70 years, before a baseball committee ruled in 1991 that it couldn't be regarded as a perfect game, since Shore hadn't started it.
Shore, who retired with 65 victories in seven seasons, went on to become a county sheriff in North Carolina, and he enjoyed the notoriety of his distinctive performance. He died in 1980, before his status as a perfect-game pitcher was downgraded to a combined no-hitter.
"Practically everyone has heard of me," Shore told _Sports Illustrated_ in 1962. "People are always asking me about that game. I can't say I really mind."
Real estate tycoon Joseph Lannin became sole owner of the Red Sox in 1914. He brought Babe Ruth to Boston, which helped the club capture World Series wins in 1915, 1916, and 1918. Lannin sold the team to Harry Frazee in 1917, setting the stage for The Curse.
That was considerably more than Fenway could accommodate, so the Sox were content to accept the Braves' offer to use their new playpen for the 1915 World Series. In case Carrigan and his teammates needed an added incentive, they had to stand and watch the Athletics raise the 1914 pennant on Opening Day at Shibe Park. Though injuries hobbled them early—the Sox were in fourth place at the end of May—their superior pitching eventually came into play. "I still believe, as do most of the other players in the circuit, that Boston is the really dangerous club," Tigers star Ty Cobb said in the _Globe_ on June 27.
After sweeping three home doubleheaders in three days from the Senators in early July, the Sox were on the move and by July 19 had taken over the lead from Chicago. By then the Braves were making another surge up from the cellar and for a couple of months the city was daydreaming about a Trolleycar Series, but the defending champions couldn't catch the Phillies.
The Sox essentially had wrapped things up by September 20 after taking three straight at home from the Tigers to go up by four games. "Goodby, Ty Cobb. You failed to show," taunted the _Globe_. Before they went on the road for the final seven games of the season, the Sox practiced for three days at Braves Field to get a feel for its supersized dimensions and found them suitable, if initially strange.
By the time the club faced the Phillies in the Series, it found its temporary autumnal home quite comfortable. Sox pitchers Dutch Leonard and Ernie Shore each baffled the visitors by 2-1 counts to give the Sox a 3-1 Series advantage. As it turned out, the Fens would have been overrun by the crowds, which numbered 42,300 and 41,096 for the two games at Braves Field, with thousands still outside when the gates were locked. When Boston closed out the Series in Philadelphia, it marked the beginning of a mini-dynasty, with three championships in four years.
Though he could have raised ticket prices in the wake of the championship, as most clubs do, Lannin actually lowered them for 1916, reducing a box seat from $1.50 to $1 and all but the first five rows of the grandstand to 75 cents. But even with Wood sitting out the season and Speaker dealt to the Indians, the Sox managed to repeat behind an extraordinary pitching staff. Ruth, who outdueled Washington legend Walter Johnson four times, posted a 23-12 mark, followed by Leonard (18-12), Carl Mays (18-13), Shore (16-10), and Rube Foster (14-7).
For the only time in franchise history, two Sox pitchers threw no-hitters at Fenway in the same season. "The Broadway tribe had about as much chance of getting a base knock off the Oklahoma farmer as they have of changing the situation in Mexico," the _Globe_ reckoned after Foster blanked New York, 2-0, on June 21. By the time Leonard squelched St. Louis, 4-0, on August 30, Boston was firmly in first place. "Their specialty for two years now has been beating pennant rivals in the pinch," Grantland Rice wrote in the _Globe_. "Their favorite dish is Crucial Series, frapped."
After the Sox held off Chicago by two games for the pennant, they returned to Braves Field for a World Series date with the Brooklyn Robins (as the Dodgers then were known). After Ruth stifled the visitors for 14 innings before 41,373 in the second game, it was clear that Boston had a bird in hand. "Only a Miracle Can Stop Sox," the _Globe_ declared after a 2-1 victory put them up two games going back to Brooklyn. When the Sox split there, it was left to Shore to stifle the Robins, 4-1, for the crown at home. Boston, Rice proclaimed, was "the unconquered citadel of the game."
With consecutive titles on his résumé and a world war on the boil, Lannin figured it was a propitious time to cash out, so he sold the club and the ballpark for $675,000 a few weeks after the season ended to theatrical men Harry Frazee and Hugh Ward. "I think I have turned over to the new owners the best team in the world," Lannin said, "and it is now up to them to keep the champions at the top."
With the same team back and Ruth coming into dominance, that seemed likely in 1917. While Ruth already was a gifted hitter—he hit two doubles and a triple to beat New York on Opening Day and also earned a win on the mound—his pitching was at least as notable. He won his first seven games with what the _Globe_ called "Ruthless warfare." Then on June 23, he provided the unwitting prelude to Shore's "perfect game" against Washington.
The 1918 Red Sox were led by Babe Ruth (back row, fifth from left), who won 13 games as a pitcher in his final season with the club and hit 11 home runs in a season abridged by World War I. The team defeated the Chicago Cubs in six games in the World Series.
After walking the first batter on four pitches, Ruth accused umpire Brick Owens of missing two of them. "Open your eyes and keep them open," Ruth shouted. "Get in and pitch or I will run you out of there," Owens replied. "You run me out and I will come in and bust you on the nose," threatened Ruth, who proceeded to clock Owens and had to be dragged off by Jack Barry, the club's new player-manager, and several policemen. On came Shore, who didn't allow a Senator to reach base. "I don't think I could have worked easier if I'd been sitting in a rocking chair," he later recalled.
As expected, Boston's pitching was superb, with Ruth winning 24 games and Mays winning 22. The club was in first place at the end of July. But punchless hitting did in the Sox down the stretch. "There is no use bewailing the fact that we cannot win this year," Frazee concluded just before the Tigers swept his club at home in September.
So he sent a congratulatory telegram to counterpart Charles Comiskey after the White Sox won the pennant by nine games, and then turned down the sixth-place Braves' offer of a consolation city series. "What Boston wants is a World Series and that is what the Red Sox are going after next season," Frazee stated.
The Sox indeed were back in the Series in 1918, but amid dramatically altered circumstances. With America at war and "Work or Fight" the popular antislacker slogan, baseball was regarded as a frivolous pastime. Fenway attendance dropped from 387,856 to a record-low 249,513 in a season that was chopped to 126 games and ended on Labor Day. But after Frazee acquired first baseman John "Stuffy" McInnis, pitcher Joe Bush, catcher Wally Schang, and outfielder Amos Strunk from the penniless Athletics, his club was the class of a league that had been depleted by military enlistments. And Ruth, the game's top slugger and an overpowering pitcher, undeniably was at the head of the class.
He was in fine shape after spending the winter chopping wood at his North Sudbury cottage. Ruth was an imposing woodsman at the plate that season also, hitting .300 and leading the league in homers and slugging percentage. "Just bust 'em," he told the _Globe_ , adding, "a base on balls is an obstacle on the path of progress. Take a good cut and bang that apple on the nose."
When Ruth jumped the club in July after a clash with Ed Barrow, the new manager who was hired after Barry went off to the Navy, the issue was hitting—specifically, Ruth's refusal to follow Barrow's orders. "I got as mad as a March hare and told Barrow, then and there, that I was through with him and his team," Ruth said. But he was back a few days later and it was his pitching that made the difference in the Series against Chicago.
"TESSIE" AND THE ROYAL ROOTERS
"Tessie," a Broadway show tune written by William R. Anderson, became the theme song of the Boston fans known as the Royal Rooters when they followed the Red Sox to Pittsburgh in the first World Series in 1903. The Red Sox were seen as huge underdogs to the Pirates, and though the song had nothing to do with baseball, it seemed to provide luck to the Red Sox in important games. Perhaps it didn't hurt Boston's chances that the song was frequently reworded to cast aspersions on the talents, manliness, and parentage of their opponents.
The Red Sox won that first World Series, and in 1904 the Royal Rooters accompanied the team to New York for the final two games of the season. The Red Sox needed but one victory in the two games to capture the AL pennant, a situation that would recur in 1949. The Rooters, led by "Chief" Johnny Keenan, along with Michael "Nuf Ced" McGreevy and Jerry Watson, hired a band to accompany them, and they wrapped up the 1904 AL championship in the first game with the help of a wild pitch by the Highlanders' ace, Jack Chesbro. Since the National League champion New York Giants refused to play the American League champion Red Sox, the Sox were declared unofficial world champions.
The song was reprised in the championship season of 1912, when the Red Sox christened Fenway Park with a World Series victory over the New York Giants. In 1914, the "Miracle Braves" borrowed Fenway for their home World Series games, and they swept the Philadelphia Athletics with the song as background. Over three more Series victories—in 1915, 1916, and 1918—Red Sox fans belted out "Tessie."
Before the 1915 World Series, the Philadelphia "Nationals" stated their opposition to the Royal Rooters "assembling as one body" at the Series games hosted by Philadelphia. On October 2, Red Sox president Joseph Lannin traveled to New York on the midnight train to confer with American League President Ban Johnson.
Lannin said, "I will move heaven and earth to see that they are accorded the treatment to which they are entitled. . . . The Boston Royal Rooters are known all over the country for their loyalty and gameness, and are considered as much a part of a World's Series in which a Boston team figures as are the players themselves. Whatever happens, the Royal Rooters and "Tessie" will have their accustomed places in the World's Series setting."
As a result of the negotiations, the Rooters received 400 seats for the Series games in Philadelphia.
As it turned out, the Red Sox hosted their own World Series games at Braves Field in 1915 and 1916 because the brand new park accommodated thousands more fans, and the more spacious playing field also played more to the Red Sox strengths of speed, pitching, and defense. The Red Sox drew more than 40,000 fans to four of their five home games in the two World Series and won all five games.
Late in the 1915 pennant race with the Detroit Tigers, a _Globe_ story headlined "Old 'Tessie' Still on Job" began: "Maybe the presence of the Royal Rooters had nothing to do with the ultimate result, and maybe 'Tessie' did not figure at all in the Red Sox victory, but it is a matter of history nevertheless that the Rooters and their beloved 'Tessie' were there just the same—very much there—and that the Rooters and 'Tessie' have yet to trail with a losing Boston team."
The story went on: "Three hundred loyal, lusty fans congregated on Commonwealth Ave. at the exit of the tunnel at 2:20, and at a word from Chief Johnny Keenan wended their way through the thousands of prospective spectators and again rendezvoused on Lansdowne Street, where the Royal Rooters' band was ready to take up the strains of the wonderful baseball campaign song."
They then marched to their customary position in the right-field bleachers, where they typically heckled the visiting teams while sunning themselves.
In April 1916, a _Globe_ article noted that demand for Fenway box seats had more than doubled from any previous season, perhaps a testament to the team's world championship the previous year. The Red Sox offered a 25-game ticket book for women at $12.50. Any number of the tickets could be used at any single game, and "the management expects that women will be more numerous at the games this season than ever before."
Indeed, when the Red Sox made the 1916 World Series, the Royal Rooters under chairman John M. Killeen announced that for World Series games at the home of the Brooklyn team (then called the Robins, but soon to be the Dodgers), "special transportation arrangements for women with escorts" would be included, a feature not part of the planning since 1912. The rate for traveling fans for that World Series was $37, which included round-trip train transportation (parlor cars $2 extra), automobile transport to the park and back to the Elks' Home at 43rd Street, pennants, souvenirs, and grandstand seats for five games, two in Brooklyn and three at home.
The Royal Rooters' band of the time featured 30 pieces, though it was augmented for the World Series by 20 jubilee singers in Red Sox uniforms.
Michael "Nuf Ced" McGreevy's 3rd Base Saloon, at Tremont and Ruggles Streets in Roxbury, opened in 1894. It was the principal gathering spot for the Royal Rooters and is generally considered the first sports bar in the country, with both the South End Grounds and the soon-to-be-built Huntington Avenue Grounds a few blocks away. The saloon closed in 1921, shortly after Prohibition had outlawed the sale of alcohol in 1920. McGreevy's bar got its name because it was the last stop before home for its patrons, and his own nickname reflected his habit of pounding his fist on the bar and announcing, "Nuf said," when he decided that an argument between patrons need not escalate any further.
In 2008, Ken Casey of the Boston band the Dropkick Murphys (which reworked and reprised "Tessie" in 2004) and baseball historian Peter Nash opened McGreevy's on Boylston Street, a tribute to the original bar and to Boston sports history.
The Royal Rooters temporarily ceded the megaphone to Red Sox bat boy Jerry McCarthy during the 1912 World Series.
Baseball didn't provide the only action on the field in 1912. Police had their hands full trying to push back Royal Rooters in the area known as Duffy's Cliff, where temporary seats were a coveted World Series vantage point.
A REGIONAL MEETING PLACE
Not long after Fenway Park opened for baseball in 1912, it became the venue of choice for all kinds of activities—from baseball and other organized sports to civic and religious meetings, including a massive turnout of 50,000 on June 29, 1919, for Irish political leader Eamon de Valera and the adoption of resolutions in favor of Ireland's independence.
In subsequent decades, Fenway was rarely opened for non-Red Sox events, which makes the first decade's schedule unusual. Among the hundreds of events held there in its first decade:
• In 1913, the Boston Braves were allowed to play several games at Fenway when the Red Sox were out of town. The holiday twin bills—featuring the Braves against the New York Giants on Patriots Day, and the Braves versus the Brooklyn Robins on Memorial Day—were expected to draw much larger crowds than their South End Grounds could accommodate. More than 22,000 fans watched each of the separate-admission holiday doubleheaders. Braves Field opened in 1915.
• In 1914, the Boston Lacrosse Club played the University of Toronto on June 1, immediately after the Red Sox played a game against the Washington Senators. It was noted that Red Sox President Joseph John Lannin, "an old lacrosse player himself," approved the use of the field, with baseball fans allowed to stay after the game to watch the lacrosse match free of charge.
• Later in June 1914, Boston College held a "baseball carnival" at Fenway, with the BC baseball squad dropping an 8-0 decision to Holy Cross, after Boston College High School had beaten Rindge Technical, 2-1, in the first game. The games were featured as part of BC Commencement Week, and two bands entertained some 3,500 fans between games and between innings.
• In July 1914, Fenway Park was the scene of dancing, acrobatics, band and orchestra music, and a parade, all put on, according to the _Globe_ story, "of the children, by the children and for the children." The hope was that some 30,000 children in the Boston area would spend 10 cents each in order to attend, thus raising $3,000 to benefit children in Salem, Massachusetts, left destitute by a massive fire there on June 25. Boston vaudeville theaters and "moving picture houses" were expected to provide several acrobatic displays and other acts.
• On August 17, 1914, the Progressive political party held a Fenway Park outing at which former U.S. President Theodore Roosevelt was to speak. More than 10,000 tickets were sold for the event, and Boston Mayor James M. Curley was to attend. The outing included track events and a baseball game, but partway through the ball game in the late afternoon, rain forced the cancellation of the rest of the athletic program, and Roosevelt's address was hastily moved to Boston Arena. About 4,000 people assembled for the address, which required "a great scurrying for trolley cars, taxis, and other means of conveyance from the baseball grounds to the arena," where Roosevelt spoke for an hour.
• In November 1914, the Dartmouth College football team thumped Syracuse, 40-0, before some 13,000 fans, in what the _Globe_ story called "a display of versatility in modern football which has never been surpassed by any eleven which the Boston public has had opportunity to see in action." The Syracuse team had earlier in the season defeated football powers Michigan and Carlisle, but was no match for Dartmouth, which went on to outscore its remaining nine opponents for the season by 359-25.
• Just one week later, on November 29, 1914, a combination of local All-Star football players, most from Harvard, defeated the Carlisle Indians, 13-6, before a crowd of 5,000 in a game to benefit the Children's Island Sanitarium. The game was called the last important game of the local season.
• In July 1915, it was "Natick Day" at Fenway Park, where the Red Sox played the Chicago White Sox, and the town of Natick, Massachusetts, feted one of its own—veteran American League umpire Tommy Connolly. Nearly 5,000 residents of Natick attended the game, which required "39 special electric cars to bring the greater part of the throng," along with autos and railroad trains. Practically all business in Natick was suspended for the afternoon, and umpire Connolly was honored in a pregame ceremony. A Natick representative "told umpire Connolly what a great umpire he is and how beloved he is by his fellow citizens," and Connolly was presented with a silver loving cup.
• In November 1915, Everett High defeated Waltham High, 6-0, before 12,000 fans at Fenway in "one of the very best played school football games ever seen in Greater Boston," thus winning the right to play Central High School of Detroit for the national scholastic football championship.
• On Memorial Day, 1916, about 5,000 Spanish-American war veterans formed in line at Copley Square and marched to Fenway Park for a memorial service featuring bands and drum corps. Two years later, on May 26, 1918, about 35,000 attended a memorial service presided over by Boston's Cardinal O'Connell for departed U.S. soldiers and sailors.
• On September 4, 1916, the Galway Men's Association hosted a field day at Fenway Park with several thousand in attendance. The event featured Irish football, foot races, and step dancing. The highlight was a hurling match between the Shamrocks of South Boston and the Cork Club of New York, which the hosts won handily.
• Some 10 days later, the Bay State Odd Fellows held a parade through the streets of Boston, followed by religious and patriotic services at Fenway Park, with some 14,000 in attendance. The story noted that, "it was neither too warm nor too cool. . . . This and the splendid music made the parade enjoyable for even the women, about 400 of them."
Ruth had spent much of the season playing in the outfield so that his bat could be in the lineup every day. After blanking the Cubs, 1-0, on September 5 in the road opener of the Series, he held them scoreless until the eighth inning of the fourth game, running his postseason scoreless streak (including his 1916 appearance) to 29⅔ innings, breaking Christy Mathewson's record.
Since attendance had tumbled, the Series was back at Fenway instead of Braves Field and the fifth game almost wasn't played after both teams initially refused to take the field as a protest against their reduced shares. "The players have agreed to play for the sake of the public and the wounded soldiers in the stands," Mayor Fitzgerald told the crowd after a settlement was reached. Although the Cubs prevailed, 3-0, the Sox took the championship a day later as Mays, who'd won both ends of the August 30 doubleheader against the Athletics that all but clinched the pennant, mastered the visitors, 2-1, on three hits.
The victory left a bitter aftertaste. "With many minds wandering in serious channels, it can plainly be seen that it was a fatal mistake for baseball men to argue over dollars," the _Globe_ observed, "creating a situation that should have been diplomatically squelched in its infancy." In retribution the national commission that oversaw the sport deprived the players of the diamond lapel pin that was the precursor to the championship ring.
It also was the last hurrah for the Sox, who didn't reach the Series again until 1946 and didn't win it again until 2004. It wasn't until 1934 that Boston even finished in the first division again. The 1919 season was a dismal downer. Mays, who'd won 72 games for the Sox in five seasons, left the club in mid-July and was dealt to New York just before the August trading deadline. Ruth, whose rambunctious roistering had become a clubhouse problem, squabbled with both Frazee and Barrow. But if he frequently acted as if he was above the team, it may have been because he was its colossus. Even as the club tumbled into the second division, eventually finishing fifth with its worst record (66-71) in a dozen years, the Big Fellow was its top drawing card.
It was clear to Ruth, if not his employer, that he was worth twice as much as the $10,000 per year he was earning. "Frazee knows what I want," Ruth declared as he flew off to Los Angeles to make a movie called _Headin' Home_. "And unless he meets my demands I will not play with the Boston club next year." But the thought of paying $20,000 to an ungovernable, if inimitable, man-child was anathema to the owner, who decided that he could make a far better deal with a certain gentleman in New York.
A panoramic view of Fenway Park in 1914.
IN THE NEIGHBORHOOD
BY RON DRISCOLL
When the team's popularity outgrew the Huntington Avenue Grounds (now the site of Northeastern University), the Red Sox built their new ballpark. With the same directness with which he baptized the team, John I. Taylor, whose family also owned the _Globe_ , said, "It's in the Fenway section, isn't it? Then name it Fenway Park."
The astute Taylors would not be hurt at all by this choice, as they also controlled the Fenway Realty Trust and were poised to directly benefit from development around the ballpark.
The Fens section of Boston was the centerpiece of the "Emerald Necklace" of parks designed by Frederick Law Olmsted, a planned environment of babbling brooks and green vistas, a design that held out a peaceful vision for urban America. But the stronger influence upon Fenway Park, wrote the _Globe_ 's Marty Nolan in 1986, was the unplanned, anti-pastoral engine of haphazard growth that butchered Boston's landscape: the railroad. Lansdowne Street necessitated the improbable left-field wall because the street was squeezed by the multi-lined pathway of the Boston and Albany Railroad.
"In some ways, the Fenway is Boston's secret little neighborhood," said Michael Ross, its longtime city council representative, in 2009. "You might not even notice it if you're not looking for it."
The Fenway begins where the Back Bay leaves off, at Massachusetts Avenue, and contains some of the city's landmark cultural, medical, and academic institutions: Symphony Hall, the Museum of Fine Arts, Harvard Medical School, Children's Hospital, Northeastern University, and the Boston Latin School.
It also has plenty of sports history besides Fenway Park, as the place where the Red Sox, Bruins, and Celtics all played their first home games. About one-and-a-half miles from the ballpark, the first-ever World Series game was played in 1903 at the Huntington Avenue Grounds. Boston Arena, now called Matthews Arena and home to Northeastern athletics, hosted the Bruins from their 1924 inception through 1928, when Boston Garden opened. The Celtics debuted at Boston Arena in 1946.
"Fenway Park, unlike other sports venues, is not an aloof stadium surrounded by a desolate tundra of parking. It's surrounded and hugged by real city streets. People and buildings, lights and signs seem to swirl and crash into one another in a visual metaphor of city vitality. This is the kind of urbanism that feels spontaneous, not like something overly planned."
—Robert Campbell and Peter Vanderwarker, _Globe Magazine_ , August 2004
You can find a statue of baseball's winningest pitcher, Cy Young, in front of Northeastern's Churchill Hall, positioned about where the pitcher's mound was on the old diamond. Young threw the first modern perfect game here for the Red Sox in 1904.
Across Lansdowne Street from the Green Monster is the House of Blues, which inherited the space that was long occupied by Avalon. The latest incarnation of the music club chain got off to a rollicking start in 2009 when the hometown Dropkick Murphys played six sold-out shows around St. Patrick's Day. The venue is but one of several restaurants and nightspots within a long fly ball of the Fenway bleachers.
A short distance from Fenway Park's clamor, you enter an area of three- and four-story walk-ups known as the West Fenway. "There are days when you could be in the West Fenway and not know there's a ball game going on a block away," said Ross. "It's somewhat tucked away, a little bit of an enclave."
At the end of the West Fenway's Kilmarnock Street are Park Drive and the Fens. More than just open space, the area includes the Fenway Victory Gardens, originated in 1942 as part of the war effort, and Roberto Clemente Field for athletics. Across the way is Simmons College, which straddles Avenue Louis Pasteur beside Emmanuel College.
Emmanuel, founded in 1919 as the first Roman Catholic women's college in New England, has benefited from a partnership with Merck, and it's impossible to miss the gleaming 12-story lab building that opened in 2004 on campus. Simmons has also built on its legacy as a women's college founded in 1899. In 2009, it opened one of the first green college buildings in the area, the $17 million, five-story School of Management and Academic Building.
A bit farther down Avenue Louis Pasteur is the Boston Latin School, the oldest school in the United States, having been founded in 1635. The current building dates to 1921, with an addition in 2000. Part of school lore is that Harvard University was founded so that Latin's first graduates would have a college to attend. Alas, Benjamin Franklin was a dropout.
Harvard is represented in impressive fashion at the avenue's end, where it meets Longwood Avenue. Harvard Medical School was founded in 1782, making it the third-oldest in the country, and it moved to the "great white quadrangle" of five marble buildings and a center quad in 1906.
Northeastern had a goal: to crack the top 100 in the _U.S. News & World Report_ college ratings, and it embarked on that decade-long quest in the mid-1990s. It became more selective, strengthened its faculty, and spent more than $400 million on buildings and campus enhancements. Nowhere is NU's transformation more striking than in the area of Centennial Common, just off Huntington Avenue.
The Museum of Fine Arts began in 1876 on the site of what is now the Copley Plaza Hotel in the Back Bay, and it moved to Huntington Avenue in 1909. For its centennial, the MFA embarked on a $500 million expansion and renovation that started with the reopening of its entrance on The Fenway.
Long known as one of Boston's most beautiful spots, the Gardner Museum's courtyard remains an idyllic setting for contemplation. The Gardner came into being in 1903 as one of the first buildings on the Fens, the vision of Isabella Stewart Gardner.
Lately, reminders of what is missing—13 priceless works stolen in March 1990 in the largest art heist in history—have overshadowed the 2,500 works that remain. Many of the frames still sit empty, and though dozens of leads have been tracked, the whereabouts of the stolen art, by Rembrandt, Vermeer, Degas, and Manet, remains a mystery.
## 1920s
Fronted by the city's official seal, Boston Mayor James Michael Curley threw out the first pitch at the 1924 home opener. The season was the sixth in a streak of 16 consecutive non-winning years for the Red Sox from 1919 to 1934.
As the Roaring Twenties progressed, those associated with the Red Sox must have kept telling themselves that it couldn't possibly get worse—but it did, again and again. When the Fenway grandstand caught fire in May 1926, destroying a huge swath of seats, a _Globe_ story said of team owner Robert Quinn, who had bought the club from Harry Frazee, "The Boston baseball public realizes what a difficult task he has had and has a world of sympathy for him." There was no such sympathy for Frazee, who, upon selling Babe Ruth to the Yankees in January 1920, said of the New Yorkers, "I do not mind saying I think they are taking a gamble." If it was a gamble, the Yanks hit the lottery several times over, as Ruth hit 54 home runs in his first season and led the team to six pennants in the 1920s alone. The trade was really just the first act in a continuous shunting of Boston's talent to the Yankees by Frazee. For Quinn's part, once he bought the club from Frazee in July 1923, he continued where Frazee had left off by making all the wrong moves. By 1923, the Sox didn't have a single player remaining from their 1918 world championship team; and seven of those traded played for the Yankees in the 1923 World Series game that clinched New York's first title. From the start, Boston fans flocked to Ruth's return engagements at Fenway. On his first trip back, 28,000 turned out for a doubleheader; the _Globe_ called it "one of the largest crowds . . . ever packed into the park for any game except a World's Championship contest." The story went on to say, "They saw what many of them went to see: the 'Swatting Babe' pole out a home run." One wonders at what point the sight became tiresome.
The decade barely had begun when Harry Frazee made the deal that would be credited—and cursed—for elevating one franchise while eviscerating the other. "You're going to be sore as hell at me for what I'm going to tell you," the owner informed Manager Ed Barrow. "You're going to sell the Big Fellow," Barrow figured. The price for Babe Ruth was massive for the time—$100,000 from the Yankees, plus a $300,000 loan from New York owner Jacob Ruppert with Fenway Park as security. Frazee insisted that he would have preferred getting players in return, "but no club could have given me the equivalent in men without wrecking itself."
While critics then and now claimed that Frazee wrecked his own club for more than a quarter-century, the fact was that the Sox already were headed south in the wake of their worst finish in a dozen years. For all his boisterous brilliance, Ruth hadn't seemed likely to change that. "What the fans want, I take it, and what I want, because they want it, is a winning team," said Frazee, "rather than a one-man team which finishes in sixth place."
Although the reaction from many journalists and fans ranged from shock and depression to anger and betrayal, those feelings weren't universal. "Men who have been in the baseball business generally conceded that Frazee was justified in making the sale," James O'Leary wrote in the _Globe_. The sellee, however, complained he'd been made the goat for his former club's failings. "I am going to return to Boston in the near future," Ruth proclaimed in a telegram printed on the front page of the _Globe_ , "and at that time the fireworks will start."
But it was the Sox who provided the pyrotechnics for Ruth's return, bashing New York, 6-0 and 8-3, in their Patriots Day doubleheader at Fenway and going on to win 10 of their first 12 games of the season. "I do not predict a pennant winner, but surprising things have happened in baseball and I may have a 1920 miracle crew in the present Sox," said Barrow, whose club was in first place in late May. "Who knows?"
But when New York returned to sweep their hosts in four games just before Memorial Day, it was the start of a 4-14 slump during which Boston tumbled into fourth place and never recovered. Yet even without the Big Fellow, the Sox still managed a modest upgrade, finishing one place higher than they had the previous year. And Ruth remained immensely popular in the city where he'd made his name. More than 33,000 fans turned up for a "Babe Ruth Day" doubleheader on the Saturday of Labor Day weekend when the Knights of Columbus gave him a set of diamond cuff links between games, each of which Ruth punctuated with a home run.
The Hope Diamond itself wouldn't have been enough to lure the man back to Boston, though, and his exodus was only the first in a procession of departures for the Bronx. Next was Barrow, who left after three years to become the Yankees' business manager and one of the fiscal architects of what would become the game's greatest dynasty.
Following him out the door before Christmas were pitchers Waite Hoyt and Harry Harper, second baseman Mike McNally, and catcher-outfielder Wally Schang. Thus continued what Boston faithful still call "The Rape of the Red Sox." Then outfielder Harry Hooper, who thought he'd be offered the manager's job instead of Hugh Duffy after wearing the uniform since 1909, departed for the White Sox. Hooper played in Chicago for another five seasons and went on to make the Hall of Fame. "One by one the old stars of the Red Sox leave us," the _Globe_ observed ruefully.
By then Boston undeniably was a second-division club. League president Ban Johnson pointedly ignored the Red Sox in his preseason evaluation for 1921 and only 7,500 fans turned up for the Fenway opener on April 21 to see the hosts blank the Senators, 1-0.
One of the season's few home highlights was a mid-June sweep of the Tigers, punctuated by the ejection of the combustible Ty Cobb in the ninth inning of the finale. Cobb, who'd been arguing pitch calls from the on-deck circle, berated the umpire, dropped a bat on the arbiter's foot, and then stepped on the man's heels as he followed him around the plate and, as the _Globe_ reported without specifying, "did things for which there was no justification whatever, and which led up to another incident as deplorable and more disgusting than anything that Cobb had done."
Herb Pennock won 240 games in the major leagues, and was part of two world championship teams with the Red Sox and four with the Yankees.
THE CURSE IS BORN
BY BOB RYAN
"Let me tell you this. You're going to ruin yourself and the Red Sox in Boston for a long time to come."
—Ed Barrow, Red Sox manager, when owner Harry Frazee told him he was selling Babe Ruth to the Yankees
The shocking news was delivered in the dead of winter.
The good people of New England awoke on the morning of January 6, 1920, to discover that the most beloved baseball player in town was now a Yankee.
Thus was born "The Curse of the Bambino."
Now, we all know there is no such thing. It is a whimsical hypothesis, the idea being that selling Babe Ruth was the equivalent of an original sin from which there can never be an absolution. Good throwaway line, that, and nothing more.
The truth is that it wasn't just the sale of Ruth to the Yankees that plunged the Red Sox into the abyss (nine last-place finishes, a seventh and a sixth between 1922 and 1933), a situation that wasn't remedied until Tom Yawkey bought the team and began spending tremendous sums of money.
It was the sale of Ruth and the subsequent sales and/or trades of catcher Wally Schang, shortstop Everett Scott, and pitchers Waite Hoyt, "Bullet Joe" Bush, "Sad Sam" Jones, and Herb Pennock to the Yankees that enabled the heretofore impotent team in New York to exchange places with the team that had been a four-time world champion between 1912 and 1918. When the Yankees clinched their first world championship by defeating the Giants in Game 6 of the 1923 Series, all seven played in the game.
That's the crime. It wasn't just the idea that owner Harry Frazee sold Ruth to the Yankees. It's the complete package. What he did was provide New York with the complete foundation of a dynasty. Burt Whitman of the _Boston Herald_ called it "The Rape of the Red Sox."
But selling Ruth to New York was the hot-button move for all time. For as big as Ruth was in Boston, over the next 15 years in New York he became more than just a successful baseball player. He became a true American icon.
Sixty-five years after his last game, he remains the biggest baseball star of them all. He was the sports embodiment of the Roaring Twenties, swashbuckling his way through both American League pitching and life itself. No sports reality has galled Bostonians over the past 80 years as much as the fact that We created him and They—and not just any They but the most hated They of them all—reaped the full benefits, both short- and long-term.
People were upset at the time, of course, but the feeling wasn't universal. The soon-to-be 25-year-old Babe was a highly flawed diamond. He was already starting to put on weight, he had a bad knee, and, most of all, he was a thoroughly undisciplined brat whose self-absorption, some said, was running the risk of harming the team. In fact, Frazee employed this last argument as a major rationale for selling his star, pointing out that despite Ruth's individual heroics in 1919 (a record 29 home runs while leading the league in runs batted in with 114, runs with 103, and total bases with 284—all with the dead ball), the team had finished sixth.
Citing Ruth's defiant behavior, which included missing the final game of the season in order to play a lucrative exhibition game in Connecticut, Frazee said, "It would have been impossible for us to have started the next season with Ruth and have a smooth working machine, or one that would have had any chance of being in the running."
Baseball historians have spent the past eight decades debating the incident. Was Frazee in serious debt? Did he really, really need the $450,000 he received from the Yankees (there was also a $350,000 loan involved, with Fenway Park as collateral) in order to finance a Broadway musical called _No, No Nanette_? Or was he to be taken at face value when he said he had made a decision based on what he honestly believed to be in the best interests of the team? One thing is for sure: the musical in question didn't open until 1925, so it wasn't the catalyst for the trade.
Still, Frazee is hard to defend. Ruth was a handful, but rather than spend the Yankee money to acquire more talent, as he promised, all Frazee did was sell, sell, sell. The record is clear. The Red Sox hung on with fifth-place finishes in both 1920 and 1921, but in 1922 they took up what was almost permanent residence in the league basement for the next decade, as the Yankees, utilizing all the aforementioned Red Sox stars, were winning pennants in 1921, 1922, 1923, 1926, 1927, 1928, and 1932.
Despite his future success in New York, Ruth fondly recalled most of his time in Boston, where he first lived in Mrs. Lindbergh's rooming house on Batavia Street (now Symphony Road); where he met his first wife, Helen Woodford, in a Copley Square coffee shop; where he bought an 80-acre farm in Sudbury in 1916; and where he became established as a star and a World Series hero.
As for Frazee, he left town on the midnight train for New York, a gesture of infinite symbolism for millions of Sox fans not yet born.
CHAOTIC KENMORE SQUARE
When the Red Sox play a home game, Kenmore Square is the conduit for the lion's share of the more than 35,000 fans who converge on the ballpark. And like the Fenway neighborhood itself, Kenmore Square didn't exist until the late 19th century.
The land the square sits on was then called Sewall's Point, and it was pretty much surrounded by tidal salt marsh. Sewell's Point was connected to downtown Boston by a narrow road (later to become Beacon Street) that ran atop a dam along the Charles River. When the Back Bay was filled in the late 19th century, the former dam road became Beacon Street, which connected to Brookline Avenue. A short time later, Commonwealth Avenue was constructed, and the three roads converged at what became known as Governor's Square.
Governor's Square (renamed Kenmore Square in 1932) became an important local transportation hub. The Peerless Motor Car Building on the west side of the square now houses Boston University's Barnes and Noble Bookstore, but its main claim to fame is the Citgo sign on its roof; it was born in the 1950s as a Cities Services billboard, and it became the landmark neon beacon in the 1960s.
Because of the local college-student population, Kenmore Square and the streets close to the ballpark feature plenty of restaurants, cafes, and music venues. The most famous jazz club was Storyville, which was located at the Hotel Buckminster starting in 1950. Legends such as Dave Brubeck, Louis Armstrong, Charlie Parker, and Sarah Vaughan played the club, which was in the ground-floor space of the Buckminster now occupied by Pizzeria Uno. The legendary rock club The Rathskeller, a.k.a. "The Rat," played grimy host to some of rock music's great bands in the 1970s and 1980s as they paid their dues, including The Police, the B-52s, R.E.M., U2, the Ramones, Tom Petty, Blondie, and Sonic Youth. It closed in 1997, and its site is now occupied by the Hotel Commonwealth, which opened in 2003.
On Patriots Day, more than 20,000 official runners pass through the square on the home stretch of the Boston Marathon, the world's oldest annual marathon. Hundreds of thousands of spectators root the runners on, and the square offers a convergence of race day and Red Sox fans spilling out of the traditional 11 a.m. holiday game.
The square was once noted for its hotels, including the Buckminster, at the corner of Beacon Street and Brookline Avenue, which was designed by Stanford White. It was the site of the first network radio broadcast, and it also played a part in the infamous "Black Sox" baseball scandal. On Sept. 18, 1919, the same day that the Chicago White Sox defeated the Red Sox, 3-2, at Fenway, bookmaker and gambler Joseph "Sport" Sullivan went to the hotel room of Arnold "Chick" Gandil, White Sox first baseman. There they hatched a plot to fix the 1919 World Series, which was to start 13 days later.
In 1915, the Kenmore Apartments building opened at the corner of Kenmore Street and Commonwealth Avenue. It later became the Hotel Kenmore, an elegant, 400-room operation that was once Boston's baseball headquarters—at one time in the late 1940s when the Braves still played in Boston, all 14 visiting major-league clubs stayed there. Countless trades were made, managers hired and fired, and post-game parties featured celebrities of the day.
"As I grew up, I knew that [Fenway Park] was on the level of Mount Olympus, the Pyramid at Giza, the nation's Capitol, the czar's Winter Palace, and the Louvre—except, of course, that it is better than all those inconsequential places."
—Former Major League Baseball Commissioner Bart Giamatti
After key injuries led to a ruined July, the club rallied with a strong finish and flirted with third place before slipping back to fifth. But that would be the best showing for more than a dozen years as the talent exodus to the Bronx continued during the offseason. Frazee swapped shortstop Everett Scott, who'd played nearly 1,100 games for Boston, plus pitchers "Sad Sam" Jones and "Bullet Joe" Bush for shortstop Roger Peckinpaugh and three hurlers. While critics lambasted the owner for continuing his yard sale, his money had paid for the furnishings. "It is Frazee's team," the _Globe_ 's James O'Leary reminded readers, "and if he has goldbricked himself he is the one who will suffer."
By Opening Day in 1922, nobody from the 1918 champions remained on the Red Sox roster. Even the stockings had been changed to ones with a dark stripe. "Picking red socks for the boys must have been left to someone who is color-blind," Mel Webb observed in the _Globe_.
Though the club won four of five from the Yankees at home in late June, Boston couldn't replace Jones and Bush, who won 39 games for New York that year. After their rotation fell apart, the Sox quickly sank from sight in July, dropping six in a row to the Indians and Tigers (the last by a 16-7 count). The club ended up losing 93 games, its most in a season since 1905, and finished in the cellar 33 games behind the Yankees. Since 1915, that had been the residence of the Athletics. Except for one season, Boston would be the new annual tenant there until Tom Yawkey bought the club in 1933.
That was the end for Duffy, who was kept on as a scout and "general all-round man." In came Frank Chance, the "Peerless Leader," who as player-manager had led the Cubs to world championships in 1907 and 1908 and to the National League pennant in 1906 and 1910. Chance, who had no illusions about what he was inheriting in the Hub, reckoned that it would take at least three years to transform the Sox into contenders.
With only a handful of regulars returning, the roster obviously was a reconstruction zone in 1923. And while Frazee predicted that the club "will be the finest, smartest lot of youngsters ever hired by a major-league ball club," Boston essentially had become a minor-league franchise. After losing the first four games in New York, the Sox had dropped to the bottom by May 12 and never inched higher than sixth for the duration. After a brutal 27-3 loss at Cleveland on July 7—when reliever Lefty O'Doul gave up a club-record 13 runs in the sixth inning—Chance knew that the task exceeded his enthusiasm and endurance. "I have a one-year contract and that is enough," he told a former Cubs director.
Michael William "Leaping Mike" Menosky played left field for the Red Sox after Babe Ruth was sold to the Yankees. He was with the Red Sox for four years, during which time he hit a total of nine home runs.
HARRY FRAZEE: THE MAN BEHIND THE CURSE
BY DAN SHAUGHNESSY
On Monday, January 5, 1920, the Harvard University football team, still celebrating its New Year's Day, 7-6, Rose Bowl victory over Oregon, rolled eastward into Chicago on the California Limited. In Washington, D.C., in a 5-4 decision rendered by Justice Louis D. Brandeis, the Supreme Court upheld the right of Congress to define intoxicating liquors, sustaining the constitutionality of provisions in the Volstead Act. Elsewhere, the last of the U.S. troops in France made their way home across the Atlantic, and a New York Supreme Court justice ruled that it was not immoral for women to smoke cigarettes.
There was one more bit of news that day. Late in the afternoon, Harry Frazee held a press conference and announced that slugger-pitcher George Herman "Babe" Ruth had been sold for cash to the New York Yankees.
"The price was something enormous, but I do not care to name the figures," said Frazee that day. "No other club could afford to give the amount the Yankees have paid for him, and I do not mind saying I think they are taking a gamble."
Prohibition was 11 days away when Frazee made this move, which would drive Sox fans to drink.
There was some outrage when the Ruth transaction was announced but none of the hysteria that would accompany such a transaction in today's age of media overkill. The sale of Babe Ruth to Gotham was front-page news in all the Boston papers. John J. Hallahan of the _Evening Globe_ led his story with: "Boston's greatest baseball player has been cast adrift. George H. Ruth, the middle initial apparently standing for 'Hercules,' maker of home runs and the most colorful star in the game today, became the property of the New York Yankees yesterday afternoon." A newspaper cartoon showed Faneuil Hall and the Boston Public Library wearing "For Sale" signs.
In his autobiography, Ruth admitted, "As for my reaction over coming to the big town, at first I was pleased, largely because it meant more money. Then I got the bad feeling we all have when we pull up our roots. My home, all my connections, affiliations and friends were in Boston. The town had been good to me."
Frazee's name was mud in Boston, just as it is now. One night he was out in Boston with character actor Walter Catlett. In an attempt to impress a pair of young ladies, Frazee had a cab driver take the group to Fenway Park. He got out of the cab and proudly displayed his baseball empire. The cab driver overheard the boasts and asked if this passenger was in fact Harry H. Frazee, owner of the Red Sox. Frazee said he was, and the driver decked him with one punch.
On July 11, 1923, Frazee sold the Red Sox to Robert Quinn for $1.25 million. The 1923 Red Sox did not have one player left from the championship season of 1918. The man who did the dirty deed didn't care anymore. While the Sox stumbled through the Roaring Twenties, Frazee finally hit the mother lode in 1925 with No, No, Nanette. It had a New York run of 321 performances and was one of the most successful shows of the 1920s, earning more than $2.5 million for Frazee.
But Frazee didn't have much time to enjoy his money. The shows after Nanette didn't do as well, and on June 4, 1929, four weeks shy of his 49th birthday, Harry H. Frazee, or "Big Harry" as he is known in family lore, died of kidney problems at his home in New York City. Frazee always said that the best thing about Boston was the train to New York, and New York City Mayor Jimmy Walker was at Frazee's bedside when he died.
Ruth went on to establish himself as his sport's greatest performer. He set a major-league record with 60 home runs in 1927 and was still the idol of millions of Americans when he died of cancer in 1948. His record of 714 career home runs stood until Hank Aaron passed him in 1974. He was one of the original five players enshrined in baseball's Hall of Fame.
Red Sox owner Harry Frazee (left) and manager Frank Chance huddled at Yankee Stadium in 1923. Frazee sold the franchise and park in July of that year.
RUTH'S TRIUMPHANT RETURN
Were Bostonians anxiously awaiting Babe Ruth's return after his trade to the Yankees in January 1920? The day before the Yankees and Red Sox squared off at Fenway, Harry Hooper won a game for the Red Sox against the St. Louis Browns with a home run in the last of the 11th inning. One headline said, "Babe Ruth missed? Not when Harry Hooper decides to settle game with his own bat."
When the Yankees arrived on May 27, Ruth hit a pair of home runs, but was nearly upstaged by a brawl that started with an umpire and a pitcher squaring off. (The headline read: "Player and umpire fight; Ruth hits two home runs.") Obviously, there was nary a dull moment, although surprisingly, only an estimated 11,000 attended.
Those fans witnessed "[Pitcher Bob] Shawkey of the visitors attack umpire [George] Hildebrand, who sideswiped the player on the head with his mask. The fans had then watched with consuming interest a regular football scrimmage in which 10 or 15 would-be peacemakers were trying to restrain the belligerents; they had seen Ruth, the demon home run swatter, knock the ball out of the lot twice . . . and altogether, they had had a great day."
Regarding Ruth, "many in the big crowd had come to see him make a home run, one having been his limit each of the previous three days." In the sixth inning of New York's 6-1 win, Ruth homered into the right-field bleachers, "a mighty drive." In the eighth inning, Ruth "knocked the ball over the left-field fence, an unusual hit for a left-handed batter against a right-handed pitcher. The ball hit the top of the fence and bounded 50 feet beyond, across Lansdowne Street. It was about the same kind of a hit he made on Ruth Day last season." Ruth would go on to hit 54 homers in 1920; the player with the next-highest total in the league only managed to hit 19.
The next day at Fenway, the story was the assembled crowd, though Ruth again played a prominent role, according to the Globe's James C. O'Leary: "Babe Ruth and the Yankees, but mostly Ruth, made a cleanup of the Boston-New York series by winning both games of the doubleheader at Fenway Park in the presence of 28,000 fans, one of the largest crowds—and certainly the biggest money crowd—ever packed into the park for any game except a World's Championship contest. They saw what many of them went to see: the 'Swatting Babe' pole out a home run. With one on, he hit the ball high over the clock which tops the left-field fence."
The Yankees won the games, 4-3 and 8-3, on a day in which the Red Sox fielding "was far below their standard, and altogether they had a decidedly off-day." The large crowd necessitated roping off the field, with any hit going into the crowd that ringed the outfield against the fence being ruled a ground-rule double. "The overflow of 5,000 or 6,000 spread out onto what ordinarily is used as the playing field, stretching from back of third base around by way of the terrace in left field (Duffy's Cliff), across center to and across the right foul line. Many of those in the 50-cent seats hopped the fence and became a part of this mass of humanity, which included many women, who were unable to get even standing room in the grandstand."
Ruth hit his homer in the fourth inning of Game 1: "With a runner on base, two out and himself crippled with two strikes—though that does not appear to be a disability so far as 'Babe' is concerned—Ruth developed two runs in the twinkling of an eye by sloughing the ball over the left-field fence."
The first game ended pitifully for the Red Sox, as they had the bases loaded with none out in the last of the ninth, trailing by one run, but could not push even the tying run across. Yankees' pitcher Jack Quinn induced two force plays at the plate, and the game ended on another force play, this time at second base.
In the second game, Ruth banged out a double, but the story noted, "It looked as if Ruth's double to right was going over into the bleachers, but [Harry] Hooper would easily have captured it if he could have played where he usually sets himself with Ruth at bat."
After seven years as owner, Frazee wanted out as well. On July 11, he agreed to sell the franchise and the park for a reported $1.25 million to Robert Quinn, the St. Louis Browns business manager who headed a Columbus syndicate and immediately asked fans for a rain check on the rest of the season. "I bespeak patience on the part of the Boston baseball public with my efforts," he said.
League President Ban Johnson, who despised Frazee, was delighted with the change. "Quinn will have a good team here before you know it," he predicted. But the manager knew otherwise. "The new owners of the Boston club have the franchise and the park, but they must get a ball club," Chance said in late August, and then told his players that he'd been misquoted.
The Yankees, who pulverized Boston, 24-4, at Fenway on September 28, with Ruth producing a three-run homer, two doubles, and two singles in six at-bats, now had most of the old club, as more than a dozen former Sox won World Series rings that year with New York. What the Sox had for 1924 was a ball bag full of optimism fueled by Quinn's promise to spend hundreds of thousands of dollars on players, and a new skipper in the person of Lee Fohl, the former catcher for Pittsburgh and Cincinnati who'd produced winning seasons managing the Indians and Browns. "That Quinn and Fohl and time are a winning combination has already been demonstrated," O'Leary wrote, "and there is no reason to anticipate any reversal of form."
The Sox blossomed early that year and found themselves in first place in early June but they went to seed in July, losing nine straight at Fenway, including an 18-1 battering by the Tigers. While the seventh-place showing was a slight upgrade from the basement and attendance doubled to nearly 450,000, the season was an outlier—a misleading prelude to the three worst consecutive years in franchise history.
In the wake of the previous season's fade, Quinn was circumspect about the club's prospects for 1925. "All I can say is that we have hopes," he stated in February. But by the time the Sox played their home opener against Philadelphia, they already had dropped five of their first six games en route to a 2-12 start that essentially interred them. "Once again we hear, 'What's the matter?'" Mel Webb observed in the _Globe_. "Just now everything is the matter."
The Sox ended up losing 105 games, finishing nearly 50 behind the Senators. They were last on merit. Boston had the worst offense (640 runs scored) and worst defense (921 runs allowed) in the league and absorbed scoreboard-busting home beatings—the Athletics lashed their hosts by 15-4, 15-2, and 12-2 margins.
FIRE SWEEPS THROUGH THE GRANDSTANDS
As if their on-field woes were not enough, the Red Sox were dealt a huge blow in May of 1926 when a fire swept through Fenway's third-base grandstand and briefly threatened the entire ballpark. The fire, on the night of May 8, had followed a spate of small fires under the grandstand during a game the previous day, and caused an estimated $26,705 in damage.
The headline in the _Globe_ of May 10 described the third-base "bleachers" as a mass of ruins and told of plans to put a concrete stand in place later. The team co-owner and president, Robert Quinn, already financially strapped, never implemented the plan, and until Tom Yawkey bought the team in 1933, the park had a gap where the burned stands had been torn out. This meant that for the next several years, the area occupied today roughly by grandstand sections 29 to 33 was an open area that was in play. This made for a huge foul territory, and the left fielder often disappeared from view chasing balls, with the bases umpire scurrying after him to rule on the play.
The _Globe_ story by James O'Leary also said: "The destruction of the bleachers was complete, only charred timbers and boards remaining. The boardwalk on the roof leading to the press box caught fire in two or three places, and a small blaze started under the floor of the press box; only fine work by the fire department saved the grandstand. The burnt section of the bleachers will be roped off today, and when the debris has been cleared away, a concrete stand will be erected. . . .
"Since he bought the club a little more than two years ago, Pres. Quinn has spent between $65,000 and $70,000 in renovating the plant and bringing it up to date. Mr. Quinn appeared yesterday to be more concerned over the poor showing, thus far, of his ball club than he was about the fire loss. . . .
"President Quinn had been tendered the use of Braves Field for as long as he wished to make use of it by the Boston National League club, but finds that he will not have to avail himself of the courtesy."
The story ended: "The Boston baseball public realizes what a difficult task he has had and has a world of sympathy for him."
BC DEFEATS HOLY CROSS
More than 30,000 football fans turned out on the last day of November 1929 to watch the archrival Boston College and Holy Cross squads square off in their annual contest, with BC taking the victory, 12-0. The _Globe_ stories of the next day described the scene in the ballpark and in the streets around Fenway.
In the game story, Melville E. Webb Jr. wrote: "In freezing weather for player and spectator alike, the Eagle[s], superior on attack, unpassable on defense, twice scored on its Worcester rival. One touchdown, a fierce, unbroken march for more than 25 yards followed the recovery of the ball on a flubbed kick. A second touchdown scored as the shadows were dark upon the flinty field during the closing moments, when Michael Vodoklys seized a Crusader forward pass, hurled from behind the Purple's goal, and tore relentlessly back to a point behind the Worcester posts. . . .
"The chill numbed but did not drive many from the hard-fought game. . . . No more grueling, slashing, desperate football fight ever has been waged between the forces of Boston College and Holy Cross than that on the Fenway battleground yesterday. No game ever was more productive of exciting thrills or more frequently marked by error, for the most part almost instantly redeemed. . . .
"Two hours freezing play, yet in all save less than five minutes of it a Worcester team threatening ever but never coming through. On Boston's door the Crusader was thumping all day long—but the stronghold never once gave way."
The postgame scene was described in another story: "Fewer than 10,000 persons who attended the game parked cars in the nearby streets, and when the flood of shivering fans rolled out after the game there was no long delay or jam. About 200 policemen helped enforce the rules, which prohibited parking on Brookline Avenue, Jersey Street, and Commonwealth Avenue, near Governor Square (known as Kenmore Square starting in 1932). Jersey Street and Brookline Avenue were made one-way streets during the game.
"After the game, about 500 BC students and friends who had secured a parade permit to march from the field to the Common via Commonwealth Avenue started a celebration and snake dance. The lines of cheering students were admirably handled by the police so as not to conflict with traffic, but the victors' enthusiasm wore off quickly and none of the group got much farther than Massachusetts Avenue before seeking shelter.
"Some careless motorists found their radiators had frozen during the game, and nearby garages did a rushing business thawing them out."
"I never believed in 'crying over spilled milk,'" Quinn declared as the campaign was winding down in September with attendance nearly halved. "And there isn't any use in alibiing a team which has finished last in every department of play." Only a rainout of the season's finale with Washington prevented the club from finishing with the worst record in franchise history.
But the bad news didn't end with the season. In November, the state board of appeals ruled that Quinn had to pay an additional $27,575 in taxes on the club's profits from the sale of Ruth and Mays to New York.
Since it didn't seem possible that things could be worse in 1926, even Kenesaw Mountain Landis sensed at least a possibility of improvement. "You seem to have a fine lot of athletes here and I wish you all kinds of luck," the commissioner of baseball told Quinn at the club's New Orleans training camp, saying that he'd wager "a golf stick or two" that Boston would pull itself out of the basement.
When the Sox came from 10 runs down in the fifth inning to nearly catch the Yankees on Opening Day in the Fens, it seemed that they might at least quicken heartbeats. "If you are inclined to apoplexy, heart trouble, shocks or faints don't spend your afternoons at Fenway Park," the _Globe_ 's Ford Sawyer cautioned after the riveting 12-11 loss.
But after the hosts went down, 11-2, to the Indians on May 7, the distress signals were unmistakable. "The Sinking of the Ship. A Farce-Comedy in Nine Acts directed by T. Speaker," read the _Globe_ headline the next morning. A day later, the theater itself was charred by a three-alarm fire that ravaged the third-base bleachers.
The front office, which had far more seats than it needed for a last-place enterprise, simply roped off the area and continued as before. The owner was more concerned about his ramshackle club, which was a far more extensive and expensive renovation project. "We get players who have been highly recommended, and who were desired by other clubs," said Quinn, "but when we get them they do not seem to hold up for us."
By Memorial Day, all that his club was holding up was the rest of the league. The Sox were 18½ games behind, en route to their poorest campaign ever (46-107). After they lost their final 14 games, all at home, Quinn was looking for a savior who actually had seen a pennant flying over the premises.
The obvious candidate was Bill Carrigan, the former catcher-manager who'd led the club to its 1915 and 1916 championships. Carrigan, however, was content with his life in Maine and had no desire to return to the dugout. "I shall not get back in the game," he declared in October after Fohl had resigned. But by December, after visiting with Quinn, Carrigan found himself back on the payroll. "I got talking baseball," he said, "and before I knew it I was manager of the Red Sox again."
No longer a member of the Red Sox, the Babe still went to bat for some Boston causes, including this Girl Scout Day promotion at Fenway in 1923.
He was, Sawyer wrote, "the Moses who is expected to lead a downcast Boston aggregation out of the wilderness of defeat and disappointment." But Carrigan would have had a better chance of parting the Charles River than of leading the bedraggled Sox to the first division, much less the promised land. His club, a 50-1 long shot to win the 1927 pennant, was dead on arrival by Opening Day.
The Sox dropped the season's first game by four runs at Washington, as the skipper "exhorted, wheedled, commanded, coaxed, bullied, and pleaded." They were swept by the Senators, and then lost three of four games at New York, including a 14-2 blasting. By Labor Day, the only reason to come to the park was to watch Ruth, Lou Gehrig, and the rest of the Yankees' "Murderer's Row" launch baseballs skyward. Nearly 35,000 showed up for the doubleheader with New York, the largest Fenway crowd since 1915, with the _Globe_ reporting that "other thousands crashed the barriers, broke them down and swept into the grounds."
So fans were startled to see the Sox outlast the Yankees, 12-11, in the 18-inning opener, with Red Ruffing pitching the first 15 innings for the home side. The game lasted so long—four hours and 20 minutes—that the nightcap had to be shortened to five innings, which were completed in a brisk 55 minutes with New York winning, 5-0. The clubs split another twin bill the next day, with Ruth clouting three homers. The first, which cleared the center-field fence to the right of the flagpole, was deemed the longest hit at Fenway up to that point. "Nobody at the park could tell where it landed," wrote O'Leary, "but when it disappeared it was headed for the Charles River Basin."
The Yankees went on to claim the pennant by a whopping 19 games and swept the Pirates in the World Series, as Boston was buried 40 games behind in the cellar. Still, Carrigan was optimistic about his men's chances for 1928, predicting that they'd be "a better ball club in many ways." Indeed, the Sox climbed out of the basement amid a May blossoming that had them in fourth place after six home victories in a row. "The time-worn theory that there are only eight basic jokes will have to be revised," the _New York Post_ suggested. "The Boston Red Sox have climbed into the first division in the American League."
Even Quinn was intrigued by that novelty, announcing that he would spend a million dollars to double-deck the ballpark and increase capacity to more than 52,000—as soon as his club was good enough to make it necessary. Acrophobia and gravity proved a fatal combination, though, and the Sox soon began free falling. Before the end of July they were back at the bottom of the standings and the _Globe_ provided an early obituary ("A Sad Decline") in early September. "On the diamond we produce, instead of a succession of championship teams, a perpetuation of tailenders," the editorial concluded.
Until Terry Francona duplicated the feat in 2007, Bill Carrigan was the only manager to have won two World Series titles with Boston (in 1915 and 1916). He returned to manage the Red Sox in 1927, but couldn't do better than last place in each of his three seasons.
After four years of viewing the league from an upside-down perspective, the Sox were justifiably reserved about their prospects for 1929. "I am predicting nothing, but I am hopeful," Carrigan said before the season. An Opening Day triumph over the Yankees at home was a splendid start, but Boston already was sending out distress signals on May Day when the Athletics dispensed a 24-6 drubbing that at the time set a Fenway record for offense—by the visitors. The _Boston Globe_ tallied what it described as "a terrific cyclone of bingles of all descriptions"—29 hits, 44 total bases (11 by Jimmie Foxx), three homers, and six doubles, with 10 Philadelphia runs coming in the sixth inning.
By mid-month, the season already was a lost cause for Boston's two baseball teams, which both finished eighth in their respective leagues. "In a postseason series between the Braves and the Red Sox, which would win?" the _Globe_ mused in October. "Don't you mean post-mortem series?" retorted the _Brockton Enterprise_.
## 1930s
Lefty Grove, who won 105 games for the Red Sox between 1934 and 1941, watches the action from the dugout.
By the time the 1930s were in the rearview mirror, Fenway Park itself and its major inhabitants had undergone a transformation. The team and the ballpark got a new owner in 1933, and the Red Sox made a slow climb from being cellar dwellers in nine out of 11 seasons through 1932 to a pair of second-place finishes in 1938-39. This effort was no doubt helped along by Tom Yawkey's inclination to spend his considerable wealth on players he thought could help. The park itself also benefited from the infusion of Yawkey's cash and enthusiasm. Although a five-alarm fire undid many of the off-season renovations that had cost nearly $1 million to complete by January 1934, Sox General Manager Eddie Collins vowed that the team would still open the season in a retooled Fenway. It required a massive additional commitment of money and manpower, but Collins was correct. For probably the first time, the Red Sox attempted to tailor their home field to take advantage of the presence of a slugger. Rookie Ted Williams had hit .327 with 31 home runs and an amazing 145 RBI in 1939. So bullpens were constructed in right and right-center field during the next off-season, with the expectation that Williams would be able to reach the seats (or at least the bullpens) more often en route to a potential Triple Crown-winning season. Williams, feeling the weight of expectation, hit fewer homers in 1940, and his on-again, off-again relationship with fans—and his open feud with newspaper writers—hit a significant rough patch. "There were 49 million newspapers in Boston, from the _Globe_ to the _Brookline Something-or-Other_ , all ready to jump us," said Ted years later. Indeed, it must have seemed that way. The Boston Braves (later Redskins) of the National Football League debuted in this decade, but the NFL failed to rouse enough local support, so the Redskins left for Washington after the 1936 season.
A Red Sox-Yankees doubleheader brought an early-morning crowd to Fenway on August 12, 1937.
As the thirties began, the Depression in the Fens already had been underway for a decade. Defeated and dispirited after three dreary years in the league basement, the skipper, who'd been used to pennants flapping during his days as player-manager, threw up his hands. "After handling three tailenders Bill Carrigan decided that he would not try again to start the Red Sox on an upward journey," Mel Webb wrote in the _Globe_.
That task fell to Charles "Heinie" Wagner, Carrigan's assistant and former teammate who'd seen enough of his players to be realistic. "All we need is a little quiet discipline and a little time," he said after inheriting the job just before Christmas of 1929. "You can't bring a ball team up to the top in a minute."
It should have been an omen when President Herbert Hoover threw out the first pitch for Boston's initial game at Washington and bounced the ball. Before the next day's home opener against the Senators, which the Sox dropped by a 6-1 count before 7,500 chilled fans, Wagner was presented with an enormous floral horseshoe that required three men to shoulder to the plate.
It was intended as a good luck gesture, but it might as well have been a funeral wreath as his club was all but buried by Memorial Day, falling back into the cellar after dropping 14 straight games. The most fortunate man on the premises was pitcher Charles "Red" Ruffing, who'd led the league in losses for two years. He was dealt to the Yankees that spring and went on to earn a half-dozen World Series rings and make the Hall of Fame.
Despite 102 losses and a sixth consecutive last-place finish, owner Bob Quinn was ebullient about his club's chances for 1931 under new skipper John "Shano" Collins, a Charlestown native who'd already been knocked around enough for a lifetime. He'd been named the victim in the case against the Chicago Black Sox, eight of whom were accused of conspiring to throw the 1919 World Series, which cost Collins $1,784 according to court documents, and then he had been traded to Boston just in time for the Sox downward spiral.
So even though the Red Sox were at the bottom of the standings and 41 games out at the beginning of the Labor Day weekend, Collins was optimistic. "I am confident that some other club will finish in last place," he predicted before his men dropped a doubleheader on September 4 to the Athletics. By then the only bit of suspense was whether Earl Webb would break the record for most doubles in a season.
Webb, who'd mined coal before he made the majors, was a journeyman outfielder whose erratic glove cost his team nearly as many two-baggers as he produced. But he banged out 67 doubles that year with only three triples while knocking in 103 runs and hitting .333. Though Webb was suspected of deliberately holding up at second, Collins said that "The Earl of Doublin" simply was "too darned slow on the bases to get to third."
The Sox managed to crawl upward to sixth place, their best finish in a decade. But by the middle of the 1932 season Collins had concluded that they were past the point of mending and quit on June 19. "I have worked unusually hard with the Red Sox," he said. "I have learned, however, that I was more or less on a treadmill and not going any place in particular."
So Quinn tapped infielder Marty McManus to take over what he deemed "the most thoroughly demoralized ball club that ever existed." Quinn, who'd been struggling financially even before the Depression, was demoralized as well. His club soon fell back to last place and ended up a whopping 64 games behind the Yankees after losing 111 games, the most in franchise history. Attendance plunged by half to 182,000, roughly a third of what the Braves were drawing on Commonwealth Avenue.
What Quinn needed was either a savior or a sucker who would take the club off his hands for a price, as he put it. The man was Thomas Austin Yawkey, a 30-year-old Yale grad who'd just inherited a fortune from his family's lumber and mining interests. On February 25, four days after his birthday, Yawkey purchased both Fenway Park and the Sox for $1.2 million. Tom's uncle and adoptive father, Bill Yawkey, had owned the Detroit Tigers from 1903 to 1919 and saw them win pennants in 1907, 1908, and 1909. Tom, who had the same aspirations, immediately began a refurbishment of both the park and the club. "Painters, plasterers and carpenters were scattered about the plant and soon everything will be in order," the _Globe_ reported a few days before the 1933 opening game of a city series with the Braves on April 8 (which the Sox won, 7-0).
Yawkey's first priority was expanding the bleachers, where he believed the real fans congregated. "I may be mistaken but I think the grandstand fan is a casual—he comes to the game in much the same mood and manner that the theatre-goer goes to a popular hit," he said. Renovating the roster was a more daunting challenge, but Yawkey was quick to start upgrading the Sox from 500-1 long shots to contenders, paying top dollar and over-thetop dollar for anyone he could grab.
SUNDAYS IN THE PARK
The legendary Massachusetts Blue Laws, which set aside Sunday as a day of worship and rest, prohibited Boston's professional baseball teams from hosting Sunday home games from their very beginnings. Boston was not alone in banning Sunday baseball, but by 1918, all but three American League cities—Boston, Baltimore, and Philadelphia—had allowed it. The state law was amended in 1929 to allow for Sunday baseball, but the Red Sox were still stymied from playing at Fenway Park because of its proximity to a church. They played their Sunday contests at Braves Field for the next few seasons, until they caught a break from legislators.
In May 1932, the Massachusetts House of Representatives sponsored a bill that would loosen the restriction and allow the Sox to play at Fenway on Sundays. Specifically, the bill would lower the required church buffer zone from 1,000 feet to 700 feet, and the Church of the Disciples, at Jersey and Peterborough Streets, was about 850 feet away from Fenway. The church raised no objection, and the bill passed the state Senate on May 19, 1932. The Sox hosted their first Sunday game at Fenway on July 3.
If they were hoping to come out winners in their first-ever Fenway game on the Sabbath, the Sox might have chosen a more fortuitous season and a less daunting foe than the Bronx Bombers. In a game that took only 2 hours and 28 minutes to complete before a crowd of 7,000, the Yankees trounced the Sox, 13-2. As a result, the Yankees improved their 1932 record to 50-21, while the Red Sox were comfortably settled in last place at 14-57. Boston would go on to finish 43-111 that season, their worst record ever—64 games behind first-place New York.
The headline of the story said: "Ten Hits in Sixth, in which 14 Hostiles Go to Bat, Convert Game into Parade." Dave "the Colonel" Egan, the _Globe_ baseball writer, wrote that the nine-run Yankee inning was filled with "carnage and sabotage and rioting."
Egan went on to write, "Ivy Paul Andrews, late of the Yankees, was the unfortunate youth upon whom the wrath of the New Yor-curs fell. He seen his duty and done it noble for the first five innings, but when the smoke of battle had cleared in the sixth, nine runs had been scored, the Messrs Andrews and Pete Jablonowski were weeping on each other's shoulders in the showers, and Bob Kline was pitching and ducking.
"In that sordid sixth, 14 of the visitors paraded to the plate, assumed a battling posture, and collected 10 hits, thus reaching a new high for the year and convincing the experts that the Depression is over. And George Pipgras upset all the fine traditions of the pitching industry by making two singles in that one stretch. There should be a law against it."
LADY WITH A MEGAPHONE
Mrs. Lillian Hopkins was known to thousands of Red Sox fans and players simply as "Lolly." A lifelong resident of Providence, she was awarded a lifetime pass to Fenway Park as the team's No. 1 fan. She made the 100-mile round-trip hundreds of times over 27 seasons between 1932 and shortly before her death in September 1959 at age 69.
Lolly occupied Seat 24 in Row 1 of Section 14, and she always came to Fenway with a megaphone and a scorebook. Many fans never met her but recognized her voice as she hollered advice and encouragement to players, managers, and umpires. According to a 1959 feature story in the _Globe_ , "Lolly had become a baseball expert through the years and never hesitated to prove it. Many have felt the good-natured wrath with which she would set them straight."
One woman asked in 1958, "Why doesn't she go over to third base, so Williams can hear her better?"
"It's habit, a habit I developed when I was a little girl and my father used to bring me up from Providence to see games at the Walpole Street and Huntington Avenue Grounds," Lolly explained. "I always tried to get seats in that section if possible."
The late Smoky Kelleher, a sports official and a Red Sox fan to rival Lolly in loyalty, used to sit in a box in front of her, where he became accustomed to her hollering. One day in 1938, he gave her a megaphone, telling her it would save her voice. Lolly hollered as loud as ever; the megaphone merely multiplied her range.
Today, Lolly's passion for the Red Sox is preserved in the Baseball Hall of Fame. A life-size figure of her, megaphone and all, is front and center in an exhibit of some of the game's most beloved fans.
His biggest blockbuster came shortly before Christmas of 1933 when he dispatched two players and $120,000 to the penniless Athletics for pitchers Lefty Grove and Rube Walberg and second baseman Max Bishop. "Yawkey appears to be Boston baseball Santa," a _Globe_ headline declared. "Has quite an array of Sox to hang up for local fans."
Yawkey had more than enough cash to fund his horse-hide hobby. He owned a massive South Carolina plantation, a New York apartment, and soon acquired a suite at the Ritz in Boston. He also supplied fare for the Yuletide groaning board, sending up duck, quail, and venison that he and a few of his new Sox employees had shot during a hunting trip on his land.
There would be a new manager as well. Though McManus had nudged the Sox up from the cellar, Yawkey and new Sox General Manager Eddie Collins, the future Hall of Famer who'd been one of Yawkey's boyhood heroes at the Irving School in New York, wanted a bigger presence. "If we could find a second edition of Connie Mack, that would be our idea of the perfect manager," Collins remarked.
He hired Stanley "Bucky" Harris, the "Boy Manager" who'd directed the once-woeful Senators to two pennants and a Series championship before he'd turned 30 and who'd just been cut loose by Detroit. Harris inherited an expensive new lineup for 1934 and an even more expensive playpen that Yawkey had to renovate twice after a January fire turned much of his new handiwork into cinders. Though the construction costs were soaring for what would be the city's biggest private building project of the decade, Yawkey had cash to burn. "Hang the money," he declared in early January. "What is the use of having money unless you do something with it?"
More than 30,000 fans turned up for the home opener against defending champion Washington and marveled at the Fenway improvements, which included an electronic scoreboard at the bottom of the left-field wall that was a baseball version of a traffic signal with its red and green lights. "They did everything in christening the new Fenway but crack a bottle of champagne over the prow of home plate," John Barry remarked in the _Globe_.
When the Yankees came to town five days later, nearly 45,000 people jammed into the park and another 10,000 were turned away. As attendance for the season ballooned past 600,000, the Sox ascended to the first division for the first time since Babe Ruth's departure and broke even with a 76-76 record. Had Lefty Grove not been sabotaged by abscessed teeth and a sore shoulder, they might have finished even higher than fourth place. "We were all set to shoot for third place," Harris said in April, "and I believe we could have made it until this terrible thing happened."
Park improvements in the early 1930s included a new dugout for the visiting team.
DAVE EGAN: "THE COLONEL" HELD COURT
In 1977, _Boston Globe_ columnist Mike Barnicle wrote about Dave Egan, who covered the Red Sox for the _Globe_ and later, for the _Boston Record_. Barnicle told the story of a boy who spotted Egan sitting alone at the old Hotel Kenmore bar.
"Excuse me," the boy stuttered as he approached. "Are you Dave Egan?"
"I am," Egan said, looking older and smaller than the boy ever imagined.
"I write you letters all the time. I think you're wrong a lot," the boy said, his knees shaking.
"I probably am, kid," Egan replied. "But I never look back. It takes too much time."
"It must be great, knowing all those ballplayers and everything like you do," the boy said.
"Listen, kid," Egan replied. "They're lucky to know me."
Egan, who apparently awarded himself the "Colonel" nickname, was never shy about displaying his ego in print. "[He] would put these awful things in the paper—these awful, outrageous, untrue, fascinating, interesting things," wrote Barnicle. "Ted Williams was always 'T. W'ms Esq.' in one of The Colonel's pieces. . . . Egan was so irritating that he probably sold 100,000 copies a day on his own."
Ray Flynn, who would go on to become the mayor of Boston, told Leigh Montville, "I used to sell the Record at the ballpark when I was a kid. Ted Williams was my idol. Whenever The Colonel would write something bad about him, I'd go through all my papers and rip out the page that had The Colonel's column on it. It was my own little tribute to my hero. I swear on my mother I did this."
When Egan died in 1958 at age 57, Boston's archbishop, Richard J. Cushing, lauded him as a man "blessed with a great natural talent. . . . While all the people who read his columns didn't agree with him, they all appreciated him."
An example of Egan's prose was his story about the 1932 home opener at Fenway Park, in which Senators left fielder Henry "Heinie" Manush hit a game-winning, three-run homer with two outs in the ninth inning. The piece began: "Perhaps he is not a varlet. Probably he is not even a viper. But Henry E. Manush of the Washington Senators (born in Tuscumbia, Ala., in 1901, by actual count) was the ruination of the Red Sox yesterday afternoon at Fenway Park.
"The ninth inning froze the chilled crowd that sat through the harsh April day. Jack Russell, who threw them for the Red Sox, had staggered through the game quite well, viewing everything in a large and statesmanlike manner, and entered the last inning with a comfortable lead of three runs. But woeful events transpired, and Manush lashed out his home run, and so the Red Sox lost their second successive game by the margin of one run.
"The Chowder and Marching Organization, headed by the Messrs Bob Quinn, Nick Altrock and Al Schacht, and accompanied by Jimmy Coughlin's 10th Infantry Band plus the athletes, gave a swank parade to the flagpole in center field, where Walter Johnson and "Shano" Collins raised the American flag on high.
"Hon. James M. Curley, Mayor of Boston, emulated Jimmy Walker, Mayor of New York, by arriving on the scene at a late hour, and it is to his discredit that he sneaked away before the blazing finish of the game. I suppose it proves that Hizzoner has no end of brains, for it was an Antarctic afternoon, more suitable for football than baseball.
"But let us get around to the ball game, if you please . . ."
In an October 9, 1976, op-ed story, the _Globe_ 's Robert Taylor lamented departed writers, including the late Egan: "The Colonel was the maestro of sprung syntax ("And this I tell you . . ."), whose prose made you feel that you were in the back room of a dingy smoke shop, with the cops pounding on the door and the proprietor swallowing the betting slips."
A TASTE FOR PIGSKIN
Boston Redskins vs. New York Giants, 1933.
When the Newark Tornadoes of the National Football League folded after the 1930 season, the franchise was sold back to the NFL. The players and the berth in the league would eventually go to George Preston Marshall, who secured a league charter to place a team in Boston.
In an April 21, 1932, _Globe_ story headlined "Pro Football Plans for Boston Outlined," Marshall noted that pro football attendance had risen 35 percent in the previous season. The story went on to say, "A canvass of this section where football is thoroughly understood has led the promoters to place a club in Boston." Though they may have understood the game, Boston sports fans were slow to embrace it.
Marshall named his team after the baseball Braves, with whom it shared Braves Field in its first season. The football Braves made their debut on October 2, 1932, losing at home to the Brooklyn Dodgers. A week later they secured their first win, 14-6, over the New York Giants. They completed their first season with a 4-4-2 record under head coach Lud Wray, but the games were so poorly attended that Marshall moved his team to Fenway Park in 1933.
Since his team was no longer playing in the same park as the Braves, Marshall changed the nickname to the Redskins—reportedly to honor their new head coach, William "Lone Star" Dietz, a Native American. The new nickname also allowed Marshall to continue to use the uniforms from the previous season.
In their first two years at Fenway, the Boston Redskins continued to play .500 football, finishing with records of 5-5-2 and 6-6. In 1935, the Redskins suffered from a punchless offense, scoring just 23 points during a seven-game losing streak en route to a 2-8-1 record.
The 1936 season—their final one in Boston—would also prove to be the Redskins' most successful on the field in Massachusetts. They won their final three games to capture the NFL's Eastern Division with a 7-5 record, outscoring their opponents, 74-6, in those three games. The Redskins featured a pair of future Hall of Fame players in running back Cliff Battles and offensive tackle Turk Edwards. But when they routed the Pittsburgh Pirates, 30-0, in their next-to-last game of the season, only 4,813 fans showed up at Fenway Park.
Marshall was so outraged by the meager turnout that he gave up the home field for the NFL championship game, choosing to face the Green Bay Packers at New York's Polo Grounds. The Redskins lost, 21-6, and citing the lack of fan support in Boston, Marshall moved the club to Washington, D.C.
On December 6, 1936, the _Globe_ 's Paul V. Craigue was at the NFL title game in New York, and he wrote of the change of venue, "Nobody could offer a satisfactory excuse for the minor-league move by a 'major-league' club." Marshall claimed that he moved the game for the players' sake. "They get 60 percent of the playoff gate, with 20 percent going to the league and 10 percent to each club. We'll get a much bigger gate here than we would in Boston." Marshall went on to say, "We don't owe Boston much after the shabby treatment we've received. Imagine losing $20,000 with a championship team."
Craigue defended the fans' indifference, noting that the Redskins "gave their worst exhibition of the season before their largest crowd and lost three of their five home games before starting their title surge against the lowly Brooklyn Dodgers. Maybe, after all, Boston would turn out for a real attraction."
He went on to ask, "Can you imagine the Red Sox winning the American League pennant and shifting their World Series games to Yankee Stadium in the interest of the gate?"
The franchise's move proved fortuitous for Marshall. The Redskins drafted a star college quarterback named Sammy Baugh before their first season in D.C., where they drew nearly 25,000 fans for their first home game. Baugh guided them to an 8-3 regular-season record in his rookie year and then threw three touchdowns to lead the Redskins over the Chicago Bears in the championship game. Baugh went on to a Hall of Fame career, and the Redskins had nine straight winning seasons in D.C. en route to becoming one of the NFL's most successful franchises.
Eight years after the Redskins' departure, another NFL club made an unsuccessful foray into Fenway Park. Ted Collins, a former recording executive who had become the manager and partner of popular singer Kate Smith, wanted to put a team in New York City and call it the Yanks. He was forced to settle for Boston, though he kept the Yanks nickname. Perhaps that name choice for a Boston team doomed it from the start.
The Yanks never had a winning season in Boston, going 2-8, 3-6-1, 2-8-1, 4-7-1, and 3-9. Collins was given permission to move the franchise to New York in 1949, though for some reason, he renamed his team the Bulldogs. They played three seasons in New York (the latter two, again, as the Yanks), before moving to Texas where they played as the Dallas Texans for one season. The franchise was sold and became the Baltimore (now Indianapolis) Colts in 1953.
The other early Boston NFL franchise was the Boston Bulldogs, who played just one season (1929) when the former Pottsville (Pa.) Maroons were sold to a New England-based partnership that included George Kenneally, a standout player for the Maroons. They played their home games at Braves Field, chalked up a 4-4 record and folded after one season in Boston.
A fire during renovations on January 5, 1934, destroyed much of the ballpark seen in this vintage postcard, though the original facade endures. The workforce was bolstered and repairs and improvements that included replacing wooden grandstands with steel and concrete were completed in time for the 1934 season.
Grove, who'd won 172 games over the previous seven years, ended up going 8-8 and Mack, the Philadelphia owner, felt so badly that he offered to give Boston back its money. But Yawkey was in for the long term. Grove pitched another seven seasons in Boston, won nearly 100 more games and made the Hall of Fame.
The franchise clearly was on the rise and its new owner had no problem opening his overflowing wallet to fuel the ascent. But Yawkey's $250,000 bid for Senators shortstop Joe Cronin, twice what the Yankees had paid for Babe Ruth, was so extravagant that even Washington owner Clark Griffith couldn't believe he was serious. "Take it or leave it," Yawkey told him to his face. "I'll not be back."
Trading the man who'd just married Griffith's adopted daughter wasn't a bad way of shedding a son-in-law, one wag observed. And Cronin, who'd led the Senators to the pennant two years earlier as player-manager, was delighted to go to Boston in the same role. "A fellow with an Irish name like mine ought to get along there," he said.
The Sox were already in first place when they knocked off the Yankees in the Fenway home opener in 1935, giving rise to a rare case of spring fever in the Hub. "One hundred and forty seven more games before the World Series," Hy Hurwitz observed in the _Globe_. While the Fall Classic would remain a pipe dream for another decade, the fans became entranced by a spirited and volatile club that could go from fifth to third to fourth place in three days and produce astonishing moments.
The massive reconstruction, directed by new owner Tom Yawkey, dramatically upgraded Fenway Park before the start of the 1934 season.
None was more bizarre than the triple play that Cronin hit into against the Indians on September 7 at Fenway, when his line drive skipped off the glove of third baseman Arvel Odell "Bad News" Hale, bounced off his forehead and into the glove of shortstop Bill Knickerbocker, who doubled up Bill Werber with a toss to second baseman Roy Hughes, whose relay to first nipped Mel Almada. "Nobody in all probability ever saw one like it," James O'Leary wrote in the _Globe_ , "and nobody is likely to see another."
Even when the Sox were out of contention they were well worth the admission price. A record 49,000 turned up for a Sunday doubleheader with the Yankees in late September with another 10,000 refused entry. As it was, hundreds of fans stood behind a rope in the outfield where they were pelted with bottles by "bleacherites" whose view was blocked. "Boston was established beyond all doubt yesterday as the greatest baseball city in the universe," Gerry Moore declared in the _Globe_.
Yet for as many millions as the new regime had pumped into the park and the roster, it only had brought the Sox up to mediocrity. The club won just two additional games that year and still finished fourth. So Yawkey did what he would do for the next four decades—he brought in a right-handed wallbanger, sending $150,000 and two bodies to the Athletics for brawny Jimmie Foxx, whose 58 homers in 1932 were just two shy of Ruth's single-season mark. "Don't be surprised to see the Ruthian record fall," predicted the burly first baseman.
Though Double X whacked 41 homers and knocked in 143 runs, the Sox sagged badly in 1936. After sitting in first place in early May, the club went into a June swoon, going 4-13 on a 17-game road trip that was so horrific Yawkey was billed for the damage that his players did to clubhouses in Chicago and New York. In mid-July, only 2,500 witnesses turned up for an 11-3 home loss to Cleveland amid what the _Globe_ termed "the unprecedented nose-dive of the most expensive collection of talent in the history of baseball."
Boston's boys of summer in the 1930s included (left to right) Jack Wilson, Jimmie Foxx, and Joe Cronin.
They were the "Gold Sox" and the "Millionaires" who had a perverse attraction to Skid Row. A national poll declared Boston's sixth-place showing as the year's biggest sports "floperoo," far ahead of the knockout of Joe Louis by Max Schmeling and the U.S. tennis team's defeat by Australia in the Davis Cup. Yet Yawkey insisted that he still liked his club's chances for 1937. "People were beginning to think that nobody liked the Red Sox but their mothers," wrote John Lardner, who dubbed their owner a "dealer in second-hand ivory."
What Yawkey most liked was the future and, after spending seven figures on the expensive present, he now began looking to his embryonic farm system to deliver. While the Cardinals and Yankees had been growing their own talent, the Sox owner had been buying overpriced and overripe produce from his competitors. So he looked at what seedlings were available on their minor-league affiliate in San Diego and came up with 19-year-old Bobby Doerr, who started immediately at second base in 1937 as the club's only field player under 23.
But Doerr wasn't quite ready for prime time—he played only 55 games—and though the Sox were clearly improving, they weren't anywhere near a match for the pinstriped champions. "You better warn the Yankees next time you see them," Foxx had advised a visitor in spring training. But when Boston had a chance to draw within grappling distance in August after climbing from fifth place to second, New York grabbed three of four games in a turbulent pair of midweek doubleheaders at Fenway that essentially ended the season for Boston.
The first of them, played before a crowd of more than 36,000 (with another 15,000 wanna-sees turned away), consumed just under six hours and after the opener went 14 innings, the nightcap was called after seven because of darkness. "It looks like Tom Yawkey will still be among the scant few who isn't on W.P.A.," Hy Hurwitz observed in the _Globe_ after more than 20,000 fans turned up the next day.
FIRE AND ALL, IT WENT YAWKEY'S WAY
In its first 21 years—through a series of owners and the disastrous wholesaling of Red Sox talent—Fenway Park remained virtually unchanged, except for the effects of a fire on May 8, 1926, that destroyed the wooden grandstand past third base. Those seats were not replaced for years, leaving a wide gap in foul territory that remained until the sale of the team and the ballpark to Thomas A. Yawkey on February 25, 1933.
Yawkey immediately ordered a renovation of Fenway that reports of the day estimated would cost between $750,000 and $1.25 million. Yawkey promised a new team and a new image, and instantly began investing in his two assets—though those plans were jeopardized by yet another fire.
The Osborn Engineering Co., the Cleveland-based firm that designed Fenway Park in 1911, was hired for the reconstruction, which began in the fall of 1933. Fenway was largely unchanged in the area that horseshoed behind and around home plate. However, all the areas that had originally consisted of wooden grandstands, including left field, right field, and center field, were given a concrete base—and the outfield bleachers were expanded.
But on January 5, 1934, in the middle of renovations, a heater being used to dry the concrete overturned. A nearby canvas caught fire and the new left-field grandstand went up in flames. The fire quickly spread to the center- and right-field bleachers, and then jumped Lansdowne Street and engulfed five buildings across the street. The effort to fight the five-alarm fire was hampered by dense smoke and icy footing, but none of the roughly 700 workers on site or the more than 100 firefighters was seriously hurt. Damages from the fire were estimated at $250,000.
Red Sox general manager Eddie Collins admitted, "We may have to rebuild a large part of the bleachers we had just finished," but he was confident that the team would still open its season on time. Though the Braves offered the use of their field as a temporary home, the number of construction workers was boosted to nearly 1,000 and the work was completed on schedule. Yawkey, who was on a hunting trip in South Carolina when the fire broke out, did not cut his trip short.
The deepest region of Fenway Park, beyond the flagpole in center field, originally measured 468 feet and the right-field foul line extended 358 feet. In the renovation, center field was trimmed back to 425 feet and right field brought down to 326.
The look of the left-field wall and Duffy's Cliff also changed considerably in the makeover. The 10-foot cliff was all but eliminated, with only a very slight grade remaining (and that would disappear completely in the years to come). The now famous scoreboard that helps give the left-field wall its distinctive look was constructed at the base of the towering barrier, which was brought up to its current height of 37 feet.
The netting above the left-field wall wasn't installed until two years later, in 1936. The net, which rose at an angle over Lansdowne Street for another 23 feet 4 inches, not only saved on lost baseballs, it also prevented damage to cars and buildings along the street, before it was dismantled in favor of the Monster seats nearly 70 years later.
As part of the 1934 makeover, 15,708 new seats were installed at a cost of $45,556.10, or $2.90 each. The maximum attendance figure was increased to nearly 38,000. (By 1995, with more restrictive fire laws and allowances for fans to stand on the perimeter of the field a thing of the distant past, Fenway's capacity had been reduced to 33,583—the smallest in the major leagues.)
Like the original Fenway opener in 1912, the Red Sox went 11 innings on April 17, 1934, the day the renovated park was unveiled. With 30,336 watching, the Sox suffered a 6-5 setback to the Washington Senators.
Seated in the stands that day was George Wright, a Hall of Famer and one of the game's pioneers. Wright could remember playing baseball in 1871 at Boston's Walpole Street grounds, a ballpark that seated 5,000 spectators.
"As I looked around," Wright, then in his 80s, said that day in Fenway, "I thought how wonderful all this was, and how baseball had advanced in every respect. That is, every respect except ability. On that point, the players of my day were just as good."
Tom Yawkey and his first wife, Elise, at Fenway Park. They divorced in 1944.
Jimmie Foxx of the Red Sox slid safely back into first base ahead of the throw to Washington Senators first baseman Joe Kuhel.
ENTERING WILLIAMSBURG
On September 24, 1939, the Red Sox announced that the home and visiting bullpens would be moved from foul ground into right and right-center field at Fenway in time for the 1940 season. The common expectation was that Ted Williams would propel home run after home run over the shortened fence. As the _Globe_ put it, Williams's "batting and home-run marks are expected to soar with the new layout."
"Everyone thought," Williams recalled years later, "that I was supposed to break Ruth's record."
After all, Williams was coming off a prodigious rookie season in which he had hit .327 with 31 home runs and a league-leading 145 RBI. There was little doubt that the reason for the ballpark reconfiguration ordered by Sox owner Tom Yawkey was to allow the Splendid Splinter, a pull hitter, to boost his number of round-trippers. The new warm-up pens were quickly dubbed "Williamsburg."
The bullpens were constructed at the base of the existing bleachers, taking more than 20 feet off the home-run distance for the start of the 1940 season. The modification also led to the construction of extra seating at the bottom of the right-field grandstand. The rounded portion of the right-field wall—often called the belly—was another offshoot of the changeover.
In announcing the change, Sox GM Eddie Collins estimated that 600 to 700 box and grandstand seats would be added and provide sorely needed revenue for the team. Fenway's home-run distance to the right-field foul pole shrunk from 325 feet to 302, while the right-field distance dropped from 402 to 380. As the Globe's Harold Kaese wrote in a story in 1952, "Some optimistic experts predicted [Williams] would hit 75 home runs, or at least break Babe Ruth's record of 60 with the park changed." However, Kaese contrasted Williamsburg's 380-foot distance in straightaway right field to the bullpens that had been constructed in the late 1940s in Pittsburgh, the so-called "Greenberg Gardens"—which required only a 335-foot poke from Pirates slugger Hank Greenberg. "Williamsburg," Kaese concluded, "is no joke, but Greenberg Gardens . . . is a big laugh for a slugger." Even with the new configuration, Fenway remained the longest right-center field fence to reach in the American League.
Kaese's analysis in 1952 led him to pronounce that Williams had gained a total of 48 home runs over nine seasons (1940-42; 1946-51), or just over five a year from the reduced distances. Indeed, in 1940, the first season of the change, Williams actually hit fewer homers overall (23, down from 31) and fewer to right field than he had in his rookie season of 1939 (seven in 1939; five in 1940—four of which landed in the pens, one of which reached the bleachers beyond). Perhaps, Kaese wondered, Williams had pressed because of the expectations.
If nothing else, Williams made shrewd adjustments over his career. Kaese noted that in his first five seasons, Williams had not hit a single home run over the left-field wall, but in his succeeding five seasons, he hit 15. Kaese noted that the uptick coincided with the introduction by Lou Boudreau, the Cleveland Indians' player-manager, of the Williams Shift, a realignment that left only one fielder to the left of the pitcher's mound.
Twelve years later, Williams recalled the inflated expectations for 1940. "I didn't hit the home runs that I had my first year," he said. "I got a lot of catcalls and criticism. That just irked me enough, so I got a little sour on everything and everybody."
That would not be the last time that Williams soured on the fans or the press in his brilliant, tempestuous 19-year career.
HERE'S TO YOU, MISS ROBINSON
Ted Williams called her "Sunshine." Joe Cronin trusted her to babysit his sons. And Nomar Garciaparra never forgot to send her flowers.
For more than 60 years, Helen Robinson was the no-nonsense Red Sox switchboard operator who controlled access to the team's decision makers, guarded its most explosive secrets, and ultimately created a legacy of loyalty and longevity.
"Helen Robinson was a legend, really," said Red Sox GM Dan Duquette after Robinson, of Milton, Massachusetts, died of a heart attack in 2001 at age 85.
Robinson had witnessed it all with a telephone in her hand, all the sadness and glory that was Red Sox history for 60 years. She fielded condolence calls after the deaths of Thomas and Jean Yawkey, Tony Conigliaro, Joe Cronin, and one Sox great after another. And she endured the profanity-laced protests from fans after the most devastating losses on the field.
Robinson's greatest admiration was for Tom Yawkey. "He wasn't just a boss," she said, "he was also a friend." Although the switchboard went crazy whenever the Sox made a trade or were in a pennant race, Robinson remembered July 9, 1976, as her busiest day. That was the day the Red Sox announced Yawkey's death.
"She has lived through some of the greatest and most trying moments in Red Sox history," then-CEO John Harrington said at a ceremony marking Robinson's 60th year on the job—one month to the day before she died, "and she will forever be entwined with Red Sox lore."
A seamstress, Robinson knitted sweaters for Sox players who went off to World War II and Korea. Decades later, she helped sew tiny American flags on the backs of uniforms after the 9/11 terrorist attacks.
Robinson was working for New England Telephone in 1941 when she learned of an opening with the Red Sox. She was interviewed by Hall of Famer Eddie Collins, then the general manager, and landed the operator's job.
"I was the only non-uniformed personnel he ever hired," she proudly told people.
Those were the days of the telephone circuit board, when the operator needed to monitor the line to know when both parties were connected and disconnected on a call. Thus, she knew more about the inner workings of the Sox than almost anyone, but she never publicly whispered a word of it.
Nor did she betray the secrets she gathered by handling personal calls for players from Williams's early years to Garciaparra's heyday. And though the proliferation of cell phones curbed her contact with contemporary players, it was taken as much more than a minor observation when she first met Manny Ramirez and declared him "a fine young man."
Robinson never married ("The Red Sox were her life," Duquette said), and she counted Ted Williams among her closest friends. When they were young and single in the early 1940s, Robinson and Barbara Tyler, Collins's secretary, often socialized with Williams, Johnny Pesky, and Charlie Wagner.
"She loved Ted," Pesky said. "Ted always has called her 'Sunshine.'"
When Elizabeth Dooley, generally considered the greatest Sox fan, died in 2000, Robinson was on the phone trying to contact Williams. "I knew Ted would want to know," she said.
At her switchboard on the third floor at 4 Yawkey Way, Robinson worked from 9 a.m. until well after a home game ended, day or night. She left precisely at 5 p.m. when the team was on the road. And during weekend homestands, she arrived promptly on Saturdays and after church on Sundays.
In 60-plus years, Robinson was rarely absent. She kept working more than 20 years after she beat cancer in the 1970s. And she made no secret that she intended to work until she no longer was able.
"She went out doing what she wanted to do," Manager Joe Kerrigan said when she died. "She worked till the last day. . . . I hope she gets her due. I hope people realize what she meant to the Red Sox."
Although New York ran away with the pennant and the World Series, and the Sox ended up fifth, they still won more games (80) than they had since 1917 and their bankroller finally had a glimpse of a payoff. Boston would win a pennant "if it takes 1,000 years," Yawkey vowed before the 1938 season, but his checkbook had a limit. "I'm through playing Santa Claus," he declared when four players still were holding out on the eve of spring training.
As long as Yawkey had "The Beast," he had a potent holiday punch, particularly at Fenway, where Foxx's right-handed bat had the power of a cudgel. He had a monster campaign on the premises on the way to winning the league's most valuable player award, hitting .405 at home with 35 homers, 104 RBI, and an .887 slugging percentage. His loudest thunderclap came at the end of an August 23 doubleheader against Cleveland, when Foxx crashed a grand slam with two out in the bottom of the ninth to give his team a 14-12 triumph and a sweep after they'd trailed, 6-0.
"Who cares if the Yankees win the pennant after this?" crowed Cronin's wife, Millie, after Foxx had circled the bases with what the _Globe_ 's Moore called "a grin as wide as the Sahara Desert." By then New York had already run away with the league en route to a third straight world championship. But Boston's second-place showing was its best since Ruth had departed.
Another potential Ruth arrived in 1939 in the form of Ted Williams, a goofy and gangly 20-year-old out of San Diego who was so skinny that he eventually was dubbed the Splendid Splinter. He'd arrived at spring training a year earlier full of braggadocio that was deemed the prerogative of a veteran like Foxx, whose slugging credentials were beyond dispute. "Foxx ought to see me hit," he proclaimed.
Doc Cramer, Joe Vosmik, and Ben Chapman, who'd had the outfield jobs locked up, mocked Williams mercilessly. "Tell them I'll be back," Williams told clubhouse man Johnny Orlando as he was shipped up to Minneapolis for seasoning, "and I'm going to wind up making more money in this game than all three of them put together."
There was no keeping Williams down the following season, although Cronin, who'd originally dubbed him, "Meathead," was quick to sit the rookie when he threw a ball over the grandstand roof in Atlanta after misjudging an outfield fly in an exhibition game. "I'll continue to crack down on him until all the 'bush league' is out of him and he begins to act like a major leaguer," the skipper vowed.
Raising the backstop—an April ritual.
There was nothing bush about his bat, though. "If he puts it there again, I'm riding it out," Williams promised after Red Ruffing had struck him out twice on Opening Day in New York, and then hammered a ball more than 400 feet that just missed going over the fence. In his third game at Fenway, Williams went 4 for 4 and hit his first homer. In Detroit, he launched the two longest homers ever hit at Briggs Stadium. "This kid can hit a baseball as far and as hard as any ballplayer that ever lived," a rival told sportswriter Grantland Rice. "And I'm not even barring the Babe."
"The Kid" was Ruthian in both his power and his personality, which was unapologetically adolescent, marked by what Moore described as "his constant boyish chatter, seldom possessing any meaning" and his "screwball acts." But his rookie numbers were irrefutably adult—.327 with 31 homers, 145 RBI, and 107 walks from pitchers reluctant to see their offerings sailing into the seats.
Even before Williams earned the nickname "Thumping Theodore," his colleagues had been optimistic about dethroning the Yankees, who'd been picked by their manager Joe McCarthy to win a fourth straight crown. "Perhaps the Fates will cross him up on the prediction," Cronin said. As always, though, it was a futile and frustrating chase that ended with a fiasco at the Fens on Labor Day weekend.
After winning the opener of the Sunday doubleheader by squeaking by the Yankees, 12-11, the Sox hoped to salvage the nightcap by stalling as the clock approached the 6:30 p.m. curfew with New York leading, 7-5, in the eighth. If the game couldn't be completed by then the score would revert to 5-5 (the score at the end of the seventh inning), so Cronin ordered an intentional walk to Babe Dahlgren to load the bases. But the Yankees countered by playing hurry-up, as George Selkirk and Joe Gordon trotted home and let themselves be put out.
After the fans littered the field with soda bottles, straw hats, and rubbish, umpire Cal Hubbard called Fenway's first forfeit and awarded a 9-0 victory to the visitors, a decision that later was rescinded by the league president, Will Harridge, who ordered a replay that was scrubbed by rainouts. Not that it mattered. Boston already was hopelessly in arrears and ended up 17 games behind.
Ted Williams at spring training in March 1938. The "Splendid Splinter" made his Red Sox debut in 1939, hitting .327 with 31 home runs and 145 RBIs.
HATS OFF TO ARTHUR D'ANGELO
BY STAN GROSSFELD
In blazing sunshine directly across the street from Fenway Park in 2006, Arthur D'Angelo, 79, was slowly and methodically pressure-washing the stale beer off the sidewalk in front of his souvenir store. By the time he finished, one side of Yawkey Way was clean enough to eat an _El Tiante_ Cuban sandwich off it.
Inside the megastore, D'Angelo's 65-person staff was enjoying the air conditioning. But D'Angelo, a short man with a sweet smile, didn't want to delegate the cleaning chore.
Arthur and his twin brother, Henry, arrived in Boston's North End from Italy with their family in 1938, fleeing the dictatorship of Benito Mussolini. "I was 14 when we came to Boston," he said. "I couldn't speak a word of English."
The brothers started hawking newspapers, the _Daily Record_ and the _Boston American_. The two eventually wandered from the North End to Dorchester to Fenway Park.
"We saw these crowds," D'Angelo said. "We didn't know what baseball was. We snuck into the ballpark. The game started at 2 p.m. and we thought, 'What are these idiots doing with a baseball bat?' But then it caught on and we loved the game. Why not capitalize on it?" They did, in a big way.
D'Angelo said that the Souvenir Store, run by Twins Enterprises, Inc., is the largest of its kind. Most locals refer to it as "Twins" even though Arthur D'Angelo's twin brother Henry died in 1987.
"We sell more caps than anybody in the world," said D'Angelo, who operates Twins Enterprises with his four sons. "We make them for all the major-league teams. We also have licensing for 200 colleges."
After a stint in the Army, D'Angelo was discharged in 1946 and returned to Fenway, hawking pennants. Interest in the team was high, as the 1946 Sox won their first American League pennant since 1918 and played to a record 1.4 million fans.
"Ted was back from the service," D'Angelo said. "They had Dave Ferriss, Bobby Doerr, Dom DiMaggio, and Johnny Pesky. Pennants were 25 cents. We had buttons with the players' pictures on them. We used to buy 'em for 12 cents and sell them for a quarter. You didn't need a license back then."
The D'Angelo brothers rented space outside Fenway until 1965, when they borrowed $100,000 and bought the building on Jersey Street (now Yawkey Way) that still houses the Souvenir Store. It almost folded.
In 1965, the Sox suffered through a 100-loss season, and the next season they drew just over 800,000 fans to Fenway to watch them finish ninth. "Nobody wants to remember a loser," said D'Angelo.
But then came the Impossible Dream team of 1967. "My favorite team was 1967," said D'Angelo. "The Sox won the pennant, then lost the World Series in the last game. Besides Yaz, there were no real standout players, but they all charged into it."
Arthur D'Angelo surveyed the scene on Yawkey Way from his office.
## 1940s
(left to right) Manager Joe Cronin, third baseman Jim Tabor, and left fielder Ted Williams at the Fenway Park batting cage.
The 1940s were tumultuous times. With the United States pulled into a world war on two fronts, baseball was greatly affected, even though the games went on. Because of wartime enlistments, the Red Sox only had their full complement of players early and late in the decade, and they ended the 1940s with one pennant, a seven-game loss in the World Series, and four runner-up finishes in the American League—including one that resulted from a one-game playoff loss. (The 1970s would produce exactly the same totals, although the American League had been split into two divisions by then.) In a decade that brought victory gardens, rationing, and the GI Bill, Fenway Park also became an important meeting place. It hosted military, civic, and religious gatherings, and it became the "neutral" setting for New England's biggest college football rivalry of the era, Boston College vs. Holy Cross. Since politics has always been a wildly popular spectator sport in the land of the Kennedys, James Michael Curley, and _The Last Hurrah_ , it is fitting that four-term President Franklin Delano Roosevelt gave the final campaign speech of his life from a platform at Fenway on the weekend before the 1944 election. Among 10-year epochs during their tenure in Fenway Park, the Red Sox seem to have developed a strong pattern of one good decade followed by two bad decades. They were at their most successful with four world titles in the 1910s, and then suffered through two mostly disastrous decades before returning to form as contenders through most of the 1940s. As the remaining chapters of this book will testify, it was a rinse-and-repeat cycle that kept going to the dawn of the 2000s.
(left to right) Star players Bobby Doerr, Ted Williams, and Dominic DiMaggio obliged the cameraman by posing for this 1940 image.
As the Thirties ended, the Red Sox franchise had made the transition from a losing proposition to a winning proposition, even though the Yankees still were a mile ahead of Boston in the standings. The lineup, too, was in flux with Jimmie Foxx, Joe Cronin, and Doc Cramer nearing the end of their careers, and Ted Williams, Dom DiMaggio, and Bobby Doerr at the beginning of theirs.
Whether Williams, his teammates, or the fans liked it, he clearly was the future of the franchise. After the front office redesigned the park to suit him, the Kid was feeling the pressure in 1940, particularly since the homers were slow in coming. Even though the Sox were in first place (thanks to a 17-6 start and a torpid spring in the Bronx) and Williams was hitting well over .300, he felt unappreciated and was vocal about it.
What especially irked Williams was his $12,500 salary, given what he felt was expected of him. "Here I am hitting .340 and everybody's all over me," he complained in early June. "I shoulda been a fireman." When Boston next played in Chicago, White Sox manager Jimmy Dykes, a notorious bench jockey, outfitted his players in papiermâché helmets, cranked up a siren, and had them howl, "Fireman, save my child!"
Once the Sox lost seven games in a row on the road in June—four of them to the fifth-place St. Louis Browns—and then dropped another eight straight in July, all of the engine-and-ladder companies in the Hub couldn't save them and they ended up in fourth place. Yet Williams, for all his angst, had an exceptional season, hitting .344, clouting 23 homers, and knocking in 113 runs. He even pitched the final two innings in a mop-up role against Detroit in late August. Williams held the Tigers to one run on three hits and struck out Rudy York on three pitches, though the Sox lost, 12-1.
That season was prelude to an achievement that has been unmatched since, as Williams hit .406 in 1941, even while opposing hurlers were pitching around him, walking him 147 times in 143 games. The biggest challenge for the club at first was to find him, as Williams absented himself from spring training, phoning in once to assure the front office that he hadn't been abducted. "Theodore says he has been so busy shooting wolves in a country so wild that no paths led from there to civilization," Mel Webb wrote in the _Globe_.
When Williams finally turned up, he was both relaxed and reflective, eager to put the tumult and torment of the previous season behind him. He even "shook hands all around" with newspaper reporters, the _Globe_ reported on March 10. "They said I was a heel and I'll admit I had a lousy attitude," he later conceded in June. "But I don't think I deserved all they wrote about me, even though I have to admit the start of it all probably was my fault."
Nobody could find a flaw in his performance, as Williams went on a 23-game hitting streak during which his average soared to .436. "The Kid has grown up," the _Globe_ declared.
When he hit the game-winning, three-run homer with two out in the ninth inning in Detroit to win the All-Star Game for the American League, there was no doubt about who was baseball's most electrifying player. By then the Sox had fallen out of contention after being swept at New York. The only two story lines keeping Boston fans intrigued were Ted Williams's quest to hit .400 (something no one had done for a full season since 1930) and Lefty Grove's push to reach 300 career wins.
It had been 16 years since his first victory and Grove was at the end of his career at age 41. When he took the mound against the Indians on Ladies Day in Fenway, he hadn't won a game in more than three weeks. The thermometer read 90 degrees and Grove fell four runs behind. "The toughest game I ever sweated through," he said. But Foxx's two-run triple and Jim Tabor's second homer gave the Sox a 10-6 triumph and Grove, who never won another game, his 300th career victory. Grove was elected to the Hall of Fame in 1947.
An American Legion memorial Mass on May 20, 1940.
FOOTBALL, FATE, AND THE COCOANUT GROVE
In 1896, Boston College and Holy Cross played the first game in what was to become one of the longest-running rivalries in college football. In 1916, the schools squared off at Fenway Park for the first time, and in 1920, BC capped an 8-0 season with a 14-0 victory over Holy Cross before 40,000 fans at Braves Field.
BC hired Frank Leahy as coach in 1939, and he turned the Eagles into a national power that won 20 of 22 games over the next two years, including an undefeated season in 1940 and two straight victories in the Sugar Bowl in 1940 and 1941.
The Eagles were routinely dominating their rivals from Worcester, and entering the 1942 matchup at Fenway Park, undefeated BC had routed its eight previous opponents that season by a combined 249-19 and was nearly assured of a berth in the Sugar Bowl. Holy Cross was 4-4-1, and the Crusaders were expected to provide little resistance. BC had already scheduled a victory party at the swanky Cocoanut Grove nightclub in Boston.
A crowd of 41,350 packed Fenway Park on November 28, and, led by halfback Johnny Bezemes, who scored three touchdowns and passed for another one, Holy Cross proceeded to rout the vaunted Eagles, 55-12. Seeing its expected spot in the Sugar Bowl vanish, BC canceled its victory party. That night, a deadly fire swept through the glitzy Cocoanut Grove nightclub, killing nearly 500 people in the worst nightclub fire in U.S. history. A BC victory surely would have resulted in the deaths of some of its football players, as a large table had been reserved for the team in the club's main dining room.
William Commane, a fullback on the 1942 BC squad, had planned to be at the Cocoanut Grove that night, but like his teammates, he switched plans after their loss and went to the Statler Hotel. "The next day, I was listening to the radio at home with my family," Commane recalled. "They read out the names all day; it was a terrible tragedy. The football game didn't mean much after that."
The teams' annual clash in 1956, a 7-0 Holy Cross victory, would be their final contest at Fenway Park, as the Red Sox banned football from the ballpark for several years. Shortly thereafter, the schools took different athletic paths, with Holy Cross deciding to de-emphasize the sport. In 1986, after losing 17 of the previous 20 games to BC, Holy Cross terminated the series.
LIB DOOLEY: ULTIMATE FAN
Elizabeth "Lib" Dooley got her first season ticket during World War II and didn't miss a Red Sox home game for the next half-century, attending more than 4,000 consecutive ballgames at Fenway. She had a direct link to the franchise's beginnings in 1901 through her father, and she sat in the first row behind the Red Sox on-deck circle in Box 36-A. Mickey Mantle would stick out his tongue at her when he homered for the Yankees, and she was Ted Williams's favorite fan.
"Forever, she's the greatest Red Sox fan there'll ever be," said Williams in 1996.
Dooley, who retired after 39 years as a physical education teacher in Boston, said that she attempted to guide her students in the right direction by telling them, "I don't drink, I don't smoke; I go to ballgames."
Dooley fell in love with the sport in an era when she was one of the comparatively few female baseball fans. She grew up hearing the men in her family share stories about the game and looked forward to days when she could accompany her father to the ballpark.
Dooley decided to buy her own season pass in 1944 and "make baseball my hobby." One of her sisters had taken up bridge, but Dooley didn't have the stomach for that: "I despise bridge and I hate gossip. I wanted something where I could go by myself without anyone ruining it by saying, 'I can't do it this week.' It gets you out in the fresh air and keeps you from talking about your neighbors."
Dooley's father, John Stephen Dooley, helped the upstart American League ballclub get a home at the Huntington Avenue Grounds in 1901. He was also the founder and president of a boosters group known as the Winter League, a forerunner to today's BoSox Club. According to family lore, Dooley attended every Boston baseball opener from 1894 until his death at the age of 97 in 1970.
Lib Dooley went on to become a member of the board of directors of the BoSox Club, the team's official booster club. In 1956, she moved to Kenmore Square so she could walk to Fenway. "It's seven minutes to the ballpark, a little more now as I get older and the crowds get bigger," she said in 1995.
Williams lived nearby, and during his playing days he would visit her apartment on his way to the park, stopping by for "her words of wisdom for the day."
"Theodore trusts me," Dooley explained, "because he knew I was a schoolteacher and I would talk to him straight." Dooley recalled telling Williams that he'd never know who his real friends were until he left the game.
For his part, Williams called her "Goddammit Red," as in: "Goddammit Red, how do you know that?"—even as her red hair became tinged with silver.
Along with Williams, her favorite players were Bobby Doerr, Dom DiMaggio, and Johnny Pesky. She called them "my four baseball brothers."
"She was just a perfect lady," said DiMaggio in June 2000, after Dooley died at 87. "I remember so well when she presented us with our American League championship rings after we won in 1946. She was so thrilled. The Red Sox were practically her whole life."
Dooley took at least one road trip with the team every year, and she claimed she'd seen five no-hitters. Despite her prime location, she fielded only one foul ball—a pop-up off the bat of Sox outfielder Carroll Hardy in the early 1960s. She broke two fingers in the effort and never attempted to catch another.
Jim Rice was a favorite in the 1970s and 1980s, and later on, she took to Nomar Garciaparra. "Nomar would give her a wink when he came into the on-deck circle," said David Leary, her grandnephew.
Her strongest negative feelings were reserved for the team in pinstripes.
"I despised the Yankees," she said, "because they gave us so much trouble. They always beat us, always." Joe DiMaggio, Phil Rizzuto, and Mantle "were the only three Yankees who I tolerated, because of their great talent."
"The fans are really baseball, not the players," Dooley said when a strike marred the 1994 and 1995 seasons. "You can always get someone to play ball—it doesn't have to be someone who gets $4 million to work two hours a day." A moment later, she added with a grin, "I wonder if I'm going to need a bodyguard to get back into that park."
Dooley once explained her position thusly: "I do not consider myself a fan. I am a friend of the Red Sox." An anecdote from her nephew, Owen Boyd, backed up that statement.
Boyd was at a game with Dooley when she directed him to dial a number for her on his cell phone. To his astonishment, "Happy birthday, Theodore" were the next words out of her mouth.
"You know Ted Williams?" Boyd asked in amazement.
"More importantly," she replied, "Ted Williams knows me."
Daily life for grounds crews in the '40s included pulling tarps and pushing lawn mowers.
HOMESTEAD GRAYS COME TO TOWN
The Homestead Grays, the two-time defending champions of the Negro League who featured Hall of Famers such as Josh Gibson and Buck Leonard, defeated the Fore River Shipyard team of the New England Industrial League, 1-0, in a game played at Fenway Park on May 26, 1944.
The Grays, who were based in Homestead, Pennsylvania, near Pittsburgh, played many of their games at Forbes Field in Pittsburgh while also using Washington's Griffith Stadium as a second "home" park. The team also barnstormed and often played two or three games a day against local amateur and professional teams.
The Boston _Globe_ had published a one-paragraph story under the headline: "Negro Home Run King at Fenway Tonight." Second only to the legendary Satchel Paige among Negro League players in terms of fame and popularity, Gibson was generally considered to be the greatest hitter in the history of black baseball and was nicknamed the "Black Babe Ruth."
The story also reported, "A United States Marine band will furnish music. Jack Burns, Skinny Graham and Charlie Bird will be in the Fore River lineup." The game account the next day showed the Grays out-hitting Fore River, 11-5, on their way to a 1-0 win. Some 3,000 fans attended the game.
Gibson reportedly won nine home-run titles and four batting championships during a 17-year career that ended in 1946. Paige called him "the greatest hitter who ever lived." Hall of Fame pitcher Carl Hubbell noted, "Any team in the big leagues could use him right now."
While Gibson certainly could have helped any major-league team that elected to sign him, the unwritten rules of baseball kept him out of the majors his entire career. Frustration over his lack of opportunity, various illnesses, and despondency over the death of his wife in childbirth caused Gibson to die of a stroke at the age of 35 on January 20, 1947, just three months before Jackie Robinson broke Major League Baseball's color barrier.
Two days after the Grays played at Fenway, the ballpark hosted a military memorial Mass that was attended by an estimated 6,000 people. In his sermon, the Reverend John S. Sexton, an American Legion chaplain, spoke out against racial and religious hatred, saying, "Jim Crow-ism and anti-Semitism are un-American, because in the first instance they are anti-God. Let's put America ahead of our own little cheap individual, and group interests."
On June 3, 1947, the _Globe's_ Hy Hurwitz wrote a story that said the Red Sox were considering signing Negro League star Sam Jethroe. The story noted that Jethroe, who was playing for the Cleveland Buckeyes of the Negro American Baseball League, was considered the best hitter in the Negro Leagues since Gibson. Jethroe had led the league in both batting average and stolen bases as the Buckeyes won the Negro League title in 1945, beating the Grays in the championship series.
Had they signed Jethroe, as they were reportedly considering, the Red Sox would have been the only Major League Baseball team other than the Brooklyn Dodgers, who had signed Jackie Robinson in 1945, with a black player on its roster. Jethroe ended up signing with the crosstown Boston Braves, and he would go on to be named the National League's Rookie of the Year in 1950. It would be more than a decade before the Red Sox promoted Pumpsie Green to their big-league club, making the Sox the final MLB team to integrate its roster.
The Williams melodrama went until the last day of the season. His average was .406 after his final home game against the Yankees, which he punctuated with a home run. Going into the concluding doubleheader at Philadelphia, it had dipped to .3995. Though Cronin offered to let him sit out the final two games so that his average would be rounded up to .400, Williams was adamant about playing _both_ games. "A batting record is no good unless it's made in all games of the season," he declared. Then he proceeded to go 6 for 8 with a homer against the last-place Athletics and finish the season at .406. "Ain't I the best goddamn hitter you ever saw?" Williams proclaimed in the clubhouse.
Joe DiMaggio, who strung together a record 56-game hitting streak as the Yankees breezed to the pennant, might have been the Most Valuable Player, but even he wouldn't challenge the Kid's primacy at the plate. Williams had a more than reasonable reprise in 1942, leading the league in batting average (.356), runs (141), RBI (137), and homers (36). But with the country at war after the bombing of Pearl Harbor, there was some dispute as to whether he should be playing at all.
Bob Feller, Hank Greenberg, and Sox teammate Mickey Harris already had been called to duty and Williams's draft board in Minneapolis had reclassified him 1A in January. But since he was supporting his mother, Williams was switched to 3A on appeal and turned up at training camp. "If Uncle Sam says fight, he'll fight," Cronin said in February. "Since he has said play ball, Ted has a right to play."
But only for one more year, Williams vowed at the end of March. "No matter what happens during the coming season, I'm going to enlist when it's over," he said. Williams began it auspiciously, lofting a three-run homer into the bleachers in his first at-bat in the Opening Day victory over the Athletics at Fenway. But despite the emergence of a brace of promising rookies (right-hander Tex Hughson and shortstop Johnny Pesky), the Sox couldn't catch the Yankees and finished nine games behind their rivals despite winning 93 games, their highest season total since 1915.
The season finale in the Fens, a symbolic 7-6 victory over New York, was telling. The story of the day was that the fans donated more than 46,000 pounds of scrap metal to the war effort. By 1943, the Sox would be donating a third of their lineup to the cause.
Williams and Pesky had enlisted with the Navy Air Corps and DiMaggio with the Coast Guard. So Cronin, who'd been easing out of his role as player-manager, pressed himself back into service at 36. The 1943 season figured to be a lost cause anyway after Boston dropped 10 of its first 14 games (seven of them to the Yankees) and sank into the cellar after the first week in May. So the skipper chose Bunker Hill Day, which commemorated a gallant but ultimately losing battle by American rebels against the British in the Revolutionary War, to make a bit of history himself.
The decade's renovations included a television and radio coop, perched atop the screen behind home plate.
His attitude wasn't always so sunny, but Ted Williams smiled for the camera when he left the Fenway Park locker room in July 1942.
"I guess the old man showed them something today," Cronin crowed after he'd pinch-hit home runs in both ends of a doubleheader against the Athletics at Fenway, becoming the first major-league player to manage the feat. By the end of the season he'd hit .429 with five homers coming off the bench. But there was no way to salvage a lost campaign. The Sox ended up in seventh place—29 games behind New York—for their worst finish in a decade as attendance dwindled to its lowest level (358,275) since the Depression.
Nothing was close to normal during the war. Had it been, the Browns never would have won their only pennant in 1944, displacing the depleted Yankees with a "collection of misfits, 4Fs, brawlers and drunks," as they were described in _The Boys Who Were Left Behind_ , by John Heidenry and Brett Topel. Still, Boston managed to stay in the pennant race until Labor Day, even after military enlistment took catcher Hal Wagner and Doerr, who'd been leading the league in batting in late August, and Hughson, who'd won 18 games.
Their departures, plus a killer schedule that put the Sox on the road for their final 17 games, finished them off, as the front office didn't bother taking World Series requests. The club lost 10 in a row and finished in fourth place, a dozen games astern. "Maybe it's just as well," mused a _Globe_ editorial, "because local fans are so badly out of practice they wouldn't know what to do with a championship."
By 1945, Uncle Sam had spirited away virtually all of the former regulars and the Sox were happy to consider anyone who might be capable of playing 154 games without tripping over himself. So the day before the season opener in New York, the club ran a Fenway tryout for three black players—Jackie Robinson, Sam Jethroe, and Marvin Williams—who were performing in the Negro League. Roxbury city councilor Isadore Muchnick had badgered the club to break baseball's color line and the players suspected that the workout was perfunctory. Though scout Hugh Duffy proclaimed them "pretty good ballplayers," the front office didn't sign them. "Not for one minute did we believe the Boston tryout was sincere," Robinson would remark. "We were going through the motions."
Several players who had enlisted as naval aviation cadets joined Lt. Cmdr. E.S. Brewer at Fenway Park on April 27, 1943, before the Red Sox-Yankees game. Front row, from left: Johnny Pesky of the Red Sox, Brewer, Buddy Gremp of the Boston Braves, and Joe Coleman of the Philadelphia Athletics. Back row, from left: Ted Williams, and Johnny Sain of the Boston Braves.
FDR'S ROAD SHOW
When U.S. President Franklin Delano Roosevelt's motorcade arrived at Fenway Park on the night of November 4, 1944, there wasn't a bare patch of ground between Kenmore Square and the ballpark, or for blocks around. It was three days before the presidential election, and FDR's race against Governor Thomas Dewey of New York had taken a nasty turn. Factor in that Roosevelt was preceded on stage by two enormous talents of music and film, and the ballpark and the surrounding neighborhood were rollicking in a boisterous but far different fashion than if the Red Sox were staging a ninth-inning rally.
Sound wagons blared campaign tunes and patriotic songs, and hawkers sold buttons promoting Roosevelt and Maurice J. Tobin, the Democratic candidate for governor of Massachusetts. Inside the park, red, white, and blue bunting and banners were draped along the infield wall, around uprights, and along the edge of the grandstand roof. More than 250 newspaper reporters were instructed by Secret Service men to meet in a garage on Ipswich Street, and they entered the park in columns of three at 8 p.m., an hour ahead of Roosevelt's entrance.
One of the key questions in voters' minds was the health of the three-term president, compelling Roosevelt to undertake a whirlwind tour of several cities, including a four-hour appearance in New York City a couple of weeks earlier in bitterly cold temperatures. When he got to Boston on this Saturday night, the estimated 40,000 people inside the park and thousands more outside were treated to opening acts by Frank Sinatra, who sang the national anthem, and writer-director Orson Welles. Just three years removed from Citizen Kane, Welles boomed out an oratory as only he could. "The audience delightedly applauded each at frequent intervals," wrote _Globe_ reporter Leslie G. Ainley.
When FDR's open car entered the park and he took the stage at the center of the field, the crowd was in a frenzied state. Announcers tried to still the tumult as he began to speak, but audience members continued their roaring welcome until the president himself reminded them in a shout over the loudspeakers that radio time costs money. He spoke for 36 minutes, firing back at charges leveled by Dewey, the Republican candidate, that Roosevelt had sold out to communists. FDR accused his opponent of "a shocking lack of faith in America."
The _Globe_ 's Louis M. Lyons called the appearance "a thundering windup" to FDR's campaign for reelection and reported that the crowd was "almost delirious in their wild cheering" for him. Just over five months later, after winning an unprecedented fourth term handily, FDR died of a cerebral hemorrhage in Hot Springs, Georgia, on April 12, 1945, at the age of 63. The appearance at Fenway Park turned out to be the final campaign speech of his life.
Jack Matheson of the Detroit Lions (right) stopped Johnny Grigas of the Boston Yanks for no gain in the second quarter of a November 4, 1945, game at Fenway Park. The Lions eked out a 10-9 victory in the game played during a heavy snowstorm.
Robinson went on to play in six World Series with the Dodgers and was elected to the Hall of Fame; Jethroe was Rookie of the Year for the Boston Braves. But the Sox didn't suit up a black player for another 14 years and went on to finish seventh that year, 17½ games behind the Tigers. By then the war had ended and Williams and his fellow veterans would be coming home, ready for a renaissance.
Once their stars had exchanged one uniform for another, the Sox enjoyed a spring flowering in 1946. The return of Williams, DiMaggio, Doerr, and Pesky revitalized the lineup, which was further bolstered by the arrival from Detroit of slugging first baseman Rudy York, who knocked in 119 runs.
Williams, who'd missed three full seasons since enlisting, dramatically announced his return with a homer in his first at-bat against the Braves in spring training. From Opening Day until the season finale, Boston owned the league, spending all but one day in first place. The Sox won their first five games and 21 of 24, including a 15-game streak that began with a 12-5 belting of the Yankees at Fenway.
When their tally reached a major-league record 41-9, the pennant race essentially was over. But the Fenway turn-stiles kept spinning as attendance more than doubled to over 1.4 million. One of the attendees, an Albany construction engineer named Joseph A. Boucher, found himself sitting in what became the most famous seat in the park on June 9 when Williams cracked a 450-foot homer that drilled a bulls-eye through his straw hat. "How far away must one sit to be safe in this park?" wondered Boucher, who was sitting in the 33rd row of the right-field bleachers, where the ball's landing place later was memorialized by a red seat.
Williams hit 38 homers that season, not counting his two in the All-Star Game at Fenway. But the most important and least expected was his unorthodox round-tripper at Cleveland on Friday the 13th of September, the day the Sox won the pennant. Williams had been frustrated yet challenged by Cleveland manager Lou Boudreau's "Williams Shift" that stationed everyone but the left fielder to the right of second base. "I've been planning for weeks to beat the crazy shift that Cleveland used against me," he said. So Williams knocked the ball deep to left and legged it out for the only inside-the-park homer of his career and the only run of the game.
IN RIGHT FIELD, A SEAT OF POWER
BY DAN SHAUGHNESSY
It sits in a sea of green, a single red chairback in the outer limits of Fenway Park's right-field bleachers. It is Seat 21 in Row 37 of Section 42. It is known simply as the red seat, and it marks the spot where Ted Williams hit the longest home run in Fenway history.
Like a fleck of red paint on a lush green canvas, the commemorative chair draws the eye. Someone is almost always sitting in it, even when just a few patrons are in the bleachers. New fans ask about the red seat, and citizens of Red Sox Nation are happy to relay the Fenway folklore.
Teddy Ballgame's mighty clout was struck in the summer of 1946, on a windy, sun-splashed Sunday afternoon in the first inning of the second game of a doubleheader against the Tigers.
"Hell, I can tell you everything about that one," Williams said from his Florida home in 1996—50 years later. "I hit it off Fred Hutchinson, who was a tough [righty] who changed speeds good. He threw me a changeup and I saw it coming. I picked it up fast and I just whaled into it."
Indeed. The ball sailed over the head of right fielder Pat Mullin, and then carried beyond the visitors' bullpen and kept on going. And then it crashed down onto the head of Joseph A. Boucher. More accurately, it landed on Boucher's straw hat, puncturing the middle of the fashionable skimmer.
Boucher was an Albany, New York construction engineer who kept an apartment on Commonwealth Avenue when he worked in Park Square during the week. He loved baseball and the Red Sox. But sitting more than 30 rows behind the bullpen, he wasn't expecting to catch any home-run balls.
Boucher spoke with the Globe's Harold Kaese after the game and asked: "How far away must one sit to be safe in this park? I didn't even get the ball. They say it bounced a dozen rows higher, but after it hit my head, I was no longer interested. I couldn't see the ball. Nobody could. The sun was right in our eyes. All we could do was duck. I'm glad I didn't stand up."
Boucher went to the first aid room briefly, where he was treated by a doctor. He returned to watch the Sox complete their sweep of the Tigers.
The next day's _Globe_ featured a Page One photo of Boucher holding his hat, his finger stuck through the hole. The caption read, "BULLSEYE!"
Newspaper accounts claimed Williams's homer traveled 450 feet, but the Red Sox measured the distance in the mid-1980s and arrived at an official distance of 502 feet.
"I got just the right trajectory," said Williams. "Jeez, it just kept going. In distance, it was probably as long as I ever hit one."
The bleachers were replaced with chairback seats in 1977 and '78. In 1984, Sox owner Haywood Sullivan decided to commemorate Williams's clout by putting a red chairback in the spot where Boucher sat on June 9, 1946.
If you find yourself sitting in Section 42, Row 37, Seat 21, don't bother to bring a glove. There was only one man who could hit a ball that far, and he's no longer with us.
TED'S CLOUT; ALL-STAR ROUT
If he threw it today—when it could be shown endlessly on TV replays and dissected on the Internet—Pittsburgh Pirate Rip Sewell's trick pitch would hardly rank as a novelty. But in 1946, Sewell's "eephus" pitch, an exaggerated lob that made it look like it was shot out of a mortar, was known only to a limited number of baseball fans.
That was before Ted Williams gave the eephus lasting fame when he hit one of them into the home bullpen in Fenway Park during the 1946 All-Star Game, won by the American League, 12-0. The July 9 victory, before 34,906 fans, still stands as the most lopsided in All-Star Game history.
"Yes, that was my first look at the eephus or oophus or whatever you call it," Williams said after putting on one of All-Star history's most memorable batting shows, a performance that included a second home run, two singles, a walk, and five RBI.
"That's the greatest exhibition of hitting I've ever seen," said National League manager Charlie Grimm, whose 1946 squad had fewer hits (three—all singles) than Williams did.
The Red Sox, who would go on to win the 1946 pennant, placed eight players on that All-Star team. One of them was Johnny Pesky, who said Williams was not even in the batter's box when he hit Sewell's eephus.
"The first one Sewell threw was high," Pesky recalled. "The crowd went 'Oooooh.' That pitch went up like 25, 30 feet in the air. Williams then moved up in the box, and the next one he hit it out of the park, but he was out of the batter's box."
Indeed, a front-page photo in the _Globe_ shows that Williams's right foot had crossed the white line marking the front of the batter's box.
Sewell said he warned Williams in advance that he would throw the pitch to him.
"Before the game, Ted said to me, 'Hey, Rip, you wouldn't throw that damned crazy pitch in a game like this, would you?'" Sewell recalled afterward. "Sure, I said. I'm gonna throw it to you. So look out."
Sewell actually threw three of the bloopers to Williams. On the first, Williams swung mightily and fouled it off the tip of his bat. A second blooper pitch was outside, and then came the third.
"It was a good one, dropping right down the chute for a strike," Sewell said. "He took a couple of steps on it—which was the right way to attack that pitch, incidentally—and he hit it right out of there. And I mean he hit it."
The ball landed in the American League bullpen.
"Well, the fans stood up and they went crazy," Sewell recalled. "I told him, the only reason you hit it is because I told you it was coming. He was laughing all the way around the bases."
Truett Banks "Rip" Sewell broke into the big leagues in 1932 and won 143 games before retiring in 1949. He would later say that Williams was the only batter ever to hit a home run off his eephus pitch.
Elaborate religious and military demonstrations brought crowds to the park in the 1940s.
When New York's Joe DiMaggio later hit a homer to beat the Tigers, Boston earned its first pennant since 1918. By then, though, the players had dressed and scattered and Yawkey, who'd been lugging champagne around the Midwest for days and spent millions for one pennant, sent Tom Dowd, the club's road secretary, around town for more than four hours to collect them.
Williams, who was visiting a dying veteran at an Army hospital, couldn't be located and Pesky was hanging out with former Navy buddies. But everyone else came to Yawkey's suite at the Hotel Statler to pop corks and toast what had appeared inevitable. "We ran away from 'em, didn't we?" said Del Baker, a Sox coach. "That's the best way to do it. Beat their brains out."
So Boston, which had won 104 games and left its pursuers a dozen games behind, was favored to win the World Series over a St. Louis Cardinals club that survived a best-of-three pennant playoff against the Dodgers. "Why they've made the Red Sox such heavy favorites is beyond me," Williams wrote in the _Globe_. "Those Cardinals have two arms and two legs like we have."
MR. RED SOX, JOHNNY PESKY
BY BOB RYAN
Momma knew best. Johnny Pesky was ready to grab the cash—such as it was. He had been leaning toward Boston after Red Sox scout Ernie Johnson made his living-room sales pitch, but the St. Louis Cardinals moved in at the 11th hour and were offering a little more money than the $500 the Red Sox were discussing. It was 1939, Depression time. Any extra dollar would have meant a lot to the Paveskovich family of Portland, Oregon. But Maria Paveskovich did not want to hear it.
"She said, 'No, no, no, Johnny,'" Pesky recalled. "'I don't care about the money. You go with Mr. Johnson. He will look out for you.'"
And thus was born a 70-plus-year relationship in which Pesky would be so closely associated with the ball club that his name would become attached to Fenway Park itself.
About that right-field foul pole. . . .
"[Sox pitcher] Mel Parnell started that," Pesky explained. "I won a game with a home run down the right-field line against the Athletics. But I didn't have any power, and I knew it."
The pole is a mere 302 feet from home plate and Pesky recalled wrapping eight of his 17 career homers down the line and around it. It was officially named "Pesky's Pole" in 2006. A lifetime .307 hitter, Pesky was a classic table-setter who led the American League in hits each of his first three seasons, a feat that has not been surpassed.
Pesky was an integral member of the Red Sox in those notable but frustrating post-World War II years between 1946 and 1950, when they won more games than any team in the league and only had one pennant to show for it.
"This is a game that can break your heart or build you to the heavens," Pesky mused. "The Red Sox have not been a lucky club."
Who better knows how cruel baseball fate can be than Johnny Pesky, who for so long was accused of "holding" the baseball on a relay throw from Leon Culberson while Enos Slaughter, running with the pitch, scored the winning run in Game 7 of the 1946 World Series?
He did, of course, no such thing. "Bobby Doerr defended me," Pesky said. "Ted Williams defended me. Even Slaughter defended me."
Johnny Pesky went away for a while. The Sox traded him in 1953 and he spent time managing in the Pittsburgh organization. But he always remained a Bostonian and now it's as if those years away from the Red Sox never happened.
Pesky remains a viable member of the Red Sox family into his 90s, having served the club as a player, coach, manager, broadcaster, and general all-around goodwill ambassador. He was one of the first inductees into the Red Sox Hall of Fame in 1995. In 2008, the team retired his No. 6.
HARRY AGGANIS: A TWO-SPORT STAR
In life, Harry Agganis was the perfect model for the great American sports novel. In death, he was the ironic portrait of a Greek tragedy.
The son of poor immigrant parents from nearby Lynn, Massachusetts, Agganis had vaulted from the wrong side of the tracks to stardom, first as an All-American quarterback at Boston University and then as a .300-hitting first baseman for the hometown Red Sox. When he died suddenly at age 26 in 1955, the American sports scene was stunned and eulogies echoed around the world.
As George Sullivan wrote in a _Globe_ retrospective in 1980, you had to have seen Harry Agganis to believe him. He was born Aristotle George Agganis. His mother called him "Ari" but his friends Americanized it to "Harry." At the age of 14, Agganis played for a local semipro baseball team that often took on teams made up of servicemen, and he would routinely lace hits off major-league pitchers.
As a football player, Agganis led Lynn Classical High School to a 30-4-1 record in three seasons, including a national high school championship. He threw for 48 touchdowns and ran for 24 more, and more than 20,000 fans routinely filled the Lynn stadium to see Agganis play.
Frank Leahy, Notre Dame's famed football coach, declared, "That boy is the greatest prospect I've ever seen," and virtually every major college power in America recruited Agganis, to no avail.
He chose Boston University because he didn't want to attend college away from his widowed mother. He went on to become an All-American player on both offense and defense, and in 1951, Paul Brown, the famed head coach of the Cleveland Browns, shocked the football world by making Agganis Cleveland's No. 1 draft choice. Though Harry was only a junior and would not be able to play professionally for another year, Brown saw Agganis taking over for Otto Graham as his next star QB.
Crowds of 40,000 and more were common for games at Fenway Park when BU played college powers of the era. In Agganis's sophomore season, he led the Terriers to six straight victories before they lost a 14-13 heartbreaker at Fenway to Maryland.
While playing at BU, Agganis was the subject of feature stories in Sport magazine and the _Saturday Evening Post_ , yet he was so unselfish a player that his coaches had to order him to call more plays for himself.
When the Greek community in Lynn held a dinner in his honor, Agganis refused the money raised by the dinner. Instead, he sent it to the Greek village in Sparta where his father had been born to buy soccer balls and uniforms for youngsters there. In 1955, the Varsity Club of Boston gave him a new car, but again, Agganis refused the gift, using the money to bestow a scholarship at BU for Greek-Americans.
He ended up spurning football for a $40,000 bonus from the Red Sox ("I've already proved myself in football," was his explanation) and, after a year with Boston's top farm team in Louisville, he was the Red Sox starting first baseman in 1954.
Still a few credits shy of his BU degree, Agganis took courses during his rookie season. One memorable Sunday after getting three hits against the Tigers—including the game-winning home run—he rushed up Commonwealth Avenue to BU Field to receive his diploma at commencement. The June 7 headline in the _Globe_ read: "Agganis the Hunter Bags Two Trophies: Tiger Hide, Sheepskin."
Just a year later, in May 1955, Agganis was stricken by viral pneumonia and hospitalized for 10 days, but he rushed back into the lineup against doctor's orders. In his last game with the Red Sox, Agganis batted cleanup behind Ted Williams at Chicago's Comiskey Park and got two hits to boost his average to a team-leading .313.
He was stricken again on a train trip to Kansas City, and the severe infection was complicated by phlebitis. Just when he appeared to be recovering, he died suddenly of a massive pulmonary embolism at Sancta Maria Hospital in Cambridge on June 27. He was 26.
More than 25,000 paid their respects at St. George Greek Orthodox Church in Lynn. Every Greek Orthodox church in North and South America, more than 300 of them, held memorial services, an honor customarily reserved for Greek royalty and statesmen.
On a fall day in 1991, 36 years on, Agganis' nephew, George Raimo, visited his grave in Lynn's Pine Grove Cemetery and found a baseball placed there, with the simple inscription: "In everlasting respect."
A crowd gathered on the roof of the post office building overlooking Fenway Park during the 1946 World Series.
NIGHT BASEBALL DEBUTS AT FENWAY
The Red Sox played their first night home game on June 13, 1947, before 34,510 fans. And although some ominous signs were noted beforehand—it was played on Friday the 13th, as the opener of a 13-game home stand—it turned out well for Boston. The Globe's Bob Holbrook reported the next day that the ball club "inaugurated night baseball at Fenway Park in a fashion pleasing to Bostonians as they tipped over the Chicago White Sox, 5-3, before a near-capacity crowd."
The Red Sox were the third-to-last of the 16 major-league teams of the time to install lights in their home park, and owner Tom Yawkey was clearly reluctant to do so. As a preview story in the _Globe_ by Hy Hurwitz noted: "No special festivities will accompany the arc-light premiere at Fenway. If not for public demand, Tom Yawkey would never have installed the giant towers over his orchard. Like Walter O. Briggs of Detroit and Phil K. Wrigley of the Chicago Cubs, Yawkey believes that baseball should be played under sunlight." Indeed, the only ceremony accompanying the first night game was the lowering of the colors at sunset, five minutes before the 8:15 game.
The Red Sox teams of the era were the franchise's best in nearly three decades, and in 1947 they were coming off a league championship. Certainly, they didn't need gimmickry to fill the ballpark. As Hurwitz noted, "Yawkey is strictly in the baseball business. He doesn't believe in fashion shows, nylon hosiery door prizes and other nonsense. As for fireworks, he hopes they will be provided by Ted Williams, Bobby Doerr, Rudy York & Co."
The Red Sox provided little offensive firepower on opening night, as they scored all five of their runs in the fifth inning on a combination of walks, errors, and infield hits. The Sox and starting pitcher Dave Ferriss held on to win and the _Globe_ story the following day noted, "You couldn't get breathing space in the park." It also said that the new lighting system—the brilliance of which "startled the capacity throng"—made it one of the two best-lighted stadiums in the world, equaled only by Yankee Stadium.
Yet the Sox filched Game 1 at Sportsman's Park as York deposited one of Howie Pollet's slow curves into the left-field seats with two out in the 10th inning. "I just shut my eyes and swung," said York after the 3-2 triumph. And after Harry "The Cat" Brecheen blinded them 3-0 in Game 2, Boston's Dave "Boo" Ferriss, who'd won 25 games, produced a dazzler of his own in the Fenway opener, which was all but decided in the first inning when York hit a three-run homer to left. "The ball which he hit struck the Cardinals right over the heart although it lodged in the fish nets many yards away," Nason wrote in the _Globe_.
Never had the citizenry been quite so febrile about Yawkey's paid performers. Nearly a half-million fans applied for Series tickets and hundreds of them jammed the Hotel Kenmore lobby. When standing room tickets went on sale, fans mobbed the ticket windows, nearly creating a riot.
The Cardinals, though, were unwilling to collaborate in the entertainment and they battered Boston in Game 4, matching the Series record of 20 hits set by the 1921 Giants in a 12-3 shelling. "I'd much rather lose a game 12 to 3 than 2 to 1 or 3 to 2," shrugged Williams, whose mates _Globe_ columnist Nason said had played like married men at a church picnic. "We just had our ears pinned back."
Joe Dobson righted things in Game 5 with his "atom pitch," an explosive curve that all but vaporized the Cardinals and staked the Sox to a 6-3 victory, sending them back to Missouri with two chances to win the championship. But Brecheen confounded Boston again in Game 6 as St. Louis knocked out Sox starter Mickey Harris early. "Well, the Cat not only skinned us again, but this time he feasted on our flesh and scratched on our bones," Williams conceded in the _Globe_.
So for the first time since 1912, the Sox were pushed to the limit in the Series. After falling behind by two runs in the fifth inning of Game 7, Ferriss was sent to the showers. Boston drew even in the eighth when DiMaggio doubled home two men. But when he twisted an ankle rounding the bag, it proved a most costly misstep. Leon Culberson replaced him in center field. With Enos Slaughter on first base and two out, Culberson didn't see DiMaggio motioning from the dugout for him to move more to the left.
TEN MEN OUT
For once, someone else stole the spotlight with Ted Williams at the plate.
During a game at Fenway with the Cleveland Indians on August 26, 1946, Lou Boudreau, the Indians player-manager, employed his typical Williams Shift, moving the shortstop into short right field and the third baseman to second base. The Indians' Pat Seerey, who stood in short left field, was their only player on the left side of the diamond.
Suddenly, a midget jumped out of a box near the visitors' dugout and walked onto the field. The man, who was later identified as Marco Songini, a vaudeville performer, picked up the glove that had been left on the field by Red Sox third baseman Mike "Pinky" Higgins and took a fielder's stance, pounding the glove for effect. It was customary for fielders of that time to leave their gloves on the field between innings. It wasn't until 1954 that the practice was banned.
The players, umpires, and the 28,082 fans in the park that day stared in disbelief, and then began to laugh. The laughs continued when Songini was eventually ordered off the field. He got a boost over the infield fence by Buster Mills, an Indians coach, and then climbed atop the visiting dugout and struck a fighting pose before the game continued.
The headline in the next day's _Globe_ read, "What Next at Fenway? Midget Plays Third, Indians Lose Anyway." The Red Sox won, 5-1, and as Gerry Moore's story noted: "Although a 10th man in the form of a dwarf-sized character tried to assist them by playing third base, Cleveland's Indians couldn't escape mathematical elimination from the American League pennant race."
Later in the game, Cleveland pitcher Bob Lemon and shortstop Boudreau combined to pick Williams off second base. Williams was so upset with himself that he kicked his glove all the way out to left field to start the next inning. Moore noted that, "The victory, the 10th man, and Ted's being picked off second seemed to satisfy the clients."
"I believe that these temples are our secular cathedrals and they tell us as much about what we care about as anything in our environment."
—Ken Burns, filmmaker
In August 1946, Red Sox fans' thirst was quenched by root beer, and by their team's pennant run.
One happy Boston fan scored World Series tickets, while another wrote the local newspaper to complain that many would not be as fortunate.
SAME OLD STORY
Letter to the editor [of the _Boston Globe_ ], August 27, 1946:
As a Red Sox fan of the last 25 years, I should like to make a suggestion now for sale of tickets to the forthcoming World Series.
Preliminary inquiries seem to indicate that only by luck, going to the ticket scalpers, or knowing someone who knows someone will the average fan be able to get a ticket to even one game.
So-called celebrities from Hollywood and New York will occupy seats that rightly belong to Boston fans.
Wouldn't it be fair to let Boston followers have first chance at available seats, even if it takes personal interviews with Eddie Collins and Mr. Yawkey?
—James Detrich, Brighton
FOWL PLAY
BY HAROLD KAESE
August 4, 1947—The Red Sox are really in a bad way. Not only did the Tigers claw them, 10-3, yesterday, but fans no longer bothered to ask, "What's the matter with the Red Sox?"
Instead they wanted to know, "What's the matter with the pigeon?"
Baseball indeed plunged to a new low at Fenway Park yesterday. The bird, identified as Parsley P. Pigeon, parked, perched, or roosted himself on the screen's western edge about 20 feet above the backstop. He remained there somewhat longer than three hours, turning this way and that, now and then fluttering his wings, but giving no great display of worry or anguish.
The major mystery was, how could he stand it? How could he sit there enduring such a tedious game when blue skies and green fields beckoned? Obviously, he was not a homer. He was stuck, a foot or toe caught in the wire mesh. Either that or he was a Detroit pigeon and was enjoying himself heartily.
Less than 15 minutes after game's end, a tall ladder was set up against the screen by Fenway Park's Ladder Company No. 4. No sooner had an MSPCA agent started climbing the ladder when Parsley flew away, briskly in the direction of South Williamsburg.
Fans calling up sports departments that evening asked first, "How did the pigeon make out?" and then, "What about the Red Sox?"
According to one press box worker, Parsley was not stuck at all. "He's just a lazy lout, that's all. I've seen him before sit all afternoon on that screen, even when there wasn't a game to watch."
A little boy with his mother was overheard to say, "Maybe he's going to build a nest, mama, and lay an egg."
That probably wasn't Parsley's intention at all. He knew that his wouldn't be the first egg laid at Fenway this season. Not the biggest. Nor the last.
Quoth the pigeon: "Bobby Doerr."
When Culberson had to chase Harry Walker's looping liner, Slaughter took off on a mad dash around the bases. Culberson's throw to Pesky was weak enough ("He threw me a lollipop," the shortstop said decades later) that Slaughter decided to risk going all the way. "All I kept seeing was the World Series ring on my hand," he said. How long Pesky held the ball has been barstool debate fodder ever since. The _Globe_ 's Nason said that Pesky "froze momentarily" and Pesky readily took responsibility. "I'm the goat," he said. "It's my fault. I'm to blame. I had the ball in my hand. I hesitated and gave Slaughter six steps. When I saw him, I couldn't have thrown him out with a .22."
After Brecheen shut down Boston in the ninth for his third victory of the Series, he was borne aloft by his triumphant birds of a feather. The Sox dressed quietly in their clubhouse. "We lost to a great team," concluded Cronin. But Williams, who wept in the shower and sat staring into his locker for a half hour, was inconsolable after hitting .200 for the Series. Boston Mayor James Michael Curley canceled his planned welcome-home reception for the club. "I guess the boys just simply aren't in the mood for a reception, anyway," he said. The railroad ride back to the Back Bay was somber. "This wasn't just an ordinary train," Harold Kaese observed in the _Globe_. "It was the Red Sox Special. It was a shield, bringing back to Boston a Red Sox corpse."
The renaissance in 1947 came from the Yankees, who hadn't won the pennant in four years and came out of the war disorganized and distracted, going through three managers in 1946 and finishing 17 games behind Boston in third place. Though the Sox were in second for most of the season, they dropped eight of nine games to fall eight games back on Independence Day and never were in contention again. With Ferriss, Hughson, and Harris all ruined by arm problems (they combined to win just 20 games), the rotation came apart and even Williams's Triple Crown season (.343, with 32 homers and 114 RBI) couldn't keep the Sox in contention.
As it was, early in the season Yawkey had come close to trading Williams to New York for Joe DiMaggio, which would have been the biggest one-for-one blockbuster in baseball history. But Boston also wanted catcher Yogi Berra, so Yankees owner Dan Topping nixed the swap. Yawkey did acquire a former Yankee icon, though, when he hired Joe McCarthy at the end of the season to succeed Cronin, who replaced Eddie Collins as general manager.
Workmen made sure everyone knew that Fenway would be host to Game 4 of the 1946 World Series.
"I'm going up with the real Irish now," cracked McCarthy, who'd resigned from the Yankees for health reasons in May of 1946. McCarthy was 60 years old, but his résumé was top-drawer—nine pennants and seven Series championships with the Yankees and Cubs. "Now the Red Sox have as their manager a man who converses with gremlins," observed Kaese, "instead of one who converses merely with knives and forks."
So the front office soon made a deal with the St. Louis Browns, whose mascot resembled a gremlin, and brought in starting pitchers Jack Kramer and Ellis Kinder and shortstop Vern Stephens for cash and scrubs. Kramer and Kinder won 28 games between them in 1948, and Stephens led the club with 29 homers. Yet it wasn't until after Memorial Day, when they were in seventh place and nearly a dozen games out, that the Sox came alive, winning 15 of 16 at Fenway in late July to move into first. "It took the Red Sox 98 days of the season to chin themselves into first place," Kaese observed. "They have 70 left in which to elevate the rest of the body."
Boston wound up in an enthralling pennant chase that came down to the final weekend with three teams in contention. The Indians, who were two games up with four to play, had the edge and the Sox needed to beat the Yankees twice at Fenway to force a playoff. "Nevertheless, if you can manipulate an Oriental abacus, you can still juggle the little wooden pegs and come out with a Red Sox victory," Hy Hurwitz calculated in the _Globe_.
The Sox took care of one variable on Saturday when Williams, who'd been bothered by a head cold and had hit only one homer in six weeks, crushed a two-run shot in the first inning, spurring his mates to a 5-1 triumph and eliminating New York from the pennant chase. Then, as the Tigers were rocking ace Bob Feller en route to a 7-1 decision at Cleveland on Sunday, DiMaggio and Stephens each hit homers to rally the Sox to a 10-5 victory over the Yankees and set up the first pennant playoff in American League history—a single elimination game against the Indians at Fenway the following day, October 4.
"We were counted out in the spring," remarked McCarthy, as thousands of fans were mobbing the ticket windows for reserved seats and thousands more were lining up overnight for bleacher spots. "We were counted out as late as last Wednesday. But those players never gave up."
The speculation was that Kinder, the most rested Sox starter, would face Cleveland's Bob Lemon for the pennant. But Boudreau opted for left-hander Gene Bearden, a 20-game winner. Bearden would be pitching after just one day of rest, but his knuckleball was unhittable when it was behaving for him. McCarthy not only skipped over Kinder, he also bypassed Mel Parnell, who'd won 15 games and had the staff's best ERA at home. Parnell was a southpaw and a rookie, which McCarthy concluded was a dangerous combination at Fenway with the pennant on the line and the wind blowing to left field. "Sorry, kid, it's not a day for left-handers," the manager told him. McCarthy instead went with a most unlikely starter in right-hander Denny Galehouse, a 36-year-old journeyman who had pitched only once since September 18.
Baffled, Boudreau had someone check to make sure McCarthy wasn't warming up his real starter beneath the stands. Galehouse served up two homers, a solo shot by Boudreau and a three-run blast by Ken Keltner, while the knuckleballer Bearden bollixed the Boston hitters, allowing only five hits as Cleveland won, 8-3. "It's pretty tough when you know what's coming and still can't hit it," said DiMaggio.
Instead of the Sox facing the Boston Braves in the first Streetcar Series in the city's history, the Indians went on to win their first championship since 1920, and Galehouse, who pitched only two more innings in his career, became a synonym for Sox failure for more than a half-century.
It didn't seem possible that the Sox could lose a pennant in a more wrenching fashion, but they did just that in 1949. Once again, they staged a second-half surge. Once again, they played with the pennant on the line in the season's final weekend against the Yankees.
That had seemed unlikely at the end of June when Joe DiMaggio, who'd been sidelined all season after heel surgery, swept Boston all by himself at Fenway in what he called the greatest series of his career. He hit a two-run homer and snagged Williams's long ball for the final out in a 5-4 victory in the Tuesday opener. He lifted up his pinstriped colleagues from a six-run hole with a three-run shot, and then hit another two-run homer in the eighth for a 9-7 revival on Wednesday. Then in the Thursday wrap-up game, DiMaggio hit a monster three-run blast off the left-field light tower in a 6-3 conclusion that brought a standing ovation from Sox fans who'd always claimed that Dom was the better DiMaggio. "You swing the bat and hit the ball," Joe explained after he'd scored five runs, knocked in nine more and batted .455 for the series.
In April 1947, the pennant was hoisted at Fenway by (from left) MLB president Will Harridge, Red Sox manager Joe Cronin, and Ossie Bluege, manager of the Washington Senators.
After dropping a doubleheader in the Bronx on Independence Day, Boston seemed all but finished, sitting 12 games behind in fifth place. But the Sox went 42-13 in August and September to come storming back into contention. The season turned when the Sox won two pivotal games at Fenway to draw even with the Yankees. They took the first on Williams's 42nd homer, a 410-foot launch into the right-field stands, and a daring dash by Al Zarilla, who scored from second after catcher Yogi Berra bounced a throw to first trying for a double play.
"It was a chance and I took it," said Zarilla, who'd been picked up from the Browns in May. After Parnell had mystified the visitors for a 4-1 victory on Sunday, the Sox proceeded to New York, where they claimed first place after Doerr squeezed home the winning run in a 7-6 bleeder.
They were back at the Stadium for the endgame, with the Sox ahead by a game with two to play and hundreds of their fans already lining up on Jersey Street for Series tickets. But the Sox squandered a 4-0 lead in the opener of the double-header and lost on a homer to left by light-hitting Johnny Lindell that just went fair. Then, with his club trailing, 1-0, after seven innings in the finale, McCarthy pinch-hit for Kinder, came up with nothing, and then watched New York score four in the eighth on a solo homer and a three-run bloop double by Jerry Coleman. The Yankees went on to win, 5-3, and take the pennant. "Too bad we had to do it to you," his former Yankees players told McCarthy when he went over to his old clubhouse to congratulate them.
The train ride back to Boston was a funeral cortege and the players were greeted at the station by mourners who'd been at Fenway hoping to snap up World Series tickets. "If we can't win one out of two, we don't deserve it," Yawkey said. But it would be another 18 years before his players would come that close again.
The bullpen had plenty of company on October 4, 1948, when the Fenway faithful packed the bleachers to watch the Red Sox face the Indians in a one-game playoff. They were stunned when Cleveland won the game, 8-3.
## 1950s
As the baby boom got into full swing, so did Boston's bats. The Red Sox put up one of the top offensive seasons of all time in 1950, scoring more than 1,000 runs for the only time in their history and batting over .300 as a team. But characteristically, the pitching staff had a bloated ERA (4.88), and the Sox finished four games behind the Yankees in third place. This combination of powerful offense and mediocre pitching kept the Red Sox in the top half of the league through most of the decade, but they never placed higher than third. So it went, with the Sox either staying in contention long enough or reviving late enough to keep interest alive among their saddle-shoed, flattop-wearing fandom. In 1955, for example, they crept up from sixth in May to fifth in June to fourth in July and were only three games back on Labor Day before losing 12 of 14. Boston's two major-league clubs, the Sox and the Braves, collaborated to host clinics for aspiring ballplayers in 1952, and when the Braves left town for Milwaukee at the end of that season, the Sox continued the practice, which lasted well into the 1970s. The 1950s at Fenway also featured the Harlem Globetrotters' hoop shtick on a basketball court set up in the infield, and some not-so-funny antics from Sox outfielder Jimmy Piersall, who overcame mental illness to eventually earn a spot in the team's Hall of Fame. Meanwhile, the team was overcoming its sad racial legacy by adding a pair of African–American players near the decade's end to become the last Major League Baseball club to integrate its roster.
The tease resumed at Fenway Park in the 1950 opener when Boston ran up a 9-0 lead, imploded, and lost, 15-10. Scoring runs wasn't a problem for a bunch of bashers who led the league in batting average (.302), runs (1,027), and homers (161). In one June week, the Sox pounded out 104 runs, 49 of them in a two-game pummeling of the St. Louis Browns at Fenway, during which the hosts broke five major-league records.
"Hot lava bubbled in the batter's box when the Red Sox had their innings," the _Globe_ 's Harold Kaese reported after a 20-run explosion. There was an even bigger volcanic eruption the next day when Sox second baseman Bobby Doerr hit three homers and drove in eight runs in a 29-4 win in which Boston was up, 20-0, after four innings, having batted around three times.
Yet Browns Manager Zach Taylor was unimpressed. "Pitching wins pennants," he declared, "and pitching is what the Red Sox will need more of if they aren't going to have another battle on their hands this season." St. Louis bashed the Sox, 12-7, the next day, and then Detroit laid on an 18-8 whipping. After subsequently being banged around in Cleveland, Detroit, and Chicago, the club had dropped to fourth place.
It was enough to drive a manager to drink, and Red Sox Manager Joe McCarthy, a notorious tippler in the best of times, had become all but legless and soon was gone for what were described as health reasons. "When a man can't help a ball club any more, it's time to quit," said McCarthy, who was replaced by third-base coach Steve O'Neill and never managed again.
The Sox were world-beaters at Fenway, their angular playpen. When his club was out of town, Yawkey delighted in putting on a threadbare team jacket and wrinkled khakis and taking personal batting practice with new balls. "I hit the Wall eight times today," he'd brag. But Joe Cronin, who'd played shortstop for a decade and managed for another dozen years before becoming the Sox general manager in 1947, hated the cozy confines. "Trying to build a ball club here is almost impossible. You build it for here, you lose on the road; you build it for the road, you lose here," said Cronin, who called it a "pissy ballpark."
The Sox were 39-38 away from home that season but 55-22 in their yard, where they went 16-1 during an August home stand to draw within a game of the lead, even though Ted Williams was sidelined with a splintered elbow that he'd sustained by running into a wall at Comiskey Park during the All-Star Game. But Boston struggled on the road, including an 8-0 blanking at New York on September 23, and was extinguished at Fenway by the Senators, who swept them in a Wednesday doubleheader as Boston was shut out at home for the first time in two years. "The ignominy of the Red Sox demise yesterday will not soon be forgotten in these precincts," Kaese wrote as the club went on to finish third behind New York.
Yet the fandom happily succumbed to temporary amnesia in 1951 when the Sox surged past New York and into first place just after the All-Star break. "Hey, looks like we'll be in the World Series THIS year, baby," clubhouse attendant Johnny Orlando taunted Joe DiMaggio as the Yankees quietly filed through the home clubhouse on their way to the bus after having been swept in early July. "Long ways to go, John," DiMaggio reminded him. "Don't count your money. Long ways to go."
Boston remained aloft, thanks to Clyde Vollmer, who'd apparently struck a Faustian bargain on Independence Day. "Dutch the Clutch" crammed an entire career into one month, hitting 13 homers and driving in 40 runs and hitting safely in 16 straight games in July. But when he returned to earth in August, so did his teammates, and the season ended with a brutal five-game sweep in New York, where fireballer Allie Reynolds no-hit the Sox in the series opener to clinch the pennant. Boston ended up third, 11 games out.
Mel Parnell made his pitching delivery wearing skis after the Red Sox home opener against the Washington Senators was postponed by a snowstorm on April 14, 1953. Parnell went on to win 21 games for the Red Sox that season. He pitched 10 seasons, all with Boston, and threw a no-hitter in 1956.
Clyde Vollmer crossed home plate after delivering a big hit for the Red Sox in July 1951.
Yawkey, who'd never seen his club held hitless, didn't stick around to watch Yogi Berra catch Williams's foul pop-up for the final out just after the catcher had dropped one. "Even he couldn't stand the sight of the once proud team stumbling all over the stadium," Bob Holbrook wrote in the _Globe_.
Terminal summer siestas had become the norm for the Sox and after Williams was recalled to duty by the Marines at the end of April 1952 to fly a fighter jet in the Korean War, it seemed possible that they might go dormant before the solstice. Yet one combustible personality took up for the other as rookie Jimmy Piersall put the club in first place in early June with a bit of clownish agitation that was as worrisome as it was entertaining.
His target was Satchel Paige, the St. Louis pitcher whose age (allegedly 46) was little more than a random number. "Satchmo, I'm going to bunt on you. And then watch out!" Piersall shouted as he came to the plate in the ninth inning with Boston trailing, 9-5. After he'd reached safely and advanced to second, Piersall began flapping his arms and making porcine grunts. "Oink, oink, oink," he snorted. After a walk moved him to third, Piersall resumed his taunting: "You look awful funny to me. Oink, oink, oink. Gosh, but you look funny out there on the mound."
Paige, who hadn't allowed a run in more than 26 innings and was renowned for his imperturbability, clearly was rattled. He walked Billy Goodman to force in Piersall, and then served up a grand slam to Sammy White to give his hosts a most improbable 10-9 triumph as White crawled across the plate on all fours. "I don't care what anybody thinks about what I was doing," Piersall proclaimed. "We won the ball game, right?"
HE WAS "DUTCH THE CLUTCH"
One July, a journeyman ballplayer had a month that rivaled nearly any that the game has ever seen. It wasn't just that Clyde Vollmer had a lot of hits for the Red Sox in July 1951, but that nearly every one seemed to win a game for Boston.
"I've never seen anybody come through like Vollmer—and I mean nobody," said then-Red Sox first-base coach Earle Combs, who had once teamed with Babe Ruth and Lou Gehrig in the Yankees' legendary Murderer's Row lineup.
Vollmer would never again come close to duplicating the heroic performances of those four weeks, during which he would become known as "Dutch the Clutch." Vollmer clubbed as many home runs as singles—13 of each—along with a triple and four doubles, while compiling a 16-game hitting streak. Though his batting average during the spree was a less-than-stunning .298, almost every hit was a key one—including a pair of grand slams off two of baseball's top pitchers.
"That month belonged to Vollmer as no single month has ever belonged to a major leaguer before or since," Red Sox broadcaster Curt Gowdy later recalled in a _Globe_ story by George Sullivan.
Vollmer, 29, was obtained from Washington a year earlier as a backup outfielder and pinch hitter. During his memorable month, he drove in 40 runs and scored 25 more to almost single-handedly lift the Sox into first place.
Fittingly, his explosion began on July 4, when he homered in the first game and singled in the nightcap as the Sox swept a doubleheader from the Philadelphia A's.
On July 7, Vollmer ripped a first-inning slam off Yankee ace Allie Reynolds to ignite a 10-4 rout of New York at Boston. From July 12-14, the Red Sox took over first place from the White Sox by winning three out of four at Chicago in a series that featured Vollmer's clutch hitting in all four games.
The binge continued for two more weeks and included a three-homer, six-RBI game at Fenway on July 26. On July 28, he had a single in the 15th inning to tie the score against Cleveland at Fenway, and then rocked a grand slam off Bob Feller, baseball's top pitcher of the time, to win it an inning later. But at month's end, Vollmer dramatically struck out in the last of the ninth inning as the Sox lost to the Indians, 5-4.
As Vollmer faded, so did the Red Sox. They finished third, 11 games behind the Yankees and five behind the Indians. By season's end Vollmer had reverted to his lifetime average of .251. And by early 1953 he was sold back to Washington, where he finished his major league career a year later.
"Those were my happiest years in baseball," Vollmer said later. "Coming to the Red Sox was the greatest break I ever got."
Birdie Tebbetts, Mel Parnell, and Vern Stephens posed in the Fenway clubhouse after a game in 1950.
A clubhouse attendant tidied up the home team locker room in '53.
YOUTH BASEBALL CLINICS
Over more than two decades, between 1952 and 1974, tens of thousands of youngsters attended baseball clinics with Red Sox coaches and players at Fenway Park. The clinics—billed as a one-day "spring training" for players from Little League to high school age—were sponsored by the Red Sox and the Boston _Globe_. Said Red Sox General Manager Joe Cronin on the announcement of the inaugural clinic in 1952, "Who knows but someday one of the pupils may be wearing a major-league uniform."
Indeed, at least two players who attended clinics as high school students went on to become instructors: Bill Monbouquette of Medford and Wilbur Wood of Belmont. In May 1960, Monbouquette threw a one-hit shutout for the Red Sox over the Tigers in front of thousands of youths who had attended a clinic earlier in the day.
The first clinic, in April 1952, featured players from the Red Sox and the Boston Braves, who would leave town a year later for Milwaukee. The event brought an estimated 5,000 young ballplayers and coaches into the park, and they stayed for the fourth game of the exhibition city series between the Red Sox and Braves. The youngsters were treated to a display of new Red Sox skipper Lou Boudreau's famous pickoff play, and instruction from the Braves' Tommy Holmes on hitting to the opposite field.
In a _Globe_ article promoting the first clinic, Boudreau directed clinic attendees: "If you get your heart set on becoming a Major League Baseball player, nothing will stop you. . . . Don't shirk your work. No manager likes a loafer. Play hard and for keeps, even if it's only a practice game."
Among the players who participated in the youth clinics over the years were Warren Spahn, Dom DiMaggio, Johnny Pesky, Jackie Jensen, Jimmy Piersall, Frank Malzone, Carl Yastrzemski, Tony Conigliaro, George Scott, and Carlton Fisk. In later years, some of the youngsters were brought out to work with the players on the field, and the clinic always concluded with a crowd-pleasing home-run contest.
THE KID BIDS ADIEU FOR THE FIRST TIME
In 1960, Ted Williams closed out his career at the age of 42 with a home run in his last at-bat, a moment celebrated by John Updike in his essay, "Hub Fans Bid Kid Adieu." But more than eight years earlier, Williams had batted for what many then thought would be his final time, on April 30, 1952—the day before he left for active duty as a fighter pilot in the Korean War. Ted was nearly 34 years old, and no one knew how long his military duty would last, or whether he would return.
Williams was brash and outspoken, and he had a tempestuous relationship with the Boston press. But he knew better than to complain when he was called to active military duty in early 1952. In 1941 Williams had been classified 3-A by his draft board, due to the fact that his mother was totally dependent on him. When his classification was changed to 1-A following U.S. entry into World War II, Williams appealed to his draft board, and the board agreed with him. He announced that he intended to enlist once he had built up his mother's trust fund, but the press and some fans were merciless in their complaints that Williams was ducking his duty. He enlisted in the Navy in 1942, and unlike many wartime ballplayers who played for service teams while in uniform, Williams was awaiting orders as a Marine pilot when the war ended.
Fast-forward to early 1952: Williams was now 33 years old, married with a child, and had not flown in eight years. But the Korean War was raging and he was called to active duty. After completing refresher training, he returned to the military service that would cost him almost five seasons of his career and very nearly take his life. He flew 39 combat missions over Korea, including one that ended in a crash landing and escape from the flaming wreckage of his crippled aircraft after it was hit by enemy fire. Soon after that, he contracted pneumonia and developed an inner-ear infection. He was discharged in July 1953.
The pregame stories on that April day in 1952 called it Williams's Fenway farewell, and the Globe's Jack Barry wrote the following day, "Fittingly climaxing a 14-year career, in what may have been his last appearance at bat . . ."
The Red Sox had declared it "Ted Williams Day" and the 24,764 fans, the Tigers players, and Ted's teammates joined hands and sang "Auld Lang Syne" during a pregame ceremony in which Williams received a new Cadillac. A day later, he would report to Willow Grove, Pennsylvania, for training, but not before putting an exclamation point on his career to that point.
The Red Sox and Tigers were tied, 3-3, in the bottom of the seventh inning when Teddy Ballgame came to the plate. Barry wrote, "In this most dramatic and tense moment, with every fan in the ballpark cognizant of what Williams has done under pressure in the past, the amazing Kid delivered."
Williams ripped a curveball from the Tigers' Dizzy Trout eight rows deep into the right-field grandstand. It was his 324th career homer.
After the game, former teammate Rick Ferrell, then a coach with the opposing Detroit Tigers, speculated that Williams could come back, even if he missed two seasons, because "he keeps in great shape."
Although Williams would miss nearly all of the next two seasons, he would return to hit nearly 200 more home runs. And before his third and final "curtain call" at-bat in 1960, he had a second one. In April 1954, Williams told the _Saturday Evening Post_ that the coming season would be his last, as most of his contemporaries had retired and the team was emphasizing its younger players. On September 26, in the season's final game, Williams came up in the seventh inning and homered into the right-field stands. However, the Sox batted around and he came up again in the eighth, this time popping out. Sure enough, after the game he confirmed his plan to retire, saying, "That's it."
But by May of 1955, he was back again, returning for six more seasons and a final adieu.
But the Browns were convinced that the zany provocateur was unbalanced. "Want to know something? I believe that man's plumb crazy," said catcher Clint Courtney. "Yuh, he's nuts altogether." A few weeks earlier, Piersall had brawled in the dugout at Fenway with Billy Martin, the Yankees' own time bomb. Sox Manager Lou Boudreau worried that Piersall was becoming unhinged and forbade photographers to take posed photos of him without the skipper's consent.
By the end of the month, the club had sent Piersall to its minor-league team in Birmingham, where his erratic behavior continued. "He just got worse every day," Cronin observed. A month later, Piersall was being treated in a Massachusetts hospital for what was described as nervous exhaustion, but, in fact, was mental illness. He recovered, returned the next season and played six more in Boston and 15 more years in the majors. But his teammates faded without him during the 1952 stretch run, losing 20 of 27 games in September and finishing sixth, 19 games behind the Yankees.
That was enough for the front office, which decided to go with bonus babies in 1953. Except for Williams, who was on sabbatical above the 38th parallel, the 1946 pennant winners were gone. Doerr had retired after the 1951 season and Johnny Pesky had been dealt to Detroit. When Boudreau elected to go with rookie Tom Umphlett in center, Dom DiMaggio called it a career in early May after playing only three games that season. "After all the good years I've had," said the 36-year-old, "I'm not going to be sitting on the bench."
Boston had finished in the second division for the first time since the war, attendance had dropped by nearly a half-million in three years and the Yankees had won four more rings. Investing in the past was a losing proposition, Yawkey concluded. "We take the long view," said Boudreau. "We can't expect to be a pennant contender for at least two years."
Yet the "Green Sox"—the league's youngest bunch in 1953—still managed to be entertaining, winning 35 one-run games and clubbing the Tigers by 16 and 20 in a two-day June outburst at Fenway. In the second bashing, Boston rolled up a record 17 runs in the seventh inning alone, batting around twice. "Who made the three outs?" cracked pitcher Skinny Brown in the wake of the 23-3 victory.
Not that the sound-and-light show counted for much. The Sox already were in fourth place, 14 games behind New York, and were still there when Williams returned from Korea at the end of July and promptly whacked a pinch-hit homer against the Indians. "There's no doubt about it—flying jet planes improves the eyesight," Yankees Manager Casey Stengel cracked after Williams knocked in four runs during a Labor Day weekend doubleheader at Fenway. "Baseballs now look as big as grapefruits to Williams."
Red Sox catcher Lou Berberet took time out to mingle with the Topsfield girls' softball team during its Fenway visit in 1958.
Ted Williams launched yet another Fenway home run on September 26, 1954, against the Washington Senators.
"I can't wait to see the new park when it's done. I want Boston to have the best. If any city needed a new park, it's Boston. I won't shed a tear."
—Ted Williams, Hall of Fame Red Sox slugger
ECHOES OF "SWEET GEORGIA BROWN"
The Harlem Globetrotters made a pair of stops at Fenway Park in the mid-1950s, and the antics of the famed troupe included star dribbler Reece "Goose" Tatum punching a basketball into the crowd behind the third-base dugout, part of a faux-baseball skit performed in deference to the ballpark surroundings.
The exhibition on July 29, 1954, was part of a 12-city "Summer in America" tour organized by Globetrotters founder and promoter Abe Saperstein. The famed basketball road show was in its 27th season of play and fresh off a tour of South America. The hoop doubleheader drew a crowd of 13,344. Preview stories alluded to the possibility of attracting the largest basketball crowd ever in New England, but the attendance turned out to be smaller than a typical sellout crowd at Boston Garden.
Still, Francis Rosa of the _Globe_ reported that fans "thrilled to the gyrations of Tatum, Leon Hillard, and Sweetwater Clifton" of the Trotters, who incidentally won the game, 61-41, over a collection of NBA players and draft picks that included future Hall of Famers Paul Arizin and Frank Ramsey and local stars Togo Palazzi and Ronnie Perry of Holy Cross. A preliminary game between the Boston Whirlwinds and the traveling House of David squad ended in a 47-46 win for the Whirlwinds, although Rosa noted that none of the winning team's players were actually from Boston.
The doubleheader was played on the Globetrotters' own portable court, which measured 80 feet by 50 feet and covered much of the infield, stretching from just in front of home plate almost to second base. The six-ton court was cutting-edge for the time and included a skid-proof surface developed by the U.S. Navy that would allow the Trotters to play in driving rain and other daunting conditions.
A small crowd of 3,332 watched the Globetrotters defeat the Honolulu Surfriders, 45-38, the following August at Fenway. The Trotters "clowned and capered their way through another victory," though the team was in transition and did not feature their longtime stars Tatum or the retired Marques Haynes. Among the opponents, the best-known player was Clyde Lovellette of the Minneapolis Lakers. The evening also featured a variety of vaudeville performers before the game and at halftime.
HOSTING THE BRAVES
The Boston Braves built the 40,000-seat stadium known as Braves Field in 1915, and they were its primary tenants until the end of the 1952 season when the team left for Milwaukee. The Braves played second fiddle to the Red Sox for nearly all of their Boston years, even though their National League franchise began in 1876, a quarter-century before Boston's American League entry was founded in 1901.
Ironically, when the "Miracle Braves" rallied to win their only World Series in 1914, they played their World Series home games at nearby Fenway Park because Braves Field, just off Commonwealth Avenue about a mile away, was under construction. It would not be ready until the following season, and in 1915 and 1916, the Braves returned the favor and allowed the Red Sox to host their own World Series games at new Braves Field, now the much larger stadium. The ballpark would host its only Braves' postseason games in 1948, when the Braves lost the World Series in six games to the Cleveland Indians. The Indians had beaten the Sox in a one-game playoff the previous week to spoil the prospect of Boston's only two-team "trolley" World Series.
Though the Boston Braves were marginal at best on the field (with only 11 winning seasons in 38 years at Braves Field), they had their share of historic feats. The longest major-league game in history was played at Braves Field on May 1, 1920, when they battled the Brooklyn Dodgers to a 26-inning, 1-1 tie before the game was called because of darkness. It was also the site of several Boston baseball firsts, including the Hub's first night game and the first televised game, and the Braves also had the first black player to wear a Boston uniform: Sam Jethroe in 1950, who was named NL Rookie of the Year that season. The National League's longest hitting streak, 37 games, was compiled by the Braves' popular Tommy Holmes in 1945, though Pete Rose broke the record years later.
The Braves' most successful seasons came too late—from 1945-48. Beyond that, they were a tough draw, attracting only 245,000 in 1943. It wasn't until 1947 that the team drew a million fans in a season.
Meanwhile, local football fans had an equally fickle relationship with another Braves team. The NFL's Boston Braves played their inaugural season (1932) at Braves Field, and then moved to Fenway Park, changing their name to the Boston Redskins. The team headed to Washington four years later because of a lack of support in Boston.
A sold-out Braves Field on Gaffney Street in Allston. Though crowds of 43,000-plus at one time packed Braves Field to watch the likes of Warren Spahn pitch—including 1.4 million in 1948—owner Lou Perini moved the team to Milwaukee in 1953 because of dwindling attendance.
WHEN FEAR STRUCK OUT: JIMMY PIERSALL
No one who lived in Boston in the 1950s can forget Jimmy Piersall, the center fielder for the Red Sox. A swift, graceful, handsome athlete, he would be off at the crack of a bat, racing toward the wall at Fenway Park, running, straining, and then timing his leap to spear the ball at the last moment.
One day in the summer of 1953, Roger Birtwell, a _Globe_ sportswriter not given to excesses, was so moved by Piersall's fielding that he wrote: "35,000 persons sat spellbound in Cleveland Municipal Stadium yesterday and watched the greatest exhibition of outfielding in major-league history as the Red Sox beat Cleveland, 2-0, 7-5, and went into third place."
But Piersall suffered from mental illness, reportedly bipolar disorder. In his rookie season of 1952, he was involved in fights with the Yankees' Billy Martin, and teammates Mickey McDermott and Vern Stephens. Finally, a series of bizarre demonstrations on and off the field led to several ejections from games and culminated in a breakdown. Piersall was confined to Westborough State Hospital for electroshock therapy and psychotherapy. To everyone's surprise, he recovered and the next year he returned to the Red Sox and became a star. Piersall was selected to the American League All-Star team in 1954 and 1956, thanks in great part to his outfield play. In 1956, he posted a league-leading 40 doubles, scored 91 runs, drove in 87, and had a .293 batting average. The following year, he hit 19 home runs and scored 103 runs. He won a Gold Glove Award in 1958, but that winter he was traded to the Cleveland Indians for first baseman Vic Wertz and outfielder Gary Geiger. Piersall earned a second Gold Glove with the Indians in 1961, also finishing third in the batting race that season with a .322 average.
In June 1963, while playing with the New York Mets, Piersall famously ran the bases while facing backward after hitting the 100th home run of his career.
He described his breakdown in a 1955 book, _Fear Strikes Out: The Jim Piersall Story_ , which became a movie in 1957 starring Anthony Perkins and Karl Malden. (After seeing Perkins play Piersall in the film, director Alfred Hitchcock signed the actor to portray Norman Bates in _Psycho_.) Though in his autobiography Piersall blamed much of his condition on pressure from his father, he later disavowed the film, saying it distorted the facts.
One afternoon in the early 1980s, _Globe_ writer Jack Thomas, who idolized Piersall as a youngster, visited him in South Carolina and accompanied him to a psychiatric ward at a Charleston hospital. When Piersall stepped off the elevator, there was a bustle among the patients, who loved Piersall, not as a ballplayer, but as a symbol of hope that perhaps they too could overcome mental illness. Piersall shook hands with the doctors, admonished two nurses for smoking, and then settled into an easy chair to chat with the children hospitalized for psychiatric care.
"Have you read my book or seen the movie about me?" he asked. "It will let you know that we all have problems. . . . I don't have a college degree, but I've got a PhD in some other things. Don't ever let anyone tell you there's no stigma to mental illness. You're going to have to prove yourself all over again. I know how tough it is to be alone. I'm a graduate of a mental institution."
Suddenly, when Piersall said he was opposed to women as umpires, there was a sharp exchange between the Sox legend and a 13-year-old patient named Cynthia. It was brief, but acrimonious, and both seemed hurt. As Piersall and Thomas were leaving, Cynthia's voice called out, "Jimmy?" He turned and walked to her quickly, knelt, and the two embraced. He kissed her cheek, and she squeezed him hard, turning her face so he wouldn't see her tears.
"You're a doll," he said, "and you're going to be OK. You're going to make it."
On September 17, 2010, Jimmy Piersall was inducted into the Boston Red Sox Hall of Fame.
Fenway Park played host to numerous boxing matches over the years. In July 1954, Tony DeMarco, left, a welterweight from Boston's North End, fought lightweight George Araujo of Providence. DeMarco won by a TKO in the fifth round.
In June 1958, park personnel prepped for the arrival of television personality Ed Sullivan, who came to host the annual Mayor's Charity Field Day.
JOHN KILEY: A THREE-SPORT STAR
Radio station WMEX was on Brookline Avenue, adjacent to Fenway Park, and John Kiley was the station's musical director and studio organist from 1934 to 1956. When Kiley played in the studio in the early 1950s, he had a regular visitor. Unbeknownst to Kiley, that listener was Red Sox owner Tom Yawkey, and Yawkey wanted to hire him.
"He was a very delightful, pleasant man," Kiley once recalled, "and he complimented me a lot when he was around the studios, but I'd just tell him to be quiet [because Kiley was on the air]. I didn't know who he was until he offered me the job."
Yawkey brought Kiley on board in 1953 when he decided to add music to the Fenway experience. Kiley was also the house organist at Boston Garden from 1942 to 1984 and was proud to be the answer to the trivia question: "Who played for the Bruins, the Celtics and the Red Sox?" Kiley was quick to point out that he also played for the Boston Braves before they left town—in fact, the Braves became his first sports gig in 1941.
Kiley began taking piano lessons when he was 6 and he later quit Dorchester High School to enroll in the Boston Conservatory. He landed jobs playing for the silent movies at the old Criterion Theater in Roxbury and several other theaters, starting at age 15, before taking over at the opulent Keith Memorial Theater downtown. When _The Jazz Singer_ ushered in talking pictures in 1927, Kiley found himself out of work before taking over at WMEX and the sports venues.
Kiley started at Fenway in an era when the goal was to please the owner's wife with songs like Rodgers and Hart's "Where or When?" (Tom and Jean Yawkey's favorite song). He was known to pump out "White Christmas" on a scorching day, and his 20-minute pregame medley of songs might include "Embraceable You," "Clarinet Polka," "Misty," and "The Way We Were."
Kiley was not a flamboyant organist, but he did add the occasional flourish, including his spirited playing of the "Hallelujah Chorus" when Carlton Fisk hit his historic Game 6 home run in the 1975 World Series. Kiley retired in 1989 and died at age 80 in 1993.
These days show tunes on the ballpark organ are fading out in favor of loud rock on expensive sound systems. Fenway is one of a handful of ballparks that still feature organ music. "We try to do either of two things at every game: make a child fall in love with baseball, or remind an adult where, when, and why they fell in love with baseball," Red Sox Vice President Charles Steinberg explained in 2005. "That's why it's essential to use the organ and pop music at Fenway."
Ted Williams enjoying himself sitting in a miniature car, tooting its horn to the amusement of Fenway Park fans, August 22, 1958. The small car was one of several used by the Shriners in a pregame ceremony.
Said Yogi Berra: "He don't look like he used to; he looks better."
The Splinter, who ended up hitting .407 with 13 homers and 34 RBI in his abbreviated season, immediately jacked up Fenway attendance by more than 3,000 a game. Not that Boston fans had a better alternative. Once the Braves decamped for Milwaukee just before the season, the Sox literally were the only game in town.
Even though the club was in seventh place by mid-May (losing Williams with a broken collarbone on the first day of spring training didn't help), and ended up 42 games behind the Indians in 1954, nearly a million spectators came through the Fenway turnstiles, with more than 85,000 turning up for a weekend series with the Yankees in late August that the Sox swept.
So it went for the next several years with the Sox, who remained just competitive enough to justify the price of a ticket, either staying in contention long enough or reviving late enough to keep their adherents interested. In 1955, they crept up from sixth in May to fifth in June to fourth in July and were only three games back on Labor Day before losing 12 of 14. "Our tank went dry when we needed gas," Piersall concluded, "and we just couldn't get a refill."
There was always enough wall-banging to keep the customers reasonably satisfied, always a new prospect—a Harry Agganis, a Jackie Jensen, a Gene Stephens—for them to check out. And, always, there was Williams—prodigious and profane, invigorating and infuriating. He vowed that the 1954 season would be his last, saying, "You think I'm kidding, but I'm not." But he came back by mid-May of 1955 (too late to qualify for the batting title) and hit .356 to lead the league.
His chilly relationship with "the knights of the keyboard" had turned acerbic by 1956 when he twice spat in the direction of the press box. Soon after came "The Great Expectoration," when Williams, after being booed for muffing a two-out fly ball in the 11th inning against the Yankees, reeled in a blast by Yogi Berra and, on his way in to the dugout, directed several saliva shots at a record crowd of 36,350. "I'm not a bit sorry for what I did," later declared Williams, who'd walked to produce the winning run in a 1-0 decision. "I was right and I'd spit again at the same fans who booed me today. Some of them are the worst in the world."
Yet they were cheering him a day later after Williams, who was fined a record $5,000 by Yawkey (who never bothered collecting), smashed the game-winning homer, and then theatrically covered his mouth as he rounded the bases. "Atta boy, Ted," one grandstand denizen applauded. "We're all with you." Homers always had been an instant remedy at Fenway. But there were other ways to please a crowd. Spectators were startled three weeks earlier when former ace Mel Parnell, spurned after two injury-ruined seasons, became the first Sox hurler in 33 years to pitch a no-hitter.
Parnell, who'd been knocked around by the Yankees 10 days earlier on Independence Day and hadn't appeared since, baffled the White Sox with sinkers, facing only one man over the minimum. "That makes up for a lot of those days the past few years when things didn't go so good," said the 34-year-old left-hander after he'd done on July 14 what no Boston pitcher had done since Howard Ehmke in 1923—and no Sox pitcher had done at home since Ernie Shore in 1917. "Boy, it felt good to hear those people cheering me."
"There are two places that I've played in my entire career that you can actually feel momentum change: Fenway Park and Yankee Stadium. You can actually feel it change."
—Tim Wakefield, Red Sox pitcher
TED AND GLADYS
As the Globe's Bud Collins recalled it, "Gladys Heffernan was one hardheaded woman. You can bet your Louisville Slugger on it. Sweet, congenial, grandmotherly, but—lucky for her and for Ted Williams that September Sunday afternoon in 1958—she was topped by a skull that could have gone 10 rounds with Gibraltar."
Collins was in the Fenway press box when Williams struck out against the Washington Senators' Bill Fischer (who would go on to become the Red Sox pitching coach in the 1980s). Williams's temper was legendary, even at age 40, and he was so furious at striking out that, as he strode away from home plate, he took a vicious cut at the air—but the bat slipped from his hands.
There was no time for the 69-year-old Heffernan, sitting a few feet away in the first row near the Sox dugout, to react. Like a guided missile, the speeding bat beaned her. A communal gasp of horror and concern swept the park, followed by choruses of boos, as attendants rushed to the felled woman. A distraught Williams was quickly over the low wall and beside her.
As Collins told it, by the time he had run down the staircase from the press box and along an aisle to where she had been sitting, he was told by an usher that she was in the first aid room. Was she dead? The usher didn't know.
Outside the room, Red Sox General Manager Joe Cronin greeted Collins with, "She's fine. Nothing to worry about."
"Fine?"
"Of course," Cronin said with a smile. "Ted has talked to her. Apologized. She told him she knows he didn't mean it, that it was an accident. Just a little bump."
"But maybe a big lawsuit, Joe?" "No, no, no. There'll be no lawsuit," he said firmly.
Collins frowned. "This happened five minutes ago. How can you be so certain there'll be no suit?"
"Because," Cronin smiled again, "Gladys happens to be my housekeeper."
Williams reportedly cried in the dugout after the accidental beaning. In his next at-bat, he doubled to drive in the second run in a 2-0 Boston win. Williams would go on to finish with a .328 average that season to win his final AL batting title.
Cronin's daughter, Maureen, who was 13 years old at the time, recalled the incident years later. She said Heffernan spent a week in the hospital, and Williams felt awful about it. "He went to see her every day in the hospital and bought her a diamond wrist watch," Cronin said. "She loved Ted and forgave him, but she never went to another game."
Joe Cronin decided the family's seats were a little too close to the action and moved them back a few rows, much to his four children's displeasure.
"That moment, when you first lay eyes on that field—the Monster, the triangle, the scoreboard . . . the left-field grass where Ted [Williams] once roamed—it all defines to me why baseball is such a magical game."
—Jayson Stark, ESPN analyst
Workmen at Fenway Park strain to assemble giant speakers, part of a stereophonic system used for the Boston Jazz Festival.
It was the last great moment for Parnell, who tore an elbow muscle and retired after the season. For nearly a decade, he'd been the mainstay of a staff that never had enough pitching, which is why Boston perennially ended up double-digits behind New York. In 1958, when Williams and teammate Pete Runnels finished 1-2 in the batting race and Jackie Jensen was Most Valuable Player, the Sox still finished 13 games out.
Had Jensen been able to play all of his games in the Fens, he might well have been a Hall of Famer. But his acute fear of flying exhausted him. When his teammates were heading for Logan Airport, Jensen often was jumping into a car and driving all night to the next city. He could handle Bob Lemon but not Rand McNally, so he quit the game a year later.
For Williams the most difficult opponent had become Father Time. He turned 41 in 1959, when a pinched nerve in his neck wrecked him for the season. The Sox sank with him and when they were in the cellar in July, Yawkey decided to ax Manager Mike "Pinky" Higgins, the former infielder who'd taken over for Lou Boudreau at the end of the 1954 season.
Yawkey dispatched General Manager Bucky Harris to Baltimore to give Pinky the pink slip. "The little one (Harris) keeps saying, 'You gotta quit, you gotta quit,'" a barmaid told two Boston sportswriters who'd paid her to provide a report. "And the big one (Higgins) keeps telling the little one to go bleep himself."
Higgins was given a scouting job and Billy Jurges, a Washington coach, was brought in to supervise the remainder of the season, which was most notable for the arrival of infielder Elijah Jerry "Pumpsie" Green and pitcher Earl Wilson, the franchise's first two black players. Green made his Fenway debut on August 4, 1959, two weeks after he first played for the Red Sox on the road in Chicago. Ever since the club had turned up its nose at Jackie Robinson at a 1945 tryout, the Sox had shown little interest in signing African-Americans. Perhaps it was a coincidence that the owner was from South Carolina and the manager from Texas, but Boston was the last team to integrate and its all-white roster all but assured continued mediocrity.
Elijah "Pumpsie" Green made his first major league start for the Red Sox in 1959. Green, the first black player for the Sox, was honored during the team's annual Jackie Robinson Day ceremony at Fenway Park on April 18, 2009.
GREEN GIANT
**The Wall, now known affectionately as the "Green Monster," is unquestionably the defining feature of America's most beloved ballpark. It stands 37 feet tall and 240 feet long, and its legend has been building since 1934.**
**Fenway Park was created in 1912, when then-owner John I. Taylor moved the home of the Red Sox from the Huntington Avenue Grounds to an undeveloped piece of land he owned in the Fenway area. With Lansdowne Street already established, architect James McLaughlin was left with no option but to truncate the field boundaries.**
**The short distance to the left-field boundary from home plate was compensated by the dead-ball era of the time and the height of the wall, and though much has happened to it since then, the large, storied structure remains formidable to hitters, pitchers, and fielders today.**
BELLY OF THE MONSTER
**Inside the Wall, it's scorching in the summertime and cold in the spring and fall, but scorekeepers get spectacular front-row seats and a chance to chat with outfielders, who can enter through a door that opens onto the field, during breaks in play. Manny Ramirez was particularly fond of ducking inside, sometimes barely making it back out before the game resumed. There is no permanent bathroom, although portables have been used.**
HISTORY OF THE WALL
**When Fenway Park was built in 1912, there was a 10-foot-tall, sloping embankment in front of a wooden wall in left field. The incline, which served as lawn seating and a picnic area for overflow crowds during the dead-ball era, as well as support for the wall itself, was tough on outfielders. However, Boston's Duffy Lewis mastered it and the hill became known as "Duffy's Cliff."**
**When Thomas A. Yawkey bought the Red Sox in 1933, Duffy's Cliff was scaled down, elevating the importance of the 37-foot metal fence behind it.**
**FISK POLE:**
The pole on the left-field foul line atop the Green Monster is known as the Fisk Foul Pole, in honor of Carlton Fisk's game-winning homer that struck the pole in the 12th inning of Game 6 of the 1975 World Series.
**310:**
At the foul pole, the Wall is only 309 feet, 3 inches from home plate, but for most of the century the Red Sox posted a sign that read "315." Club officials refused to allow an independent measurement of the distance, but when a Boston _Globe_ reporter snuck into Fenway and came up with the new figure, the Sox grudgingly changed the sign to read 310 feet in 1995. Major League rules today stipulate that no fence in any new park be closer than 325 feet to home plate.
**96:**
Metric distances were added to the outfield walls in 1976, when it was thought that the U.S. would adopt the metric system; thus the 315-foot marker had a smaller accompanying 96-meter marking in yellow. The metric figures were painted over during the 2002 season.
**THE LADDER:**
A ladder runs from above the scoreboard to the top of the Wall. It was once used to retrieve home-run balls, but it is no longer needed with the advent of the Monster seats. A ball is in play if it hits it.
**CITGO SIGN:**
Every time a player hits a home run over the Green Monster, the CITGO sign is seen by fans at the ballpark and on television. The computer-operated sign is double- faced and measures 60 feet by 60 feet. In early 2005, the sign received a major restoration and technology upgrade from neon light to LEDs.
**"FENCE GREEN"**
The green paint used on the wall is a 100 percent acrylic made by California Paints, which was founded in Cambridge, Mass. in 1926 and is now based in Andover, Mass. The color, called "Fence Green," is considered proprietary by the Red Sox; it is not sold publicly, and the formula of colorants is a secret. The hue of Fence Green has apparently been the same since the wall was first painted in 1947, although California Paints didn't start producing the color until the 1970s. It takes about 35 gallons of paint to cover the wall. Other green hues are used around Fenway Park, including Scoreboard Green, Box Green, and Special Green.
FENWAY'S LEFT FIELDERS
Red Sox players who played most games in left field for each of the last 100 years.
## 1960s
Carl Yastrzemski follows the flight of his second home run of Game 2 of the 1967 World Series, a 5-0 Red Sox victory. Yaz supplanted Ted Williams in left field for the Red Sox and created his own legacy, including an MVP season in '67 and his stature as the first American League player to have at least 400 home runs and 3,000 hits.
In the early 1960s, the fledgling Boston Patriots of the American Football League began playing at Fenway Park. While there would also be soccer games and wrestling matches played at Fenway during the 1960s, what the ballpark didn't have for the first part of the decade was much of a baseball club. When Ted Williams retired after hitting a home run in his final at-bat in 1960, he took most of the drama surrounding the team with him. The Red Sox weren't just a boring club, they were also inept. The Sox of the 1960s echoed the franchise's teams of the 1920s by finishing in the second division for eight straight seasons, including ninth-place finishes in 1965 and 1966. Little wonder that owner Tom Yawkey ceded to the Patriots' wishes to play in his park, lifting a ban on football at Fenway partly so that he could derive some income from the newcomers, since Sox fans were staying away in droves. The nadir for the Red Sox came on September 28 and September 29 of 1965, with the team en route to a 100-loss season; the attendance for consecutive games was 461 and 409 fans, respectively. The Sox changed things up in 1967, hiring a brash new manager named Dick Williams and giving its young nucleus a chance to play, and to flourish. Everything changed for Boston and the Red Sox in that Summer of Love; while young people throughout the country flush with "Flower Power" were being warned not to trust anyone over 30, New Englanders were learning to count on a man in his late 20s called Yaz. The Sox captured one of the most exciting pennant races in history, jostling past the White Sox, Twins, and Tigers for their first AL title in 21 years. And Boston baseball would never be the same.
UPDIKE HIT IT OUT OF THE PARK, TOO
BY BOB RYAN
On September 28, 1960, John Updike, 28 years old and, though raised in Pennsylvania, a Ted Williams fan since childhood, decided it was a good idea to attend the afternoon game between the Red Sox and Baltimore Orioles. He, like all members of the public, knew only that it would be the final home game of Ted's career. Not until the game was concluded did people learn that it would be Ted's last game, period, that he had decided before the game he would not be making a season-ending trip to Yankee Stadium.
The times were different. The word "hype" had barely entered the language. Today, there would be special editions, minted coins, and live shots galore.
"The world was a simpler place," Updike noted.
But Wednesday, September 28, 1960, was a dank, dreary day. And the Red Sox, Ted Williams aside, were a dank, dreary team on their way to a 65-89 record and a seventh-place finish. Accordingly, a mere 10,454 fans showed up. And it could very easily have been 10,453. Updike's first choice that day was to visit a lady on Beacon Hill. Fortunately, the lady was not home.
Updike explained in a 1977 epilogue: "I took a taxi to Beacon Hill and knocked on a door and there was nothing, just a basket for mail hung on the door. So I went, as promised, to the game and my virtue was rewarded."
The resulting "Hub Fans Bid Kid Adieu," published in the October 22, 1960 edition of the _New Yorker_ , is the most spellbinding essay ever written about baseball. Some, like critic Roger Dean, go even further. "It is simply the greatest essay I have ever read," he said.
"It influenced me in a big way," said Roger Angell, who would become the foremost baseball writer of the late 20th century, but in 1960 had yet to publish a word about it. "And it has influenced just about every sportswriter who followed. The great thing is that he went expecting something amazing and incredible—and it happened. Only baseball provides in any number those totally unexpected turns."
"My one effort as a sportswriter," explained Updike. "It's had a longer life than I would have expected."
We all know how the story ends. In the eighth inning, battling horribly adverse weather conditions that had already cost him one shot at a homer, Williams hit a 1-1 pitch from Jack Fisher onto the canopy covering a bench in the Red Sox bullpen. He ran the bases hurriedly amid relentless applause and did not tip his cap. He took his place in left field at the start of the ninth and was replaced by Manager Mike Higgins with Carroll Hardy, in the hopes that Williams would acknowledge the crowd, and again he did not tip his cap. He had not tipped his cap since 1940 and he had no remote intention of deviating from his policy.
Wrote Updike, "No other player visible to my generation concentrated within himself so much of the sport's poignance, so assiduously refined his natural skills, so constantly brought to the plate that intensity of competence that crowds the throat with joy."
Williams liked the piece. At least, that's what was conveyed to Updike by a third party. And Ted even suggested Updike be a collaborator on a biography, an offer that Updike, a longtime resident of Boston's North Shore who died in 2009, politely declined.
"I'd said all I had to say on the subject," he said in the epilogue.
Of Ted's stubborn refusal to tip his cap that day, despite being given three separate opportunities to do so (coming to the plate, rounding the bases, and trotting in after being removed from the field), Updike sagely noted, "Gods do not answer letters."
But they sometimes leave behind epic accounts of epic events.
"Everything is painted green and seems in curiously sharp focus, like the inside of an old-fashioned peeping-type Easter egg."
—John Updike, from "Hub Fans Bid Kid Adieu"
Teammate Jim Pagliaroni offered congratulations after "Teddy Ballgame" hit his 521st home run in the final at-bat of his career.
"We knew almost all the fans in the stands by name."
—Dick Radatz, the top closer in baseball when he pitched for the woeful Red Sox teams of the early 1960s
There were only 2,466 fans in the stands on April 11, 1962, when the Red Sox played the Cleveland Indians, winning the contest 4-0 on a Carroll Hardy grand slam in the 12th inning.
Bill Monbouquette, a native of Medford, Massachusetts, was one of the few bright spots for the Red Sox of the early 1960s. A four-time All-Star, he won 96 games over eight seasons in Boston, and he no-hit the White Sox at Comiskey Park in 1962.
As mediocre as most of the fifties had been for the Red Sox, the club at least had been finishing in the money at a time when that meant that most of its players wouldn't have to spend the off-season selling tires. But by 1960, the Sox were bouncing around the bottom of the league and their owner was growing exasperated. "Want to buy a ball club?" Tom Yawkey asked broadcaster Curt Gowdy after a 12-3 home loss to the Indians. "I'll sell it for seven million. Seven million will take it."
It was early June and Boston already was stuck in eighth place, nearly a dozen games out of first. To shake things up one game, manager Billy Jurges switched Bobby Thomson, who'd hit the "Shot Heard 'Round The World" to win the 1951 pennant for the Giants, from the outfield to first base, where Thomson made two of his team's four errors in the fourth inning.
"I can't believe what I saw here tonight," declared Yawkey, sipping bourbon in his office after the game to blot out the memory. "A guy playing first base in the major leagues with a finger glove on. A finger glove at first base. I'll sell this club. Take it off my hands. This is the major leagues? THIS is the major leagues?"
And yet more than a million spectators came through the Fenway turnstiles that year, most of them to watch Teddy Ballgame play his final season. After his miserable 1959 campaign, by far the worst of his career, Ted Williams actually demanded that the club chop $35,000 from his $125,000 salary. Though Yawkey had suggested during the off-season that the slugger retire, Williams wanted one more .300 season and to reach 500 career home runs before he put away his bat. He easily accomplished both, batting .316 and hitting 29 homers to finish with a lifetime average of .344 and 521 career homers.
While everyone in the park knew that the midweek game against the Orioles at the end of September would be Williams's last at Fenway, nobody but he, Yawkey, Gowdy and the clubhouse denizens knew that it would be the last of his career. Williams had a nasty cold and had asked the owner if he could skip the final series in New York. So when he deposited Jack Fisher's fastball atop the Sox bullpen in the eighth inning, it made for a grand finale and a farewell that was completely in character.
"I thought about tipping my hat, you're damn right I did, and for a moment I was torn," Williams later confessed. "But by the time I got to second base I knew I couldn't do it. It just wouldn't have been me." Yet Manager Mike Higgins, who'd supplanted Jurges on June 12, wasn't going to let him get away without one last ovation. "Williams, left field," he declared at the top of the ninth—then sent Carroll Hardy out after him once Williams had been saluted for a final time.
The club already had Williams's successor—there was no replacement—in mind. Carl Yastrzemski, who'd grown up on a Long Island potato farm and whose father had spurned the Yankees' lowball offer for his son, had signed a $100,000 deal while he was at Notre Dame. "We're paying this kind of money for THIS guy?" said skeptical general manager Joe Cronin when he met his skinny bonus baby.
The Sox gave Yastrzemski the jersey No. 8 because it was nearest to his predecessor's legendary No. 9, and assigned him the locker adjacent to Williams's at spring training, where Williams, now a batting instructor, gave Yaz a thorough tutorial. But by midsummer, when he was stuck in a prolonged slump and haunted by Yaz-vs.-Ted comparisons, the overwhelmed rookie went to see the owner. "I feel guilty about not giving you your money's worth," Yastrzemski recalled telling Yawkey in his autobiography. "Could Ted come in for a day or two and take a look at me?"
'PATRIOTS DAY' AT FENWAY
Almost from the time it opened, Fenway Park served as a professional and collegiate gridiron for several teams. But Red Sox owner Tom Yawkey banished football from his ballpark in the late 1950s and early '60s because he wanted to protect the grass for baseball. When Sox attendance plummeted, however, Yawkey welcomed the Boston Patriots of the American Football League to Fenway in 1963.
It's hard to fathom today, but the Patriots operated on pretty much a shoestring budget for their first decade. The team offices were in a basement in Kenmore Square, and when it came time to draft players, reporters sat alongside as team officials flipped through the _Street & Smith_ football guide to make their choices. Former Patriot star Gino Cappelletti would typically rush from training camp to anchor WBZ television sportscasts five nights a week to supplement his $7,500 salary. The Patriots played their first three seasons at Boston University's football field (the former Braves Field) before the Red Sox offered their ballpark.
"We had played at BU and that was appreciated," said Cappelletti. "But once we got to Fenway Park, it immediately gave us a feeling of having arrived."
Fenway's football configuration put one end zone on the third-base line and the other in front of the bullpens in right field. Gil Santos called the games from a makeshift booth atop the first-base grandstand, and temporary stands were erected in front of the Green Monster. Both benches were initially situated in front of the temporary stands to avoid blocking the sight-lines of fans sitting behind the first-base dugout.
"Funny thing about those side-by-side benches," said Cappelletti. "During the games, we would get closer and closer to the other team and start eavesdropping. I remember one game when we heard [Chiefs' head coach] Hank Stram calling for a screen pass. The next year, they put us over on the first-base side."
A receiver/placekicker, Cappelletti booted a lot of footballs into the stands. "The bullpens in right were pretty close to the end zone so most of my extra points went over the bullpens and into the bleachers," he recalled.
One summer, because of the uncertainty over where they would be playing, the Patriots printed tickets for three potential locations. Patrick Sullivan, the son of Patriots' founder Billy Sullivan and the team's former general manager, said his favorite venue was Fenway, where the Pats played for much of six seasons, until 1968.
"It was a great place to watch a football game, for precisely the same reason it is for baseball," he said. "If you were in the temporary seats that we set up against the Green Monster—it was a big grandstand with about 5,600 seats—you were right on the action. The conversations that occurred between some of the coaches and our fans were hysterical. I was a ball boy, so I would listen in on them."
Sullivan also remembered how players and fans would mingle on the field after games, which created a connection that doesn't exist today.
"Romances were started, business relationships were formed, and guys got jobs during those times," he said. "The visiting players would hang out, too, and one time I bumped into [Bills quarterback] Jack Kemp. One of the things he later told me was that it was a Patriots' season ticket holder who convinced him to later go into politics: Tip O'Neill." Kemp went on to a successful political career after his playing days, which included a run for the presidency in 1988.
Cappelletti and quarterback Babe Parilli were two of the team's big stars of the time, and running back Jim Nance's appearance on the cover of Sports Illustrated gave the Patriots national recognition in the mid-1960s. The AFL gradually gained acceptance, and later, a merger with the NFL.
"When we first got there, I don't think many of us realized that pioneering was what we were doing," said Cappelletti. "The original dream was survival. First, you had to survive to make the team. Then, you hoped for franchise survival. And then you hoped that the league would survive."
"Get behind by four runs, no problem. Ahead by four in the eighth, delay the champagne. Nothing was—or is—certain, not even a pitcher sailing along. One little hit, an error maybe, can open a door to a pop-fly homer in the net. . . . That's the magic of Fenway Park. That's why people love it so."
—Ned Martin, longtime Red Sox announcer
So Williams abandoned a fishing trip to give Yastrzemski a refresher course that made for a respectable debut season highlighted by 80 RBI and 11 homers. His teammates, few of whom would deign to speak to a rookie, didn't do nearly as well, and the Sox finished in sixth place, far enough behind that they would have needed a spyglass to locate the Yankees. The highlight of the season came against the last-place Senators on a Sunday in mid-June when the Sox staged the greatest comeback in franchise history, coming from seven runs down with two out in the ninth to win the opener of their doubleheader, 13-12. They then claimed the nightcap in 13 innings, 6-5.
"You had a real bad day at the plate today," an observer jokingly told Jim Pagliaroni, who hit the killer grand slam in the first game and the walk-off homer in the second. "You were only 2 for 11." "Gee, that's right," the catcher realized. "My batting average is going to blazes."
In a season when the Sox finished 33 games out of first place, novelties of any kind were welcome. The most notable came in the season's finale in the Bronx when Tracy Stallard proffered the ball that Roger Maris launched for his record-breaking 61st homer of the season. "I have nothing to be ashamed of," concluded the rookie, who pitched only one more inning for the club, but ended up as a Cooperstown footnote, albeit with an asterisk. "Maris hit 60 other homers, didn't he?"
It was otherwise a forgettable summer that engendered several more of the same. "There may be more broken beer bottles than broken hearts in the trail they leave behind them," concluded _Globe_ columnist Harold Kaese on the morning of the club's Fenway finale.
Boston fans (they were not yet a Nation) derived their satisfaction from rare and wondrous moments that season. For instance, who would have predicted that the club's first no-hitter in a half-dozen years would be pitched by an African American?
Earl Wilson, who'd grown up in Louisiana, had been signed in 1959 as Boston's second black player after Pumpsie Green, as much for his demeanor as his ability. "Well-mannered colored boy," said the club's first scouting report on him. "Not too black, pleasant to talk to, well-educated, very good appearance." Wilson had come up from the minors before the 1962 season, but he was flawless when he faced the Angels on June 26, feeding them a steady diet of fastballs. "It's hummin', man," catcher Bob Tillman assured him as Wilson strung together a row of zeros and won the game, 2-0, with a homer off Bo Belinsky, who'd thrown a no-no of his own in May.
"All I can say is the Good Man was with me tonight," proclaimed Wilson, the first black pitcher to toss a no-hitter in the American League, and the first Red Sox pitcher to hurl one since Mel Parnell in 1956—and the first right-hander to pitch a no-hitter at Fenway since Ernie Shore in 1917. Fenway's man upstairs was suitably appreciative, as Yawkey made a rare visit to the clubhouse, gave Wilson a $500 bonus, and bumped up his salary by $1,000.
Little more than a month later, Bill Monbouquette threw another no-hitter in Chicago. It was the first time a pair of Boston pitchers had managed the feat since Dutch Leonard and George Foster both pitched no-hitters in 1916. By then, though, the season had been long lost. The Sox were entombed in eighth place, 17 games out, and Higgins, who'd morphed from caretaker to undertaker, was dismissed.
Succeeding him was Johnny Pesky, the former Sox shortstop and manager of their Triple A affiliate in Seattle. He was determined to put a stop to the "country club" culture that had become embedded over the previous decade. So during spring training in Scottsdale, Arizona he established a midnight curfew, banned swimming (because "It dulls the reflexes") and discouraged golf. "I'm not a slave driver," the new skipper insisted. "I'm just trying to be helpful."
What the 1963 club needed was something close to unearthly intervention. It arrived in the form of Dick "The Monster" Radatz, a hulking closer who was coming off a percussive rookie season during which he'd led the league in relief appearances, victories, and saves and was voted "Fireman of the Year." His repertoire consisted of one pitch—a fastball—but it was delivered from a 78-inch frame that weighed 245 pounds.
Helicopters were employed to help dry out the field in March 1967 as preparations for the season got underway. Later in the year, veteran groundskeeper Jim McCarthy used a riding mower to keep the grass in check.
Ned Martin broadcast—on radio and TV—for 31 Red Sox seasons, partnering first with Curt Gowdy and last with Jerry Remy.
Carl Yastrzemski supplanted Ted Williams in left field in 1961 and went on to create his own Hall of Fame legacy.
During a season when victories were again hard won, Radatz's was the only number on Pesky's bullpen speed dial when the game hung in the balance. In June and July alone, Radatz made 30 appearances, once working six times in six days and closing out both ends of consecutive doubleheaders. In one 15-inning victory over Detroit on June 11, he pitched the final nine innings. "God bless Dick Radatz," proclaimed Pesky. "He's our franchise." The _Globe_ 's Kaese avowed that Radatz was to the Red Sox "what (Pablo) Casals is to music, the Prudential Tower is to Boston's skyline."
In the All-Star Game at Cleveland, Radatz pitched two innings and struck out five men, including future Hall of Famers Willie Mays, Willie McCovey, and Duke Snider. But his 15 victories and 25 saves couldn't rescue a club that went into a summer swoon, falling from second place to seventh and finishing 28 games behind the Yankees. The Sox poster boy for the season was first baseman Dick Stuart, nicknamed "Dr. Strangeglove" for his mystifying fielding. While Stuart led the club with 42 homers and 118 RBI, he also struck out a franchise-record 144 times and made 29 errors.
Wood always had been a higher priority than leather around Fenway, so when a 19-year-old rookie from Swampscott cracked a 450-foot homer against the Indians in spring training, he immediately was plugged into the 1964 starting lineup, despite having only played a year of "A" ball. The only challenge was getting Tony Conigliaro into, and then out of, bed.
When the Sox scheduled a workout for Yankee Stadium after their season opener with New York was rained out, the rookie still was dozing at the hotel. "What a way to start a career," he moaned. "I can hear my kids asking me some day, 'What did you do on your first day in the big leagues, Dad?' And I'll say, 'I slept.'"
Tony C., as he immediately was dubbed, was wide awake on Opening Day in the Fens though, launching the first pitch he saw from Chicago's Joel Horlen over the left-field wall. He hit 22 more homers before the season was done, even though he missed about six weeks with various injuries, including a broken wrist and forearm. Broken curfews also were an issue, as Conigliaro was casual about bedtimes. "I am not a playboy," he insisted after Pesky fined him for being AWOL in Cleveland.
Not that it would have set Conigliaro apart on a club that usually played as if it had been up all night. The Sox snoozed through the rest of 1964 and Pesky was dismissed two games before the end with third-base coach Billy Herman, who hadn't managed since 1947, inheriting a club that finished eighth. They were the Dead Sox now, flatlining by May. In 1965 Boston was buried in seventh place after 14 games, en route to 100 losses with foul balls clanging off empty seats.
"When I was with Kansas City we played in Fenway one day and I was in right field, counting people in the stands," recalled Ken Harrelson, who ended up in Boston in 1967. "There couldn't have been more than a couple hundred." Only 1,274 turned up on September 16 when Dave More-head, a 23-year-old right-hander who'd lost 16 games that season, pitched a no-hitter against the Indians, missing a perfect game on a full-count walk in the second inning.
THE PATRIOTS FLOP IN FIRST MEANINGFUL GAME
Gino Cappelletti peeked through his bedroom curtains and couldn't believe his eyes. Snow? Can't be, he said to himself.
The date was December 20, 1964, and the Patriots would be taking on the Buffalo Bills at Fenway Park in the biggest game in team history.
The game was for the AFL's Eastern Division championship. Buffalo, led by Jack Kemp, Cookie Gilchrist, and Elbert Dubenion, came to Boston with an 11-2 record. The Patriots were 10-2-1 and had halted the Bills' 10-0 start by beating them at Buffalo. With a win, the Patriots would capture the division and face the San Diego Chargers for the AFL title at Fenway. The Patriots sought revenge for the 51-10 whipping the Chargers had administered in San Diego a year earlier in the Patriots' only championship game appearance.
For the first time in history, a Patriots' game not only was going to sell out Fenway Park, but it would also be the national TV game of the day. The surprise snowstorm that had hit Boston the night before meant that the game had to be delayed for 45 minutes while the grounds crew finished clearing the field. And with 38,021 fans converging on the snow-covered streets around Fenway, the entire area was in chaos.
Said Cappelletti, "I got stuck on Route 9 on the way in [from his home in Wellesley]. Thank goodness the game was delayed. The rumor in the locker room was that I got kidnapped by gamblers. No kidding. That was the story going around, and some of the guys believed it."
Much of the crowd stood throughout the game because the seats were never cleaned. The Globe's Bud Collins described Fenway as a glacier: "There were snowball fights and fist fights and drawing from the hip flasks." When the game finally started, it was all Buffalo. On the first play from scrimmage, Gilchrist, a powerful fullback, leveled Patriot cornerback Chuck Shonta, who was shaken up but stayed in the game. Two plays later, Dubenion beat Shonta for a 57-yard touchdown pass.
"You could see the Bills get a lift after that," said Cappelletti. "They also had an excellent defensive game plan."
The Bills won, 24-14, and went on to beat the Chargers for the AFL championship. For the Patriots, it would be 21 more years before they would play in a league championship game, and it took them 37 years to win their first league title in Super Bowl XXXVI.
The Fenway Park gridiron was bathed in white after an unexpected storm dumped four inches of snow on December 20, 1964. The Patriots (led by Gino Cappelletti, above) played the Buffalo Bills later that day for the AFL's Eastern Division title, falling to the eventual AFL champions, 24-14.
Tony Conigliaro worked on his batting stance in the Fenway Park dressing room under the watchful eye of then-Red Sox vice president Ted Williams on July 21, 1966.
Even that feat wasn't enough to keep Yawkey from dumping Higgins as his general manager in the middle of the game—Higgins's second sacking by the Sox in six years. The former skipper, who'd been the primary target of fans' displeasure, had advised the owner to fire him. "You'd be better off with somebody else," Higgins said. "I'm not popular in this town."
Yawkey first said the change would be made during World Series time. But he lowered the boom in the fifth inning ("I'd like to make that change now"), giving Morehead 45 minutes to savor his no-hitter before announcing the decision to the public. Replacing Higgins was Dick O'Connell, who quickly shook things up by moving spring training from Arizona to Florida and promoting farmhands like George Scott, Joe Foy, Reggie Smith, and Mike Andrews.
But the club still finished ninth in 1966, dropping its first five games and 20 of 27. By mid-May, the Sox were in last place. "You're always optimistic in spring training," Yastrzemski wrote in his memoir, _Yaz_ , "but that optimism's gone after the first two months of the season when you're 25 games out of first place and looking at 2,000 people in the stands in springtime."
It was no consolation that Boston finished the season ahead of New York. The franchise was going sideways and attendance still was well under a million. So O'Connell brought in Dick Williams, who'd been the Triple A manager in Toronto and who insisted on two things—a one-year contract and absolute autonomy in the dugout. "I decided if I'm going to go down, I'm going down my way," said Williams, who'd been a reserve infielder on the Sox soporific 1963 and 1964 teams where "nobody really cared about winning."
Given the dismal denouement of the previous campaign, expectations for the coming season were modest by the players and less-than-modest by oddsmakers, who listed the Red Sox as 100-1 long shots. "If our pitching holds up, we'll finish fifth," reckoned Yastrzemski, the club's perennial cockeyed optimist. "No kidding. I think we can make it to the first division."
That's precisely where the Sox found themselves in mid-July after a 10-0 home loss to the Orioles. Though none of them predicted what was about to occur—a 10-game winning streak that lifted Boston to second place, a half-game behind Chicago, and bestirred a fandom that long ago had been lulled into somnolence.
"Our mouths were open," shortstop Rico Petrocelli said after 15,000 Sox fans turned up at Logan Airport to salute the players upon their return from a four-game sweep of the Indians in Cleveland. "We were shocked. That's when we started believing."
The summer of 1967 was a revival, a daily salvation show for both the Sox and their supporters, whose souls had been deadened by decades of disappointment. It was the wildest ride in memory as the club went from first place to ninth to first to eighth, all before Memorial Day. Attendance was on its way to doubling to more than 1.7 million—the franchise's highest ever.
In the annual Father-Son Game, on July 24, 1966, Mike Yastrzemski, 4, threw a pitch as dad Carl offered support. Vin Martelli, 5, godson of Sox outfielder Tony Conigliaro, was the batter, while George Thomas of the Red Sox caught and umpire Hank Soar called the balls and strikes.
BALLPARK BACKDROP ENDURES
For nearly five decades, Red Sox fans have watched home-run blasts over the Green Monster soar against the backdrop of a Boston landmark. When the Citgo sign in Kenmore Square was first illuminated in 1965, it replaced a 1940 neon sign that displayed the shamrock logo of Cities Services, the predecessor to Citgo.
The pulsating, computer-controlled design of the 60-by-60-foot sign was an immediate hit in the psychedelic age, and an avant-garde filmmaker made a critically acclaimed three-and-a-half-minute film in 1967 called _Go, Go Citgo_ , in which he set the sign's display to music by the Monkees and Indian sitarist Ravi Shankar.
The sign became so renowned that even opposing managers knew it well. In 1978, with Sox slugger Jim Rice on a particularly torrid stretch, Royals manager Whitey Herzog deployed a four-out-fielder shift. "What I'd really like to do is put two guys on top of the Citgo sign and two in the net," Herzog quipped. Rice beat the shift by hitting one over the left-field wall and the netting for a home run.
Bostonians have taken their skyline pretty seriously ever since Robert Newman placed two lights in the tower of the Old North Church in 1775 as a beacon to Paul Revere. This may explain why plans to tear down the Citgo sign caused an uproar in 1982.
In September 1979, at the urging of state officials, Citgo had turned off the sign to set an energy-conservation example, even though it cost only $60 a week to light. By November 1982, Citgo was preparing to tear it down.
The public was outraged. The sign's supporters even urged the Boston Landmarks Commission to make it an official landmark. A man who testified before the commission said, in complete seriousness, "Paris has the Eiffel Tower, London has Big Ben, and Boston has its Citgo sign." On November 16, the Landmarks Commission issued a cease and desist order.
"We had no idea it would receive such a response. Now we know how much people in Boston love that sign," Citgo spokesman Kent Young later said. By August 1983, the sign was again aglow.
In 2005, more than 1.7 miles of LED lights replaced the existing 5,878 glass tubes of neon, which saved thousands of dollars in energy costs. But when those LED lights went out of production in 2010, crews replaced the sign's 218,000 lights with brighter, more weather-resistant versions.
Citgo is a division of Venezuela's government-owned oil company, and in a 2005 relighting ceremony, Mayor Thomas M. Menino of Boston flipped the ceremonial switch with Juan Barreto, then mayor of Caracas, and former Red Sox shortstop Luis Aparicio, the only Venezuelan in the Baseball Hall of Fame.
The sign continues to glow, part symbol of roadside culture, part Boston icon, and some fans still call out "See it go!" (C-IT-GO) when a Red Sox player blasts a home run in its general direction.
Jim Lonborg laid down the bunt for a base hit that triggered the winning Red Sox rally in the final game of the 1967 "Impossible Dream" season. The Red Sox trailed the Minnesota Twins, 2-0, in the sixth when Lonborg set the table for a five-run inning and an eventual Boston victory. The Red Sox edged the Twins and Tigers by one game for their first pennant in 21 years.
WHEN RED SOX NATION WAS BORN
They listened on transistor radios in the clubhouse for the crackling report from Detroit's Tiger Stadium. It was October 1, 1967, and the Red Sox had held up their share of the deal by defeating the Twins earlier in the day, clinching no worse than a tie for the AL pennant. Fans had stormed the field to hoist pitcher Jim Lonborg onto their shoulders, and players bathed in adulation from success-starved Bostonians who couldn't quite believe that their longtime American League doormats could be going to the World Series.
The Red Sox had started that 1967 season with a victory on Opening Day—before a mere 8,234 fans. It turned out to be the Summer of Love nationally, but in New England, it was the summer when the lovable losers of Fenway became a team that mattered, a team that played meaningful games in late summer and fall. _Globe_ columnist Dan Shaughnessy, who was 13 years old that year and had never seen a decent Sox team, years later wrote: "The Red Sox of the new century are still beholden to the Impossible Dream crew."
Red Sox teams of that era didn't fall out of the pennant race in the final weeks—no, they were usually out of any consideration by the Fourth of July. The Boston clubhouse was known as a "country club" where star players had only to run to the owner, the benevolent Tom Yawkey, if they thought a manager wasn't treating them fairly. They had coasted to eight consecutive losing seasons, finishing eighth, seventh, eighth, ninth, and ninth in a 10-team league the previous five seasons.
Now, as the "Age of Aquarius" dawned, a rookie manager named Dick Williams was in charge, and it seemed he had some backing from above. The Sox, a collection of retread players and youngsters led by two stars who were having the seasons of their lives, had reawakened a generation of Red Sox fans. By July, they were in first place, a totally unexpected perch. They hung on with improbable determination and entered the season's final weekend with two home games against the Minnesota Twins, who led them by one game in the standings. Win two, and the Sox could do no worse than tie for the pennant with the Detroit Tigers.
Boston fans were wild for "Gentleman Jim" Lonborg's pitching, and yet he triggered a game-changing rally with—of all things—a bunt single, as Boston defeated the Twins on the season's final day. It was Lonborg's 22nd victory of the season, and it would help earn him the Cy Young Award, the first such honor for a Red Sox pitcher.
The tale of Don Quixote was all the rage on Broadway that season, and, thus, the Red Sox rebirth was tagged—forever and always—as the Impossible Dream season. They won their first pennant in 21 years in the culmination of baseball's best-ever pennant race, which was finally settled when the radio told the tale and the champagne was uncorked: the Tigers had lost, and no playoff game was necessary. They would finish the season one game behind the Sox with Minnesota. With 44 homers, 121 RBI, and a .326 batting average, Carl Yastrzemski won the Triple Crown (no one has done it since, or even come close). He capped the year with a 7 for 8 hitting performance in the last two games. That season was the launching pad for all of the team's success since then. After eight straight losing campaigns, the Red Sox would go on to have 16 straight winning seasons and establish the base of Red Sox Nation.
The annual Bat Day promotion at Fenway Park was just one more reason to love 1967.
October '67, as seen through a fish-eye lens.
For one terrifying moment on August 18 at Fenway, when Angels pitcher Jack Hamilton hit Tony Conigliaro in the face with a fastball that left him motionless in the dirt, the magic vanished. "I thought I was going to die," said Conigliaro, who suffered a fractured cheekbone, a dislocated jaw, and a damaged retina that prevented him from playing again until 1969. "Death was constantly on my mind."
Yet his shaken teammates won, 12-11, that night and swept the Sunday doubleheader, coming from eight runs down to claim the nightcap, 9-8. From then until the end of the season, Boston was never more than a game out of the lead. "We were doing the impossible," said Petrocelli, "and we were doing it together."
It was the Impossible Dream, a quixotic adventure that swept up all of New England in its giddy unlikelihood. No ball club ever had come from ninth place to finish first in one season. But as Boston remained in contention into September, the possibility created both anticipation and anxiety. "I can remember people saying, 'How can you stand the pressure?'" said Yastrzemski, who found himself on the cover of _Life_ magazine. "I'd say, 'Pressure? This is fun. This is what the game is about.'"
"The Yaz Song," radio humorist Jess Cain's ditty, became the summer's refrain as the slugger defied the game's laws of probability. "I was in the zone," Yaz recalled decades later. "You usually stay in it for 10 days, but I was in it for a month."
As the season came down to its final week, the Sox still were very much in the chase with Minnesota, Detroit, and Chicago. The standings shuffled by the hour. "You were in first place or fourth, depending on the time of day," said Williams.
Scoreboard-watching became an obsession, particularly in the home dugout. "At Fenway we had the best way of keeping score—the guy in the Wall," ace pitcher Jim Lonborg noted. "You would see that number disappear and wait for the next one to come up. It wasn't like it was being blurted out on a Jumbotron."
Even the improbable numbers worked for Boston in the last few days. After the club was beaten, 6-3 and 6-0, by Cleveland at home, the players figured they were out of the race. "We thought it was over," conceded Yastrzemski. "Everyone was saying, 'Well, we had a great year.'"
But when the Twins lost at home to the Angels that same day and the Athletics swept the White Sox in a doubleheader, the Red Sox realized they were still alive, tied with the Tigers and only a game behind Minnesota, with the Twins coming to town for the final two games. "We got up the next morning and said, 'You know, we still have a chance,'" Yastrzemski said.
What the hosts needed was for Detroit to lose two of its four at home to the Angels while Boston took both games from a Minnesota club that had beaten them in 11 of 16 meetings and that had aces Jim Kaat and Dean Chance scheduled to pitch. The Sox won the first meeting, 6-4, on a three-run homer by Yastrzemski, and then pinned their hopes on the gentlemanly and scholarly Stanford grad who'd won 21 games for them.
"This is the first big game of my life," Lonborg mused Saturday evening. "I haven't seen a big one until tomorrow. Never." To prepare for it, he borrowed teammate Harrelson's room at the Sheraton-Boston and fell asleep reading _The Fall of Japan_.
After such an enchanted campaign, it was inconceivable that the finale would be without drama or quirkiness. With his team trailing, 2-0, in the sixth, Lonborg began the comeback with a leadoff bunt that ignited a five-run rally marked by a two-run single by Yastrzemski, two wild pitches, and an error. With victory imminent, Fenway organist John Kiley played "The Night They Invented Champagne" and after Rich Rollins popped up to Petrocelli to end things, Fenway erupted in joyous disbelief. As youngsters tried to scale the backstop screen, the crowd rushed the diamond and hoisted Lonborg atop its shaky shoulders on a hero's ride, ripping off parts of his uniform for souvenirs.
Since Detroit had won the opener of its doubleheader, the celebration in the clubhouse was exuberant yet restrained as the Sox drank beer and smeared each other with shaving cream. If the Tigers won the nightcap, there would be a one-game playoff at Detroit for the pennant.
Carl Yastrzemski launched a three-run homer off Minnesota Twins pitcher Jim Merritt in the seventh inning of the penultimate game of the 1967 season. The Sox won their first pennant in 21 years the next day.
YAZ MANAGED TO OUTLAST HIS DREAM SEASON
BY BOB RYAN
"Tris Speaker may have done it, or Duffy Lewis, or some other Red Sox giant of long ago, but Ted Williams didn't, nor Jimmie Foxx. If any player in baseball history ever had a two-week clutch production to equal Carl Yastrzemski's, let the historians bring him forth."
—Harold Kaese
Most baseball careers are measured in years. Carl Yastrzemski's was measured in epochs. He was in left field the day Roger Maris hit his 61st homer in 1961. He was still playing the day _M*A*S*H_ aired its final episode in 1983 (not that there was much chance he had ever heard of Hawkeye Pierce).
No one has ever played as long for one team, and one team only, as Carl Yastrzemski. He is one of the select few with 3,000 hits and 400 home runs. He was handed the thankless task of replacing Ted Williams in left field in 1961, and by the time he retired 22 years later, he had found a way to create his own distinct legend.
But perhaps Yastrzemski's most significant achievement was that he managed to overcome the bizarre handicap of having that one transcendent season. He could have been Orson Welles, never able to top _Citizen Kane_. He could have been Don McLean, still waiting for the appropriate follow-up to "American Pie." He could have let 1967 engulf him, but in due time, he allowed it to define him.
1967. If you weren't there, you'll just never know. You won't understand what Boston was like, when every night in June, July, August, and September you could follow every pitch with Ken Coleman and Ned Martin from stoplight to stoplight and front porch to front porch and business to business, because the entire city was recaptured by baseball and the Red Sox, thanks to Yaz and his teammates. Before Yaz won the Triple Crown in 1967, before the Impossible Dream season, you could pretty much have any Fenway seat you wanted at any time.
In 1967, Carl Yastrzemski showed us what grace and determination under athletic pressure could produce, and while he never again had an all-around season like that, neither has anyone else. With his bat, glove, arm, and will, he personified the idea of the Most Valuable Player.
Hyperbole? OK, you judge. In the final 12 games of a sizzling four-team pennant race, Yaz was 23 for 44 (.523), with five homers, 16 RBI, and 14 runs scored. Throw in the Gold Glove Award and underline the 7 for 8 hitting in the final two games of the season, and then put the exclamation point on the performance by throwing out Bob Allison trying to stretch a single to squelch an eighth-inning Minnesota rally on the season's final day. In the ensuing years, we have not seen a more valuable Most Valuable Player.
Manager Billy Herman told him after the 1966 season, "You can't be a leader the way you played this year. You can be a great ballplayer if you'll work at it."
Yaz hired a Hungarian immigrant fitness trainer named Gene Berde and said, "I'm yours." No baseball player had ever done such a thing. He reported to training camp with a new body and a new sense of purpose in 1967 to play for Dick Williams, a new, energetic, no-nonsense manager. But as heroic as Yaz was in leading the Sox to the pennant, he could not win the World Series all by himself. He certainly did his part, batting .400 with three home runs, but the Sox lost in seven games to Bob Gibson and the St. Louis Cardinals.
Yaz would go on to have good years, but never anything like 1967. What he did was last long enough at a high enough level to construct a new image, that of the ultimate grinder. His toughness, his consistency, and his unmatched work capacity—along with that unforgettable 1967 campaign—remain his legacy.
So the players waited several more hours, listening to the play-by-play from Michigan. "For us to be sitting around a radio instead of a TV, it reminded me of an old-time movie where you were listening for news of some important event," Lonborg said. When the Tigers grounded into a double play with the tying run at the plate to complete an 8-5 loss, Williams leaped up. "It's over, it's over," the skipper proclaimed. "It's unbelievable!"
Now it was time for champagne and tears. "This is the happiest moment of my life," declared Yawkey as he sipped Great Western from a paper cup, relishing the end of two desiccated decades. "THE SMELL OF THE PENNANT . . . THE ROAR OF THE CROWD" declared the headline on the front page of Monday's _Globe_.
Just getting to the World Series was such a miracle that few fans had time to ponder the upcoming challenge—the formidable St. Louis Cardinals club of Lou Brock, and Bob Gibson, Curt Flood, and Orlando Cepeda that had won the National League flag by a mile over the San Francisco Giants. Gibson, its glaring ace, had missed nearly two months after his right leg was broken by a line drive in July, but he was overpowering in the opener at Fenway, yielding only a solo homer to counterpart Jose Santiago in the third inning while holding hitless Yastrzemski, Petrocelli, and Harrelson, all of whom took batting practice after the 2-1 loss.
Yastrzemski responded with two homers and four RBI the next day, but his team needed just the bare minimum as Lonborg befuddled the Cardinals, taking a perfect game into the seventh inning and coming within four outs of a no-hitter before Julian Javier smacked a double, in a 5-0 whitewash. "I hope we can end the Series on Monday," Williams said with brash optimism.
On Monday, though, his club was teetering on the edge of extinction after St. Louis had administered 5-2 and 6-0 tutorials at Busch Stadium. It was left to Lonborg to bring the Sox back home alive and he delivered with a 3-1 decision. "Same lineup, same result," Williams decreed for the sixth game in Boston. But his pitching choice was startling. Gary Waslewski had started only four games all season and had been sent down twice to the minors.
Nobody with that little experience had ever been tapped for a decisive World Series game, but Waslewski performed superbly until the sixth inning and Boston hitters worked over eight St. Louis hurlers. The Sox set a post-season record with three homers in the fourth as Yastrzemski, Reggie Smith, and Petrocelli chased starter Dick Hughes. Boston then broke things up with four more runs in the seventh for an 8-4 triumph.
The outfield wall got a thorough scraping in March 1968.
REMEMBERING TONY C
Mike Higgins, the general manager, thought he was too young. But Johnny Pesky, the manager, had already seen enough of Tony Conigliaro in spring training in Arizona in 1964 to know he had a natural power hitter in camp, and he brought him up to stay at age 19.
On Opening Day at Fenway Park that season, Conigliaro, a year out of St. Mary's High School of Lynn, Massachusetts, stepped to the plate and in his first at-bat unloaded a home run off White Sox pitcher Joel Horlen.
Conigliaro went on to become the youngest American League player to reach the 100-homer mark, and the youngest ever to lead the majors in homers (with 32 in 1965, at age 20). At the height of his popularity, the handsome, dark-haired Tony C also did a little singing, and he performed on _The Merv Griffin Show_ , among other places.
Conigliaro was hitting .287 with 20 homers and 67 RBI in 1967 as the Red Sox chased their first pennant in 21 years. But on Friday, August 18, at Fenway, his life changed forever. Conigliaro "was never one to back off from an inside pitch," said Pesky, and Tony C was struck in the face by a fastball from Angels pitcher Jack Hamilton.
The ball shattered Conigliaro's cheekbone and cracked the orbital bone encasing his left eye. The impact also severely damaged the retina of his left eye. The beaning was so severe that Conigliaro dropped to the ground face first, bleeding from the nose and eye.
Later, Conigliaro described the impact: "His first pitch came in tight. I jumped back and my helmet flew off. There was this tremendous ringing noise. I couldn't stand it. . . . I kept saying to myself, 'Oh, God, let me breathe.' I didn't think about my future in baseball. I just wanted to stay alive."
Conigliaro sat out the entire 1968 season, and after two comeback attempts, he retired in 1975 at age 30. Teammate Rico Petrocelli remembered the comebacks, the first of which produced a remarkable 36-homer, 116-RBI season in 1970, before Conigliaro was traded to the Angels.
"He wouldn't quit. He made the greatest comeback, I think, in the history of baseball. He was the most courageous player I ever saw," said Petrocelli.
Tony C remained a popular figure in Greater Boston, running a nightclub with his brother Billy, who had also played for the Red Sox. While he was being driven to the airport by his brother on January 3, 1982, Conigliaro suffered a massive heart attack; his heart stopped for several minutes, and he suffered a stroke and lapsed into a coma.
Conigliaro remained in a vegetative state until his death on February 24, 1990. He was 45 years old.
The Tony Conigliaro Memorial Award is presented annually by the Boston Baseball Writers Association to the major-league player who has overcome adversity with the spirit and courage demonstrated by Conigliaro.
On August 18, 1967, Tony Conigliaro's life changed forever when he was struck in the face by a pitch from the Angels' Jack Hamilton. "Tony C" had led the league in home runs in 1965 with 32, at age 20 the youngest to do so at the time. His injuries from the beaning kept him out of baseball for almost two years, and after two comeback bids, he retired from the game in 1975.
Nuns' Day, which was a park tradition in the '60s, brought an added dimension to the term "Fenway faithful." Boston Archbishop Richard Cardinal Cushing (seated) was often in attendance.
Fenway has played host to lots of soccer matches. In this 1968 contest, Ruben Sosa (17) of the North American Soccer League's Boston Beacons fired on Chicago Mustangs goalkeeper Gerd Langer.
"Lonborg and champagne," Williams predicted for the finale even though his ace would be dueling Gibson on two days' rest. "CINDERELLA TRIES ON THE SLIPPER," read the _Globe_ headline on the morning of Game 7 as all of New England anticipated a fairy-tale finish. "This is a story that has to have a happy ending," Kaese declared. If the Sox lose "it will be the wolf gobbling up Little Red Riding Hood."
The wolf licked its chops.
The Sox couldn't solve Gibson, who struck out 10 and hit a homer off the depleted Lonborg, who departed after six innings with St. Louis leading by six runs. "Lonborg and champagne, hey!" the Cardinals chanted in the clubhouse after they'd claimed their second world championship in four years by a 7-2 count. "Now it's our turn to pop off," shouted outfielder Flood as he pried open a bottle of Mumm's. "Pop this."
As the club packed for the winter, hundreds of Red Sox fans lingered in their seats, unwilling to let go of the most enthralling season of their lives. "The laurels all are cut, the year draws in the day, and we'll to the Fens no more," Roger Angell wrote in his _New Yorker_ account.
While the Dead Sox days were over, it would be another eight years before the Sox suited up for a playoff game. Even before the club convened for spring training in 1968, the chances for a reprise had lessened markedly when Lonborg tore up his knee skiing at Lake Tahoe before Christmas. That was the prelude to a procession of problems that sabotaged the season. Lonborg returned, but posted a losing record. Santiago hurt his arm and never won another game. Scott saw his average plummet from .303 to .171 and failed to hit a homer at Fenway. And Conigliaro's comeback bid ended before it began, undone by blurred vision.
The Sox, who never were in first place after Opening Day, were out of contention by Flag Day. "We'll win it next year," declared Yastrzemski, after Boston had finished fourth, 17 games behind the Tigers. "There's no doubt in my mind about that."
When Conigliaro smashed a two-run homer and scored the winning run in the 12th inning of the opener at Baltimore in 1969, Boston fans saw it as a harbinger of a restoration. But nobody that year was going to catch the Orioles, who won 109 games and claimed the East Division in the first season of baseball's four-division format. When Boston slipped from second to third in August, player discontent grew with Williams and his whip-hand ways. The friction peaked in Oakland when the skipper yanked Yastrzemski after the first inning for not hustling, bawled him out, and fined him $500.
With nine games to go in the season, Yawkey dumped Williams, who went on to win two championships with Oakland and be enshrined in the Hall of Fame. Memories of the Impossible Dream had been soured by an irremediable dyspepsia. "The days of wine and roses didn't last long, which should have been predictable," _Globe_ columnist Ray Fitzgerald mused.
## 1970s
It is one of the most famous images in all of sports: Carlton Fisk willing (and waving) his fly ball to stay fair for a walk-off home run in Game 6 of the 1975 World Series.
By the time the 1970s began, a new attitude pervaded baseball in Boston. Smiley faces were everywhere, and not just on those ubiquitous T-shirts. Fans no longer hoped for a team that could play meaningful games after the Fourth of July, they had suddenly come to expect it. In the 1970s—through Vietnam, Watergate, the oil embargo, and the Bicentennial—the Red Sox had a winning record every season, extending their streak of consecutive winning seasons that began in 1967 to 13 by the end of the decade (then ultimately to 16—the most in their history). With that success, however, came seemingly inevitable heartache. They performed heroically in a World Series that many proclaimed the best ever played, which helped to take Bostonians' minds off divisive court-ordered school busing. However, the Sox lost in the ninth inning of Game 7 on a bloop single to what many considered one of the best teams ever assembled, the 1975 Cincinnati Reds. Fenway got a facelift after that series, as an electronic message board—a first for baseball—and a padded, resurfaced left-field wall debuted in 1976. Two years later, the Sox seemed poised to run away with a pennant, only to have it all come crashing down in the most wrenching fashion. Getting 99 wins in the regular season—their second-highest victory total in 63 years—only got them into a one-game playoff with the Yankees at Fenway, where they lost by a run with the tying man on third in the bottom of the ninth. The Green Monster, which had given them so much during a teamrecord display of home–run hitting the previous year, took from them this time, in the form of a seeming pop fly ball by a light-hitting shortstop. The sage who had spray-painted "No hope!" in foot-high letters across the street from the ballpark in late September 1978 had been right, but barely.
Manager Eddie Kasko had a few words for his team in the clubhouse on April 5, 1973.
After his vocal and volatile predecessor, the bespectacled Eddie Kasko seemed decidedly buttoned-down and dialed-back. But the new Red Sox manager was not averse to the occasional outburst if provoked. "I've been known to throw furniture around," he said. Since most of the key members of the 1967 squad still were in place, the clubhouse mood was optimistic going into the 1970 season. "We'll win the pennant," Carl Yastrzemski declared during spring training.
They did win the opener at New York. But the Sox never spent another day in first place. After they went 9-17 in May, they were stuck in fifth. The low point came on June 25 when the club lost, 13-8, to Baltimore in 14 innings at home, despite having led, 7-0. They gave up six runs with none out in the final inning. Before long even Yastrzemski, the team's only All-Star starter, was being booed at Fenway.
Kasko's Mr. Chips mien was fraying, too. One day in Anaheim, he stormed into the press box after the organist had played "Tiptoe Through the Tulips" during one of his mound visits.
"We need players," general manager Dick O'Connell told Tom Yawkey in the press room one August evening when the Sox were stuck in fourth. "We've only got eight players."
"We do?" the owner replied. "Who are they?"
After Boston finished almost exactly as it had in 1969—21 games behind in third place, with an 87-75 record—changes were inevitable. The most startling was a trade that sent Tony Conigliaro, who had led the team in RBI the previous season, to the Angels for three players. "Honest, I never thought I'd be traded. [I thought] that being a hometown boy meant something," said Conigliaro, who'd spent all of his previous seven major-league seasons in Boston.
Mike Andrews was shipped out, too, dealt to the White Sox for Luis Aparicio. "The fans want a winner and I think we've got one now," Kasko declared before the 1971 season. The club was in first place by the end of April, and won 13 of 15 in late April and early May, putting them up by four games. But the Memorial Day weekend was followed by a slump in which they dropped 11 of their next 14 outings to fall five games off the pace. A lifeless 6-1 loss at New York in early June prompted Kasko to close the clubhouse door and upbraid his underperforming crew. "I'm sick of watching guys hang their heads around here just because we're in a rut," he barked.
A vendor sold popcorn and ice cream on Van Ness Street outside Fenway Park in August 1974.
THE SPACEMAN COMETH
William Francis Lee once told Bowie Kuhn, the commissioner of baseball, that he sprinkled marijuana on his pancakes. Lee was an outspoken nonconformist who famously called his Red Sox manager, Don Zimmer, a gerbil. He jogged five miles to the park on the days he was scheduled to pitch, and he once staged a 24-hour walkout when the Red Sox released friend and teammate Bernie Carbo. He said, "Baseball's a very simple game. All you have to do is sit on your butt, spit tobacco, and nod at the stupid things your manager says."
Lee also won 119 games over his 13-year career, appearing in the most games ever by a Sox left-hander (321), and recording the third-most victories (94) by a Sox lefty.
Lee won 17 games in three consecutive seasons for the Red Sox and started two games in the 1975 World Series. Lee's loathing for the Yankees endeared him to Sox fans. In 1976, a collision at home plate resulted in a bench-clearing brawl and Yankee third baseman Graig Nettles threw Lee to the ground in the melee. After the game, Lee called Yankee manager Billy Martin "a Nazi" and the team "Steinbrenner's Brown Shirts." Lee missed two months with torn ligaments in his shoulder.
For much of his time in Boston, Lee also feuded with Zimmer, as Lee's attitude and lack of respect for authority clashed with Zimmer's old-school personality. Zimmer relegated Lee to the bullpen, and at the end of 1978, the Sox traded Lee to the Montreal Expos for Stan Papi, a utility infielder. The furious Lee bade farewell to the Red Sox by saying, "Who wants to be with a team that will go down in history alongside the '64 Phillies and the '67 Arabs?" Lee went on to win 16 games for the Expos in 1979, but the team tired of his antics as well, releasing him in 1982 after he staged a one-game walkout over the release of infielder Rodney Scott.
As _Globe_ columnist Ray Fitzgerald once said, "He marches to a drummer no one else would let into the ballpark."
As for himself, Lee said he was either long before or long after his time. He tried to keep the game in perspective. "I think about the cosmic snowball theory," Lee explained. "A few million years from now, the sun will burn out and lose its gravitational pull. The earth will turn into a giant snowball and be hurled through space. When that happens it won't matter if I get this guy out."
Bill Lee and young fan Tammy Patterson surveyed a pumpkin carved in the likeness of Luis Tiant.
Still, the Sox lost to the Royals for the fifth straight time in mid-June, and slipped into third place. They never regained the lead, finishing 18 games behind Baltimore. That gave management the green light to clean house and by the time the team convened for spring training in 1972, only four players remained from the Impossible Dream team.
The biggest part of the exodus came with a 10-player deal that sent six Boston players, most notably Jim Lonborg and George Scott and Billy Conigliaro (younger brother to Tony), to Milwaukee. "I'm sick of listening to some of those people," said general manager Dick O'Connell, who'd been annoyed by clubhouse grumbling.
When the season started, Tommy Harper was in center, Danny Cater at first and imposing rookie Carlton Fisk behind the plate. It took until late summer for the new group to coalesce and after losing 10 of their first 14 games, the Sox were buried in fourth place and had slipped to fifth as June was coming to an end. But as the pitching improved with Luis Tiant and Marty Pattin, Boston won 19 of its next 25, including a dozen in a row at Fenway.
The midsummer run peaked with two extra-inning home victories over the Athletics, both wins coming from Oakland miscues. After A's skipper Dick Williams intentionally (and inexplicably) had Doug Griffin walked to load the bases in the 11th inning so that Darold Knowles could face Yastrzemski, his pitcher walked in the winning run to give Boston a doubleheader sweep. The next night, with 1967 notable Gary Waslewski on the mound for Oakland in the 14th inning, Yastrzemski bounced a grounder off the glove of A's third baseman Sal Bando to score Griffin from first base.
By Labor Day, the Sox had taken over first place, beating the Yankees, 10-4, on three-run homers by Harper and Rico Petrocelli as the fans chanted "We're Number One!" The season came down to a three-game series on the final weekend in Detroit after the Tigers had swept the Brewers (bashing ex-Sox Lonborg in the opener). "I wish the Red Sox a lot of luck and hope they win," said former Sox team-mate Joe Lahoud after Detroit had run up a 30-10 aggregate on Milwaukee over three games. "But God help them."
Fans scaled a billboard to peer into Fenway Park in 1971.
Boston needed divine intervention after losing, 4-1, in the series opener against the Tigers. Beating Tigers ace Mickey Lolich, who posted 15 strikeouts, would have been challenging enough. But the Sox literally tripped over themselves as Aparicio, who would have scored the lead run, stumbled rounding third on Yastrzemski's third-inning double in a painful reprise of his Opening Day pratfall at Tiger Stadium that cost the club a victory. "I stepped on top of the bag instead of the corner and then I hit a soft spot on the grass," said Aparicio, who gashed his right knee with his left foot when he went down. "How dumb can I be, spiking myself?" he groaned.
Yet it was Yastrzemski who made the killer mistake, steaming toward third without noticing that his teammate had gone down and that Eddie Popowski had called Aparicio back to the bag. "There was no way then to hold Yaz to second," the third-base coach said after Yastrzemski had been tagged out. When the Tigers finished them off the next day, some Sox players wept in the clubhouse. "You came a long way and have nothing to be ashamed of," Kasko told them.
Had Boston not given up so much ground in the early going, the season might not have come down to an untimely slip. "I told Haywood Sullivan I didn't want to come to Boston next June and find the team eight or nine games back," Yawkey said. "We've tried that too often."
The 1973 season started sublimely, as the Sox swept the Yankees at Fenway for the first time in an opening series since 1933. "Ah, Tiant et Fisk et Yaz, quisque cum clangore epico ballatorum ex Iliade," _Globe_ columnist George Frazier exulted in Latin after the Tibialibus Rubris had clobbered Eboracum Novum by a count of XV-V.
The introduction of the designated hitter provided added oomph to a Sox lineup that needed more production, with Orlando Cepeda clouting the winning homer in the ninth inning of the third game. "I see about 15 guys at home plate," marveled Cepeda, who'd been signed as a free agent. "It looks like the World Series kind of homer."
But the early euphoria vanished when Boston dropped four to Detroit at home. By the beginning of May, the club was stuck in sixth place and didn't get above .500 until late June. Though four victories in New York in early July moved the Sox from fifth place to third, even an eight-game winning streak in mid-August made up no ground on the runaway Orioles, and Boston ended up eight games in arrears.
That was enough to get rid of Kasko, who was dismissed before the final day of the season and never managed again. "We are the ones who let him down," said pitcher John Curtis. "I can't look at this season and say we gave it our best shot."
RAY, LITTLE STEVIE AND ALL THAT JAZZ
On July 27 and 28 of 1973, the famed Newport Jazz Festival came to Boston. Not only that, it came to Fenway Park for two shows, with a stage set up near second base and a portion of the ballpark's seats cordoned off. Attendance was reported as 14,000 for the first night, and 21,500 for the second night of the festival, which is traditionally held in Newport's Fort Adams State Park. Concertgoers were both fascinated and disoriented. The Globe's Ernie Santosuosso admitted that "even I kept casting instinctive glances toward the board for out-of-town scores."
_Globe_ reporter William Buchanan covered the first night's five-hour-plus performances, and he was obviously unaccustomed to the ballpark comforts. "All due respect to you, Mr. Yawkey," he wrote, "those Fenway seats could test the mightiest of Spartans after an hour or two."
The featured artists on Friday were Freddie Hubbard, Billy Paul, War, Herbie Mann, the Staples Singers, and Ray Charles. Not a bad lineup for a ballpark concert, although some fans were irate when it turned out that they had bought counterfeit tickets from scalpers, while, Buchanan noted, "There is still an obvious problem with pickpockets [outside the ballpark]. Youthful teams of two or three would jostle an unsuspecting fan and woosh, the wallet is gone." Inside the park, he noted, "Boston police were polite but firm, and that made for a generally relaxed feeling."
Among the highlights were: Paul's "Me and Mrs. Jones," which had won him a Grammy Award earlier in the year; War's 50-minute set, and the "musical and spiritual togetherness" it inspired; and Charles's "What'd I Say," though Buchanan noted that the lineup of backup singers to Charles, known as the Raelettes, was "a bit weak."
The second night's lineup included B. B. King (in a silver-gray suit and accompanied by a nine-piece band), Stevie Wonder and Donnie Hathaway, plus Charles Mingus and Roland Kirk. During Hathaway's set, according to the _Globe_ 's Ray Murphy, "hundreds streamed onto the Fenway infield, but they finally returned to their seats after the earnest pleading of Hathaway himself."
Luis Aparicio, a Hall of Fame shortstop who played three seasons with the Red Sox, turned this double play on September 27, 1972.
With the new manager (and former Pawtucket skipper), Darrell Johnson, preaching defense and equality ("All of us have the same rules, nothing special for anyone"), the Sox remained near the top of the standings in 1974, moving into first place on May 22 after bashing the Yankees, 14-6 and 6-3, at Fenway and staying there for most of the summer even without Fisk, who tore knee ligaments in a home-plate crash in Cleveland at the end of June. "We're no longer a team that waits for something to happen," declared Yastrzemski. "Over 162 games, this is going to be a better team."
But as the bats lost their pop, Boston began losing ground and fell out of first place with a home loss to New York in September. When the club was starved, 1-0, in both ends of the Labor Day doubleheader in Baltimore with aces Luis Tiant and Bill Lee on the mound, the slippage began in earnest. The Sox soon fell out of first after losing to the Yankees and ended up dropping 16 of 21 games. When the soaring Orioles took two of three in their next series at Fenway, bashing Sox hurler Reggie Cleveland, 7-2, in the finale, a young man rose behind the home dugout in the ninth inning, put a bugle to his lips, and played "Taps." "We're going to have to win almost every game now," acknowledged Johnson.
When the end came, with barely 5,000 witnesses rattling around the Fenway stands for a date with the Indians, the Sox were seven games out, finishing third behind Baltimore and New York. "Well, to say that I'm not disappointed would make me a damn liar," said Yawkey as he prepared to head to his South Carolina plantation for the winter.
A hot dog vendor on Lansdowne Street awaiting the game-day bleacher crowd in August 1975.
The prospect of World Series tickets prompted crowds to camp out all night in October 1975.
Fans awaited the sale of bleacher seats for Red Sox-Yankees games in June of 1978.
JOE MOONEY: SPLENDOR IN THE GRASS
From 1971 to 2000, Joe Mooney was the keeper of the greensward, the man behind the meticulously manicured Kentucky bluegrass at Fenway Park that always leaves first-time visitors—not to mention some frequent attendees—agape. As the head groundskeeper for the Red Sox for some 30 years, he liked to point out that his career in baseball stretched from Schilling to Schilling—Chuck to Curt.
"I worked for the minor-league team in Minneapolis in the late 1950s and used to pitch batting practice to [former Red Sox] Chuck Schilling and Carl Yastrzemski," said Mooney.
A native of Dunmore, Pennsylvania, Mooney worked at Red Sox minor-league parks in Louisville, San Francisco, and Minnesota before he broke ranks and went to work in Washington, D.C. There he kept up D.C. Stadium (later renamed RFK Stadium) for professional teams that featured a pair of legendary coaches. Former Red Sox legend Ted Williams managed the Washington Senators at the time, and Vince Lombardi then coached the Washington Redskins.
"Imagine working for those two guys," said Mooney. "It didn't come any better than that."
Mooney also credits Williams for helping him land his job with the Red Sox. He had spent much of his 10-year stint in D.C. improving a poorly designed field, and Fenway was in need of an overhaul as well.
"They had tried out a different grass—rye, I think—that couldn't take the gaff, and they had a lot of fungus," he recalled. Mooney solved the Fenway woes and transformed the park into a dazzling beauty. Ask Yaz, who patrolled left field and first base while Mooney groomed the grounds: "Fenway is the best there is, at least since Joe Mooney started working on it."
Mooney was known to throw interlopers off his field, regardless of their title or temperament. His customary explanation was this: "You walk on grass too much, it gets beat up."
In 2001, Mooney ceded control of the field to Dave Mellor while he continued to supervise maintenance of much of the rest of the park. In the winter of 2004, the Fenway drainage system was renovated, leaving Mooney shaking his head.
"When I came here, we had three drainage pipes; now there will be 50," he said. Even for groundskeepers, "it's a different game."
"Why? Why should the bond between a people and their baseball team be so intense? Fenway Park is a part of it, offering a physical continuum to the bond, not only because Papi can stand in the same batter's box as Teddy Ballgame, but also because a son might sit in the same wooden-slat seat as his father."
—Tom Verducci, sportswriter
But Yawkey, who'd been the club's spiritual 26th player for four decades, was exuberant a year later as his club won the pennant for the first time since 1967 and went on to face the Reds in a World Series that featured the most dramatic game in postseason history.
Nobody would have predicted it after Boston had lolly-gagged its way through a listless spring training that prompted Yastrzemski to rip into his teammates in a closed-door meeting before the April 8 opener with the Brewers. "The worst attitude I ever saw," the captain railed. "If it keeps up, we'll finish in last place."
That opener featured Hank Aaron making his American League debut after 21 years with the Braves, and Tony Conigliaro coming back as a designated hitter for the Sox after three-and-a-half years away from the game while his injured eye healed, Fenway was awash in emotion, especially when the once-favorite son singled in his first at-bat. "The ball looked like a basketball," he said following his team's 5-2 victory. "That's what counts."
While Conigliaro's renaissance lasted only until June, the town soon was captivated by the Gold Dust twins, rookies Jim Rice and Fred Lynn, who had extraordinary debut seasons, with Lynn hitting .331 with 105 RBI and 21 home runs and Rice hitting .309 with 102 RBI and 22 homers.
With Tiant and Lee anchoring a pitching staff that received career seasons from Rick Wise (19-12) and Roger Moret (14-3), the club took over first place on May 24 after Lee, who'd mocked California's flaccid lineup ("The Angels could take batting practice in the lobby of the Grand Hotel and not bother a chandelier."), blanked them, 6-0, at Fenway. "He popped off and backed it up," conceded Angels manager Dick Williams, whose players had taken batting practice in the Sheraton-Boston lobby that day using Wiffle bats and Nerf balls, until a hotel security guard halted the antics.
Boston soon proved that it was a genuine contender, winning nine straight at home to take a five-game lead following the All-Star break. For punctuation, Rice smashed a homer off the Royals' Steve Busby that went over the center-field bleachers next to the Fenway flagpole. "I think the ball is on the New Hampshire toll road somewhere," reckoned Busby. After taking beatings of 8-3 and 9-3, the visitors had seen enough. "Get me the hell out of here," growled Royals Manager Jack McKeon.
But Boston's biggest challenge was the Orioles, who'd won 10 of 12. When they came to Fenway for two pivotal games in mid-September, Tiant and Jim Palmer hooked up in a showdown for the ages. "Lou-eee, Lou-eee, Lou-eee," the fans chanted as El Tiante befuddled Baltimore, 2-0, while Petrocelli and Fisk each rocked Palmer for homers. "We hurt them bad," concluded Johnson, whose employers immediately invited fans to mail in applications for playoff tickets. "Sure, they can beat Oakland," Palmer said. "Why not?"
While it was conceivable that Boston could win a post-season series against the Oakland A's, who had won the three previous World Series ("Four In a Row?" _Sports Illustrated_ mused in its pre-playoff cover story), nobody was predicting a sweep. But the Sox made the Athletics look like bushers in the Fenway opener as Tiant mesmerized them, conceding three hits across nine innings. "This is Lou-eee's palace," proclaimed Yastrzemski after Boston had punished the visitors with five runs in the seventh inning of a 7-1 rout. "In here, he can do no wrong." And Oakland, which had won its division by seven games over Kansas City, could do no right, committing four errors, three of them on five batted balls in the first inning. "We embarrassed ourselves," admitted A's center fielder Billy North.
Even with Vida Blue, their 22-game winner on the mound, the A's went down again the next day as Boston, sparked by Yastrzemski's two-run homer in the fourth and Petrocelli's leadoff shot off Rollie Fingers in the seventh, came from three runs down to win, 6-3, while the jubilant fans gave Oakland owner Charlie Finley a mocking farewell serenade: "So long, Charlie, we hate to see you go!"
THE
GREATEST GAME
EVER
It was 1975, Clark Booth recalled in a _Globe_ story many years later, and the nightmare of Vietnam had just ended. The country was still shaking off Watergate. In Boston, court-ordered busing had turned neighborhoods into battlegrounds. These were not the best of times. Then along came the Reds, the Red Sox, and Game 6, and we discovered that baseball could still inspire us.
In 1975, the Red Sox came into their first World Series in eight years in a familiar position: although they were not the out-of-nowhere pennant winners of 1967, they were again facing a team that had dominated the National League. In 1967, it had been the 101-win Cardinals; this time, it was the 108-win Reds, and few people gave the Red Sox much of a chance against the "Big Red Machine." They went down to defeat, but the Series went down as possibly the greatest ever played.
By the time Carlton Fisk was urging his dramatic 12th-inning home run fair in Game 6 on October 21, the Red Sox had made believers of much of the nation. And although Game 7 was decided on a ninth-inning looping single to center field, the Red Sox and Reds had compiled enough highlights to squarely put baseball back into the forefront of American sports fans' collective consciousness, after it had languished behind football for several years.
"If this ain't the national pastime, tell me what is," Reds third baseman Pete Rose asked after Game 6. Rose would go on to be named the Series MVP.
Boston's 7-6 victory in Game 6 was the first home night game in Red Sox postseason history, and it took four hours and almost 12 innings to complete. Each team seemed to have the game won at least once: the Reds were leading, 6-3, with two outs in the eighth before Bernie Carbo hit a dramatic three-run, pinch-hit homer; the Sox had the bases loaded and none out in the ninth inning of a 6-6 game, only to fail to score. The Reds seemed poised to take the lead in the 11th when, with a man on base, Joe Morgan laced a drive over Dwight Evans's head in right field. Evans managed a leaping, lunging catch at the fence, and Boston finally prevailed an inning later on Fisk's blast off the left-field foul pole, which was officially christened the "Fisk Pole" 30 years later when the Reds visited Fenway Park in June 2005.
Roger Angell of the The New Yorker wrote: "What can we say of Game 6 without seeming to diminish it by recapitulation or dull it with detail?"
Fisk remembered telling Fred Lynn going into the 12th inning that he would hit one off the wall and that Lynn, whose earlier home run had given Boston a quick 3-0 lead, should drive him home. Instead, at 34 minutes past midnight, Fisk gave us one of the most oft-replayed clips in sports history, as he leaped, waved, and coaxed his drive off Reds pitcher Pat Darcy fair.
Fisk started down the line, carrying his bat the first few steps. He dropped the bat and started to yell at the ball, "Stay fair! Stay fair!" He never actually ran. He skipped sideways, watching the ball, and then began waving his arms from left to right. The dance was immortalized by a camera inside the left-field wall. (Legend holds that NBC got the shot because a rat scared the cameraman from his chair and the camera never moved, but the camera trails Fisk's first few steps up the line.) The ball clanged off the foul pole high above the Green Monster, and Reds left fielder George Foster caught the winning home run on the fly as it ricocheted back toward the field. Ballpark organist John Kiley broke into the "Hallelujah Chorus" as Fisk tore around the bases, and in Fisk's hometown of Charlestown, New Hampshire, David Conant—who had known Fisk since the catcher was a baby—rang the bells of the Episcopal church.
Foster kept the Fisk home-run ball at his mother's house in California for the better part of the next 24 years. It sold at an auction in July 1999 for $113,273.
Despite being just 12 outs away from victory in Game 7 on October 22 (the Red Sox held a 3-0 lead as late as the sixth inning), Cincinnati's Big Red Machine won the game and the World Series, both 4-3. The world championship drought in Boston would continue another 29 years.
CLEAN PLATE CLUB?
BY RAY FITZGERALD
May 25, 1975—Carl Yastrzemski, long known as a man of the soil, tried to show Lou DiMuro yesterday at Fenway Park how they plant potatoes on Long Island, but the umpire misread Carl's intentions entirely.
DiMuro had just called Yaz out on a 3-2 pitch that the Red Sox captain judged to be somewhere in Jamaica Plain. Yaz, perhaps thinking that DiMuro should get into another line of business, proceeded to give the umpire a tip on truck farming.
Carl lovingly gathered some Fenway dirt in his hands and spread it over home plate to a depth of, oh, three inches.
He covered the middle and he covered the corners. He was nothing if not thorough.
Then Yaz flipped his batting helmet on the dirt, the way potato farmers always do back in East Hampton to protect their precious crop from grubs, cutworms and the common blight.
As a gracious afterthought, Yaz grabbed Rick Burleson's dark glasses and tossed them in DiMuro's general direction in case there should be a sudden eclipse of the sun.
DiMuro looked upon the heap of dirt not as a back-to-the-earth movement, but as a back-to-the-locker-room invitation.
And that's where he consigned Capt. Yaz. He calmly uncovered home plate with his whisk broom and said, in effect: "To the clubhouse, Euell, and don't forget your pail and shovel."
So Yaz went, to join the already banished Bernie Carbo and Manager Darrell Johnson.
Two days later at the Coliseum, the Sox were spraying each other with champagne for the first time in eight years. "I don't know if this stuff feels better when you drink it or have it poured over your head," said Fisk after Boston had closed out the series with a 5-3 decision. Next up were the Reds, who would be playing in their third World Series in six years, but were looking for their first ring since 1940.
Boston's drought was longer by 22 years. But in the wake of an effortless 6-0 victory in the Fenway opener on October 11, a bejeweled ring finally seemed possible. "In the National League we don't face anyone who throws a spinning curve that takes two minutes to come down," observed third baseman Pete Rose after the gyrating Tiant had twisted them into knots, and then touched off a killer six-run seventh with a single off a misplaced Don Gullett changeup.
The Sox came tantalizingly close to winning Game 2, carrying a 2-1 lead into the ninth before the visitors scored a pair of two-out runs off Dick Drago to escape with a 3-2 win. "Nobody said it was going to be easy," Lynn remarked as the club packed for Cincinnati. "No way."
Nobody said it would be fair, either. In Game 3, umpire Larry Barnett, misinterpreting the rules, called Fisk for interfering on pinch hitter Ed Armbrister's 10th-inning bunt. The Reds, who'd led 5-1 before a Sox comeback, won it, 6-5, on Joe Morgan's single. "This is brutal," declared Fisk. "I never saw anything like it in my life."
Fortunes were reversed the next night. It took a five-run inning and 163 pitches by Tiant, but Boston hung on for a 5-4 victory. Then came another shift in momentum: Gullett mastered his opponents, 6-2, in Game 5 and the Sox returned home on the brink.
A three-day nor'easter gave them time to regroup and allowed Johnson to start Tiant instead of Lee. "Tiant is our best," he said. "We're down 3-2. We have to win the sixth game first."
It took the Sox until after midnight to do it and Tiant was long gone when they did. But after pinch hitter Bernie Carbo's two-out, three-run homer in the eighth brought Boston back even, Fisk won it in the 12th with a homer that bounced off the left-field foul pole and set church bells ringing in triumph in his hometown of Charlestown, New Hampshire. "I straight-armed somebody and kicked him out of the way and touched every little white thing I saw," said Fisk after his celebratory gallop around the bases through fans and teammates on his way to touch the plate.
For five innings the next night, the faithful could envision a championship flag flapping as the Sox took a 3-0 lead. But Lee served up a looping, ill-timed "Leephus" pitch with two out in the sixth to slugger Tony Perez, who launched it over the Wall for a two-run homer. "Lee threw it to the wrong guy," Morgan mused years later. "He could have thrown it to me, Pete Rose, Johnny Bench, anybody. But Perez was the best off-speed hitter on our team."
ATTACK OF THE GREEN MONSTER
Game 6 of the 1975 World Series was noteworthy for, among other things, one catch that was made (by Dwight Evans, to steal a possible home run from Joe Morgan in the 11th inning) and one that was not (by Fred Lynn when he slammed into the left-field wall in pursuit of Ken Griffey's two-run triple).
As part of the modifications for the 1976 season, the wall was stripped of its metal skin and fitted with a smoother, Formica-like covering. Protective padding was added to the wall's base to prevent serious injury, a move swayed at least in part by Lynn's scary World Series collision.
The new surface made caroms off the wall more predictable, and the Jimmy Fund, the Red Sox-supported charity, was made a little richer. The green sheet metal of the old façade was turned into paperweights and sold to fans to benefit cancer research.
The biggest change, however, was the addition of a huge electronic message board and scoreboard in center field—a first in Major League Baseball. The manually operated left-field scoreboard was reduced in width, with the elimination of the section of the board devoted to National League scores (which would instead be shown intermittently on the message board).
The new scoreboard cost $1.5 million, which was more than the cost of the complete stadium overhaul that took place in 1934. The board was 40 feet wide and 24 feet high, flashed 8,640 lights and was equipped to show both film and videotape, including instant replay.
Did the new scoreboard affect the game? A flagpole stood in center field for much of Fenway history. Jim Rice was the last man to hit a ball completely out of the park in that direction when he homered to the right of the flagpole off the Royals' Steve Busby in 1975. It's nearly impossible to do now because of the center-field scoreboard, the 2011 version of which is even larger.
Bill Lee showed teammates Carlton Fisk and Rico Petrocelli the blister on his thumb that would force him to leave Game 7 of the 1975 World Series in the seventh inning. The Red Sox lost the game, 4-3, after holding a 3-0 lead.
Lee left with a blister in the seventh. Then Rose tied it up. After Johnson yanked reliever Jim Willoughby for pinch hitter Cecil Cooper and brought in rookie Jim Burton for the ninth, Morgan won the game and World Series with a two-out blooper to center. "It was a slider low and away, and I couldn't have asked for a better location," Burton said as his teammates glumly put on their street clothes. "He didn't even hit it well."
The Sox were honored the next day at City Hall Plaza by thousands of fans for their gallant run, but second place in October brings no reward. "We're going to win that Series yet," vowed Yastrzemski.
Any chance that Boston had of a reprise vanished early in 1976 when the owners locked out the players in March after the Basic Agreement had expired and Lynn, Fisk, and Rick Burleson held out. "The togetherness, it wasn't there," Johnson would say at the conclusion of the club's most turbulent season in memory. "With those men playing out their options, I could see something was different, right from the start."
"There's nothing in the world like the fatalism of the Red Sox fans, which has been bred into them for generations by that little green ballpark, and the wall, and by a team that keeps trying to win by hitting everything out of sight and just out-bombarding everyone else in the league. All this makes Boston fans a little crazy and I'm sorry for them."
—Bill Lee
Senator Ted Kennedy and nephew Joe were faces in the crowd during 1975 World Series play.
Resilient Red Sox Nation returned to its "wait till next year" philosophy in the wake of the '75 loss.
SIZING UP THE WALL
The sign at the base of Fenway Park's famous left-field wall long indicated that the fence was 315 feet from home plate. Yet, from the day that sign was posted, batters insisted that the wall was less than 315 feet from home plate.
No doubt some of this skepticism was due to the height and breadth of the wall. Its imposing dimensions make it appear closer than it really is. But it turns out there was more to it than perception—there was proof.
In 1975, _Globe_ sports editor Dave Smith was presented with aerial photos of the park, accompanied by the report of an expert who flew reconnaissance in World War II. The military man concluded that the distance from home plate to the left-field wall was 304.779 feet. Smith made a formal request to the Sox to have someone from the _Globe_ measure the foul line. The team refused to comply, and the next day it was Page One news in a story by Monty Montgomery. Original blueprints from the Osborn Engineering Co.—which built Fenway in 1912—indicate that the wall is 308 feet from home plate.
Citing various reasons, the Red Sox for years had been reluctant to let anyone measure the debated distance. It was part of the Fenway mystique. Author George Sullivan once used a yardstick and came up with a distance of 309 feet 5 inches. It turns out that Sullivan was darned close.
One day in March 1995, in broad daylight, armed with a 100-foot Stanley Steelmaster measuring tape, the Globe's Dan Shaughnessy vaulted the railing and measured the line. He found it to be 309 feet 3 inches, give or take an inch, and he said so in a story on April 25. In May, the Red Sox surreptitiously admitted to their longstanding fib. Groundskeeper Joe Mooney re-measured the distance, and after huddling with Sox management, quietly changed the numbers on the wall to "310" feet. For a while, it went unnoticed.
"I was sitting in the stands before the game wondering when someone would come up and ask about it," said Mooney later. "Nobody did."
When Shaughnessy's estimate of 309 feet 3 inches was mentioned, Mooney said, "That's about what it is. We rounded it off. It came out in that story, so why hide it?"
After all those years, why indeed?
Even at 310 feet, Fenway's dimensions would be against the rules if the ballpark were being built today. The league now stipulates that fences must be no less than 325 feet from home plate.
By autumn, Johnson had been fired, Yawkey had died, a blockbuster deal with Oakland had been voided and the Sox, who never spent a day in first place, had to make a magnificent closing surge just to finish third as they posted their fewest number of victories (83) in a decade. Ten straight losses, half of them at Fenway, buried the club in sixth place and it never recovered.
The troubles in the '76 season began early. Eight days after a brawl-filled May meeting in the Bronx that put Lee on the sidelines for two months with a torn shoulder ligament, the Sox met the Yankees at the Fens, where extra security was added in case the fans sought retribution. Though Boston dropped two of three and sank to fourth, prospects seemed markedly brighter when O'Connell made a bold deal with the cash-strapped Finley, buying A's pitcher Rollie Fingers and first baseman/outfielder Joe Rudi for $1 million apiece.
"We had to make up our minds that we're trying to win the pennant this year," said the general manager. The deal, though, seemed too good to be true, and it was. Commissioner Bowie Kuhn soon nixed the sale "in the best interests of baseball."
"Bowie Kuhn is acting like the village idiot," fumed Finley, whose sale of Vida Blue for $1.5 million to the Yankees also was voided.
Three weeks later Yawkey, who'd had no problem opening his pocketbook if it would have meant another pennant, died at 73 after battling leukemia for several years, ending an era both in Boston and in baseball. "He gave so much to this city," said Mayor Kevin White. "Over the years so much pleasure. Last year so much excitement. And always so much class."
The Fenway grounds crew caught a few Zs in July 1978.
"I love Fenway. I love it in spite of the things about it that I hate."
—Stephen King, author
The Sox soon went into free fall, losing 11 of 13 games on the road to drop to fifth. That meant the end for Johnson, who'd been Manager of the Year in 1975, but couldn't find a way to keep his club in contention in 1976. "What happened? I just ran out of answers," he said. It was easier, O'Connell conceded, to change the manager than the team, "which would be practically impossible."
So Don Zimmer, the Sox third-base coach who'd previously managed the Padres for two years, took over as skipper and the club caught fire as summer turned into fall, winning 15 of their final 18. "Third place is no big thing," Zimmer acknowledged, "but it was an outstanding finish."
Except for signing Twins reliever Bill Campbell to a seven-figure deal, the front office opted not to shop in what it deemed an overpriced bazaar during the first year of MLB free agency, instead bringing back George Scott and Bernie Carbo from the Brewers for Cecil Cooper. That created a cadre of wallbangers that rivaled the 1927 Yankees for percussive power.
The 1977 club set franchise records for homers, hitting 124 at home and 213 in all, with Rice contributing 39, Scott 33, Hobson 30, Yastrzemski 28, and Fisk 26. Its most jawdropping—and delightful—display came on a mid-June weekend in Fenway when Boston cranked 16 round-trippers off the Yankees amid a three-game sweep in which the hosts outscored their archrivals by a combined 30-9 on national television.
The shelling began immediately as the Sox battered Cat-fish Hunter for four homers in the first inning of the opener, and then added two more in a 9-4 bashing. "They beat the hell out of us tonight," acknowledged New York's Reggie Jackson. "They were sending bombs everywhere." Unfortunately, the triumph was marred by bleacher oafs who pelted New York center fielder Mickey Rivers with metal bolts. "If it happens again, I'm going to pull my team off the field," vowed Yankees Manager Billy Martin. "We won't stand for that kind of stuff. Somebody could get killed."
The Sox were doing enough damage to the visitors with horsehide, thumping another five homers (with two apiece from Carbo and Yastrzemski) in a 10-4 drubbing on Saturday that was equally notable for a dugout confrontation between Martin and one of his star players, Jackson, who chafed at being pulled from the game because the skipper thought he'd loafed on a fly ball.
The Yankees left the Fens both squabbling and reeling after dropping the finale, 11-1, on Sunday in a game highlighted by another five Boston homers. "We just had our men playing in the wrong spots," observed Martin, whose own club didn't manage a single shot. "We should have stationed them in the screen."
It was the most rewarding home stand in memory for the Red Sox, who went 9-1 and vaulted from third to first place. Nobody who'd witnessed the humbling of New York would have bet that the Yankees would come back to win the division or that the Sox soon would turn into a white-knuckle carnival ride. They lost nine in a row, and then won seven of eight. During a West Coast road trip in late July and early August, they won nine straight as part of a 16-1 surge, and then lost seven straight to fall into second place.
Despite winning 11 of 13 to start September, Boston never again was atop the pile and the season ended with an ironic twist as the Sox gave up six homers to the Orioles in an 8-7 loss that eliminated them and let the Yankees spray champagne. "Just like the horses I bet on, I came up a little short," Zimmer joked in a telegram to Martin. "Congratulations."
While the Yankees went on to beat the Dodgers to win the World Series for the first time since 1962, the Sox were left to ponder yet another near miss. "You start out in April and you hope to be in a pennant race when it ends," mused Yastrzemski, after Boston had finished in a second-place tie with Baltimore. "We were in this one until the end and I have no regrets. The better team won."
In 1978, the Sox were in the race to the end and beyond, and the payoff was the most painful October moment in three decades. There had been significant changes before the club reconvened in Florida with the Yawkey Trust shifting control of the franchise to widow Jean, former vice president and catcher Haywood Sullivan, and former trainer Buddy LeRoux. After dismissing O'Connell, management reshaped the roster, shipping out Jim Willoughby, Ferguson Jenkins and, eventually, Wise and Carbo—the bulk of the bohemian "Buffalo Heads" clique that Zimmer abhorred—and bringing in Mike Torrez and fellow pitchers Dennis Eckersley (who won 20 games), Dick Drago, and Tom Burgmeier.
GAME NO. 163
After 162 games, there was only one way to settle one of the most tumultuous playoff races in baseball history: with Game No. 163. One of the two teams—the Yankees or the Red Sox—would reach the 100-victory mark and move on to the 1978 American League Championship Series vs. the Kansas City Royals. The other would have a litany of questions to answer about a final duel that would define this six-month roller-coaster ride of a season.
That 1978 AL East battle is remembered mostly for a Red Sox collapse. What many forget is that the Red Sox actually rallied to post a 12-2 record in their last 14 games, winning their final eight, to catch New York on the season's final day and set up the one-game playoff at Fenway.
Boston was sailing along with a 2-0 lead in the seventh inning when light-hitting Yankee shortstop Bucky Dent (who had a batting average of .140 over the previous 20 games) sent a fly ball into the screen off Mike Torrez with two men on, deflating Sox fans and surprising Dent himself, who didn't think his hit would clear the wall. ("I couldn't believe it," he later admitted.) New York extended its lead to 5-2, and just as they had played it out over the long season, the Sox would be forced to rally from behind in the late going.
The collective will of more than 35,000 Sox fans, along with a couple of timely hits, brought Boston back to within 5-4, and Carl Yastrzemski, who had homered earlier in the game when the day held such promise, stepped in with two outs in the last of the ninth and Rick Burleson on third base representing the tying run. Yaz managed only a towering pop fly off the Yankees' flamethrower Rich Gossage, and the most agonizing season in Red Sox annals was complete.
After 163 games over more than six months, the Sox had come up shy by one base, the distance between third and home plate. They had won 99 games, fourth-most in their history, but finished second to the Yankees' 100 victories. New York would go on to win its second straight world title. "We have everything in the world to be proud of," said Yaz afterward, "what we don't have is the ring."
Teammates welcomed Bucky Dent after his three-run homer in the seventh inning gave the Yankees the lead in the 1978 one-game playoff. Afterward, Yankees' owner George Steinbrenner (right) consoled Red Sox catcher Carlton Fisk.
It was just another rainy June evening at Fenway Park in 1977.
The result was an explosive start (45-19, 28-4 at home) that astounded even the most pessimistic Sox fans and had Boston in first place by seven games on June 17. At the All-Star break, the Sox led Milwaukee by nine games, with New York a distant 11½ astern. But in the wake of a spate of injuries, the pitching, hitting, and defense all collapsed, and the Sox began drooping by late summer.
By the time Boston met its traditional tormentors, the Yankees, at Fenway in September, New York had replaced Billy Martin with the less acerbic Bob Lemon, healed their internal divisions and closed to within four games. By the time New York left town, the Yankees had drawn even with a four-game sweep so devastating that it was labeled "The Boston Massacre." The visitors, who'd won 12 of their previous 14 outings, teed off on Torrez, who'd won two Series games for them the previous year, and administered a 15-3 flogging that was so unsightly that thousands of fans fled early. "Maybe we tired (the Yankees) out," Zimmer said, wryly. "They had scheduled extra hitting Friday afternoon. They called in the fifth inning and canceled it."
There was no joking the following night when the Yankees breezed to a 13-2 decision (aided by seven Boston errors) with a scoreline—2600210—that prompted one press-box imp to dial that number in New York. "No one was home," Peter Gammons reported in the _Boston Globe_. "Perhaps he was out looking for playoff tickets."
After a 7-0 loss, and then a 7-4 defeat in the finale, the Sox saw a season's work vanish in a weekend. "I'm not happy and I'm not worried," proclaimed second baseman Jerry Remy. "We're tied and we shouldn't be. But we're not going to roll over and cry."
Indeed, despite dropping 13 of 16 games during their September swoon, Boston rallied heroically, winning its final eight dates with Detroit and Toronto. (Tiant shut out the Blue Jays, 5-0, on two hits at Fenway on the final day.)
When the Indians hammered Hunter and the Yankees, 9-2, Boston found itself in a one-game playoff for the first time since 1948, when it met Cleveland at Fenway. "There is no way any of us right now can appreciate or even understand how it came to this," said Burleson. "But what we do know is that this is the biggest day of our lives."
For six innings, the division title appeared within grasp after Boston had nicked Ron Guidry for two runs. But with one swing, the least likely Yankee, light-hitting Bucky Dent, lofted what seemed to be a harmless two-out fly ball to left. Torrez, assuming that the inning was over, walked off the mound. "Then I looked over my shoulder on the way to the dugout and couldn't believe it," he said. "Yaz is back to the wall, popping his glove, looking up. I said, 'What's this? What the . . .'"
It was a three-run homer that put the visitors ahead, 3-2. The Yankees would take a 5-4 lead into the final frame. With two out in the bottom of the ninth, Burleson stood on third representing the tying run. But Yastrzemski, whose homer originally put his team ahead, popped up to third and the season was gone. The Yankees went on to win the World Series again and the Sox simply went home. "We won, but you didn't lose," New York owner George Steinbrenner told the disconsolate Sox, but none of them believed him. "We just blew it, that's all," said Burleson.
It was a particularly bitter ending for Yastrzemski, who craved not only his third pennant, but also his first championship ring. "Someday we're going to get that cigar," declared the 39-year-old captain, who'd wept in the clubhouse after watching the last ball off his bat drop into Graig Nettles's glove. "Before old Yaz retires, he's going to play on a world champion."
Except for Eckersley, who went on to claim a ring with Oakland in 1989, none of the players on that Red Sox team ever did. By the time the 1979 season began, Tiant had taken his rhumba delivery to the Yankees as a free agent and Lee and his extraterrestrial aura had been dealt to the Expos. Still, optimism reigned as author John Updike (who 19 years before had penned "Hub Fans Bid Kid Adieu," the famous farewell essay to Ted Williams) wrote an Opening Day story for the _Globe_ describing the "first kiss of another prolonged entanglement."
But while the Sox started briskly, winning 12 of their first 16 games to lead the division by two-and-a-half games, they were demolished, 10-0, at home by the Yankees in their first meeting with Torrez on the mound and Dent, who was heartily hooted on sight, knocking him out of the game. "I turned on the radio this morning and heard that the season was over," said Drago, whose team-mates were booed the next day when they came out for infield practice.
Despite magnificent seasons from Lynn (a league-leading .333 batting average with 39 homers and 122 RBI) and Rice (.325, 39, 130), Boston never saw first place again and finished third, 11½ games behind Baltimore.
Opening day always drew a crowd in the 1970s.
## 1980s
Red Sox manager Joe Morgan fumed after Rich Gedman was called out for crashing into Oakland's Mike Gallego at second base during Game 3 of the AL Championship Series in 1988.
The signature moments of the decade—on the diamond at least—occurred a couple of weeks and thousands of miles apart—in Anaheim, California, and in Queens, New York. But swirling around the historic Dave Henderson homer and the haunting Bill Buckner gaffe of 1986 were plenty of Fenway Park vignettes, both celebratory and shocking. Seemingly for the want of a stamp on an unmailed contract offer, native son and stalwart catcher Carlton Fisk walked away from the team in 1981. Soon after, the luxury-box era debuted at Fenway, and ultimately 40 of the revenue-producing suites were built on the ballpark roof. In 1983, an ugly ownership rift was exposed on a June night that had been reserved for celebrating the wondrous 1967 pennant and one of its stricken heroes. In the fall of that same season, Captain Carl, who had kept his feelings in check for pretty much his entire 23-year career, said farewell with an emotional lap around the ballpark. On a cold April 1986 night, Boston's "Rocket Man," Roger Clemens, was propelled into the spotlight by a record 20-strikeout game, the opening salvo in his first Cy Young Award season. Sparked by the "Hendu" home run, the Sox made the World Series for their only time of the decade; it had been 11 years since they had been there, and it would be 18 more before they returned. When the team foundered in 1988, Joe Morgan, the bullpen coach, assumed the duties as manager on an "interim" basis. But the Sox took off on a remarkable run, winning a leaguerecord 24-straight home games. All the while, Morgan, who hailed from nearby Walpole, kept things on a "six, two and even" keel as he guided the team to the playoffs twice in three years. Once in the postseason, though, his Sox struggled mightily—their losing streak, which began with Game 6 of the 1986 World Series, would last 13 games and 12 seasons.
The club never got rolling in 1980. Stalwarts like Jim Rice, Fred Lynn, Carl Yastrzemski, and Butch Hobson were among the lame and the team was stuck in fifth place at the All-Star break. At the end of September, when attendance had dropped by 400,000, management dumped Manager Don Zimmer with a year left on his contract. "A change was needed and we made it," said General Manager Haywood Sullivan, who promoted Johnny Pesky from coach to manager for the rest of the season. "Economics has something to do with it, fan reaction, public relations, on-the-field things. Let's be fair about it, sometimes change creates attitude. I'm looking for a little different tone, that's all."
Zimmer, whose 411 victories in four-plus seasons still put him in the top seven in club annals, was sanguine about his dismissal. "I've heard a lot of boos," acknowledged the man who'd never been fired nor held a job outside of the game. "I've made a lot more friends than I have enemies in my stay here. That certainly holds true in the clubhouse. This has been a good club to manage. A damn good club."
What the front office wanted, Sullivan stated, was "a strict disciplinarian, a solid baseball man and a motivator." The choice was Ralph Houk, a decorated Army major who'd directed the Yankees to consecutive World Series championships in the early 1960s, and then returned to oversee their reconstruction before moving on to Detroit, where he took on another renovation job before retiring to Florida. "My golf game didn't get very good," Houk said upon returning to baseball. "The fish weren't biting. How often can you cut the grass?"
Red Sox manager Don Zimmer kicked at the dirt after being ejected by umpire Marty Springstead on April 14, 1980.
Houk was a transition specialist, which made him a natural fit for a 1981 Boston club that had lost three prominent regulars—Fred Lynn, catcher Carlton Fisk, and shortstop Rick Burleson—during the off-season. Burleson was traded with Butch Hobson to the Angels for Carney Lansford, Rick Miller, and Mark Clear. Fisk and Lynn took advantage of a baffling oversight by the Boston front office to switch uniforms; because management hadn't offered them new contracts by the December deadline before their option year, Fisk and Lynn were able to declare themselves free agents.
While the dispute was being heard in January, the club traded Lynn and pitcher Steve Renko to the Angels for outfielder Joe Rudi and pitchers Frank Tanana and Jim Dorsey. When arbitrator Raymond Goetz ruled for Fisk, "Pudge" exchanged red socks for white and decamped for Chicago. Through a scheduling coincidence, Fisk played in his 10th Opening Day at Fenway (but his first as a visitor), where he cracked a three-run homer in the eighth that gave his new confrères a 3-2 decision.
"I honestly wasn't concerned about how the fans were going to react to me," said Fisk, whose line-drive home run quieted a crowd that hadn't been sure all afternoon whether it should applaud or boo him. "I thought it would be mostly positive, which I feel it was. I knew there would be some hooters, but I thought the majority would be favorable because I don't think I ever gave them any reason in all of the years I played here to feel any differently."
Rich Gedman, up from the minors, was the part-time new man behind the plate with Glenn Hoffman at short and Miller in center, and the syncopated Sox were essentially out of contention by the end of April after dropping seven straight, including four games by a 28-8 aggregate during a sobering sweep by the Twins at Fenway. So a two-month players' strike actually was a blessing, since Major League Baseball went to a split-season format when play resumed in mid-August, giving everyone a clean slate.
"Well, we're only a game out," Houk reckoned after Boston had been mugged, 7-1, by Chicago in Opening Day II at Fenway. Since the Sox had been four behind when the strike began, that constituted progress. By season's end, though, the deficit was two-and-a-half games, which turned out to be the club's closest finish to the top for five years.
While there was some satisfaction in playing a full slate in 1982, the Sox experienced a peculiar season that was as disappointing as it was encouraging. Despite winning 89 games (only five times in 30 years had they won more), they finished third, six games behind the Brewers. And though they were in first place at the end of June, the Sox subsequently dropped 10 of 13 games and never recovered. "We just weren't quite good enough," observed Clear.
An electrician tinkered with some of the 95 lights on one of Fenway Park's light towers in March 1983.
In April of '83, for the 10th straight year, Jane Alden put up bunting on the Red Sox dugout and along the right-field line.
JIM RICE'S ONLY CAREER SAVE
One of the more emotional moments in Fenway history started as just another perfect summer Saturday at the ballpark.
On August 7, 1982, Tom Keane drove down to Fenway from his home in Greenland, New Hampshire, on the Granite State's sliver of a seacoast, with his sons Jonathan, 4, and Matthew, 2. Keane had scored tickets for an afternoon game against the White Sox, and these weren't just any seats. Through a friend, Keane had gotten tickets from Red Sox Executive Vice President Haywood Sullivan, and they were two rows from the field, just to the left of the Red Sox dugout.
"You were actually right there," Keane told an ESPN reporter years later. "It was a seat that everybody would dream of when they had little kids and you wanted to get them close to the action. It was just ideal."
The boys had cheered wildly when Sox second baseman Dave Stapleton—Jonathan's favorite player—came to the plate in the fourth inning. But Stapleton swung late at a pitch and slashed a foul ball into the stands. The elder Keane didn't see the ball, but he did hear a cracking sound. He thought the ball had hit the side of the dugout—until he turned and saw the blood coming down his son's face. Jonathan had been hit by the foul ball and suffered a fractured skull.
Red Sox left fielder Jim Rice, who had been perched on the top step of the dugout, waiting his turn to hit, reacted instinctively. "Jim Rice was right there with his arms immediately," Keane said, "I mean immediately."
Rice was described by the media of the day as standoffish, or even sullen. His reserved manner was fortified for a time by an impression that, as an African-American ballplayer, he wasn't as warmly received by Boston fans as a white superstar would have been. And as an eight-time All-Star and the American League's MVP in 1978, Rice certainly qualified as a superstar. But none of those perceptions mattered one bit on that day, when Rice's quick reaction may have saved Jonathan Keane's life.
Rice, a father of two young children, later said he was thinking of one thing, and one thing only. "My child," he said. "Just someone, myself, just taking care of my child, picking my child up and taking him to the clubhouse."
"I was kind of chasing Jim Rice; he was carrying Jonathan," said Keane. "There was an ambulance waiting. When we got to the hospital, they were set up for neurosurgery."
Doctors at Children's Hospital, just a mile away, relieved the pressure on Jonathan's brain and gave him medicine to guard against seizures. He was hospitalized for five days.
Eight months later, Jonathan was reunited with Rice. On April 5, he threw out the first pitch at Fenway to open the 1983 season.
"Obviously, as we sit here today, what he did saved [Jonathan's] life," Keane said. "I mean you had a young child, his left skull is fractured open, it is bleeding profusely. The worst could have happened."
Said Rice, years later, "Playing baseball was more of a talent than a gift. The reaction to save somebody's life, that's entirely different."
Boston was nowhere near good enough in 1983, when the club finished 20 games out in sixth place with its worst record (78-84) since 1966. Everything went sour in early June, when minority owner Buddy LeRoux announced a takeover (the so-called "Coup LeRoux") that provoked an immediate riposte from Sullivan. The dueling press conferences, observed _Globe_ beat man Peter Gammons, resembled "the staging of a Mel Brooks parody of post-Mao China."
The front-office turmoil sent an aftershock through the clubhouse as the Sox promptly were swept at home by the Tigers and went on to lose seven in a row, tumbling from first place to fifth in less than a week. "I can't believe what I saw," said a nonplussed Houk after a comedy of errors resulted in a 10-6 loss to the Orioles. "I've never seen so many things happen like that in my life."
The rest of the season was forgettable, except for the finale, when Yastrzemski made a farewell tour of the park where he'd performed with both dignity and distinction for 23 years. "They just kept saying, 'We love you, Yaz,' over and over," he said after the fans had saluted him. "I'll never forget it."
That was the end of an era that began with the departure of Ted Williams and ended with the blossoming of Wade Boggs, who won the batting title in only his second year with the club by hitting .361 in 1983. Yastrzemski wasn't the only familiar face missing from the starting lineup when the club convened for 1984. Gedman had taken over for Gary Allenson behind the plate; Bill Buckner (obtained from the Cubs for Dennis Eckersley) was at first; Marty Barrett stood in for the hobbled Jerry Remy at second; Jackie Gutierrez replaced Hoffman at short; and Mike Easler (obtained from Pittsburgh for John Tudor) was the designated hitter.
It was a dramatic changeover and after losing 10 of their first 14 outings, the Sox were 10 games off the pace by the end of April. Yet with a young rotation—that included Bruce Hurst, Dennis Boyd, Bob Ojeda, and Al Nipper—gradually finding its way, the club's stock clearly was on the rise, most notably with the arrival of right-hander Roger Clemens, the top draft pick from the previous year who debuted on May 15 and quickly established himself as an overpowering force.
His dominance of Kansas City in an August date at Fenway was a high-octane preview of future performances, with Clemens striking out 15 batters with no walks and only 31 balls to 33 batters. "The sound you heard was me: Owww!" testified catcher Jeff Newman. "He done hurt my hand." The Royals, who managed one run across nine innings, were duly impressed. "That was the best stuff we've seen all season and we're not going to see better," declared Mike Ferraro, a Royals coach. "He has the best stuff in the American League, hands down."
IN THE LAP OF LUXURY
The 1980s gave us the concept of corporations slapping their names on stadiums for a price, and the debut of the luxury box, including at Fenway Park.
In 1980, the Red Sox began to realize they could no longer survive with the park as it stood. With only about 34,000 seats and an escalating payroll that could not be met by ticket sales alone, Fenway Park was in danger of crumbling under the financial stresses of the day.
Before 1970, players made an average of $20,000 per year; by the early 1980s, they were making more than $100,000 on average. There once was a time when owner Tom Yawkey didn't worry about finding new ways to generate revenue, but Yawkey was gone. He had died of leukemia in 1976.
The Red Sox constructed 21 luxury boxes on the roof of the ballpark along the first-base side that debuted during the 1982 season. They later added boxes on the third-base side and behind home plate that brought their total to more than 40 luxury boxes (or suites), with an average of 14 seats per box, at a rental price of $50,000 to $70,000 per season.
From the day he took over the Sox business operations, Red Sox co-owner Buddy LeRoux set out to raise revenues. Along with constructing the luxury boxes, he increased ticket prices dramatically and eliminated Yawkey's policy of saving 6,500 bleacher tickets for sale on the day of a game. The interest on money deposited for advance ticket sales was believed to have brought the Red Sox as much as $2 million a year at that time. LeRoux's plan to add a second tier of seats, some 6,000 of them, atop the luxury boxes was never implemented.
LeRoux's approach ran counter to the Yawkey philosophy that no one should invest in sports to get rich.
Eventually, the ownership battles between LeRoux and his limited partners on one side and Sullivan and Jean Yawkey, Tom's widow, on the other, culminated in LeRoux being bought out of his ownership stake by Mrs. Yawkey in 1987.
Ted Williams stood, typically forgoing a tie, during the ceremony in which his No. 9 was retired, along with Joe Cronin's No. 4, on May 28, 1984. Team owner Jean Yawkey and former teammate Johnny Pesky joined the festivities.
YAZ SHOWS HE'S HUMAN
Carl Yastrzemski always wanted the other team to think that he was a machine. He wanted the pitcher to stare in and see a hitting robot staring back from the plate.
But on October 1, 1983, in his final game after 23 years of faithful service, Yaz broke down in full public view, before a sellout crowd at Fenway Park.
"I thought I had it under control," Yastrzemski said. "But then, when I started to hear the fans . . . that's why I put my head down when I got to first base."
For most of his final season, Yaz hadn't wanted any fuss. He just wanted to be what he always had been—the ballplayer, the worker. On this day, he endured the speakers and their kind words and the bows, until Yastrzemski himself was speaking, asking for a moment of silence for his mother and for former owner Tom Yawkey.
"New England, I love you," Yaz finally said, and then he was running around the perimeter of the field, touching as many hands as he could.
"I wanted to show my emotions," he said. "For 23 years, I always blocked everything out. I wanted to show these people that deep down, I was emotional for all that time."
When Yaz retired, he had played in more games (3,308) than anyone in baseball history, all of them for the Red Sox.
"I've always had tremendous happiness coming to this ballpark," said Yaz in a 1989 interview. "Never in my life did I think I'd make the Hall of Fame and have my number retired. I worked, and just working so hard overshadowed everything. I never enjoyed it after a game in which I did well. I was always thinking about tomorrow. I just never dwelled upon the success."
"I came to love Fenway. It was a place that rejuvenated me after a road trip; the fans right on top of you, the nutty angles. And the Wall. That was my baby, the left-field wall, the Green Monster."
—Carl Yastrzemski
Clemens went on to submit a 9-4 record, and with the outfield of Rice, Tony Armas, and Dwight Evans combining for 103 homers and 349 RBI, Boston managed to post 86 victories and climb to fourth place.
The club's transition complete, Houk retired at the end of the season and was succeeded by John McNamara, who'd been managing the Angels. "Am I too late to apply for the job?" Don Zimmer jokingly asked Sullivan over the phone after the announcement was made. Even before he'd checked into his hotel room upon arrival, McNamara discovered what Zimmer had learned—that everyone in New England was a would-be manager. "Is Rice going to hit fourth?" the Sheraton-Boston bellman inquired.
Rice batted third until August, but his 1985 season was cut short by a knee injury. Armas missed nearly two months with a torn calf muscle, and Clemens, dogged by a shoulder injury that required late-summer surgery, had only 15 starts as Boston finished fifth, more than 18 games behind the Blue Jays. "It's certainly not the kind of record I had in mind when I came here," McNamara conceded after his club was swept at home by the Brewers on the final weekend to end with a .500 record. "To me, this has been a very disappointing season."
In 1986, the disappointment was limited to two October nights in Queens, but it was painfully profound. The regular season, highlighted by Clemens's record 20-strikeout performance against the Mariners in April, proved surprisingly easy as the Sox held first place from May 16 until the end, won 95 games and claimed the division by five-and-a-half games despite losing the final four to the Yankees at home. "The usual rallying cry around here is, 'Wait until next year,'" remarked McNamara as his men prepared to take on the Angels in the league championship series. "Well, next year starts Tuesday night. We're prepared."
Boston hadn't played a postseason game since the one-game divisional playoff in 1978 with the Yankees, and the ALCS opener at Fenway was over almost as soon as it began. The visitors rocked Clemens for four runs in the second inning. "We'll be all right," Clemens insisted after California counterpart Mike Witt took a no-hitter into the sixth inning and wrung out the Sox by an 8-1 count.
THE LEROUX COUP
In the annals of bizarre incidents at Fenway Park, probably none can top the night of June 6, 1983. The Red Sox had announced plans to honor former player Tony Conigliaro, who had been in a coma for months following a stroke. Many of his teammates from the revered Impossible Dream team of 1967 were gathered, along with a large media contingent.
Co-owner Buddy LeRoux interrupted the proceedings to announce that enough of the team's partners had voted in his favor to give him control of the Red Sox over co-owners Haywood Sullivan and Jean Yawkey, with whom he had been feuding for years. A half hour later, Sullivan and John Harrington, representing the Yawkey Foundation, held a rebuttal news conference to announce that the team's chain of command was intact, and that LeRoux's announcement was "illegal, invalid and, above all, not effective."
"Le Coup LeRoux," as it was dubbed, reduced the celebration of the 1967 team and the stricken Conigliaro to an afterthought.
In the next day's paper, _Globe_ columnist Michael Madden wrote, "Buddy LeRoux and his bottom-line men did not have the patience, the class, the sensitivity, the good taste or even the most basic respect to wait another day." Madden called it "a sideshow, a shoddy, slippery coup by banana-republic generals. . . . What was to have been the Night of the Impossible Dream was transformed meanly into the Day of the Shameful Scene."
Former player George Thomas watched the competing news conferences, both held directly under a color portrait of the late, longtime owner of the Sox, Tom Yawkey.
"If he could see this now," said Thomas, "he'd be spinning in his grave."
Ironically, LeRoux had developed a strong bond with the Yawkeys as the team's trainer in the 1960s. He went on to build a real estate empire, and in 1977, with Sullivan as his partner, LeRoux led a group that sought control of the Red Sox after Tom Yawkey's death. When some questions arose about financing, Jean Yawkey joined the consortium and the sale was approved by Major League Baseball.
LeRoux took over the business end of the Sox operations, guiding the construction of luxury boxes at Fenway and signing lucrative TV and radio contracts. Before long, however, the ownership group became divided, leading to the stunning events of "Tony C Night."
The outcry over LeRoux's takeover bid was immediate and harsh. He was taken to court by Yawkey and Sullivan and eventually lost the battle for control of the team. Once he realized that he would never be able to buy the others out, LeRoux sold his shares to Yawkey in 1987.
Boston indeed bounced back the next afternoon. The sun-blinded Angels ran into outs, watched balls skip over and away from them, and made three errors in the seventh inning—when the hosts scored three runs on one hit. But the Sox dropped the next two in Anaheim, including a killer 11th-inning loss in Game 4 after they'd blown a 3-0 lead in the ninth with Clemens on the mound. It seemed that the season would end on the Left Coast, especially when the Angels took a three-run lead into the ninth inning of Game 5. Don Baylor cranked a two-run homer to draw the Sox to within 5-4, but pretty soon the visitors were down to their final strike, and the home team was preparing to rush the diamond in triumph. "I looked across the field and I could see everyone in the Angels dugout getting ready to celebrate," said Boston's Dave Stapleton. "Gene Mauch. Everyone. They had those little smiles that you get before you start hugging everyone."
THE ROCKET'S AWESOME COMING-OUT PARTY
As the 1986 Red Sox season opened, Roger Clemens was a 23-year-old question mark coming off shoulder surgery that curtailed his 1985 season. But he turned the question mark into an exclamation point on a weeknight in late April when the Red Sox game was playing a distant second fiddle to a Celtics-Atlanta Hawks playoff game that featured Larry Bird across town at the old Boston Garden.
Only 13,414 fans were on hand at Fenway Park on April 29 to see "Rocket Roger" do something that no pitcher in 111 years of major-league history had managed to do before: strike out 20 batters in a nine-inning game.
The Mariner lineup included a couple of guys who would later factor into Boston's amazing run to the 1986 World Series: Spike Owen and Dave Henderson. Owen had also been Clemens's teammate at the University of Texas. Clemens came into the game with a 3-0 record, which would eventually grow to 14-0 en route to a 24-4/MVP/Cy Young Award season. Clemens struck out the side (all swinging) in the first inning. He fanned two more in the second and one in the third. Although he went to three balls on five of the first nine batters, he ended up with zero walks to go along with his 20 strikeouts.
After giving up a single to Owen to start the fourth, Clemens struck out eight in a row, tying an American League record. But in the seventh inning, Seattle's Gorman Thomas drove a 1-2 fastball into the center-field bleachers to give Seattle a 1-0 lead. Amazingly, Clemens was on his way to history, but was in danger of losing the game. Fortunately, Dwight Evans crushed a three-run homer in the bottom of the inning to put the Sox ahead, 3-1, the eventual final score.
Clemens had 18 strikeouts after eight. Owen struck out on a 1-2 fastball to open the ninth, and when umpire Vic Voltaggio rang up Phil Bradley on an 0-2 fastball for No. 20, Wade Boggs jogged to the mound from third base and shook Clemens's hand. Ken Phelps grounded to shortstop for the final out on Clemens's 138th pitch.
Clemens would go on to throw another 20-strikeout game (no walks again) in Detroit 10 years later, but this was the night that signaled Clemens's ascendancy.
"In the clubhouse afterward," General Manager Lou Gorman said, "even the veterans were awed."
'OIL CAN' KEPT US GUESSING
In the 1980s, few people called him by his given name, Dennis. Years later, "Oil Can" Boyd told Dan Shaughnessy, "I'm blessed with this mystique. I got a nickname and I know how to pitch."
The Can started 207 games in his 10-year MLB career. He started the third game of the 1986 World Series and was lined up to pitch Game 7 before rain and Red Sox Manager John McNamara changed history. The Can went 78-77 in the big leagues with a 4.04 ERA and pitched the division-clinching game for the Sox in 1986. But his favorite baseball memories come from his younger days in Meridian, Mississippi.
"That was the most fun, around when I was 12-15 years old," he recalled. "Baseball was the epitome back then. I played a lot of baseball when I was little, so I have big reminiscences of baseball and that's how I came to be the player I came to be."
Boyd could throw a ton of innings and he always wanted the ball. Baseball was in his blood. His dad, Willie James Boyd, once faced Henry Aaron and Willie Mays at the ballpark in Meridian. Dennis Ray Boyd was nicknamed "Oil Can" because of his fondness for beer (the nickname gets a special citation from Susan Sarandon in _Bull Durham_ ), and he became one of the more delightful characters on the Boston sports scene after splashing down in 1982. Manager Ralph Houk—and later McNamara—didn't know quite what to make of the Can, but they knew he wanted the ball every fifth day and that he could pitch.
There was plenty of controversy. The Can was hospitalized with a mysterious liver ailment in 1986, and then went into a rage and temporarily quit the Sox when he didn't make the All-Star team. There were rumors of drugs, and he got into a jam with the police. The Sox subjected him to a psychological evaluation and went public with their concerns. When team-mate Wade Boggs announced he was a sex addict a few years later, Can said, "Now who needs the psychiatrist?"
In the end, blood clotting in his throwing shoulder took Boyd away from the game he loved. He pitched for the Expos and Rangers before retiring after the 1991 season.
Dennis "Oil Can" Boyd was restrained by a teammate and an umpire as he argued a call during a 1986 playoff game.
Then Dave Henderson, picked up from Seattle in August, whacked another homer off Donnie Moore to tie the score in the ninth. And it was Henderson again in the 11th inning, knocking in Baylor with a sacrifice fly to put the Sox ahead, 7-6. "If we lost this game today, it wouldn't have been fair," concluded Baylor after Calvin Schiraldi, a mid-season call-up turned closer who had blown the save in Game 4, had set down the Angels in the bottom half of the 11th to preserve the Game 5 victory.
Once back in their own yard, the Sox closed out the Angels with a pair of emphatic blowouts. In Game 6, Boston easily evened the series with a 10-4 decision, knocking out starter Kirk McCaskill by way of five runs in the second inning. "Both teams have shown they know how to play and win," Mauch observed. "Now we'll find out which one knows how to win when they have to win."
Not since the 1912 World Series had the Sox won an October postseason series that had gone the distance. But the finale was nearly a walkover. Clemens stifled the Angels while his mates laid a battering on John Candelaria, who gave up seven runs in the first four innings. The Sox outburst was punctuated by Rice's three-run shot over the left-field wall. "Yogi's quote gets better and better," exulted McNamara after his club had banished the Angels by an 8-1 margin to advance to the World Series for the first time in 11 years. "It's still never over until it's over. For me, it is a dream come true to get there."
The Sox hadn't won the Fall Classic since 1918. But after they had mastered the Mets, 1-0 and 9-3, in the first two meetings at Shea Stadium, the Fenway faithful began believing that they could see it happen in person, even though Boston had been a decided underdog when the Series started. "You look at some of the guys on our ball club," mused Buckner. "You go down the names, and then you wonder what the bookies were thinking. I think those guys are sorry now. I wish I could have made a bet."
Boston never had lost consecutive postseason games at home, but New York dealt them two sobering defeats in two days. The Mets knocked around Boyd for a 7-1 triumph in Game 3. Then Bob Ojeda, who had been traded in 1985, mastered his former teammates, as the Mets ran up a 5-0 lead in Game 4 en route to a 6-2 decision that evened the Series. But Hurst, who'd blanked the Mets in the opener, held them off again, 6-2, in Game 5 and the Sox returned to Queens with two chances to win their rings. "Our backs are not at the door," declared Rice. "Their backs are."
The Mets seemed all but finished in Game 6 after Henderson's leadoff homer and Barrett's RBI single gave Boston a 5-3 lead in the 10th inning and Schiraldi had the hosts down to their last strike with two on. But Ray Knight laced a single to score one run and Bob Stanley, who came in to close, uncorked a wild pitch with two strikes on Mookie Wilson that enabled New York to draw even. When Wilson's bouncing grounder went through Buckner's legs to score Knight from second, millions of Sox fans, many of whom had awakened children and propped them in front of the TV so that they could bear witness to history, went to bed in shock.
"I had dreamed of this moment," said Stanley after the Mets had escaped, 6-5, in the only Series game ever decided by an error. "How I would be on the mound for the clincher and how it would wipe out all the bad things. I was there tonight . . . but the dream turned into a nightmare."
Yet Boston had another chance to end 68 years of autumnal futility. A Sunday rainout enabled McNamara to start Hurst for a third time and the Sox staked him a 3-0 lead in the second inning. Redemption seemed at hand. But New York squared things in the sixth off a tiring Hurst, and went ahead 6–3 the next inning. After Boston had drawn to within a run on Evans's double, the Mets stung Al Nipper for two more in the bottom of the eighth, and soon were spraying themselves with bubbly. "It's a wrong ending to a storybook year," said Henderson as his teammates quietly packed their gear.
Young Red Sox fans bought up every souvenir in sight to cheer on the 1986 team during postseason play at Fenway.
Fenway fans saluted Jim Rice after his three-run homer sealed the Game 7 victory over the Angels in the 1986 AL Championship Series.
BUCKNER'S REDEMPTION
It was October 6, 1986, and Bill Buckner was talking to a Boston TV reporter at Fenway Park about the upcoming postseason for the Red Sox, their first since 1975. "The dreams are that you are going to have a great series and win, and the nightmares are that you are going to let the winning run score on a ground ball through your legs," said Buckner. "Those things happen, and I think a lot of it is just fate."
Fate brought Buckner to the plate in the top of the ninth inning of Game 5 of the ALCS in Anaheim against the Angels, who led the game, 5-2, and the series, 3-1. The Angels were three outs away from advancing to their first World Series. Buckner's single started a rally that concluded with Dave Henderson's dramatic, final-strike, game-tying homer that helped to propel the Red Sox into the Series.
Fast-forward a couple of weeks to Shea Stadium, to the play in Game 6 of the World Series that propelled the Mets to the world title and unfairly defined Buckner's career. We can close our eyes and see the videotape of Mookie Wilson's ground ball, we can hear announcer Vin Scully exclaim, "A little roller up along first . . . behind the bag! It gets through Buckner! Here comes Knight . . . and the Mets win it!"
Buckner played in four decades. He collected more hits (2,715) than either Ted Williams or Joe DiMaggio did. He won a National League batting title in 1980 and was an All-Star in 1981. But that one play, which has been called the Zapruder film of baseball, took on a life of its own and made him the poster boy for the Curse of the Bambino.
Years later, Buckner said, "This is the honest-to-God's truth. My first thought was, 'We lost the game.' The second thought was, 'Oh man, we get to play the seventh game of the World Series.' There was no doubt in my mind we were going to win the last game."
The Sox lost the last game to the Mets, of course, and Buckner was vilified, made the butt of hundreds of bad jokes. He played half of the 1987 season for the Red Sox, and played the final 22 games of his career in Boston in 1990, but then he moved far away, to Boise, Idaho, and for a long time he stayed away, including when the 1986 Sox team was honored in 2006.
In a 2003 interview with the Globe's Stan Grossfeld, Buckner acknowledged that all was not forgotten. "I still hear stuff," he said. "I laugh at it. Sports are for teaching young people to deal with success and failure. The saddest thing is, what are you teaching kids today? That you can't make a mistake? You make an error and you don't win, so you say, 'I don't want to play.' That's not what sports is all about."
On Opening Day 2008, after enduring years of hecklers and worse, Buckner came back to Fenway to throw out the first pitch to former teammate Dwight Evans and received an emotional, four-minute ovation.
"I didn't think I was going to do it," said Buckner later. "I told them I'd think about it, but I made up my mind I wasn't going to come. Then I prayed about it a little, and here I am. Glad I came."
Bill Buckner wiped his eyes during the extended ovation he received when he threw out the first pitch to open the 2008 Red Sox home season.
The hangover continued through the following season as Boston finished 20 games out in fifth place in 1987, posting a losing record (78-84) for only the second time in 21 years. Things had begun trending downward during spring training when both Hurst and Boyd were injured and Clemens held out; the club already was 0-3 when the pennant was hoisted at the home opener. "It made me mad," said Evans, after the players received rings for winning the American League the previous season. "There should have been 'World Championship' on that ring."
By early May, the Sox already were sagging badly after dropping eight of 10 on a trip to Texas and the West Coast. Just after mid-month, they were buried in sixth place and never rose above fifth for the rest of the season. "I'm just waiting now for October 4th to get here," McNamara remarked on September 23. The dismal season wasn't the manager's fault, Lou Gorman declared. "There's no way you can blame him for this year," the general manager concluded. "I'm sure some people will do it, but I don't think he's responsible at all for the kind of season we've had."
McNamara did get the blame in 1988 when a $6 million palimony suit filed against Wade Boggs by companion Margo Adams made national news and created clubhouse dissension. With the team barely above .500 at the All-Star break, owner Jean Yawkey concluded that McNamara had to go and fired him against the wishes of both Sullivan and Gorman. "We want to try and turn this thing around," said Gorman. "We're not saying that John McNamara didn't do a good job, but the manager is always the scapegoat—fairly or unfairly."
Stepping in with just a few hours of notice was Third-Base Coach Joe Morgan, a Walpole, Massachusetts, native who'd managed the PawSox for nine years and was the first local resident to manage the Boston Red Sox since Charlestown-born Shano Collins in 1932. "Communication is important," he acknowledged during his first day on the job. "You have to talk to players. But if they get out of line, you have to step on 'em."
The first to feel Morgan's tread was the captain. When Rice objected loudly after being pulled for pinch hitter Spike Owen, he and Morgan shouted and scuffled in the dugout and Rice was suspended for three games. "I'm the manager of this nine," Morgan declared.
By then the Sox already were up and away on their best run since 1948, a 12-game winning streak that included a six-run comeback over the Royals. "It's unbelievable, really," marveled Stanley. "I'm shocked. We're winning games we were losing before."
It was "Morgan Magic," the press declared, a spell conjured by a plainspoken citizen who channeled the catchphrase, "Six, two and even," drove a snowplow along the Massachusetts Turnpike during the off-season, and seemed unfazed by a position that had driven some of his predecessors to drink. "There is no pressure, gentlemen," Morgan declared as his nine ascended the standings, winning 19 of 20. "I had more pressure trying to hit in the big leagues than I do managing."
By Labor Day, the Sox were in first place for good and despite dropping six of their final seven games—including 11-1, 15-9, and 1-0 home losses to Toronto—Boston still won the division by a game over Detroit and earned a playoff date with Oakland. Boston had dismissed the Athletics en route to the 1975 World Series, but this edition of the A's proved to be decidedly more stubborn.
In the opener at Fenway, Hurst pitched well enough to have won most playoff outings. But Oakland pitcher Rick Honeycutt starved the hosts until old friend Dennis Eckersley entered in the eighth to finish off his former mates, whiffing Boggs on three pitches with two on and two out in the ninth. "We've got to come back tomorrow with the hammers of hell," declared Morgan. That meant Clemens firing ingots from the mound. But the visitors rallied from two runs down on Jose Canseco's two-run blast in the seventh, nicked Lee Smith for the winner with two out and two strikes in the ninth, and brought in Eckersley again to tie the Sox in knots.
THE MANIACAL ONE
"Sometimes the truth hurts," said Chuck Waseleski with a chuckle when asked about his nickname. "It's true. I can't refute it."
Waseleski, known as "The Maniacal One" for his devotion to baseball record-keeping, came along before computers and fantasy baseball exploded onto the sports scene, before situational statistics, before obscure but telling numbers became necessary fodder for sports talk shows and play-by-play announcers looking to fill air-time. He provided these statistical insights to Peter Gammons, who began to include them in his nationally recognized Sunday baseball column for the _Globe_.
"You hear that so-and-so is a good two-strike hitter," he said to the _Globe_ 's Dan Shaughnessy in 1987. "I hate that. I want to know how good. Be specific."
Maniacal Chuck was born and raised in the village of Millers Falls, Massachusetts, and he went to Turners Falls Regional High, where he was class valedictorian in 1972. Like a lot of New Englanders, he was a casual Red Sox fan until the 1967 Impossible Dream season. "That hooked me for good," he said.
In 1982, he began corresponding with Bill James, author of the annual Baseball Abstract. "It was then that I realized there was some demand for this material," Waseleski recalled. "People would ask, 'Is Jim Rice a clutch hitter?'"
Waseleski had the information. He could tell you that "Wade Boggs is hitting .488 (21 for 43) on 1-0 counts this year." Or, "Ed Romero hit the ball off the left-field wall twice in 1986." Waseleski has watched and charted every Boston game—every single pitch—since 1983, which happens to be the year Boggs won his first batting title.
"Oh, yeah," Boggs told Tampa Tribune columnist Martin Fennelly just before he was inducted into the Baseball Hall of Fame in 2005. "He was the guy that would tell everyone how many times I popped up to the infield and kept crazy stats on me, like how many times I swung and missed."
Before long, players and their agents were using such numbers to their advantage in salary arbitration cases. Waseleski compiled negotiation files for 30 players after the 1985 season, and for 70 players after the 1986 season.
In October 2004, when the Red Sox won their first World Series since 1918, many people wept. Waseleski typed into his Dell.
"I can tell you exactly what I wrote when we won it. I have it right here. I wrote that it was a 1-0 count, a fastball, and a ground ball back to the pitcher. It was Keith Foulke's 14th pitch," he told Fennelly.
Maniacal as always.
The Wave passed by Roger Clemens and Oil Can Boyd, who were focused on other things in the dugout during a 1986 regular-season game.
Morgan shook up his lineup as the series moved to the Bay Area and Boston grabbed a 5-0 lead in the second inning of Game 3. But Oakland's bashers quickly pounded Mike Boddicker for six runs and went on to a 10-6 decision that left the visitors on the brink. "We got a game left," said Boddicker, who'd arrived from Baltimore at the end of July. "We got a bullet left. It hasn't been done in baseball, but records are broken all the time."
Not this time. In the finale, Oakland's Dave Stewart scattered four runs across six innings and Eckersley came in to complete the sweep by closing out his fourth game. "All we have is some [hats and T-shirts] that say 'AL East champs,'" Boston's Todd Benzinger commented once the drubbing was done. "We never really got to enjoy it."
Joy remained scarce in 1989 as Boston finished third, a half-dozen games off the pace and with no playoff hopes. The dissolution began in late May, when the club lost six of eight games at Fenway and drifted downward from first place. By far the worst of the defeats—in fact, the biggest collapse in franchise history—was a 13-11 loss to Toronto in which the Sox blew the 10-0 lead they had mounted in six innings. "What a loss," moaned Morgan, after Ernie Whitt had crushed a grand slam off closer Lee Smith in the ninth and Junior Felix had clouted a two-run shot off Dennis Lamp in the 12th inning of the four-hour-and-36-minute fiasco. "This is the worst defeat of my managerial career in any league or city, hands down."
When Boston had climbed back to within one-and-a-half games of the lead in mid-August, the Blue Jays returned to put them out of the race with a sweep. "Is it the president?" someone asked Morgan when the phone rang after the season finale with Milwaukee. "I doubt it," the skipper said.
The video scoreboard in center field was removed on October 16, 1987. It was replaced by a version with Diamond Vision technology.
## 1990s
Catcher Mike Macfarlane (15) and outfielder Matt Stairs celebrated a win over Milwaukee that clinched the division title in 1995.
The decade started with turmoil, and the rumblings of discontent rarely abated throughout the Nineties. Roger Clemens was thrown out of Game 4 of the 1990 American League Championship Series, virtually assuring that the Oakland A's would sweep the Sox for the second time in three years. After a late-season run the following year, the Red Sox lost 11 of their final 14 games, and put an end to the mini-era of Morgan Magic. The team rushed one of its gritty heroes of the late 1970s, Butch Hobson, into the manager's seat—a position for which he seemed totally unprepared. Hobson guided the team to a 207-232 record over two-plus seasons and Boston's first basement finish in 60 years. Hobson was released from duty in 1994—while baseball was out on a work stoppage that would cancel the World Series. Amazingly, attendance at Fenway improved after the strike. That 1995 season brought an unexpected AL East title, and in 1996, attendance improved by another 150,000. In 1997, though they flirted with last place before rallying to finish third, the Sox still averaged more than 27,000 fans per game. As Dan Shaughnessy put it in 1997, "This is Boston. This is Fenway. Fans come for the ballpark and the baseball. Strikes, losing teams, and off-field transgressions don't matter much here." By the time the petulant Clemens stomped out of town, bound for Toronto at the end of the 1997 season, he had long since worn out his welcome with many. He was replaced in fans' hearts first by Mo Vaughn, then by Nomar Garciaparra, and finally by Pedro Martinez, who pitched the Sox to a playoff win over Cleveland. The 1990s and the millennium ended—fittingly, some would say—with an inspiring All-Star Game tribute to the Splendid Splinter, and a playoff defeat to the Yankees.
Following the 1989 shortfall, there was little reason to believe that Boston would be playing in October of 1990. But when Bill Buckner came back as a spring-training long shot, it seemed a cosmic sign of faith renewed. Buckner received a warm and redemptive Opening Day ovation from the forgiving, if not quite forgetting, Boston fans. Then he legged out an inside-the-park homer against the Angels on his first day in the lineup.
So it went for the Sox, who beat the Twins, 1-0, in a July game where they hit into two triple plays in five innings after conceding only two in the previous 25 years. Buoyed by magnificent pitching from Roger Clemens and Mike Boddicker, Boston built a lead of more than a half-dozen games by Labor Day. But after Clemens went down with a sore shoulder, the club dropped 10 of 12 and slipped behind the Blue Jays. The season came down to the final day at Fenway, with Boddicker on the mound; Boston needed a victory over the White Sox to avoid a playoff at Toronto, where Clemens already had been dispatched, just in case.
With the hosts ahead, 3-1, in the ninth and Chicago down to its last strike, champagne was at the ready. But Sammy Sosa singled and Red Sox closer Jeff Reardon plunked Scott Fletcher. Then up stepped future manager Ozzie Guillén to rip a line drive toward the right-field corner. Had it bounced past a diving Tom Brunansky, the game at least would have been tied. "No time to think about what I should do," said Brunansky, who raced toward the noman's-land by Pesky's Pole. "I just had to do it."
What resulted was one of the greatest catches in Fenway history, with Brunansky snaring the ball just before sliding into the wall. "Timmy, I've got the ball, I've got the ball," he shouted to Tim McClelland, the first-base umpire, one of the few people in the park who had a clear view. "I saw the play," McClelland said. "He never dropped the ball."
Thus did Boston claim the divisional title and earn a rematch with Oakland. "They called us misfits from the North Pole, castoffs," said Wade Boggs. "We were etched in stone for seventh place. But this team has got heart and desire I've never seen before. A heart as big as the Pru."
But up against A's aces Dave Stewart and Bob Welch, the Boston bats were as useless as toothpicks, managing only two runs in the first two games at Fenway. "A beautiful game turned into a horrible evening, didn't it?" mused manager Joe Morgan after the 9-1 opening loss, when Oakland bashed his bullpen for nine runs—seven in the ninth inning—after Clemens had blanked the visitors for the first six innings.
Wade Boggs rejoiced after the Red Sox clinched the 1990 East Division title with a victory over the Chicago White Sox.
The next evening was less horrid, but the 4-1 defeat sent the Sox to the Bay Area in a formidable pickle. "You don't have to tell 'em too much now," Morgan said. "They can read that easy enough." Another 4-1 stumble in Game 3 put Boston on the verge of an early winter and after Clemens was ejected in the second inning of Game 4 for yapping at plate umpire Terry Cooney, his teammates went down by a 3-1 count, swept again and bitterly criticizing their manager.
"Those things don't bother me," Morgan said. "They don't amount to a row of beans. I'll tell you this—if a guy is going to manage in Boston, that guy better have some thick skin on his body. We've won two out of three years, so we must be doing something right."
Boston would not play another playoff game for five seasons. The front office signed Clemens to a five-year contract worth more than $20 million. But before the 1991 season, there were multiple departures (Boddicker, Dwight Evans, Oil Can Boyd, Marty Barrett) and a few notable arrivals (Jack Clark, Danny Darwin, Matt Young). While Clark, acquired from the Padres, made a concussive entrance on Opening Day, clouting a grand slam in a 6-2 decision over Toronto, his new comrades were all but buried by midsummer, falling nine games behind after the Twins swept them by a 33-6 aggregate.
The Sox made a heroic late run, winning 12 of 14 in August and 17 of 21 in September, and found themselves only a half-game out of the lead on September 21 in the wake of hammering the Yankees, 12-1, at Fenway. "It's not over yet," cautioned Clark. "We're still in second place." That proved to be the summit for the Sox, who dropped 11 of their final 14 games to tumble out of contention. They came home to Yawkey Way for a somber farewell weekend. "Have a good autumn, a good winter and we'll have a better year next year," Morgan promised the fans before the Sunday finale against the Brewers.
Two days later, Morgan was gone, replaced by Butch Hobson, the Pawtucket skipper and former Sox third baseman the front office feared it would lose without a promotion. Morgan shrugged off his dismissal. "I think they just wanted a change, that's all," said the man who'd been the only manager other than Bill Carrigan to direct the Sox to the postseason twice in three years. "If they thought I wasn't doing the job, they had every right to fire me."
Hobson, who'd been in Winter Haven with the instructional team, was startled to learn he'd been upgraded. "I can't believe this is happening to me," he said following his Fenway introduction. "This is the greatest feeling in the world."
LIGHTS OUT
Red Sox outfielder Ellis Burks was standing in the batter's box, awaiting a 2-2 pitch from Chicago's Jack McDowell, when Fenway Park did its best imitation of Boston Garden.
The Garden was where Bruins fans had seen games interrupted by blackouts, fog, and an occasional stray rodent on the ice, and where Celtics fans had seen games suspended by condensation on the court and interrupted by a pigeon on the parquet. But no such incidents interrupted the national pastime at 4 Yawkey Way. At least, not until the blackout of May 13, 1991.
As the Red Sox were waging a futile battle to erase a 2-0 deficit with two outs in the third, a power outage plunged Fenway Park and its crowd of 31,032 spectators into darkness for 59 minutes.
When the park went dark at 8:45 p.m., it was as though someone had begun a New Year's Eve-style countdown. The blackout touched off a raucous ovation and created a rock concert atmosphere, complete with flickering lighters and a battery of flashbulbs.
The likely cause was a blown manhole cover that cut through a power line and knocked out electricity to several buildings in the area, including Fenway Park.
Radio station WRKO and cable TV channel NESN lost their broadcasts. Auxiliary power illuminated the seating areas, but the rest of the park remained shrouded in darkness.
Public address announcer Sherm Feller, in a partially lit press box, played to the fans as though he were doing a vaudeville routine in a dimly lit nightclub. Announcing to the crowd that the power outage was the result of a problem outside the park, Feller cracked, "Boston Edison's working on it right now. If they send us a bill, we'll pay it."
The Fenway power outage was not unprecedented, though the previous one had gone relatively unnoticed by fans. "It was a weekday day game back in April 1981," said Josh Spofford, Red Sox director of publicity. "We didn't have any power the whole game. It was back when we were in the press box that was closer to the field, and Sherm led the crowd in the anthem a cappella and did the lineups with a bullhorn."
Players and spectators killed time batting beach balls during the Fenway blackout of May 13, 1991.
SOX CARETAKER
When Jean Yawkey died in 1992, John Harrington was entrusted to run the Red Sox. Harrington said that his goal was quite simple. "I just want to honor Jean's generous legacy—and prove myself worthy of that trust."
Harrington grew up in Boston and graduated from Boston College. While working as an accounting professor at BC, he was hired by Joe Cronin, president of the American League, to be the league's controller in 1970. From there, he was hired by Red Sox owner Tom Yawkey to be the team's treasurer. He eventually returned to the Red Sox in the mid-1980s and became an important adviser to Mrs. Yawkey.
After Mrs. Yawkey died, most observers expected that minority owner Haywood Sullivan, a former Red Sox player and executive, would buy out the Yawkey trust's majority share of the team. Instead, Harrington, acting on behalf of the trust, bought out Sullivan's general partnership stake in late 1993.
Sullivan talked about the relationship between Harrington and Mrs. Yawkey. "As the years passed, he became a kind of surrogate son. Certainly she depended on him more and more . . . and when he talked, you knew he was speaking for Jean."
Harrington reveled in his role as Red Sox CEO, jesting that "every kid under 12 wants to be Nomar or Pedro, and every kid over 40 wants my job." He acted as the chief negotiator for MLB owners during the 1994-95 strike, and he helped guide baseball's divisional realignment to accommodate the wild-card playoff format in 1995. But in 2000—with the team in good financial and competitive shape, and state lawmakers apparently poised to approve funding for Harrington's new ballpark project—he decided to sell the team on behalf of the trust. It was time to say good-bye.
"It was the right time for the team and the trust, and I knew I had to sell," Harrington said.
But what had seemed a dream to the new manager, who brought back his old skipper Don Zimmer as third-base coach, turned into a summer nightmare in 1992 as the Sox went into a June swoon on the road, falling 9½ games out on the way to their first cellar finish since 1932, the year before Tom Yawkey bought the franchise. "This team isn't as good as people think," Morgan had declared on his way out the door.
With no .300 hitters and Clemens as their only dominant starter, the 1992 Red Sox lost 89 games—the most the franchise had dropped in a single season since 1966—and finished 23 games behind Toronto. It was a rude awakening for Hobson, who had never experienced a losing season as a player. "Maybe I was a little in awe of managing Wade Boggs or having Roger Clemens on my pitching staff," he said. "That hasn't sunk in with me." It was an unsettling year for the franchise, which was thrown into transition when Tom Yawkey's widow, Jean, died in February of complications from a stroke. John Harrington, Yawkey's longtime confidant, stayed on as president of JRY Corp., which owned the Sox.
After another underwhelming campaign in 1993, when the Sox finished 15 games out in fifth place, Harrington shook up both the clubhouse and the front office. Gone after 11 years and more than 2,000 hits was Boggs, who departed for the Bronx. So was Ellis Burks, who changed red stockings for white.
During the autumn, Harrington bought out Haywood Sullivan's interest in the franchise. He moved General Manager Lou Gorman to executive director for baseball operations and brought in Dan Duquette from the Expos as GM. The 35-year-old Duquette, a lifelong Boston fan from Dalton, Massachusetts who'd made the most of a shoestring budget in Montreal, arrived with a five-year mandate to make the club better by going both cheaper and younger, and to revamp a clubhouse that had 15 men who were 30 or older. "We're going to renew the roster with new life," Duquette vowed during spring training. By August, the Sox had suited up 46 players, half of them pitchers.
But the 1994 season was a lost cause by then. A 12-game home losing streak in June had mired Boston in third place, 10½ games behind. The breaking point came in a 10-4 home loss to New York during which Hobson had a meltdown and was ejected. He was later suspended five games for arguing with plate umpire Greg Kosc and bumping crew chief Larry Barnett after pitcher Sergio Valdez had been warned for throwing behind a Yankee batter. "There is a rage inside the guy [Hobson] that a lot of people don't know about," commented slugger Mo Vaughn. "If it goes, it goes. Behind this big job of being a Red Sox manager is a man."
On July 22, when his club returned to Boston 13 games behind after having been swept in Anaheim, Hobson passed out towels to his battered ballplayers. "I ain't throwing mine in," he told them. "Don't throw yours in."
Three weeks later, the towel was tossed in for baseball itself as the players went on strike for the first time since 1985. "See you at the Patriots games," Vaughn told sportswriters after the finale at Baltimore had been washed out. The 54-61 record represented Boston's third straight losing season, its worst stretch since 1966, and it marked the end for Hobson, who was summoned from his Alabama home to be dismissed in September.
"I believed in my heart that this day would never happen," said Hobson, who later resurfaced as a scout and minor-league manager in Sarasota. "I'm not going to burn any bridges. When new faces come in, they want to bring in new faces. I know that." The new face belonged to mustachioed Kevin Kennedy, who'd been Montreal's minor league field coordinator and bench coach during Duquette's time there and had just been fired by the Rangers. "This is the first and only place I wanted to be," said Kennedy, who'd called Duquette as soon as Texas let him go.
An unprecedented winter of discontent followed the first canceled World Series, with the labor dispute remaining unsettled until March of 1995. It was unclear how the players would be received by fans when they took the diamond on April 26 for Opening Day.
Knowing the importance of public relations, new Sox slugger Jose Canseco, who'd been acquired from the Rangers in December, was outside Fenway by 8 a.m., meeting and greeting ticket holders. "We can't forget what really counts," said his Sox teammate Mike Greenwell, who signed autographs for an hour after batting practice. "It's the fans."
Dan Duquette, a native of Dalton, Massachusetts, achieved his childhood dream of running the Red Sox in 1994. And though the Sox made back-to-back playoff appearances for the first time in more than 80 years on his watch, Duquette's eight seasons as general manager were ultimately more tumultuous than triumphant. Duquette was considered one of baseball's brightest young executives when then-CEO John Harrington hired him away from Montreal as the Sox GM. Though he showed some flashes of brilliance, he was fired in March 2002 when the team's new ownership group led by John Henry decided they wanted their own man. "I've never had a bad day at Fenway Park," Duquette said after his ouster.
A youngster had nowhere to go but down after making a lunge for a foul ball in a game at Fenway Park against the Chicago White Sox on April 29, 1995.
Mo Vaughn led cheers as he rode a Boston Police horse, and manager Kevin Kennedy got a ride from players Mike Macfarlane and Tim Wakefield, after the Red Sox clinched the AL East title on September 20, 1995.
And Sox fans seemed forgiving once their team had crushed Minnesota, 9-0.
The rebuilt and rededicated club took over first place on May 13 and stayed there for the rest of the season. Duquette, who'd reworked the roster (only three of the original 1994 starters were still in the lineup), kept tinkering, with 53 players (26 of them pitchers) suiting up by season's end.
Boston ran away with the AL East, winning the divisional title for the first time since 1990 and clinching at home on September 20 with a 3-2 victory over Milwaukee. For their triumphal procession around the premises, the players mounted police horses—even Vaughn, their resident Clydesdale. "Everyone got on the horse and so I had to get on the horse," Vaughn said, after John Harrington admitted that he was more worried about the horse than about his top slugger. "That's the way this team is."
The Sox were quickly unhorsed, however, in the playoffs by the Indians, who'd posted the league's best regular-season record with 100 victories. The 5-4 loss in the 13-inning opener at Jacobs Field was doubly hard to take since the Sox led, 2-0, and then 4-3, in the 11th—and since Tony Pena, their former catcher, clouted the winning homer off Zane Smith with two out. After Orel Hershiser blanked the Sox, 4-0, in Game 2, the season came down to what knuckle-baller Tim Wakefield, who'd won 16 games after being picked up from Pittsburgh, could do with his notoriously unpredictable pitch.
The visitors were so confident they would finish off the Sox that they checked out of their hotel before the game. They then tagged Wakefield for seven runs in six innings, completing the sweep with an 8-2 triumph while extending Boston's postseason losing streak to an unlucky 13. "Sometimes I wish I could throw a hundred miles an hour like Randy Johnson," said Wakefield. Still, there was no disgrace in losing to a Cleveland club that went on to play in the World Series for the first time since 1954. "There are no excuses to be made for this series," reasoned Kennedy. "We lost, they won, and we'll be back."
But it was the Yankees who were back in 1996, winning their first World Series since 1978. The Sox finished in the middle of the pack after pretty much dooming themselves by losing 19 of their first 25 games, and then falling 17 games behind in early July. Despite posting the best record in baseball over the final two months, Boston ended up in third place. Taking three of four from New York on the final weekend in Fenway provided a small bit of consolation. "It was emotional," said Kennedy after the Sox had won the finale by a 6-5 count, and then packed their bags for the winter. "Maybe it wasn't the intensity of winning a World Series game, but it was one we wanted."
TED AND JIMMY TOUCH BASE
He had been a symbol and a secret for a half-century, a New England icon frozen in time, representing all children with cancer. Then, in 1998, the Jimmy Fund logo came to life when Carl Einar Gustafson of New Sweden, Maine, revealed that he was the true "Jimmy." On July 9, 1999, in a thrill for both of them, he met Ted Williams, the Jimmy Fund's all-time biggest booster.
Anyone born in New England in the past 50 years knows of this charity dedicated to eradicating cancer in children. In 1948, Gustafson was chosen by Dr. Sidney Farber, the godfather of modern chemotherapy, to represent stricken kids everywhere on a national radio broadcast. They called him "Jimmy" to protect his privacy. The show was a hit, and the Jimmy Fund was launched.
As decades passed and treatments progressed, there was less curiosity about what happened to the original Jimmy. It made sense to assume that the child had died—as did almost all cancer patients of that era. A Maine man to the core, Gustafson never bothered to call attention to his role. "In my day, we were taught to keep things quiet," he explained. "There really was nothing to say about it. That was bragging."
In 1997, a year before the 50th anniversary of the fund's launch, Gustafson's sister sent a letter to Mike Andrews, the fund's executive director, explaining that her brother was alive and well. "We'd had a lot of leads like this before," said Andrews, who temporarily set aside the letter. "They were like Elvis sightings." An investigation based on hospital records and Gustafson's correspondence with Farber finally convinced Andrews that Gustafson was the real "Jimmy."
Williams finally met Gustafson when he visited Boston for the All-Star Weekend in 1999. They met at the Dana-Farber Cancer Institute and visited with dozens of patients—people who had a better chance of survival, in part, because of Williams's tireless efforts, in concert with an army of doctors, nurses, technicians, administrators, and fund-raisers.
"How are you, Jimmy baby?" Williams asked as Gustafson greeted him. "This is the biggest thrill of my trip, right here! Jeez, you look great! You're an inspiration to everybody!"
Gustafson appeared at numerous functions and recorded public service announcements for the Jimmy Fund. His story was featured in People magazine and Sports Illustrated. As for meeting Williams, he said, "I can't tell you how proud I feel to have met him."
Gustafson died in 2001 at age 65 after suffering a stroke.
"After 50 years, to find out he was alive was a miracle," said Andrews. "Then he turned out to be the most wonderful man. If we had tried to create a Jimmy, we couldn't have done better. And he was just thrilled to be part of it."
"Guys like George Will and Bob Costas come in and want to romanticize Fenway Park. But how many times have they had to sit in Section 1, 2, 3, 4, or 5? You sit in sections 6, 7, 8, or 9, you could get a crick in your neck from having to turn to the left all the time; you're looking straight at center field."
—Ted Sarandis, Boston radio talk-show host
On Opening Day in 1995, the woman at the center paid a scalper $50 for this obstructed-view seat behind home plate. Today, that would be called a steal.
The Fenway outfield, where the bullpens are located, has always offered a spectacular view of the park.
A day later, Kennedy was gone for good, dismissed by Duquette after taking the fall for his club's wayward start and its perceived lack of cohesiveness. "It hurts," said Kennedy, whose players were startled by his firing. "I love what I do and I poured everything I had into it. You can ask the players how much I cared and how much they wanted to do well for me." By the time successor Jimy Williams arrived, Clemens had filed for free agency and then was gone in December, off to Toronto for what was then the richest contract ($31 million over four years) in baseball history.
When Clemens returned as a Blue Jay on July 12, 1997, after having once vowed that he'd never appear at Fenway in another team's uniform, he received a decidedly mixed reaction as he strolled out to warm up. Then he blinded his former mates, striking out 16 of them in an overpowering performance that had his former supporters chanting "Roger, Ro-ger." "He came to make a point and he did," conceded Vaughn, who whiffed three times.
Clemens—who had been characterized by Duquette upon his departure from the Sox as being in the "twilight of his career"—was jubilant.
"It was a special day, a beautiful day," said the Rocket, who went on to win the Cy Young Award as the league's best pitcher. The Sox, who were 17 games out and buried in last place that day, ended up fourth despite having a quintet of .300 hitters, including rookie shortstop Nomar Garciaparra.
The club clearly needed a pitching infusion and Duquette made a trade with his old club for ace Pedro Martinez, who signed a new contract with Boston for six years and $75 million. Martinez won 19 games in his first season in Boston, but it was Vaughn's bat that set the tone for 1998 with a mighty grand slam that brought his team back from the dead after thousands of discouraged fans had left the premises on Opening Day. "It felt like the World Series," noted winning pitcher Rich Garcés after the Sox had come from five runs down in the ninth to win, 9-7.
Vaughn, who'd squabbled with management over his salary during the off-season, became a fan favorite with his game-ending shot off Paul Spoljaric. "Sign Mo Now!" fans chanted as he circled the bases. By early May, the club already had come from behind seven times to win in the final inning.
MO'S PUZZLING DEPARTURE
For six seasons, Maurice Samuel "Mo" Vaughn, a native of Norwalk, Connecticut, was the heart and soul of the Red Sox. He twice led the team into the postseason, won an MVP Award and made three All-Star teams. He hit 230 home runs and drove in 752 runs in his tenure with the Sox, and in 1998, the "Hit Dog"—as he came to be known—had a career-best .337 batting average.
But following that 1998 season, Mo was gone, signed away by the Anaheim Angels as the Red Sox made a late, low-ball counteroffer that they had to know would virtually guarantee his departure.
Vaughn's souring relationship with Sox management had caused it to look past his good work on and off the field.
Mo's supporters included the Globe's Bob Ryan, who wrote in July 1998 as the "Should they re-sign Mo?" debate was raging: "Here was a star player—a star player of color, on top of that—who was reflecting glory on the organization because he really was a man in, and of, the community. Central casting couldn't have shipped over a better player to the Red Sox. And now Mo is the bad guy? Don't believe it."
For a long time, Ryan noted, the Sox were eager to tout Vaughn's standing in Boston and his tireless work with youngsters.
But Vaughn also had gotten into a fight outside a Boston nightclub, and in 1998 he crashed his truck on the way home from a Rhode Island strip club and was arrested for failing field sobriety tests. (He was later tried and found not guilty of drunken driving.) These incidents, along with concerns about Vaughn's weight, created a growing chorus of Mo detractors.
As the 1998 season went on, his relationship with Sox management deteriorated, and his decision was made for him by the Angels when they made him baseball's highest-paid player at the time with an $80-million deal.
And though Mo appeared to be on the fast track to Cooperstown when he left the Sox, in the five succeeding seasons, he played just 466 games for the Angels and Mets before injuries forced him to retire.
Still, Vaughn later insisted he had no regrets. "It was just time to move on," he said. "If I had stayed in Boston, I would have retired as an angry player, and that's wrong, because I love this game."
While the Sox didn't come close to catching the Yankees (who won 114 games en route to another World Series ring), Boston did return to the playoffs for the first time in three years. "We made it, we made it," exulted Martinez, hugging Duquette and dousing Vaughn and Garciaparra with champagne after Boston had beaten Baltimore to clinch a wild-card berth.
Again, the opponent was Cleveland, and this time the Sox sensed that they were in for a reversal of fortune. Martinez stifled the Indians and Vaughn whacked two homers and knocked in seven runs in an 11-3 slamdown at Jacobs Field. "This is just the start," declared Dennis Eckersley after Boston had won its first postseason game since 1986. Everything was going the visitors' way when Indians Manager Mike Hargrove and starting pitcher Dwight Gooden were ejected and the Sox took a 2-0 lead in the first inning of Game 2.
But the Indians battered Tim Wakefield to salvage a split, and when they cranked three homers off Bret Saberhagen and another off Eckersley in the ninth inning of the first Fenway game, the Sox suddenly found themselves on the brink of elimination. "Please, Jimy, please pitch Pedro," Boston fans implored Sox Manager Jimy Williams as he walked toward the clubhouse.
Rather than use Martinez on three days' rest, Williams opted for a fresh Pete Schourek, who'd been scooped up from Houston in August. "The bottom line is we've got to win two games," said Saberhagen, "and Pedro can't win them both."
For seven innings of Game 4, Boston thought that it would even the series. Schourek was pitching a shutout and Garciaparra, who had a divisional series record 11 RBI, had put his team ahead with a leadoff homer in the fourth. When Tom "Flash" Gordon came in to wrap up things in the eighth, it seemed that the Sox would force a decisive fifth game with Martinez on the mound.
But Gordon, who'd strung together 43 saves in a row and hadn't blown one since April, conceded a two-run double to David Justice that gave the visitors a 2-1 decision. If Red Sox third-base coach Wendell Kim hadn't rashly waved home John Valentin for a damaging out in the sixth or if the autumnal wind hadn't reduced Vaughn's would-be tying homer to a Wall double in the eighth, Boston might have survived. "I'm pretty surprised it ended this way," Valentin remarked as the Sox cleaned out their lockers. "I anticipated that we'd go far." Instead the Indians earned a quixotic date with the Yankees. "There's nothing to watch," declared Garciaparra. "We won't be there."
THE LONG GOOD-BYE
In May 1999, the Red Sox were gearing up for their first hosting of baseball's All-Star Game in 38 years. As far as the team was concerned, that summer's All-Star festivities would also provide an opportunity to say a gracious good-bye to Fenway Park, then 87 years old.
"Fenway is a wonderful ballpark," said John Harrington, chief executive officer of the Red Sox at the time. "But the sad truth is it's economically and operationally obsolete. It just doesn't allow us to compete like teams with modern ballparks do."
What the Sox hoped to sell to fans, political leaders, and the public was a new Fenway Park built right next door to the old one, "with the intimate scale and feel of the old" but up to 30 percent larger with modern conveniences and revenue streams.
As head of the JRY Trust that controlled the team, Harrington wanted to replace the existing 33,871-seat park with a new facility for 45,000 fans. Plans called for it to be built on roughly 14 acres bordered by Boylston Street, Brookline Avenue, and Yawkey Way.
Residents of the Fenway neighborhood and ballpark preservationists demanded that the park not be razed, but renovated. Harrington said he had studied several renovation options and rejected all of them.
Said Dan Wilson, a leading member of a group called "Save Fenway Park" and a Boston lawyer: "The Red Sox will be throwing out their most important asset if they build a new ballpark."
The Sox attempted to assuage that argument by offering plans to forever preserve parts of Fenway—a portion of the left-field wall, the 1912 brick entrance, and the entire infield—as a tourist attraction and open space.
After years of wrangling with city planners and park preservationists, the JRY Trust abandoned its new ballpark bid and put the team up for sale. Among the final five bidding groups in 2002, only the New England Sports Ventures group led by John Henry was committed to retaining the original ballpark. Interestingly, several of the options proposed by the Save Fenway Park contingent were ultimately implemented by Henry and his winning group, including the expansion of walkways and concession space inside the park, and the closing off of Yawkey Way to provide a game-day fan concourse.
Red Sox CEO John Harrington pointed out details of the proposed new Fenway Park to MLB commissioner Bud Selig (center) and Red Sox General Manager Dan Duquette (right) before the 1999 All-Star Game.
Nomar Garciaparra walked down the runway toward the Red Sox dugout with Red Sox Director of Communications Kevin Shea in 1998.
Boston was there in 1999, even after Vaughn decamped for Anaheim for $80 million over six years. Despite spending time on the disabled list with an uncooperative shoulder and squabbling with Duquette, Martinez won 23 games, including a one-hitter at New York in September. Though the Yankees, as usual, proved uncatchable, Boston did snare another wild-card spot for the playoffs and another meeting with the Indians.
After the Sox dropped the first two games of the divisional series in Cleveland—the second by an ugly 11-1 margin—they returned to Fenway facing extinction yet again. But they managed to stay alive with a pyrotechnic display in the seventh inning of Game 3, scoring six runs with two out on Valentin's two-run, bases-loaded ground-rule double, rookie Brian Daubach's three-run homer, and Lou Merloni's two-on single, to secure a 9-3 triumph.
Few would have predicted the next day's result, when the Sox literally put up telephone numbers in a 23-7 T-ball contest that drew them even. "Everything we threw up, they hit and where it came down, we weren't standing," observed Hargrove. No Boston team ever had punished as much horsehide in one October day. Three homers, two of them by Valentin, who knocked in seven runs. Twenty-four hits, including a dozen for extra bases. A 10-2 lead after three innings, 15-2 after four, 18-6 after five. "It's a one-gamer now," concluded Williams.
So the Sox headed back to The Jake for the finale with a most appropriate, yet least likely figure providing their deliverance. Martinez had lasted only four innings in the opener before straining his back trying to blow a fastball past strongman Jim Thome. "I didn't know when Pedro could pitch again," admitted Joe Kerrigan, the Sox pitching coach. But after the Indians had pummeled Saberhagen and Derek Lowe for eight runs in three innings, Martinez came out of the bullpen to hurl six scoreless innings and stake his compañeros to a 12-8 triumph that put them into the league championship series for the first time since 1990.
The club's first postseason series victory since 1986 earned the Sox their first October date with the Yankees since the one-game 1978 playoff. "They better sweep us," warned Valentin. "They better sweep us, baby." By the time the Sox returned to Fenway, they were down two games after being edged, 4-3 and 3-2, in the Bronx, despite leading each night. They still had one defiant and glorious game left in them, though, and their 13-1 destruction of New York on Saturday afternoon delighted the full house, especially since it came at the expense of Clemens, who'd exchanged plumage for pinstripes in February.
A vendor offered soda (real New Englanders call it tonic) in July 1995.
Programs were hawked on Yawkey Way before the 1997 home opener.
The view from atop the Green Monster at sunset, with both the ballpark and Lansdowne Street packed.
A STAR AMONG STARS
Baseball's All-Star Game frequently fails to live up to its hype. When Fenway Park hosted its third All-Star Game in 1999, the result was a 4-1 American League victory that featured zero home runs. But there will never be another sight like that of Ted Williams, who threw out the ceremonial first pitch, being engulfed by other baseball greats, including Stan Musial, Willie Mays, Bob Feller, Hank Aaron, Bob Gibson, and Cal Ripken Jr., just to name a few.
The ovation was loud and long. "I thought the stadium was going down," said Pedro Martinez afterward.
The moment made a lasting impression on Mark McGwire of the St. Louis Cardinals. "When you have a chance to meet one of the best hitters in the game," said McGwire, "and you see tears running down his eyes because of the appreciation that the fans and all of us gave him, it becomes a very special time."
Introduced as "the greatest hitter who ever lived," Teddy Ballgame, 80, rode into Fenway on a golf cart. After a lap around the field, Williams was brought to the mound, where he was surrounded by both All-Star squads and 31 of the top 100 ballplayers in baseball history. It was without question the greatest assemblage of hardball talent ever gathered on any diamond. With a giant No. 9 stenciled into the outfield grass, and the ancient theater shaking on its foundation, Williams stood in front of the mound and delivered a strike to Carlton Fisk, the longtime Red Sox catcher and his soon-to-be fellow Hall of Famer.
The hero of the 1999 AL victory was Red Sox ace Pedro Martinez, who struck out five of the six batters to face him—Barry Larkin, Larry Walker, Sammy Sosa, McGwire, and Jeff Bagwell—and was voted the game's MVP. But even Pedro took a backseat to Williams.
"I don't think that there will be any other man that's going to replace that one," said Martinez.
"It was like something out of Field of Dreams," said Cleveland's Jim Thome.
Said a young woman in the stands, "Oh, my God. Ted Williams threw a pitch to Carlton Fisk. I'm going home happy."
Valentin's two-run homer in the first was the opening shot in a barrage that put Boston up, 6-0, after three innings and put New York starter Clemens out of the game. "Where is Roger?" the fans chanted gleefully in the seventh inning, when the lead had soared to 13-0 and Martinez still was throttling the visitors. "In the shower." It was the worst beating that the Yankees ever had taken in October and it made owner George Steinbrenner dyspeptic. "This can happen once," he told his players in the clubhouse, "but it can't happen again."
New York turned the tables emphatically on Sunday night with a 9-2 victory that put Boston on the brink. Everything turned on a bad call by umpire Tim Tschida, who allowed an inning-ending double play in the eighth with Boston trailing, even though Yankees second baseman Chuck Knoblauch missed tagging José Offerman after fielding Valentin's weak grounder. "No, I didn't make the right call," admitted Tschida. The furious crowd had agreed, littering the diamond with bottles in the ninth after Williams was ejected from the game for tossing his cap in the air.
As both teams were sent to their dugouts during an eight-minute delay, public address announcer Ed Brickley informed the spectators that the game would be forfeited unless order was restored. When play resumed, the Yankees ended things emphatically as pinch hitter Ricky Ledee hit a grand slam off Rod Beck. "It's the first one to win four and it's not over yet," Williams declared in a statement while remaining secluded in his office.
But New York made sure of it a day later, clinching the series in five games with a 6-1 victory that produced its 36th pennant and set the stage for another Series ring. "We wanted to finish it here," said Derek Jeter, who hit a two-run homer off Kent Mercker in the first inning. "We didn't want to give them any life or confidence."
While the Yankees sprayed each other with nonalcoholic champagne, the Sox, who'd had their best season in 13 years, were philosophical in defeat. "There's nothing for us to hang our heads about," Garciaparra proclaimed. "Disappointed? Of course. Any year we don't win the World Series, I'm disappointed. But I'm not going to hang my head."
Pedro Martinez argued his case to manager Jimy Williams in the Red Sox dugout on August 14, 1999. Williams had refused to let the late-arriving Martinez start the game; instead, he went with Bryce Florie. Martinez pitched the last four innings and Boston came away with a 13-2 victory over Seattle.
## 2000s
A lot more changed around Lansdowne Street and Yawkey Way in the 2000s than the millennium, as the Red Sox reached heights of success that had been seen only in the ballpark's infancy, but not before yet another wrenching October setback. The Yawkey Trust put the team up for sale in 2001, and the new ownership group led by John Henry immediately promised to keep the Red Sox in Fenway Park for the foreseeable future. With that commitment (and the $700 million price tag) quickly came imaginative expansion of the seating that retained the charm and historic character of the park. In addition, after decades of relative quiet on all but 81 days a year, the doors were thrown open, and over the course of the emerging 2000s, Fenway became the scene of concerts, family and charity events, soccer games, citizenship ceremonies—even ice hockey games. Fenway retained its starring role as the Red Sox began a record sellout streak for major professional sports, but soon it was forced to share the spotlight when a bunch of scruffy underdogs took New England and Red Sox Nation on the wildest ride in postseason sports history. The self-proclaimed Idiots of 2004 won the final eight games of the postseason, many in heart-stopping fashion, to end 86 years of often excruciating frustration. The Sox went on to capture postseason berths in an unprecedented six out of seven seasons through the decade, while adding a second World Series sweep. In the process, they emphatically abandoned the label of front-runners who ultimately lost and took up the mantle of masters of come-from-behind victory. To wit, over three American League Championship Series, the Red Sox won nine consecutive games when facing elimination, going on to win two of the series and losing the third in Game 7. Curse foiled, again and again.
With the new millennium and the club's 100th season at hand, the Red Sox made two blockbuster announcements in 2000: there would be both a new ballpark and new ownership. Despite Fenway's quirky charm and rich history, it was the oldest and smallest facility in the major leagues. "If we do nothing," chief executive John Harrington declared in a _Globe_ op-ed column on May 25, "we will be left behind."
By the end of July, the front office, state, and city had agreed on a $665 million project, with nearly half of it to be publicly funded. The site, though, was undetermined, with management preferring the South Boston waterfront and Mayor Thomas Menino preferring the Fenway neighborhood. Meanwhile, the team was engaged in its annual battle—trying not to be left behind by the Yankees.
With Pedro Martinez and Nomar Garciaparra on track to retain their Cy Young and batting crowns, the Sox were in first place as late as June 22. But they faded during the summer and essentially were finished off on September 10 after the Yankees swept them at Fenway for the first time since 1991.
So the Sox finished second in the division for the third straight time and the Yankees went on to win their third consecutive World Series. And five days after the season ended, the For Sale sign went up on a franchise that had borne the Yawkey name since the Depression.
While Harrington wanted the next owner to be "a diehard Red Sox fan from New England," management estimated that it would take at least a year to have a buyer step forward and be approved.
In the interim, the Sox were under pressure to close the gap between them and their pinstriped overlords in 2001. So the front office lured away Manny Ramirez from the Indians with an eight-year deal worth $160 million, the fattest contract in franchise annals and second only in all Major League Baseball to the $252 million deal that Alex Rodriguez signed with the Rangers that same day.
"I'm just tired of seeing New York always win," said Ramirez, who grew up in Washington Heights, not far from Yankee Stadium, but who enthusiastically chugged a symbolic cup of chowder during his Fenway introduction. Ramirez, acquired for his prodigious ability to drive in runs, knocked in three against Tampa Bay in his new team's home opener. That came just two days after right-hander Hideo Nomo had celebrated his Sox debut in Baltimore with a no-hitter, the first by a Boston hurler since 1965. So the fans began dreaming about October in April.
But as Martinez, Garciaparra, and Jason Varitek all struggled with injuries, things went sour in midsummer. Jimy Williams was replaced by Joe Kerrigan, the pitching coach, and the season came apart in late August with an 18-inning loss at Texas. The club lost 12 of its next 13 games, the death knell coming at Fenway with a weekend sweep by the Yankees. "We don't have a monkey on our back," Red Sox outfielder Trot Nixon muttered. "We've got a goddamned gorilla on our back."
A "Save Fenway Park" mural outside the State House in Boston drew signatures from fans of all ages.
A Red Sox groundskeeper scaled the ladder to the top of the Green Monster to retrieve baseballs hit into the netting during batting practice.
At season's end, Boston was 13½ games behind New York and the collapse set the stage for a massive transformation before the 2002 season that featured new ownership, a new GM, and a new manager. The biggest change came in December when a group headed by Marlins owner John Henry bought the franchise from the Yawkey Trust for $700 million, more than twice the previous record paid for a Major League Baseball team.
It was the first time since 1933 that the club had been sold.
Henry, who gave up his Florida Marlins stake, took over as the club's principal owner, while former San Diego owner Tom Werner became chairman, and former San Diego and Baltimore chief executive Larry Lucchino assumed the role of president.
When the new group took over in March, they immediately made changes, replacing General Manager Dan Duquette with assistant Mike Port on an interim basis and hiring Grady Little, a former bench coach, to succeed Kerrigan, whose tenure lasted only 43 games.
There were changes to Fenway, too, as the owners endeavored to show the faithful that they were committed to restoration instead of relocation. "When I think of Paris I think of the Eiffel Tower," Henry mused. "When I think of Boston, I think of Fenway."
Before Opening Day, the ballpark was painted, the clubhouses freshened, and 10 new concession stands and 400 new seats were added. "Fans want to be at the game," said Henry, who ensconced himself in a front-row box for the unveiling. "So we'll put them on the field, we'll put them on the rooftops, we'll even put them on the bench."
The fans soon got a close-up of the first Fenway no-hitter since Dave Morehead's in 1965 and it came from an unlikely source—Derek Lowe, who'd been booed mercilessly during his 5-10 campaign the previous year. "It's surreal," marveled Lowe, whose teammates presented him the ball on a silver platter after he'd allowed Tampa Bay only one base runner during his 10-0 masterpiece on April 27.
While Lowe went on to start the All-Star Game and have a career year (21-8), his teammates faded after a 40-17 start that helped them stay in first place until late June; they finished more than 10 games astern.
Sport's most storied rivalry heated up during the off-season when Lucchino referred to the Yankees as the "Evil Empire" and New York owner George Steinbrenner called the Sox president the game's "foremost chameleon of all time." The clubs fought each other to a near-standstill in 2003, with New York winning the season series, 10-9. "They know we are here," Little proclaimed in late July. "And they know that we are not going away."
SWEET SEATS
They have been voted the best seats in baseball by ESPN SportsTravel and _USA Today_. And even Fenway purists would have to admit that when the Red Sox added seats atop the left-field wall, they did it right.
When his ownership group took over the team in 2002, John Henry broached the idea of putting seats atop the 37-foot-tall Green Monster, which has been called the most famous wall this side of China. It wasn't the first renovation project Henry & Co. undertook, but, initially, it was the most controversial.
"Certainly there were raised eyebrows," recalled Charles Steinberg, former Red Sox executive vice president for public affairs. But by the time the project was finished, it became the most popular of the many ballpark upgrades. Henry credits Janet Marie Smith, the Red Sox senior vice president of planning and development, with the design of 269 stools with bar rails. The section has three rows, plus a fourth of standing-room-only and concession stands. The Red Sox resisted the urge to cram in as many seats as possible.
The challenge, said Smith, was "how to put seats up there without overpowering the Green Monster. We wanted to make the seats special even after the novelty wore off."
"They should've done it years ago," Marty Feeney, of Quincy, Massachusetts, told the Denver Post when he attended the 2007 World Series. "Now you get the real feel of a home-run shot. The camera's on you. You get your one minute of fame."
During a game in 2008, one fan atop the Monster was swamped with text messages from friends within seconds after he narrowly missed catching a home-run ball.
Smith was with the Red Sox for nearly eight years (she left in 2009 to help renovate the Rose Bowl), during which time the team spent roughly $150 million on improvements to the park, increasing Fenway's capacity by 5,000, while waterproofing its leaky stands and reinforcing its foundations to last another 40 years.
Another of the most popular changes is just outside the walls of the park: the Yawkey Way concourse, which debuted in 2002. The street named for longtime owner Tom Yawkey is closed to traffic and non-ticket holders on game days, creating a sort of street carnival with entertainment, food carts, vendors, and often a chance to visit with and get an autograph from a former Sox player.
**RECENT CHANGES MADE TO FENWAY:**
**2002:** | Dugout seats and Yawkey Way concourse.
---|---
**2003:** | Green Monster seats and right-field concourse.
**2004:** | Right-field roof seats.
**2005:** | Third-base concourse and Game On! Sports Bar.
**2006:** | EMC Club and State Street Pavilion.
**2007:** | Jordan's Third-Base Deck.
**2008:** | State Street Pavilion expansion, Coca-Cola Corner, Bleacher Bar under center-field bleachers.
**2009:** | Right-field roof renovations, repair of original 1912 seating bowl.
"When I was there, I always realized there was something bigger than us as players: these people that had bled, cried tears, and cheered over the years. Winning a World Series in Boston is more than an individual player winning a World Series—it was winning a World Series for these people."
—Nomar Garciaparra, March 2010
The Sox had become relentless buckaroos who rode into the late summer with a sense of urgency, sporting red T-shirts that declared "THE TIME IS NOW . . . SO COWBOY UP." Yet Boston needed a near-miracle to earn a postseason date with the Yankees after losing the first two games of their best-of-five divisional series at Oakland. "I know it can be done," said Varitek, who had played on the 1999 club that had come back to beat the Indians after spotting them a 2-0 advantage in the playoffs. "Just let us get home and see what happens."
The odds, though, were daunting, especially given that the Sox had lost 10 straight playoff games to the Athletics. "[Our fans] may be jumping off bridges," conceded Garciaparra, "but I guarantee they'll get out of the water and they'll be out there supporting us on Saturday."
It was almost Sunday morning by the time the club began its great escape with Trot Nixon's long, looping pinch-hit homer into the center-field seats for a walk-off victory in the 11th inning. "There was a little gust of wind from the good Lord," said Nixon, after his homer sealed the 3-1 triumph, "and it ended up going out of the ballpark."
There was an earthly intervention on Sunday afternoon from David Ortiz, who cranked the winning double off the Oakland bullpen in the eighth. "I've never cried at a baseball game before but I couldn't help it," said Henry after Ortiz, who'd been 0 for 16 in the series, knocked in Garciaparra and Ramirez for a most unlikely 5-4 decision. "It was an unforgettable moment."
A street-fair atmosphere took root on Yawkey Way outside Fenway Park beginning in 2002, but it's for ticket-holders only. No ticket? The Cask 'n Flagon still accepts all comers.
SO LONG, TEDDY BALLGAME
When Ted Williams died at 83 on July 5, 2002, there was no wake and no funeral. Only the makings of a circus, with his three children battling over what exactly his last intentions were, and son John Henry Williams insisting that his father had signed an agreement to be cryonically frozen in Arizona.
Though the legal battle over Williams's remains would play out for months, the Red Sox held a ceremony in his honor on July 22 at Fenway Park, where lifelong friends and former teammates found kinship and a measure of closure for the passing of "Teddy Ballgame."
"The tribute was to him and his life, what he did on and off the field," said former Sox shortstop Rico Petrocelli. "And that's the way it should have been. I think we needed this, we being Boston and the former players, needed this closure."
Jerry Coleman, a former Yankee rival and fellow fighter pilot, met Williams at the 1950 All-Star Game. He said he immediately admired Williams. "He went to the wall to make a catch and crashed into it. He broke his arm," said Coleman. A few innings later, Williams came to the plate and singled. "I was thinking, 'Geez, this guy hits better with a broken arm than most guys do with two arms.'"
Said Dan Shaughnessy: "Teddy Ballgame was our own Babe Ruth, an oversized figure who forged his way into every New England household. . . . Forget cryonics. The Kid stays alive through folklore, the telling of tall tales. He's a baseball Bunyan."
In a sad footnote to the Williams ceremony, long-time Sox announcer Ned Martin died of an apparent heart attack at the Raleigh airport after participating in the tribute at Fenway. Martin, who delighted New England with his erudition and gentle wit, described the Sox action on radio and television between 1961 and 1992.
Martin served in the Marines in World War II and saw action in Iwo Jima. He worked with Ken Coleman on Sox TV broadcasts from 1966-72. He famously cried, "There's pandemonium on the field" when Petrocelli caught the pop-up to end the final game of the 1967 regular season, as the Sox won their first pennant in 21 years.
As a closer, Lowe held off the A's to save Martinez's win in the finale on the Coast, and then he and his mates headed for the Bronx, and grabbed the ALCS opener from the Yankees with three homers and Wakefield's devilish knuckler. That was the beginning of what would be the most spirited, memorable and, ultimately, painful October meeting between the two rivals.
The first game in the Fens produced a bench-clearing brawl that included Don Zimmer, the former Sox skipper turned pinstriped Buddha. Zimmer charged Martinez, who'd thrown at Karim Garcia's head, and the 72-year-old was tossed to the ground in the scuffle. "When this series began everyone knew it was going to be quite a battle, very emotional, with a lot of intensity," said Little after the visitors had prevailed, 4-3. "But I think we've upgraded it from a battle to a war."
Though Wakefield evened things in Game 4, the Yankees countered with a 4-2 victory that sent them home with two chances to win the pennant. "The clock is ticking on us right now," acknowledged Little. "This isn't something we've never been through before. We were through this about a week ago."
Red Sox pitcher Pedro Martinez threw Yankee bench coach Don Zimmer to the ground after Zimmer accosted him during a brawl in the fourth inning of Game 3 of the AL Championship Series on October 11, 2003 in Fenway Park.
Despite his controversial antics, Manny Ramirez never lacked a following in Boston, on or off the field.
Jason Varitek grappled with the Yankees' Alex Rodriguez after Rodriguez was hit by a pitch from Bronson Arroyo on July 24, 2004 at Fenway Park. They were among four players ejected after a bench-clearing brawl. The Red Sox rallied from a 9-4 deficit to win the game, 11-10.
ROCKIN' THE PARK
This time, when Fenway Park echoed with the chords of "Take Me Out to the Ball Game," it was different.
The organist was Danny Federici of the E Street Band, and Bruce Springsteen was taking the field for the first rock concert in the park's 91 years.
It was September 6, 2003, and it had been 30 years since Stevie Wonder, War, and Ray Charles had played at Fenway as part of the Newport Jazz Festival. But this was different.
"What this park needs is a rock 'n' roll baptism, a rock 'n' roll bar mitzvah . . . a rock 'n' roll exorcism," Springsteen told the capacity crowd of more than 35,000 as he and his bandmates played a typical "Boss" show that encompassed 28 songs and three hours, ending appropriately with a cover of the ubiquitous Beantown anthem, "Dirty Water," helped along by Peter Wolf, former lead singer of the Boston-based J. Geils Band.
Bruce was starting the final month of a 14-month world tour in support of his The Rising album, a paean to America in the aftermath of the 9/11 attacks. His Boston audience was receptive to all of it, except his teasing about the Yankees rivalry. When he brought up the "evil citizens" to the South, the crowd booed lustily.
Still, Springsteen obviously got it. Toward the end of the show, he said, "There's not many places where you can walk into an empty place and feel the soul of the city, but this is one."
Since Springsteen headlined Fenway's coming-out party as a rock music venue, it has hosted at least one major rock or pop act per year, and it often plays host to the Dropkick Murphys, the backbeat of Red Sox Nation.
**21 ST CENTURY PLAYERS**
**2003:** | Bruce Springsteen and the E Street Band
---|---
**2004:** | Jimmy Buffett and the Coral Reefer Band
**2005:** | The Rolling Stones
**2006:** | Dave Matthews Band with Sheryl Crow
**2007:** | The Police
**2008:** | Neil Diamond
**2009:** | Dave Matthews Band with Willie Nelson
**2009:** | Phish
**2009:** | Paul McCartney
**2010:** | Aerosmith and the J. Geils Band
**2011:** | New Kids on the Block and Backstreet Boys
Standing atop the Red Sox dugout, the Dropkick Murphys played "Dirty Water" in October 2004.
"Other places have spectators;
Fenway has 35,000 participants."
—Bill Veeck, longtime owner and baseball executive
Teammates swarmed David Ortiz after his 10th-inning home run clinched a three-game sweep of the AL Division Series against the Anaheim Angels on October 8, 2004.
A fan was rewarded for keeping the faith in the 2004 AL Championship Series against the Yankees: David Ortiz rounded third after hitting a home run in the eighth inning of Game 5, which Boston went on to win, 5-4, in 14 innings.
Resurrections had become routine. So when Boston came from two runs down to force a seventh game with a 9-6 victory, anything seemed possible. Before the Sox took the field, the Fenway grounds crew painted a 2003 World Series logo on the grass behind home plate. "They're crazy!" remarked Yankee closer Mariano Rivera. "It's silly. . . . Maybe they want to believe they won."
The finale, with Martinez and Clemens dueling again, was more painful than the 1986 World Series nightmare in Queens. With his club ahead, 5-2, in the bottom of the eighth and a fresh bullpen in reserve, Little left Martinez on the mound as the Yankees rallied and erased the Sox lead. The season slipped away in the 11th inning when pinch hitter Aaron "Boone-bino" Boone, a light-hitting infielder, lofted a Wakefield knuckler into the left-field seats. "Go back to Boston, boys, good-bye!" New York owner George Steinbrenner crowed as the buses pulled out of the stadium lot.
That was the end for Little, who was let go before the end of the month. "Yes, we came up short of our goal," Little acknowledged in a statement, "and to the Red Sox Nation I say: I hurt with each of you. It was painful for all of us."
For his successor, the front office selected Terry Francona, who knew about working in a demanding town after managing for four years in Philadelphia. "Think about it for a second," said Francona, who'd been a big-league player like his father, Tito. "I've been released from six teams. I've been fired as a manager. I've got no hair. I've got a nose that's three sizes too big for my face and I grew up in a major-league clubhouse. My skin's pretty thick. I'll be okay."
Prospects for an autumn rematch seemed remote in 2004 after the Sox had fallen well behind the Yankees by late July. But one startling turnabout at Fenway foreshadowed what ranks as the greatest resurrection in baseball history. Aroused by a brawl touched off by Varitek shoving his mitt in Alex Rodriguez's face after he'd been plunked by Bronson Arroyo, the Sox rallied from two runs down in the ninth and won on Bill Mueller's two-run homer off Rivera.
"I hope we look back a while from now and we're saying that this brought us together," Francona said. "I hope a long time from now we look back and say this did it."
That set the stage for another October showdown with New York. But Boston first had to finish off the Angels, who arrived at Fenway on the verge of extinction in their divisional series after being hammered twice at home. What the Sox needed, on the heels of setting a dubious record by blowing a 6-1 lead in the seventh, was a coup degrace. Ortiz provided it with a two-run shot over the Green Monster off Jarrod Washburn with two outs in the 10th for an 8-6 triumph and the second Sox sweep of a postseason series since 1903.
Fenway was more than ready for World Series action as Game 1 got underway in 2004. A historic sweep was ahead.
"Now we have two more celebrations to go," observed Theo Epstein, who had become the youngest general manager in MLB history two years earlier when the Red Sox named him their GM at the age of 28.
That seemed impossible when Boston dropped the first two games of the championship series at New York, and then absorbed an ugly 19-8 smackdown at home that was the worst postseason loss in franchise history. No major-league ball club in history had won a seven-game series after losing the first three matchups. The only way to do it, Francona pointed out, was to win one each day.
His club came within one inning of a devastating sweep, trailing by a score of 4-3 in Game 4 with Rivera on the mound in the ninth. Then Kevin Millar drew a leadoff walk and the greatest comeback in MLB history was underway.
Dave Roberts, who'd been picked up from the Dodgers at the trading deadline as a speedy spare part, was sent in as a pinch runner and stole second on the next pitch. Then Mueller knocked him in with a single to center to tie the game and send it into extra innings. David Ortiz won it in the 12th with a two-run homer into the visiting bullpen off Paul Quantrill at 1:22 a.m. "This is a team that never gives up," Ortiz declared after Boston had won the five-hour duel by a 6-4 count. "Great heart."
The Sox again defied probability later that same day with another far-fetched escape. This time, it was a 5-4 triumph that required 14 innings and five hours and 49 minutes to complete, and ended with another killing blow from Ortiz, whose two-out single to center scored Johnny Damon. "To continually do it night in and night out, it's ridiculous," marveled Red Sox first baseman Doug Mientkiewicz. "It's a freak of nature."
Had Ortiz not clouted a homer in the eighth to help his teammates climb out of a two-run hole, the season likely would have ended an inning later. "Being down 3-0 and being down the last two nights shows the depth, the character, the heart, the guts of our ball club," proclaimed Wakefield, who collected the victory as Boston's seventh pitcher. "And it took every ounce of whatever we had left to win tonight's game and to win last night's game."
Still, the Sox trailed, 3-2, in the series going back to the Bronx, where the previous season had ended painfully. "We're in the same position as last year and we came awfully close," Henry observed. "But the odds are still against us."
Yet the club continued its historic resurrection in Babe Ruth's old playpen as Curt Schilling, pitching on a sutured right ankle that bled through his sock, dazzled the Yankees, 4-2, in Game 6. Then Derek Lowe starved the hosts in the finale and Damon crushed them with a second-inning grand slam. "How many times can you honestly say you have a chance to shock the world?" crowed Millar after the 10-3 knockout that put Boston into the World Series for the first time since 1986.
STANDELLS GET A SECOND ACT
BY BRIAN MACQUARRIE
Any time the Red Sox prevail at Fenway Park, you hear it: the slinky raunch of the guitar, the snide snarl of the vocals, the backhanded celebration of Boston in the lyrics. Hundreds of thousands of Sox fans recognize the Standells' "Dirty Water" as the unlikely anthem for their beloved team and city.
But for one Red Sox fan in California, "Dirty Water" means much more. For Dick Dodd of Buena Park, the song's surprising staying power has provided a link to his rock 'n' roll past, which included opening for the Rolling Stones, and a connection to a place he had never seen before "Dirty Water" came out.
Dodd sang the 1960s proto punk ode to a grungy Boston. Dodd was the drummer for the Los Angeles-based Standells when "Dirty Water" hit the airwaves, and he is the one who added the raspy opening that set the mood for the attitude-laced song.
"I'm gonna tell you a story," Dodd said one day in 2005, repeating his lyrics as he looked upward into a sunny California sky. "I'm gonna tell you about my town. I'm gonna tell you a big bad story, baby . . ."
After the Standells broke up in the early 1970s, Dodd bounced around as a restaurant manager, an office employee for a construction-equipment company, and a chauffeur. Over the years, his daydreams often drifted 3,000 miles to Boston. And now, the Standells, who reunited after all those years and perform from time to time, have played twice for Fenway fans.
Dodd was stunned by the odds-defying popularity in Boston of a song that peaked at No. 11 on the pop charts back in 1966. "When you get to be my age, you get a little choked up by this," Dodd said.
Dodd said he began following the Red Sox in the mid-1980s. But he was unaware the team had adopted "Dirty Water" until the day he heard the distinctive chords pulsating around Fenway Park at the end of an ESPN telecast of a Red Sox game. They were playing his song. "The crowd was singing every word," Dodd said, shaking his head in amazement.
Some four decades after the song's release, Dodd was still surprised that "Dirty Water" became a hit in the first place. None of the four Standells had been to Boston before creating the song. The band recorded "Dirty Water" only at the prodding of their producer, the late Ed Cobb, who wrote the hit after a visit to Boston, during which he was mugged on the Massachusetts Avenue Bridge over the Charles River, Dodd said.
Dodd and the Standells were invited by the Red Sox to perform "Dirty Water" as a Fenway Park surprise before Game 2 of the 2004 World Series. It was a dream come true for Dodd.
"Nobody knew we were going to be there, number one. And I don't care who you are, you're going to get nervous with Fenway Park sold out," Dodd said. "Then everyone went freaking crazy, and right at that moment, when I knew I wasn't singing it alone, it was just unbelievable. God, I just wanted to hug everybody."
When the Red Sox searched in 1997 for a theme to celebrate each home victory, General Manager Dan Duquette and Manager Jimy Williams chose the down-and-dirty sound of the Standells. And even though the team ownership changed, the Red Sox are committed to the song as their victory music.
Red Sox co-owner Tom Werner strolled with the ALCS trophy outside Fenway Park before the 2004 World Series.
Sox co-owner John Henry (center) played guitar with members of the Standells at a playoff rally in October 2007.
This time the opponents were the Cardinals, who'd beaten the Red Sox in the 1946 and 1967 World Series and who boasted the season's best record. Once again the Sox, who became the first team in 56 years to start a pure knuckleballer in a Series game, won in a most unorthodox fashion, surviving a blown 7-2 lead and four errors to win, 11-9, on Mark Bellhorn's two-run homer off Pesky's Pole in the eighth.
Despite making another four errors the next night, the Sox won again, 6-2, behind the sore-ankled but irrepressible Schilling. Fate seemed to be turning in their favor after decades of disappointment. And when Martinez stifled St. Louis, 4-1, at Busch Stadium to win Game 3, the Sox fans who'd made the trip to St. Louis sensed the end of an 86-year drought. "One more game," they chanted behind the visiting dugout. "One more game."
Boston won it easily, with Lowe shutting out the Cardinals, 3-0, and closer Keith Foulke flipping the ball to Mientkiewicz for the final out that sent thousands into the streets around Fenway to celebrate. "This is like an alternate reality," said Henry, as the players—now an infamous band of self-proclaimed Idiots—sprayed each other and Johnny Pesky with champagne. "All of our fans waited their entire lives for this."
When the Sox next took the Fenway diamond on April 11, 2005, it was as champions. What made the Opening Day celebration even sweeter for their long-tormented fans was that the Yankees had to watch the proceedings from the opposite dugout. "I think everybody was curious to see just what the Red Sox would do on the day they got their World Series rings," said New York Manager Joe Torre, whose players clapped politely throughout.
The ceremonies, the first on the premises since 1919, were on the scale normally reserved for coronations. After five flowing red banners commemorating the 1903, 1912, 1915, 1916, and 1918 World Series victories had been draped along the Wall, they were eclipsed by one for 2004 that stretched from end to end. "We had some grown men on our bench about to cry," Damon said.
Damon and his teammates received regal diamond and ruby rings while franchise icons Pesky and Carl Yastrzemski hoisted the championship flag atop the flagpole in left-center field. Then it was back to work. Boston began its title defense in style with an 8-1 triumph. But anyone with a sense of history and drama knew that when New York returned for the final weekend, the season likely would be on the line.
TRIUMPH AND TRAGEDY
On the night the Red Sox completed their historic comeback in the 2004 American League Championship Series at Yankee Stadium, Boston police fired pepper-pellet guns into an unruly crowd outside Fenway Park, killing Victoria Snelgrove, a 21-year-old Emerson College student, and wounding two other people. In 2005, the City of Boston reached a $5 million settlement with the Snelgrove family, the largest wrongful death settlement in city history.
In 2007, Boston Police Commissioner Ed Davis announced that the type of pellet gun blamed in the death of Snelgrove, who was struck in the eye, would never again be used by Boston police. An independent panel concluded in 2005 that Snelgrove's death was an avoidable tragedy that was caused by poor planning and "serious errors in judgment" by Boston police officers and commanders. Two officers were suspended for 45 days and other officers received demotions and written reprimands.
The Snelgrove family also reached an undisclosed settlement in a suit against FN Herstal, the maker of the weapon. Shortly after their daughter's death, Richard and Dianne Snelgrove established a memorial fund that has funded children's playgrounds in and around her hometown of East Bridgewater and scholarships at East Bridgewater High School and Emerson College.
"Why do [the Red Sox] draw two million people? Why do they get 30,000 people at the end of the season, even when they're not in it? People come to see the ballpark, to see the Green Monster, to be close to the players. Boston must balance development growth with the preservation of what makes our city so livable—our historic character, scale, and charm. We are distinct from other American cities because we view our buildings as resources, not liabilities."
—Thomas M. Menino, mayor of Boston
Despite losing Martinez to the Mets and Foulke to knee surgery and missing Schilling for nearly half the campaign after off-season ankle repair, the Sox remained in first place from late June until the final 10 days of September. But New York, which had been seven games under .500 in May, returned for the decisive series at Fenway a game ahead in the division.
It was God's plan, Damon reckoned, that the two blood rivals would go head-to-head for the title. After a split of the first two games gave New York the AL East crown, Boston needed either a Sunday victory or a Cleveland loss to the White Sox to avoid a one-game Fenway playoff with the Indians for the wild-card spot. The clubhouse message board was bluntly optimistic: "TOMORROW. PACK FOR 3 DAY TRIP."
"Our goal was to get into the postseason and our goal in the postseason is to win the World Series," said Schilling, after Cleveland's loss mooted his teammates' 10-1 triumph and earned them a trip to Chicago for the divisional series. "We got Step One done."
Step Two, though, proved a stumbling block as the White Sox, who hadn't won a home playoff game in 46 years, claimed the first two games by counts of 14-2 (Boston's worst October loss by run margin) and 5-4 (after trailing, 4-0) to send their scarlet counterparts home for an elimination game. "Our backs are truly against the wall," acknowledged Epstein. "It's the personality of this team not to do things easily."
The White Sox finished things off with a 5-3 decision in Fenway and went on to sweep Houston to win their first World Series since 1917, ending a drought that had been a year longer than Boston's. "No one can ever take away what we did last year," said Kevin Millar. "This year we fought. We just weren't the better team."
Fans soon found out there was fighting going on behind the scenes, too. On Halloween night, Epstein walked out of Fenway wearing a gorilla suit to avoid reporters' questions about why he'd just turned down a contract extension. When he returned (in street clothes) in January, he had a new contract and nothing but good things to say about Sox CEO Larry Lucchino.
Front-office drama aside, multiple additions and deletions were inevitable during the off-season. Pitcher Josh Beckett and third baseman Mike Lowell, who had both won rings with the Marlins, arrived in a seven-player deal. Departing free agents included Damon (to the Yankees), Mueller (Dodgers), and Millar (Orioles).
REMDAWG'S LONG RUN
In February 1988, former Boston second baseman Jerry Remy was named analyst for Red Sox games on New England Sports Network. Little did he know at the time that his tenure in front of the camera would stretch nearly three times longer than his playing career did.
Play-by-play partners have included local legend Ned Martin, Sean McDonough, Bob Kurtz, and Don Orsillo, with whom Remy has worked the past 11 seasons. Perhaps owing in part to Remy's local roots and his clear-cut, concise game analysis, his popularity soared—he gained a nickname ("RemDawg"), a website called the Remy Report, and even a couple of restaurants that bear his name.
Remy was chosen from among five finalists in 1988 to succeed Bob Montgomery on NESN. The other candidates were also former Sox players: Dick Radatz, Rick Miller, Jim Lonborg, and Mike Andrews. Montgomery, a former Sox catcher who began as a TV analyst in 1982, continued in that role on Channel 38 through 1995.
In July 2010, Remy and NESN agreed to a multiyear contract extension. This followed a three-and-a-half-month absence from the booth during the 2009 season while he recuperated from lung cancer surgery, along with a post-surgery infection and bout with depression. Just before his return to the booth in August 2009, he said he'd received so much support from fans that he felt guilty about not coming back sooner.
"I have boxes and boxes of cards, letters, prayers, tweets, e-mails," said Remy. "But I was crying reading them. . . . In a way, you feel like you've done something right for these people." Returning to the TV booth, he realized, was the way to thank them.
Jerry Remy acknowledged the fans upon returning to the NESN broadcast booth with Don Orsillo in August 2009, following Remy's surgery for lung cancer.
The Red Sox and Yankees lined up for the national anthem as the first Red Sox pennant in 86 years flew over Fenway Park on Opening Day 2005.
War veterans escorted the World Series trophy and rings.
Red Sox legend Johnny Pesky (6) finally received a World Series ring, along with congratulations from team executives (left to right) John Henry, Tom Werner, and Larry Lucchino.
The sight of Damon, clean-shaven and pinstripe-crisp, unleashed a cascade of boos when he made his return the following season to Fenway, where a "JUDAS DAMON" sign hung from a balcony. "People around here are born to hate the Yankees," Damon shrugged. "That's what they are booing, the uniform."
Damon came back to torment his former supporters in August when New York tore the Sox to shreds with a five-game sweep that was even worse than the infamous Boston Massacre that sent the 1978 Sox into a tailspin. The carnage began with a Friday doubleheader that produced 12-4 and 14-11 defeats. The doubleheader consumed 8 hours and 40 minutes, with the nightcap establishing a mark as the longest nine-inning game ever at 4 hours and 45 minutes. It didn't end until 1:22 a.m., after most of the faithful had departed.
Much of the damage was done by Damon, who was 6 for 12 with two homers, a triple, and seven RBI. But Saturday's loss, a 13-5 pratfall in which Beckett walked nine batters and conceded nine earned runs, was even more unsightly. "Never happened before, to get beat up this bad," concluded Ortiz after Boston had allowed a dozen or more runs in three consecutive games for the first time in franchise history. "Doesn't matter where we were, we're going in the wrong direction."
After 8-5 and 2-1 losses completed the club's first five-game sweep since 1954, the Sox had fallen six-and-a-half games behind New York. Boston ended the season 11 back in third place, their lowest finish in the division since 1997. It was, Epstein concluded, "an imperfect year" but it was followed by an extraordinary season that restored Boston to the top of the baseball heap.
MONKEY BUSINESS
It was a Monday night in October 2005, and Theo Epstein had just rejected a contract offer, seemingly ending his brilliant tenure as the 11th general manager in Red Sox history after just three years. In their basement offices, Epstein and members of his baseball operations staff saw camera crews gathering outside to capture him leaving Fenway Park. How could they get Epstein out of the building without facing the camera crush?
It happened to be Halloween, and someone had a gorilla costume handy. Epstein slipped into the suit and walked out of Gate D, past the cameras with a smile on his concealed face. Three months later, after Epstein had discussed working for the Dodgers and Sox CEO Larry Lucchino had talked with Jim Beat-tie about replacing Epstein, their feud was over. Theo was back in the fold, and his gorilla suit was being auctioned off at the annual benefit concert, Hot Stove, Cool Music—for $11,000.
After those negotiations became something of a spectacle in 2005 (Epstein called it "far too public"), he vowed that he would not reveal the end date of any future contract. Lucchino later said, "Walls have crumbled, perceptions of one another have changed, and appreciation of one another has grown."
"I regret that it came to that and I wish we all could have handled it differently—me included," said Epstein in 2007. "I'm not in this job to be recognizable or 'famous' on the local scene. I'm in this job because I want the Red Sox to have a chance to win the World Series every year and I like contributing at this level."
With an unprecedented five playoff teams in his first six seasons as Sox GM, Epstein was not just another guy in a suit.
Before a game with Kansas City on August 2, 2005, Manny Ramirez showed off a sign that he had stashed inside the scoreboard of the Green Monster.
Bobby Doerr acknowledged the fans on August 2, 2007—"Bobby Doerr Day" at Fenway Park. Doerr's No. 1 had previously been retired by the Red Sox in 1988. The number is displayed on the façade in right field behind Doerr.
Fenway denizens received an early hint that a remarkable season had arrived in 2007 when the Sox swept their first home encounter with New York in April, coming from at least two runs down to win all three games. After trailing by four in the eighth inning of the opener, Boston won, 7-6, when Coco Crisp knocked in two with a triple and Alex Cora singled him home with the winner. Then Ortiz knocked in four runs to spur his mates to a 7-5 triumph, and the Sox finished off the Yankees with four consecutive homers (from Ramirez, J. D. Drew, Lowell, and Varitek) off rookie left-hander Chase Wright in the finale.
"I haven't been part of anything like that, not even in Little League," marveled Lowell after the Sox had prevailed, 7-6, to sweep New York at home for the first time since 1990.
For the rest of the season, the Red Sox had the Yankees and everyone else in their rearview mirror, boosting their divisional lead to 11½ games on July 5 by sweeping Tampa Bay at home. The difference was a convenient convergence of inspired pitching and a couple of precocious rookies in second baseman Dustin Pedroia and center fielder Jacoby Ellsbury, the products of a beefed-up farm system.
The most intriguing newcomer, though, was Daisuke "Dice-K" Matsuzaka. The Sox lured the Japanese right-hander from the Seibu Lions with a $50 million contract after spending approximately that much for the privilege of chatting with him over sushi. Matsuzaka, known as "The Monster" back home, won 15 games to go along with 20 by Beckett and 17 by Wakefield. Jonathan Papelbon, a starter turned cold-hearted closer, chipped in 37 saves.
It was so dominant a staff that rookie Clay Buchholz, who pitched a no-hitter in September at Fenway in his second career start, didn't make the postseason roster. For a pleasant change, reaching the playoffs wasn't an arduous enterprise. Boston clinched a postseason berth with seven games to play and ended up with the best record in baseball (96-66) for the first time since 1946. More significant was the first divisional crown in a dozen years, which the Sox won in their clubhouse on the final weekend when the Orioles beat New York in extra innings after Matsuzaka had mastered the Twins.
The divisional series against the Angels went smoothly. First Beckett shut them out, 4-0, in the Fenway opener, and then Ramirez pushed the visitors to the brink with a three-run homer in Game 2, his first walk-off shot in a Boston uniform. "My train doesn't stop," declared Ramirez, echoing bash brother Ortiz after the 6-3 victory. The Angels had been rendered earthbound and the Sox finished them off two days later in Anaheim. "We didn't come out of spring training to win the first round," Lowell observed after Boston had scored seven runs in the eighth inning of a 9-1 smackdown. "We want to win the world championship."
But getting to the Series proved more difficult than he and his comrades might have estimated in the wake of clobbering the Indians, 10-3, in the ALCS opener at Fenway. The tide turned dramatically the next night as Cleveland scored seven in the 11th inning, with old friend Trot Nixon sending home the winner in a 13-6 bust-up that lasted five hours and 14 minutes.
The Indians then won the next two games at The Jake by 4-2 and 7-3 counts, forcing Beckett to save the season with a 7-1 gem in Game 5. But once the Sox returned to Yawkey Way, they dispatched the Tribe with baffling ease. Drew's first-inning grand slam fueled a 12-2 giggler to even the series, and the Sox scored six runs in the eighth inning of the seventh game on their way to an 11-2 win and trip to their second Fall Classic in four years.
The Rockies, who'd been up in the ozone while winning 21 of 22 games and powering past the Phillies and Diamondbacks to win their first National League pennant, were immediately brought to ground in the Series opener at Fenway, where they ran into hot Boston bats and icy pitching from Beckett. "You can't make any mistakes," sighed Colorado starter Jeff Francis after Pedroia had hit his second offering over the left-field wall to begin a 13-1 rout, a record margin for a Series opener.
CRIME OF THE CENTURY
It was June 2007, and Dave Roberts was joking that the signature play of his career gets closer with each viewing.
"As I watch the footage from three years ago to two years ago to when I just saw it in the clubhouse, it gets closer and closer every single time," said Roberts. "I hope five, 10, 20 years down the road that [umpire] Joe West doesn't change his mind and call me out."
On the basis of a three-second dash to second base in the ninth inning of Game 4 of the 2004 American League Championship Series, Roberts made himself into a Boston folk hero. Kevin Millar drew the walk, and Roberts ran for him. Bill Mueller got the single that drove Roberts in from second. Three people were involved in the manufacture of the run that saved the season and set the forces in motion for the greatest postseason comeback in baseball history, but somehow, Roberts's star shines brightest in the telling.
"When I was with the Dodgers," said Roberts, "Maury Wills once told me that there will come a point in my career when everyone in the ballpark will know that I have to steal a base, and I will steal that base. When I got out there, I knew that was what Maury Wills was talking about."
Roberts was a 32-year-old outfielder acquired from the Dodgers about 10 minutes before the trading deadline (for minor leaguer Henri Stanley), on the same day Nomar Garciaparra was dealt to the Cubs. Roberts was strictly fine-print material.
"That whole team, from the moment I got there, had this undeniable belief that something special was going to happen," said Roberts. When the time came, he was ready. "I was scared, excited," he says. "I can't tell you how many emotions went through me. He threw over once, and that was good because it helped settle me. He threw over again, and he almost picked me off. He threw over again, and now I was completely relaxed. I knew that after three throws they weren't going to pitch out. I got a great jump. It was close, but, thank God, Joe West called me safe."
"[Jorge] Posada made a great throw," said Giants manager Bruce Bochy, recalling the play. "It was bang-bang. It just goes to show you what a thin line there is between winning and losing. Another few inches, and he's out. And Boston's done."
Mueller's single tied the game, and then David Ortiz delivered a game-winning, series-extending homer in the 12th inning.
Almost forgotten is the postscript: Game 5 at Fenway, eighth inning, 3-2, Yankees. Millar walks. Guess who pinch runs. Pitcher Tom Gordon is so obsessed with Roberts that he goes down 3-0 to Trot Nixon, who singles Roberts to third. Jason Varitek brings him home with the tying run on a sacrifice fly, and again, the Sox win in extra innings.
Roberts never again played for the Red Sox. Not in Game 6 or Game 7 of the ALCS, or in the World Series.
Roberts's playing career was rather pedestrian. He was a .266 career hitter with 243 stolen bases in 10 seasons. He was a good teammate who could steal a base, slap a key hit, and play good defense in the outfield.
And in 2006, he was inducted into the Red Sox Hall of Fame.
When Schilling shut them down, 2-1, in Game 2—helped by Papelbon's killer eighth-inning pickoff of Matt Holliday, the first of his career—the Rockies were facing a steep climb.
The Sox adapted quickly to Denver's thin air, scoring six runs in the third inning of Game 3 without a homer and three more in the eighth for a 10-5 decision that put them on the verge of joining the Yankees and Reds as the only clubs to sweep World Series in consecutive appearances. Still, they were taking nothing for granted. "We don't want to eat the cake first before your birthday," cautioned Ramirez.
The celebration came the next day with two unlikely heroes blowing out the candles. Jon Lester, the young gun who'd begun the season in Single A and wasn't on the initial playoff roster, pitched flawlessly and pinch hitter Bobby Kielty, who'd been released by the Athletics at the end of July, smacked what turned out to be the game-winning homer in a 4-3 victory with his only swing of the Series. "I felt like I was running on clouds," said Kielty, who never played another game in a Boston uniform.
Despite the high-altitude giddiness, Sox management was quick to tamp down talk of a dynasty only three years after an 86-year curse had been lifted. "Baseball will humble you in a hurry," observed Epstein. "Just when you think you have something, it turns on you."
The odds were against Boston retaining its crown in 2008—only the Yankees had managed back-to-back World Series titles since 1993. Even repeating as division champions proved impossible as Tampa Bay, which had never had a winning season, finished two games in front. For most of the spring the Sox had been the front-runners, inspired by Lester's 7-0 no-hitter against Kansas City on May 19, less than 21 months after he'd been diagnosed with lymphoma.
But the Rays, who lost the first seven meetings at Fenway by a 45-16 aggregate, gradually made up ground and left a calling card in September when they won the final two games there, taking the finale on Carlos Pena's three-run homer in the 14th inning off Mike Timlin. With the Yankees missing the playoffs for the first time since 1994, it seemed inevitable that Boston and Tampa Bay would meet in October.
First was the customary date with the Angels in the divisional series. This one, though, proved decidedly more stressful. Though the Sox won the first two games in Anaheim to run their unbeaten playoff streak against the Angels to 11, they needed a two-run homer from Drew in the ninth to win the second encounter, 7-5, after frittering away a 5-1 lead.
Fans got a shower of champagne from Curt Schilling, who climbed on the dugout to celebrate after the Red Sox clinched the 2007 AL East Championship.
NO LOAVES, JUST FISHES
BY KEVIN PAUL DUPONT
October 13, 2007—As the Red Sox opened the 2007 American League Championship Series with Cleveland, the weather was unpredictable. For Fenway Park director of grounds Dave Mellor and his crew, that was hardly unusual.
"Weather dictates so much of what we do," said the 43-year-old Mellor, who became Fenway's lawn doctor when longtime director of grounds Joe Mooney handed over the rake just prior to the 2001 season.
When Mellor took the job at Fenway, on the recommendation of Mooney, a man he calls his mentor, the new man had to be schooled on the many different ways rain could foul up Fenway. Day No. 1 on the job, the two men sat in the stands, and Mooney told Mellor that heavy rains often flooded the dugouts.
"Not really a problem," said Mellor. "I saw plenty of that at County Stadium [in Milwaukee]."
Heavier rain, Mooney told him, would blow the drain covers off beneath the stands on the third-base side, and water would shoot up, geyser-like, from the ground.
Right there, with that bit of Old Faithful imagery, Mooney had the new hire's attention. Gushing water. Something he had never seen in Milwaukee.
But wait, Mooney warned, there was more.
"He tells me, 'And when it really, really, really rains, the camera pit over on first base will flood,'" recalled Mellor. "And then he says, 'And fish will swim out on the field.' Hey, I'm new here, right? I didn't challenge him. All I said was, 'OK, Joe, yeah, got that.'"
Fast-forward to Opening Day 2001, with the Yankees in town. It rained and rained, to the point that the tarp cradled two-and-a-half inches of water. Making his appointed rounds, Mellor swung by first base, and there it was, just as Mooney warned—a fish.
"I kid you not . . . I couldn't believe it," said Mellor. "A fish! I looked all around, figuring they had some camera in the stands or something, and I was on Candid Camera. But then I looked over to second base, and there was more. From the camera pit to second base, a total of eight fish. I only wish I had taken a picture. In fact, I should have kept one and had it mounted."
According to Mellor, when contractors ripped out the old field after the 2004 World Series, sure enough, they found a drain line that acted as the conduit that made Fenway the fish-friendliest ballpark in North America.
"Next time I saw Joe," recalled Mellor, "I said to him, 'Oh my gosh, Joe, tell me anything and I'll believe you.'"
When the scene shifted to Fenway, the Angels were determined to make a fight of it after having been swept in the previous two postseason showdowns between the clubs. It took 12 innings, and 5 hours and 19 minutes, but the visitors managed to stay alive when Erick Aybar dropped a single in front of Crisp to score Mike Napoli from second for a 5-4 triumph.
Just in case they couldn't close things out the following night the hosts packed their bags for the West Coast. But with the score tied in the ninth, Jason Bay, who'd been acquired from the Pirates in a trade that sent Manny Ramirez to the Dodgers, hit a ground-rule double into the right-field stands, and then came home on rookie Jed Lowrie's single. "When he hit that ball I was going to score," said Bay, who'd stumbled rounding third and needed to sweep the plate with his hand to produce the run that clinched the 3-2 win. "There was no question. I was gone."
After their champagne shower, Bay and his teammates headed for Florida and a championship-series date with the Rays, who'd taken down the White Sox. While Tropicana Field had been a hothouse of horrors for Boston, which had lost eight of its nine meetings there during the season, the visitors very nearly took the first two games.
First, Matsuzaka shut out the Rays, 2-0, in the opener. Then the Sox led their next meeting three times before Tampa Bay prevailed, 9-8, in the 11th on three walks and a sacrifice fly. "We're not frustrated," insisted reliever Mike Timlin, who took the loss after not having pitched in nearly two weeks. "You come down to somebody else's place and you split, we're still looking pretty good."
But the club looked marked for death after two staggering losses on Yawkey Way pushed it to the brink. The Sox, who hadn't dropped more than two home dates to the Rays in six seasons, lost that many in consecutive nights. The visitors scored 22 runs in six-and-a-half hours of play in by far the worst consecutive postseason defeats Boston had ever suffered.
The hosts could shrug off the 9-1 battering in Game 3 as an aberration. But the 13-4 loss in Game 4, ignited by back-to-back, first-inning homers off Wakefield by Pena and Evan Longoria, left the Sox reeling. "We're down 3-1 and if we lose we're going home," said Pedroia. "Hit the panic button."
His colleagues, who'd escaped from 3-1 deficits in three previous championship series, responded with the greatest playoff comeback in franchise history two nights later, coming from seven runs down in the seventh to win, 8-7. "I've never seen a group so happy to get on a plane at 1:30 in the morning," Francona noted.
The Red Sox found a way to top their 2004 world title, literally. The 2007 banner was hung on Opening Day 2008, and the flags of 62 nations were also displayed, to represent the reach of Red Sox Nation. The Sox had opened the season in Japan, then played in California and Canada before their home debut.
When the Sox won Game 6 by a 4-2 count at The Trop on a blinding performance by Beckett and a winning homer from Jason Varitek, who'd been 0 for 14 in the series, they were only one game from their third World Series appearance in five years. But Boston couldn't solve Matt Garza in the finale and its most improbable resurrection ended there. "We played as hard as we could," Pedroia said after the Rays had won, 3-1, and earned a World Series date with the Phillies. "We just ran out of magic."
Tampa Bay, crushed by the Phillies in the Series, proved a one-hit wonder.
A century's worth of experience had taught Boston that its perennial rival hadn't changed. "Year after year," Henry had said, "the Yankees are Halliburton." So they were again in 2009, even though the Sox won their first eight meetings with New York and led the division at the All-Star break.
The season essentially came undone in a bizarre Fenway series in late August. After New York had won the opener in a 20-11 bashfest and Boston had responded with a 14-1 blowout, the visitors took the Sunday finale, 8-4, crashing five homers off Beckett. "Right now, they're definitely better than we are," conceded Bay after the Yankees had grabbed a seven-and-a-half-game lead.
Though Francona's club finished eight games behind, it still earned its sixth playoff appearance in seven years during a month that fans had come to regard as Soxtober. As usual, Boston's dancing partner was the Angels.
This meeting, though, was unlike the others because Boston was forced to operate with a discombobulated rotation. The Sox, who hadn't lost a playoff game in Anaheim since 1986, dropped the first two as haloed aces John Lackey and Jered Weaver held the visitors to a total of one run. "I don't think it's the way we scripted it before we got here," observed Lowell after Boston had been stifled, 5-0 and 4-1. "We've definitely dug a pretty good hole for ourselves."
Boston, which had won nine of 11 elimination games under Francona and lost only one at Fenway since 1999, figured that it could make a last stand at home. Leading, 6-4, in the ninth inning of Game 3, the Sox seemed secure. But Papelbon, who hadn't allowed an earned run in 26 playoff innings, came unglued.
After allowing two inherited runners to score in the eighth, the star closer was one strike away from victory three times in the ninth, but conceded a single, a walk, and a double before Vlad Guerrero smacked a fatal two-run single to center. "The season doesn't wind down," remarked Francona after his club had absorbed a shocking 7-6 loss that left the faithful sitting stupefied in the stands. "It just comes to a crashing halt."
The 2010 season never got started, as the Red Sox lost nine of their first 13 games and were in first place for only one day—after beating the Yankees on Opening Day. Storm warnings were evident as early as Patriots Day weekend, when the Sox were swept at home by Tampa Bay, dropping the final two games by a combined 15-3. "When you don't show up to play, you're going to get beat," Pedroia said after he and his mates had fallen into fourth place, six games off the pace. "Doesn't matter if you play the Rays or Brookline High School."
The club made an encouraging mid-June run when the National Leaguers came to town, taking eight of nine from the Phillies, Diamondbacks, and Dodgers to climb within a game of the division lead. One of the visitors was particularly familiar as Manny Ramirez returned for the first time since being traded to Los Angeles and was given a decidedly mixed reception from fans, who continued to appreciate his role in two Series triumphs even as they disdained his lackadaisical approach. "There's no reason I should have behaved that way in Boston," Ramirez admitted to a Spanish network announcer.
Yet "Mannywood" wasn't behaving much differently in L.A., and so he was wearing a White Sox uniform by September. By then his former Boston teammates were stuck in third place and out of contention after both their lineup and rotation had been shredded by injuries, with 19 players spending time on the disabled list for a total of 1,013 man-games lost to injury.
Leadoff hitter Jacoby Ellsbury played only 18 games after fracturing his ribs. Pedroia missed the final three months after breaking a foot and Kevin Youkilis was sidelined for the season after tearing a thumb muscle. Marco Scutaro injured his rotator cuff and was switched from shortstop to second base. Josh Beckett spent two months on the DL, where Daisuke Matsuzaka was placed twice.
"It's disappointing in that we didn't get where we want to go, but there's still a lot to be proud of," Epstein said on the season's final weekend, as the Sox finished below second place for the first time in 13 years. "I'd like to rewind and start over and do 162 again and see how it turns out."
As they had during the first six years after the ballpark had opened, the Fenway faithful had become accustomed to Soxtober—watching the hometown team playing for a championship—and Sox President Larry Lucchino had promised them "the constant, unwavering commitment to winning." So as Mike Lowell, Victor Martinez, and Adrian Beltre departed, the front office moved boldly to sign stars Adrian Gonzalez from San Diego and Carl Crawford from Tampa Bay for 2011.
With a $161 million payroll and 15 All-Stars on the roster, the club was widely touted as the likely champion in spring training. "We have a lot of work to do but I can see why people are talking about going back to the World Series," acknowledged rightfielder J.D. Drew. "On paper, we have that kind of team."
A Fenway Park concession-stand worker watched Red Sox-Yankees action on a television monitor in mid-September 2007.
Fenway Franks made their way through the park concourse before a game against the Toronto Blue Jays in July 2011.
Just two years after battling cancer, left-hander Jon Lester threw the 18th no-hitter in Red Sox history—and the team's fourth of the 2000s—beating the Kansas City Royals, 7-0, at Fenway Park on May 19, 2008. Jason Varitek was behind the plate, setting a major-league record by catching the fourth no-hitter of his career.
But the fact that the opener at Texas was played on April Fool's Day might have been an omen about the historic pratfall that was to follow. Boston lost its first six games to the Rangers and Indians; it was the team's worst start since 1945 when most of its top players were wearing Uncle Sam's uniform. "It can't get any worse than this," third baseman Kevin Youkilis declared after the Sox lost 1-0 at Cleveland on a squeeze bunt in the eighth inning to go 0-6.
The 100th opening day at Fenway brought a delightful turnaround as the hosts drilled the Yankees 9-6. "I've never seen a team so happy to be 1-6," observed Francona. The upward ascent took time, though. It wasn't until May 16 that the club had a winning record and not until May 27 that it climbed past New York into first place.
Boston still was there at the beginning of September but an unsightly 10-0 home loss to Texas was the first misstep in a fatal tumble. Though the Sox still had a comfortable nine-game lead over Tampa Bay in the wild-card race, they soon went into free fall en route to the worst September swoon (7-20) in major-league history.
"It's crazy," proclaimed designated hitter David Ortiz after the players were hooted off the diamond for losing the Fenway finale to the last-place Orioles. "I've never seen anything like that around here as long as I've been here. If you would have told me in August this would happen in September, I would have laughed at you."
Going into the final six games at New York and Baltimore, Boston still had a two and a half game edge over the Rays for the wild-card spot. But the tailspin continued as the visitors lost two of three in the Bronx, needing Jacoby Ellsbury's three-run homer in the 14th to salvage the finale.
THRILLS, CHILLS, SPILLS
For years it had been easy to forget New England's hockey roots. In the first decade of the new century, the Bruins got lost in the championship seasons of the Red Sox, Patriots, and Celtics. But on New Year's Day 2010, Boston reminded North America that the Hub is still a hockey town. (And, 18 months later, they would finally bring home the Stanley Cup.)
In the third NHL Winter Classic, 38,112 stoic souls stood in the cold at Fenway Park for three hours before they were rewarded with a 2-1 Bruins victory over the Philadelphia Flyers on a tip-in goal by Marco Sturm in the second minute of extra time.
"It's the perfect day for hockey in Boston," said Bruins legend Bobby Orr, who skated to center ice for a ceremonial pregame handshake with Flyer nemesis Bobby Clarke after leading the Black and Gold onto the infield ice. "It's a thrill to see all these pros turn into kids again for one day. This is how we all started playing hockey—outdoors. And this day, here at Fenway Park, truly is a classic."
It was a day that could scarcely have been imagined when Fenway opened in 1912, a dozen years before the Bruins would take the ice for the first time.
The postcard-perfect afternoon lacked the Currier & Ives snowfall that marked the previous day's photogenic practice session, but everything else worked out the way it was sketched on the NHL blueprint.
Late in the third period, comedian Lenny Clarke came out to lead the singing of "Sweet Caroline," and the Red Sox magic took over. Mark Recchi scored a power-play goal with 2:18 remaining, and then Sturm potted the OT winner. It was time to cue up "Dirty Water." No one wanted to leave on the day Fenway put on its snow pants and we all came home to hockey.
"It's Fenway Park. It's history. It's something you're going to remember the rest of your life," said the Bruins' Patrice Bergeron. "You want to be on the good side of the outcome. You want to win."
One day later, they played the AT&T Legends Classic for charity, and 33,000 fans showed up to watch former Bruins, including Cam Neely, Brad Park, and Terry O'Reilly, skate with celebrities such as Tim Robbins, Denis Leary, Bobby Farrelly, and Kiefer Sutherland.
"I don't know how many people can actually say they skated in front of 33,000 people," said Sutherland afterward. "In the middle of a snowstorm, they stayed. I've never seen fans like that. It was pretty awesome." The game raised $200,000 apiece for the Bruins and Red Sox charitable foundations, and for a third charity, Hockey Fights Cancer.
Before they took the boards down a week later, Boston University and Boston College clashed in the college version of the Winter Classic—with BU winning, 3-2, in a battle of the previous two NCAA hockey champions before 38,472 fans.
Hockey fans in Boston were treated to a wondrous sight: a rink set up in front of the Green Monster at Fenway Park, where the Bruins and Philadelphia Flyers squared off in the NHL Winter Classic on New Year's Day 2010. Even better, team and regional icon Bobby Orr led the B's onto the ice for the start of the game, and his counterpart was longtime rival Bobby Clarke of the Flyers. The Bruins prevailed on this day, 2-1, on an overtime goal by Marco Sturm.
WHO'S OUR DADDY?
When Pedro Martinez's name came up in early 2010, nobody seemed to know what the former Red Sox pitching great was up to. He wasn't at spring training. He wasn't working on a deal to join a team at mid-season, as he had done with the Phillies the previous year. Some wondered whether he might be found under that mango tree in the Dominican Republic where he said he planned to spend his time after retirement.
But no, fittingly enough, Martinez made his first appearance of the year when he emerged from a tent in the left-field corner at Fenway to throw out the ceremonial first pitch on Opening Night vs. the Yankees. Wearing his familiar Red Sox No. 45 and blowing kisses and making hugging gestures to the fans, he strolled in from the outfield and back—at least fleetingly—into the Red Sox-Yankees rivalry that he was such a major part of. We remember his 17-strikeout effort vs. New York in 1999, when he allowed just one hit. We remember the duel in 2000 against the Yankees' Roger Clemens when Trot Nixon homered to complete a 1-0 Sox win. We remember his throwdown of Don Zimmer in Game 3 of the ALCS in 2003. We remember, too, the eighth inning of Game 7 of the 2003 ALCS when Martinez, leading, 5-2, was left in too long by Grady Little in what became one of the historic meltdowns in playoff history. We remember Pedro after a victory in 2001 snarling, "I don't believe in damn curses. Wake up the Bambino and have me face him. Maybe I'll drill him in the ass."
We remember him saying, after a loss late in the 2004 season, "I just have to tip my hat to the Yankees and call them my daddy." That comment resulted in a crescendo of "Who's your daddy?" chants from the New York faithful in his next outing there, in the 2004 ALCS. But the Red Sox busted the curse in that series, and Pedro ended his seven-year tenure with the Red Sox with a 117-37 record, a 2.52 ERA, two Cy Young Awards, and almost certain election to the Baseball Hall of Fame five years after he retires.
After the Sox split the first two in Baltimore the season came down to the 162nd game, and for seven innings everything was breaking Boston's way, with the Sox leading 3-2 in the seventh and the Yankees drubbing the Rays 7-0. Then the skies opened in Baltimore and, during the 86-minute rain delay, the players sat in the clubhouse watching Tampa come back from the dead to even the score and send that game into extra innings.
Still, a Boston victory would mean at least a one-game playoff with the Rays and with two out in the ninth, nobody on base and closer Jonathan Papelbon on the mound, victory seemed assured. But two doubles and Robert Andino's looping liner to left that Crawford couldn't glove gave Baltimore a 4-3 triumph. By the time the Sox made it from the dugout to the clubhouse Evan Longoria had cranked a walkoff homer in the 12th to put the Rays into the divisional series.
"We can't sugarcoat this," Epstein conceded after Boston had missed the post-season in consecutive years for the first time since 2002. "This is awful. We did it to ourselves and we put ourselves in position for a crazy night like this to end our season." That crazy night ended not only a season but also an era. Francona, who'd concluded that the front office wouldn't extend his contract, resigned two days later after eight years as skipper, frustrated that he couldn't get through to veterans who had a "sense of entitlement." Epstein soon followed him out the door to take on the challenge of rebuilding another ballclub that played in a storied park and that hadn't won a World Series for even longer—the Chicago Cubs.
Amid the upheaval and uncertainty, Sox owner John Henry made an impromptu and impassioned appearance on a local sports radio talk show to assure the public that stability and success would return. "We're going to be back as an organization," he vowed. "We're going to have a top-class manager and general manager and we're going to have a great team next year."
Whether or not the calendar included a Soxtober every year, management was continuing its mission to preserve Fenway for the next generation. Tiger Stadium, which had opened on the same day as Fenway in 1912, had gone under the wrecking ball in 2009. So had The House That Ruth Built, replaced by a $2 billion pinstriped pleasure dome across the street.
Boston, though, traditionally has been reluctant to toss its architectural treasures into the trash bin. A city that still has its original 18th-century State House and 19th-century City Hall has seen no reason to dismantle a 20th-century playground that still attracts more than three million ticket holders a year and has sold out every game since early in the 2003 season.
Fenway has undergone annual makeovers in recent years, with ownership spending $40 million in enhancements before the 2011 season—including the addition of three high-definition video screens. The total tab was 60 times more than the $650,000 that John I. Taylor had spent to build the ballpark a century earlier.
Over the past decade, John Henry and his colleagues have underwritten $285 million in improvements—from Monster seats atop the left-field wall to expanded concourses to a new playing surface—designed to carry the "lyric little bandbox" comfortably into the middle of the 21st century. Yet for all of the updating, America's Most Beloved Ballpark remains essentially as it was in 1912. If Duffy Lewis were to return today, hunting for his misplaced glove, he wouldn't need to ask directions to left field.
Game announcements and updates were made through a megaphone when the Red Sox played the Chicago Cubs on May 21, 2011—one feature of a series that marked the first meeting of the two teams at Fenway Park since the 1918 World Series. To commemorate the occasion, both teams wore throwback uniforms.
POP CULTURE
OK, so maybe the ballpark isn't always the star. But it has certainly been a major player in movies and popular culture for much of its history.
Kevin Costner's character saw old-time player Moonlight Graham's name and hometown flash on the Fenway Park message board in the ultimate baseball movie _Field of Dreams_ (1989), when he attended a game with Terrance Mann, played by James Earl Jones.
Drew Barrymore as Lindsey dropped from the center-field bleachers and sprinted toward boyfriend Jimmy Fallon in the box seats as hapless security personnel pursued her in _Fever Pitch_ (2005), which required a last-minute revamp when the Red Sox confounded the scriptwriters and won it all in 2004.
Ben Affleck and his homeboys stole millions in receipts from a just-concluded Sox-Yankees series in _The Town_ (2010), with final scenes filmed just inside Gate D of the park. In _Moneyball_ (2011), Brad Pitt as Oakland A's general manager Billy Beane discusses a job offer from the Sox on location at Fenway. The park also had cameos in _A Civil Action_ (1998), _Blown Away_ (1994), _Little Big League_ (1994), _Major League II_ (1994), and numerous documentaries.
In July 2011, New Hampshire native Adam Sandler built a replica of the Green Monster on a Cape Cod Little League field to be used in filming a comedy due out in 2012 and tentatively titled, I Hate You, Dad. Several ballparks and Wiffle-ball fields throughout New England pay homage to Fenway, and another replica park, dubbed "Little Fenway," hosts the Bucky Dent Baseball School (how dare he?) in Delray Beach, Florida.
On the small screen, in addition to being woven into the fabric of Cheers, Fenway occasionally played a part in Boston-bred producer David E. Kelley's _Ally McBeal_ and The Practice. It was also the setting for comedy sketches by Jimmy Fallon and Rachel Dratch, as Sully and Denise, on Saturday Night Live.
Novelist and Sox diehard Stephen King came up with the plotline for his novel _The Girl Who Loved Tom Gordon_ while at Fenway. A young girl lost for days in the Maine woods keeps up her hopes by tuning in Sox games on a Walkman.
Longtime Boston musician Jonathan Richman released a song in 2005, "As We Walk to Fenway Park in Boston Town," that appeared in Fever Pitch. Other music associated with Fenway and its team includes the victory song "Dirty Water" by the Standells, Neil Diamond's eighth-inning staple "Sweet Caroline," "Tessie"—the Royal Rooters' anthem that was resurrected by the Dropkick Murphys in 2004, "The Red Sox Are Winning" by Earth Opera, "Losing" by Pondering Judd, and the "Hot Stove Cool Music" series of benefit concerts spearheaded by Peter Gammons and Theo Epstein.
PHOTOGRAPHY CREDITS
THE _BOSTON GLOBE_ :
Back cover(bottom): David L. Ryan/Globe Staff
1: _Globe_ file photo
2-3: Jonathan Wiggs/ _Globe_ staff
4-5: David L. Ryan/ _Globe_ staff
6: Jim Davis/ _Globe_ staff
7-9: David L. Ryan/ _Globe_ staff
10-11: _Globe_ file photo
12: David L. Ryan/ _Globe_ staff
20: _Globe_ file photo
22-23: Stan Grossfeld/ _Globe_ staff
27: _Globe_ file photo
29: _Globe_ file photo
31 (bottom): _Globe_ file photo
32 (top): _Globe_ file photo
35-37: _Globe_ file photos
40: _Globe_ file photo
51-52: _Globe_ file photos
67 (top): _Globe_ file photo
70: _Globe_ file photo
74-75: _Globe_ file photos
80-81: _Globe_ file photos
84-85: _Globe_ file photos
87: Stan Grossfeld/ _Globe_ staff
89 (top center): _Globe_ file photo
93: _Globe_ file photo
95-97: _Globe_ file photos
103: Stan Grossfeld/ _Globe_ staff
104: _Globe_ file photo
106-109: _Globe_ file photos
111: _Globe_ file photos
113: _Globe_ file photo
115: _Globe_ file photo
117 _Globe_ file photo
118: _Globe_ file photo
123: Louis Russo/ _Globe_ staff
125: Charles Dixon/ _Globe_ staff
129 (bottom): _Globe_ file photo
134: _Globe_ file photo
135 (top): _Globe_ file photo
144: _Globe_ file photo
146: _Globe_ file photo
150 (top): Harry Brett for the _Globe_
150 (bottom): Jack Sheehan/ _Globe_ staff
151 (left): Gilbert E. Friedberg/ _Globe_ staff
151 (right): Dan Goshtigian/ _Globe_ staff
152: _Globe_ file photos
155 (top): _Globe_ file photo
155 (bottom): John Tlumacki/ _Globe_ staff
158: Dan Goshtigian/ _Globe_ staff
159: Ted Dully/ _Globe_ staff
162: Dan Goshtigian/ _Globe_ staff
163: _Globe_ file photos
164: Ted Dully/ _Globe_ staff
165: Charles B. Carey/ _Globe_ staff
166 (center): _Globe_ file photo
167 (top left): Frank O'Brien/ _Globe_ staff
167 (top right): Tom Landers/ _Globe_ staff
167 (bottom left and right): Dan Goshtigian/ _Globe_ staff
170: Dan Goshtigian/ _Globe_ staff
171: George Rizer/ _Globe_ staff
172: Don Preston/ _Globe_ staff
173: Dan Goshtigian/ _Globe_ staff
175: Dan Goshtigian/ _Globe_ staff
176: George Rizer/ _Globe_ staff
177 (top): Dan Sheehan/ _Globe_ staff
177 (bottom): Paul Connell/ _Globe_ staff
178 (left): Frank O'Brien/ _Globe_ staff
178 (right): David L. Ryan/ _Globe_ staff
187 (top): David L. Ryan/ _Globe_ staff
187 (bottom): Jack O'Connell/ _Globe_ staff
188: Frank O'Brien/ _Globe_ staff
190: Ulrike Welsch/ _Globe_ staff
191 (left): _Globe_ file photo
191 (right): George Rizer/ _Globe_ staff
192: David L. Ryan/ _Globe_ staff
193: Don Preston/ _Globe_ staff
194 (top): Frank O'Brien/ _Globe_ staff
194 (bottom): _Globe_ file photo
195 (top center): Frank O'Brien/ _Globe_ staff
195 (top right): George Rizer/ _Globe_ staff
195 (bottom): Janet Knott/ _Globe_ staff
198: John Blanding/ _Globe_ staff
199 (top): David L. Ryan/ _Globe_ staff
199 (bottom): Frank O'Brien/ _Globe_ staff
200: Ted Gartland/ _Globe_ staff
201: _Globe_ file photo
202-203: Jim Wilson/ _Globe_ staff
204: Stan Grossfeld/ _Globe_ staff
206: Joanne Rathe/ _Globe_ staff
207: George Rizer/ _Globe_ staff
208: John Blanding/ _Globe_ staff
209 (top): Suzanne Kreiter/ _Globe_ staff
209 (bottom): George Rizer/ _Globe_ staff
210: Stan Grossfeld/ _Globe_ staff
211: Jim Davis/ _Globe_ staff
212: Stan Grossfeld/ _Globe_ staff
213: David Molnar for the _Globe_
214: Bill Brett/ _Globe_ staff
215: Tom Herde/ _Globe_ staff
216 (top): Mark Cardwell/ _Globe_ staff
217 (top left and right): Bill Greene/ _Globe_ staff
217 (bottom left): John Blanding/ _Globe_ staff
217 (bottom right): Suzanne Kreiter/ _Globe_ staff
218: Jim Davis/ _Globe_ staff
221: Pat Greenhouse/ _Globe_ staff
223-224: Jim Davis/ _Globe_ staff
225: Stan Grossfeld/ _Globe_ staff
226: John Tlumacki/ _Globe_ staff
227: Stan Grossfeld/ _Globe_ staff
228: Jim Davis/ _Globe_ staff
229: Barry Chin/ _Globe_ staff
230: Stan Grossfeld/ _Globe_ staff
231 (top): Dominic Chavez/ _Globe_ staff
231 (bottom): Bill Greene/ _Globe_ staff
232-233: Stan Grossfeld/ _Globe_ staff
234: Max Becherer/ _Globe_ staff
235: Barry Chin/ _Globe_ staff
236 (bottom): Barry Chin/ _Globe_ staff
237 (top left): George Rizer/ _Globe_ staff
237 (top center): Jim Davis/ _Globe_ staff
237 (top right): Mark Wilson/ _Globe_ staff
237 (bottom): Barry Chin/ _Globe_ staff
238: Jim Davis/ _Globe_ staff
240: Wendy Maeda/ _Globe_ staff
241: David L. Ryan/ _Globe_ staff
242: John Tlumacki/ _Globe_ staff
243: Jim Davis/ _Globe_ staff
244: David L. Ryan/ _Globe_ staff
245 (bottom): Jim Davis/ _Globe_ staff
246: Barry Chin/ _Globe_ staff
247: Bill Greene/ _Globe_ staff
248 (top): Stan Grossfeld/ _Globe_ staff
248 (bottom): Yoon S. Byun/ _Globe_ staff
249 (top): Barry Chin/ _Globe_ staff
249 (center and bottom): Jim Davis/ _Globe_ staff
250 (bottom): Jim Davis/ _Globe_ staff
251: Stan Grossfeld/ _Globe_ staff
253: John Tlumacki/ _Globe_ staff
254: Stan Grossfeld/ _Globe_ staff
255: Jim Davis/ _Globe_ staff
256 (top): Bill Greene/ _Globe_ staff
256 (left): Stan Grossfeld/ _Globe_ staff
256 (right): Jim Davis/ _Globe_ staff
257-258: Jim Davis/ _Globe_ staff
259: Stan Grossfeld/ _Globe_ staff
260: Jim Davis/ _Globe_ staff
262-263: Jim Davis/ _Globe_ staff
264: Yoon S Byun/ _Globe_ staff
265-268: Jim Davis/ _Globe_ staff
269: Barry Chin/ _Globe_ staff
271 (top left, bottom left): Stan Grossfeld/ _Globe_ staff
271 (top right): Barry Chin/ _Globe_ staff
272 (top): Stan Grossfeld/ _Globe_ staff
272 (bottom): Jim Davis/ _Globe_ staff
273 (top left): David L. Ryan/ _Globe_ staff
273 (top center): Bill Brett/ _Globe_ staff
273 (top right): Aram Boghosian for the _Globe_
273 (center left): Stan Grossfeld/ _Globe_ staff
273 (center right): Jim Davis/ _Globe_ staff
273 (bottom): Barry Chin/ _Globe_ staff
278: David L. Ryan/ _Globe_ staff
ADDITIONAL PHOTOS BY:
WERNER H. KUNZ
Front Cover
BOSTON PUBLIC LIBRARY
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CORBIS
INDEX
A
Aaron, Hank, , , , ,
Adams, Margo,
Aerosmith, ,
Affleck, Ben,
Agganis, Aristotle George "Harry," , , 136–37
Agnew, Sam,
Ainley, Leslie G.,
Alden, Jane,
Alexander, Grover Cleveland, 38–39
Allen, Frank,
Allenson, Gary,
Allison, Bob,
All-Star Games, , , 116–17, , , , ,
_Ally McBeal_ ,
Almada, Mel,
Altrock, Nick,
American Soccer League,
Anderson, John,
Anderson, William R.,
Andino, Robert,
Andrews, Ivy Paul,
Andrews, Mike, , , ,
Angell, Roger, , , ,
Aparicio, Luis, , , 174–75
Araujo, George,
Arizin, Paul,
Armas, Tony,
Armbrister, Ed,
Armstrong, Louis,
Arroyo, Bronson, ,
Aybar, Erick,
B
B-52s,
Backstreet Boys,
Bagwell, Jeff,
Baker, Del, ,
Baltimore Orioles, , , , , , , 173–75, , , , , , , , , , , ,
Bando, Sal,
Barnett, Larry, ,
Barnicle, Mike,
Barreto, Juan,
Barrett, Marty, , ,
Barrow, Ed, , , , , ,
Barry, Jack, , ,
Barry, John,
Barrymore, Drew,
_Baseball Abstract_ ,
Battles, Cliff,
Baugh, Sammy,
Bay, Jason, , ,
Baylor, Don,
Beane, Billy,
Bearden, Gene,
Beattie, Jim,
Beck, Rod,
Beckett, Josh, , 257–58, 262–63
Bedient, Hugh,
Begeron, Patrice,
Belinsky, Bo,
Bellhorn, Mark,
Beltre, Adrian,
Bench, Johnny,
Benchley, Robert,
Benzinger, Todd,
Berberet, Lou,
Berde, Gene,
Berra, Yogi, , , , ,
Bezemes, Johnny,
Bierhalter, Bits,
Bird, Larry,
Birtwell, Roger,
Bishop, Max,
Blondie,
_Blown Away_ ,
Blue, Vida, ,
Bluege, Ossie,
Bochy, Bruce,
Boddicker, Mike, , 220–21
Boggs, Wade, , 206–7, 212–14, 216–17, , ,
Boone, Aaron,
Booth, Clark,
BoSox Club,
Boston Arena, ,
Boston Beacons, ,
Boston Braves, , , 41–43, , , , , , , , , , , , , , 116–17, , , ,
Boston Braves (football), , ,
Boston Bruins, , , , 266–67,
Boston Bulldogs,
Boston Celtics, , , , ,
Boston College, , , , , , ,
Boston College High School,
Boston Garden, , , ,
Boston Jazz Festival,
Boston Landmarks Commission,
Boston Marathon,
Boston Patriots, , , , ,
Boston Redskins, , , 88–89
Boston University, ,
Boston Whirlwinds,
Boston Yanks, , , ,
Boucher, Joseph A., 101–2
Boudreau, Lou, , , , , , , 135–36
Boyd, Dennis Ray "Oil Can," , 207–8, , ,
Boyd, Owen,
Boyd, Willie James,
_Boys Who Were Left Behind, The_ (Heidenry & Topel),
Bradley, Hugh,
Bradley, Phil,
Braves Field, , , 46–47, , , , , , , ,
Brecheen, Harry "The Cat," ,
Brewer, E.S.,
Brickley, Ed,
Briggs, Walter O.,
Briggs Stadium,
Brock, Lou,
Brooklyn Dodgers, , , ,
Brooklyn Dodgers (football),
Brooklyn Robins, , , ,
Brown, Paul,
Brown, Skinny,
Brubeck, Dave,
Brunansky, Tom, ,
Buchanan, William,
Buchholz, Clay, ,
Buckner, Bill, , , , 210–11, , ,
Buddin, Don,
Buffalo Bills, ,
Buffett, Jimmy, ,
_Bull Durham_ ,
Burgmeier, Tom,
Burke, Billie,
Burks, Ellis, 221–22
Burleson, Rick, , , , ,
Burns, Ken,
Burton, Jim,
Busby, Steve, ,
Busch Stadium, ,
Bush, Joe "Bullet Joe," , , ,
C
Cabrera, Orlando,
Cain, Jess,
Campbell, Bill,
Campbell, Robert,
Candelaria, John,
Candlestick Park,
Canseco, Jose, ,
Cappelletti, Gino, , ,
Carbo, Bernie, , , , , ,
Carlisle Indians,
Carrigan, Bill, , 41–42, 50–51, , , , ,
Casey, Ken,
Cater, Danny,
Catlett, Walter,
Celtic F.C., , 272–73
Centennial Common,
Cepeda, Orlando, , ,
Chance, Dean,
Chance, Frank, 58–59, ,
Chapman, Ben,
Chara, Zdeno,
Charles, Ray, ,
Charles Logue Building Company,
_Cheers_ , ,
Chesbro, Jack,
Chicago Bears,
Chicago Cubs, 42–43, , , , , , ,
Chicago Mustangs,
Chicago White Sox, , , , , , , , , , , , , , , , , , , ,
Children's Hospital, ,
Children's Island Sanitarium,
Cincinnati Reds, , , 180–81,
Citgo sign, , ,
_Civil Action, A_ ,
Clark, Jack,
Clarke, Bobby, 266–67
Clarke, Lenny,
Clear, Mark, ,
Clemens, Roger, , , , 205–6, , , , 216–17, 219–22, , 235–37, ,
Cleveland, Reggie,
Cleveland Browns, ,
Cleveland Buckeyes,
Cleveland Indians, , , , , 77–78, , , , , , 113–15, , , 127–28, , 146–47, , , 166–67, , 193–94, , , , , , , ,
Clifton, Sweetwater,
Cobb, Ed,
Cobb, Ty, , ,
Cocoanut Grove, ,
Coleman, Jerry, ,
Coleman, Joe,
Coleman, Ken, ,
Collins, Bud, ,
Collins, Eddie, , , , 83–84, , ,
Collins, John "Shano," , ,
Collins, Ted,
Colorado Rockies,
Combs, Earle,
Comiskey, Charles,
Comiskey Park, , ,
Commane, William,
Conant, David,
Conigliaro, Billy, , ,
Conigliaro, Tony, , , , , 153–54, , , 165–67, , , ,
Connolly, Tom, ,
Cooney, Terry,
Cooper, Cecil, ,
Coors Field,
Cora, Alex,
Coral Reefer Band,
Cordero, Will,
Cork Club,
Costas, Bob,
Costner, Kevin, ,
Coughlin, Jimmy,
County Stadium,
Courtney, Clint,
Coyle, Harry,
Craigue, Paul V.,
Cramer, Doc, , ,
Crandall, Otis,
Crawford, Carl, , , ,
Crips, Coco, ,
Cronin, Joe, 77–78, 84–86, 88–90, , , , , , , , , , , , , , ,
Cronin, Maureen,
Cronin, Millie,
Crow, Sheryl,
Culberson, Leon, ,
Culp, Ray,
Curley, James Michael, , 51–52, ,
Curse of the Bambino, , , , , ,
Curtis, John, 174–75
Cushing, Richard J., ,
Cy Young Award, , , , , , , , ,
D
Dahlgren, Babe,
Dallas Texans,
Damon, Johnny, , , , , 270–71
D'Angelo, Arthur,
D'Angelo, Henry,
Darcy, Pat,
Dartmouth College,
Darwin, Danny,
Daubach, Brian,
Dave Matthews Band, ,
Davis, Ed,
Davis, Piper,
Davis Cup,
D.C. Stadium,
de Valera, Eamon, , ,
Dean, Roger,
DeMarco, Tony,
Dent, Bucky, , , ,
Detrich, James,
Detroit Lions,
Detroit Tigers, , 42–44, , , , , , 101–2, , , , , , 124–25, , , 156–57, , , , 173–74, , , , 216–17
Dewey, Thomas,
Diamond, Neil, , , ,
Dietz, William "Lone Star,"
DiMaggio, Dom, , , , , , 113–15, , , 136–37, ,
DiMaggio, Joe, , , , , , 114–15, , ,
DiMaggio, Vince,
DiMuro, Lou,
"Dirty Water," , , ,
Dobson, Joe,
Dodd, Dick,
Dodger Stadium,
Doerr, Bobby, , , , , , , , , , , , , , , , ,
Donovan, Timothy,
Dooley, Elizabeth "Lib," ,
Dooley, John Stephen,
Dorsey, Jim,
Dowd, Tom,
Drago, Dick, , 192–93
Dratch, Rachel,
Drew, J.D., , ,
Dropkick Murphys, , , , ,
Dubenion, Elbert,
Duffy, Hugh, , , 66–67,
Duffy's Cliff, 38–39, , , , ,
Duquette, Dan, , 222–24, 227–30, 236–37, , ,
Dykes, Jimmy,
E
E Street Band, ,
Earth Opera,
Easler, Mike,
Eckersley, Dennis, 192–93, , , , ,
Edwards, Turk,
Egan, Ben, ,
Egan, Dave "the Colonel," ,
Ehmke, Howard,
Elias, Christian,
Ellsbury, Jacoby, , ,
Ellsworth, Dick,
Ely, Joseph Buell,
Emmanuel College,
Engle, Clyde,
Epstein, Theo, , , 257–58, , , 268–73
Evans, Dwight, , , , , , ,
Evers, Hoot,
F
Fallon, Jimmy,
Farber, Sidney,
Farrelly, Bobby,
_Fear Strikes Out_ ,
Federici, Danny,
Feeney, Marty,
Felix, Junior,
Feller, Bob, , , ,
Feller, Sherm,
Fence Green,
Fennelly, Martin,
Ferraro, Mike,
Ferrell, Rick,
Ferriss, Dave "Boo," , , ,
_Fever Pitch_ , , ,
_Field of Dreams_ , ,
Fingers, Rollie,
Finley, Charlie, ,
Fischer, Bill,
Fisher, Jack, ,
Fisk, Carlton, , , , , , 173–75, 179–81, , , , , 194–95, 197–99, , , ,
Fisk Foul Pole, ,
Fitzgerald, John "Honey Fitz," 30–32, , ,
Fletcher, Scott,
Flood, Curt,
Florida Marlins,
Florie, Bryce,
Flynn, Frank J.,
Flynn, Ray,
Fohl, Lee, , , 66–67
Forbes Field,
Foss, Eugene,
Fossum, Casey,
Foster, George "Rube," , , ,
Foulke, Keith, , ,
Foxx, Jimmie, , 77–78, , , , ,
Foy, Joe,
Francis, Jeff,
Francona, Terry, , 250–51, , , , ,
Franklin, Benjamin,
Frazee, Harry H., 42–43, , , 53–54, , 58–59, ,
Frazier, George,
Frisch, Frank,
G
Gaffney, James, 41–42
Galehouse, Denny,
Gallego, Mike,
Gammons, Peter, , , , ,
Gandil, Arnold "Chick,"
Garcés, Rich,
Garcia, Karim,
Garciaparra, Nomar, , , , 227–28, , 235–36, , , , , ,
Gardner, Isabella Stewart, , ,
Gardner, Larry,
Gardner Museum, , ,
Garza, Matt,
Gedman, Rich, , ,
Gehrig, Lou,
Geiger, Gary,
Gibson, Bob, 161–62, ,
Gibson, Josh,
Gilchrist, Cookie,
_Girl Who Loved Tom Gordon, The_ ,
Goetz, Raymond,
Gold Glove Award, , ,
Gonzalez, Adrian, ,
Gooden, Dwight,
Gordon, Joe,
Gordon, Tom "Flash," ,
Gorman, Lou, , , , ,
Gossage, Rich,
Gowdy, Curt, , , ,
Graham, Moonlight,
Graham, Otto,
Green, Elijah Jerry "Pumpsie," , , ,
Green Bay Packers,
Green Monster, , , , , , , 138–41, , , , , , , , 232–33, 241–42, , , , , ,
Greenberg, Hank, ,
Greenwell, Mike, ,
Gremp, Buddy,
Griffey, Ken,
Griffin, Doug,
Griffith, Clark,
Griffith Stadium, ,
Grigas, Johnny,
Grimm, Charlie,
Grossfeld, Stan,
Grove, Lefty, , , , , ,
Grugg, Vean,
Guerroro, Vlad,
Guidry, Ron,
Guillén, Ozzie,
Gullett, Don,
Gustafson, Carl Einar,
Gutierrez, Jackie,
H
Hale, Arvel Odell,
Hall, Bill,
Hall of Fame, , , , , , , ,
Hallahan, John J.,
Hamilton, Jack, ,
Hardy, Carroll, , , 146–47
Hargrove, Mike, ,
Harlem Globetrotters, , ,
Harper, Harry,
Harper, Tommy, ,
Harrelson, Ken, , ,
Harridge, Will, ,
Harrington, John, , , 222–24, , , ,
Harris, Joe,
Harris, Mickey, , ,
Harris, Stanley "Bucky," , , , ,
Harvard College, , 29–30, , 49–50
Harvard Medical School, 48–49
Hatcher, Billy,
Hathaway, Donnie,
Haynes, Marques,
_Headin' Home_ ,
Heidenry, John,
Henderson, Dave, , , , ,
Henry, John, , , , 241–42, , 251–53, , , 268–71
Herman, Billy, , , , 166–67
Herrernan, Gladys,
Hershiser, Orel,
Herzog, Whitey,
Higgins, Mike "Pinky," , 135–37, , , , , , 166–67
Hildebrand, George,
Hillard, Leon,
Hobson, Butch, , , , 221–23,
Hoffman, Glenn, ,
Holbrook, Bob, ,
Holliday, Matt,
Holmes, Tommy, ,
Holy Cross, , , , ,
Homestead Grays,
Honeycutt, Rick, ,
Honolulu Surfriders,
Hooper, Harry, , 38–39, ,
Hooper, Henry,
Hoover, Herbert,
Hopkins, Lillian "Lolly," 72–73,
Horlen, Joel, ,
Horton, Willie,
"Hot Stove Cool Music," , ,
Houk, Ralph, , , , ,
House of David squad,
Hoyt, Waite, ,
Hubbard, Cal,
Hubbard, Freddie,
Hubbell, Carl,
Hughes, Dick,
Hughes, Roy,
Hughson, Tex, , ,
Hunter, Catfish, ,
Huntington Avenue Grounds, 24–26, , , , , ,
Hurst, Bruce, , ,
Hurwitz, Hy, , , , , ,
Hutchinson, Fred, ,
I
_I Hate You, Dad_ ,
Impossible Dream season, 156–57, , , , , ,
J
J. Geils Band, ,
Jablonowski, Pete,
Jackson, Reggie,
Jacobs Field, ,
James, Bill,
Janovitz, Bill,
Javier, Julian,
Jenkins, Ferguson,
Jensen, Jackie, , ,
Jeter, Derek,
Jethroe, Sam, , , , ,
Jimmy Fund, , ,
Johnson, Ban, , ,
Johnson, Bob,
Johnson, Darrell, , , , , 188–89, 194–95
Johnson, Ernie,
Johnson, Randy,
Johnson, Richard A.,
Johnson, Roy,
Johnson, Walter "Big Train," , ,
Jolley, Smead,
Jones, James Earl, ,
Jones, Leslie,
Jones, "Sad Sam," ,
Jurges, Billy, , , ,
Justice, David,
K
Kaat, Jim,
Kaese, Harold, , , 112–13, , , , ,
Kansas City Athletics, ,
Kansas City Royals, , , , , , , ,
Kasko, Eddie, , , 173–74,
Keane, Tom,
Keenan, "Chief" Johnny,
Kelleher, Smoky,
Kelley, David E.,
Keltner, Jen,
Kemp, Jack, ,
Kenneally, George,
Kennedy, Joe,
Kennedy, John,
Kennedy, John F.,
Kennedy, Kevin, 223–24, , 236–37
Kennedy, Ted,
Kerrigan, Joe, , , 240–41,
Kielty, Bobby,
Kiley, John, 130–31, ,
Killeen, John M.,
Kim, Wendell,
Kinder, Ellis, 113–15
King, B.B.,
King, Stephen, ,
Kirk, Roland,
Kline, Bob,
Knickerbocker, Bill,
Knight, Ray, ,
Knoblauch, Chuck,
Knowles, Darold,
Kosc, Greg,
Kowalski, Killer,
Kramer, Jack,
Kuhel, Joe,
Kuhn, Bowie, ,
Kurtz, Bob,
L
Lackey, John,
Lahoud, Joe, 173–74
Lamp, Dennis,
Landis, Kenesaw Mountain,
Langer, Gerd,
Lannin, Joseph John, 41–42, , , 50–51
Lansford, Carney,
Lardner, John,
Larkin, Barry,
Lazor, Johnny,
Leahy, Frank, ,
Leary, David,
Leary, Dennis,
Ledee, Ricky,
Lee, Bill, , , , ,
Lee, William Francis,
Lemon, Bob, , ,
Leonard, Buck,
Leonard, Hubert "Dutch," , ,
LeRoux, Buddy, , , , 216–17
Lester, Jon, , ,
Lewis, George "Duffy," 33–34, , 38–39, , 140–41,
Lindell, Johnny,
Little, Grady, , , , , , 270–71
_Little Big League_ ,
Lolich, Mickey,
Lombardi, Vince,
Lonborg, Jim, , , 156–57, , , , , ,
Longoria, Evan, ,
Los Angeles Dodgers, , ,
Louis, Joe,
Lovellette, Clyde,
Lowe, Derek, , , , , ,
Lowell, Mike, , , 262–63
Lowrie, Jed,
Lucchino, Larry, , 255–57, ,
Lynn, Fred, , , , 184–85, , , ,
Lyons, Louis M.,
M
Macfarlane, Mike, ,
Mack, Connie, ,
Madden, Michael,
_Major League II_ ,
Malden, Karl,
Malone, Sam "Mayday,"
Malzone, Frank, ,
Mann, Herbie,
Mann, Terrance,
Mantle, Mickey, ,
Manush, Henry "Heinie," ,
Maris, Roger, ,
Marshall, George Preston,
Martelli, Vin,
Martin, Billy, , , , ,
Martin, Ned, , , , , , ,
Martinez, Pedro, , , 227–28, , 234–35, , , , , , , , 270–71,
Martinez, Victor,
Matheson, Jack,
Mathewson, Christy, , , ,
Matsuzaka, Daisuke "Dice-K," , , ,
Matthews Arena,
Mauch, Gene,
Mays, Carl, 42–43, ,
Mays, Willie, , ,
McAleer, Jimmy, , , ,
McCarthy, Jim,
McCarthy, (Senator) Eugene,
McCarthy, Joe, , 112–15, , ,
McCartney, Paul, , , ,
McCaskill, Kirk,
McClelland, Tim,
McCovey, Willie,
McDermott, Mickey,
McDonough, Sean,
McDowell, Jack,
McGraw, John,
McGreevy, Michael T. "Nuf Ced," , , 44–45
McGwire, Mark,
McInnis, John "Stuffy,"
McKeon, Jack,
McLaughlin, James E., , ,
McMahon, Patrick,
McManus, Marty, , ,
McNally, Mike,
McNamara, John, , 207–8, , 216–17
McRoy, Robert,
Mellor, Dave, ,
Menino, Thomas M., , ,
Menosky, Mike,
Mercker, Kent,
Merritt, Jim,
_Merv Griffin Show, The_ ,
Meyers, Chief,
Mientkiewicz, Doug, , ,
Millar, Kevin, , , , ,
Miller, Rick, ,
Mills, Buster, ,
Milwaukee,
Milwaukee Braves, , ,
Milwaukee Brewers, , , , , 217–18,
Mingus, Charles,
Minneapolis Lakers,
Minnesota Twins, , 156–57, 159–61, 220–21, ,
Moeller, Danny,
Monbouquette, Bill, , , ,
_Moneyball_ ,
Monkees,
Montgomery, Bob,
Montgomery, Monty,
Montreal Expos, ,
Montville, Leigh, ,
Mooney, Joe, , ,
Moore, Donnie,
Moore, Gerry, , 85–86,
Morehead, Dave, , , ,
Moret, Roger,
Morgan, Joe, , 184–86, 196–97, , , , 220–22,
Morgan, Ray,
Moses, Wally,
Moulter, Nate,
Muchnick, Isadore,
Mueller, Bill, 250–51, ,
Mullin, Pat,
Munson, Thurman,
Murnane, Tim H., 29–30, ,
Murphy, Ray,
Museum of Fine Arts, 48–49
Musial, Stan,
N
Nance, Jim,
Napoli, Mike,
Nash, Peter,
_Natural, The_ ,
Neely, Cam,
Negro Leagues, , ,
Nelson, Willie,
Nettles, Graig, ,
New England Industrial League,
New England Patriots,
New Kids on the Block,
New York Bulldogs,
New York Giants, , 32–38, 41–42, , ,
New York Giants (football),
New York Highlanders, , , , ,
New York Mets, , , , ,
New York Yankees, , , , , 53–54, , 58–61, , , 70–71, , 77–78, 85–86, , , , , 112–15, , , , , , , 172–73, , , 191–93, , , , , 227–28, , , , 240–41, , , , , 255–58, 262–63, , ,
New York Yankees (soccer),
Newark Tornadoes,
Newman, Jeff, ,
Newport Jazz Festival, , ,
Niles, Harry,
Nipper, Al, ,
Nixon, Trot, , , 258–59,
_No, No, Nanette_ , ,
Nolan, Martin F., ,
Nomo, Hideo, ,
North, Billy,
Northeastern University, 48–49
Novello, Don,
Nuns' Day,
O
Oakland Athletics, , , , 195–96, , , , , , ,
Oakland Raiders,
O'Brien, Buck, ,
O'Connell, Cardinal William, , ,
O'Connell, Dick, , , , , 189–90, ,
O'Doul, Lefty,
Offerman, José,
Ojeda, Bob, ,
O'Leary, James, , , , 60–61, ,
O'Leary, Troy,
Olmsted, Frederick Law,
O'Neill, Steve, ,
O'Niell, Tip,
O'Reilly, Terry,
Orlando, Johnny, ,
Orr, Bobby, 266–67
Orsillo, Don,
Ortiz, David, , 250–51, 257–59, ,
Ostermueller, Fritz,
"Outlaw League,"
Owen, Spike, ,
Owens, Brick, ,
Ozawa, Seiji,
P
Pagliaroni, Jim, ,
Paige, Satchel, , ,
Palazzi, Togo,
Palmer, Jim,
Papelbon, Jonathan, , , , ,
Papi, Stan,
Parilli, Babe, ,
Park, Brad,
Parker, Charlie,
Parker, Kristin,
Parnell, Mel, , 114–15, , , , , , ,
Patterson, Tammy,
Pattin, Marty,
Paul, Billy,
Paveskovich, Maria,
Peck, Hal,
Peckinpaugh, Roger,
Pedroia, Dustin, , 261–63,
Pena, Carlos, 260–61
Pena, Tony,
Pennock, Herb, ,
Perez, Tony,
Perini, Lou,
Perkins, Anthony,
Perry, Ronnie,
Pesky, Johnny, , , , , , , 104–5, , , , , , , , , 178–79, , 216–17, , , 271–73
Pesky's Pole, , , , , ,
Petrocelli, Rico, , , 162–63, , , ,
Petty, Tom, ,
Phelps, Ken,
Philadelphia Athletics, 29–30, , , 41–43, , , , , , , , , , , , ,
Philadelphia Flyers, , 266–67,
Philadelphia Phillies, , , , 262–63
Philpott, A.J.,
Phish, ,
Pierce, Paul,
Piersall, Jimmy, , , , , ,
Pipgras, George,
Pitt, Brad,
Pittsburgh Pirates, , , , , ,
Pittsburgh Pirates (football),
Police, The, ,
Pollet, Howie,
Polo Grounds,
Pondering Judd,
Popowski, Eddie, ,
Port, Mike, ,
Posada, Jorge,
Pottsville Maroons,
_Practice, The_ ,
Progressive Party,
Prohibition,
Q
Quantrill, Paul,
Quinn, J.A. Robert, , , , , 65–66, 70–71,
Quinn, Jack,
Quinn, John,
R
Radatz, Dick, , , , ,
Raimo, George,
Ramirez, Manny, , , 140–41, , , 257–58, 260–61, , ,
Ramones,
Ramsey, Frank,
Ranana, Frank,
Reardon, Jeff,
Recchi, Mark,
_Red Sox Country_ ,
Reder, Johnny,
R.E.M.,
Remy, Jerry, , 192–93, ,
Renko, Steve,
Reynolds, Allie,
Rice, Grantland, ,
Rice, Jim, , , , , , , , , , , 208–9, 212–13,
Richman, Jonathan,
Ripken, Carl, Jr.,
Ritzgerald, Ray, ,
Rivera, Mariano, 250–51
Rivers, Mickey,
Rizzuto, Phil,
Robbins, Tim,
Roberto Clemente Field,
Roberts, Dave, , , ,
Robinson, Helen,
Robinson, Jackie, , , , , ,
Rodriguez, Alex, , ,
Rolling Stones, , , ,
Rollins, Rich,
Romero, Ed,
Roosevelt, Franklin Delano, , , , ,
Roosevelt, Theodore, , ,
Rosa, Francis,
Rose, Pete, , , ,
Ross, Michael, 48–49
Rothrock, Jack,
Royal Rooters, , 24–25, , , , 44–45,
Rudi, Joe, ,
Ruffing, Charles "Red," , ,
Runnels, Pete, ,
Ruppert, Jacob,
Russell, Jack,
Russell, Lefty,
Ruth, George Herman, Jr. "Babe," , , 38–39, 41–43, , 50–51, 53–54, , 59–61, 63–66, , ,
Ryan, Bob,
S
Saberhagen, Bret, ,
Sain, Johnny,
Sammartino, Bruno,
Sandler, Adam,
Santiago, Jose, ,
Santos, Gil,
Santosuosso, Ernie,
Saperstein, Abe,
Sarandis, Ted,
Sarandon, Susan,
Sarducci, Guido,
Sargent, John Singer,
_Saturday Night Live_ ,
Sawyer, Ford, ,
Scarritt, Russ,
Schacht, Al,
Schang, Walter "Wally," , , ,
Schilling, Chuck,
Schilling, Curt, , , , ,
Schiraldi, Calvin,
Schmeling, Max,
Schourek, Pete,
Scott, Everett, ,
Scott, George, , , , ,
Scott, Rodney,
Scully, Vin,
Scutaro, Marco,
Seattle Mariners, , 205–6, 216–17, ,
Seaver, Tom,
Seerey, Pat,
Seibu Lions,
Selig, Bud,
Selkirk, George,
Sewall's Point,
Sewell, Truett Banks "Rip,"
Sexton, John S.,
Shamrocks,
Shaner, Wally,
Shankar, Ravi,
Shaughnessy, Dan, , , , , , , ,
Shawkey, Bob,
Shea, Kevin,
Shea Stadium, ,
Shibe Park,
Shonta, Chuck,
Shore, Ernie, 41–43, , ,
Siebert, Sonny,
Sinatra, Frank, ,
Slaughter, Enos, ,
Smith, Dave,
Smith, Janet Marie,
Smith, Kate,
Smith, Lee,
Smith, Reggie, , ,
Smith, Zane,
Snelgrove, Victoria, ,
Snider, Duke,
Snodgrass, Fred, ,
Soar, Hank,
Songini, Marco,
Sonic Youth,
Sosa, Ruben,
Sosa, Sammy, ,
South End Grounds, , 45–46
Spahn, Warren, ,
Speaker, Tris, , 33–34, , 38–39,
Spofford, Josh,
Spoljaric, Paul,
Springstead, Marty,
Springsteen, Bruce, , , 270–71
St. Louis Browns, , , 60–61, , , , ,
St. Louis Cardinals, 104–5, , , 161–62, , , , ,
Stahl, J. Garland "Jake," , , , ,
Stairs, Matt,
Stallard, Tracy,
Stallings, George "Miracle Man,"
Standells,
Stanley, Bob, , ,
Stanley, Henry,
Staples Singers,
Stapleton, Dave, , ,
Stark, Jayson,
Steinberg, Charles, ,
Steinbrenner, George, , , , ,
Stengel, Casey,
Stephens, Gene,
Stephens, Vern, , ,
Stewart, Dave, ,
Storyville,
Stout, Glenn,
Stram, Hank,
Strunk, Amos, ,
Stuart, Dick,
Sturm, Marco, 266–67
Sullivan, Billy,
Sullivan, Ed,
Sullivan, George, , ,
Sullivan, Haywood, , , , , , 200–01, , , , ,
Sullivan, Joseph "Sport,"
Sullivan, Patrick,
Sutherland, Kiefer,
"Sweet Caroline," , , ,
Symphony Hall,
T
Tabor, Jim, ,
Tampa Bay Rays, 240–41, , 260–63, , ,
Tatum, Reece "Goose,"
Taylor, Benjamin,
Taylor, Charles, ,
Taylor, James,
Taylor, John I., , , 26–27, , , , , ,
Taylor, Robert,
Taylor, Zach,
Tebbets, Birdie,
"Tessie," , , 44–45,
Texase Rangers,
Thomas, Bud,
Thomas, George, ,
Thomas, Gorman,
Thomas, Jack, ,
Thome, Jim, ,
Thomson, Bobby,
Tiant, Luis, , 172–75, , ,
Tiger Stadium, , ,
Tillman, Bob, ,
Timlin, Mike, 260–61
_Titanic_ , 24–25,
Tobin, Maurice J., ,
Topel, Brett,
Topping, Dan,
Toronto Blue Jays, , , , , , 219–22, ,
Toronto University,
Torre, Joe,
Torrez, Mike, 191–93
_Town, The_ , ,
Tropicana Field, 261–62
Trout, Dizzy,
Tschida, Tim,
Tudor, John,
Tyler, Barbara,
U
U,
Umphlett, Tom,
Updike, John, , , , 144–45, ,
V
Vache, Tex,
Valdez, Sergio,
Valentin, John, , ,
Vanderwarker, Peter,
Varitek, Jason, , , , , , 258–59, , ,
Vaughan, Sarah,
Vaughn, Maurice Samuel "Mo," , 236–37
Vaughn, Mo, , 222–24, 227–28,
Veach, Bobby,
Veeck, Bill,
Verducci, Tom,
Vodoklys, Michael,
Vollmer, Clyde "Dutch the Clutch," 120–21
Voltaggio, Vic,
Vosmik, Joe, , ,
W
Wagner, Charles "Heinie," , , , ,
Wagner, Hal,
Wakefield, Tim, , , , , , 250–51, , ,
Walberg, Rube,
Walker, Chet,
Walker, Harry,
Walker, Jimmy, ,
Walker, Larry,
Wally,
Walsh, David Ignatius, ,
Waltham High,
Ward, Hugh, , ,
Waseleski, Chuck,
Washburn, Jarrod,
Washington Redskins,
Washington Senators, , , , 41–43, , , , , , 74–75, , , , , 120–21, , , ,
Waslewski, Gary, ,
Watson, Jerry,
Weaver, Jered,
Webb, Earl,
Webb, Melville E., Jr.,
Webb, Melville E., Jr. "Mel," , , ,
Welch, Bob,
Welles, Orson, ,
Werber, Bill,
Werner, Tom, , , ,
Wert, Vic,
West, Joe,
White, Kevin,
White, Sammy,
Whiteman, George,
Whitman, Burt,
Whitt, Ernie,
Will, George,
Williams, Dick, , , , , 161–62, 165–67, ,
Williams, Jimy, 227–28, , , , , ,
Williams, John,
Williams, John Henry,
Williams, Ken,
Williams, Marvin, ,
Williams, Ted, , , , , 83–87, 89–90, 92–94, 97–99, 101–2, 104–6, , , 112–13, , 117–18, 120–21, 124–26, 132–37, 140–45, , , , , , , , , 202–3, , 216–17, , , , , 270–71,
Willoughby, Jim, ,
Wills, Maury,
Wilson, Dan,
Wilson, Earl, , ,
Wilson, Jack,
Wilson, Mookie, ,
Winter Classic, , 266–67,
Wise, Rick, ,
Witt, Mike,
Wolf, Peter,
Wonder, Stevie, ,
Wood, Howard "Smoky Joe," , 32–34, ,
Wood, Wilbur,
Woodford, Helen,
World Series, , , , 32–39, 41–45, , 50–51, , , , , , , 110–13, , , , 140–42, 167–69, , , 180–81, 184–87, 194–95, , , , , 251–53, , , 271–2
Wray, Lud,
Wright, Chase,
Wright, George,
Wrigley, Phil K.,
Y
Yankee Stadium, , , ,
Yastrzemski, Carl, , , , 141–43, , , , 153–54, , 159–61, 165–67, , 173–75, 178–79, , , , , 193–95, , 201–2, 216–17, ,
Yastrzemski, Mike,
Yawkey, Bill,
Yawkey, Elise,
Yawkey, Jean R., , , , 178–79, , , , , , , , ,
Yawkey, Thomas Austin, , , , , , , , , , , 77–79, 83–85, , , , , , 120–21, , , 135–36, , , 147–49, , , , , , 174–75, , 189–90, , 201–3, , ,
Yawkey Way,
Yerkes, Steve, ,
York, Rudy, , , ,
Youkilis, Kevin, ,
Young, Cy, ,
Young, Kent,
Young, Matt,
Z
Zarilla, Al,
Zimmer, Don, , , , , , , , , , ,
Zurbaran, Francisco de,
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,592 |
\section{Introduction}
Several fields such as industrial, military, scientific and civil have chosen to make use of computer vision in order to recognize the existence of objects and their location, among other features; most of these systems need a personal computer and the execution of the software that processes the image data.
Applications such as unmanned vehicle systems, autonomous robots, among others, have limitations of space, consumption, robustness and weight, making the use of a personal computer to be impractical, or requiring complex and expensive methods to transmit the image to a fixed station that processes the image and re-transmit the data interpretation.
Reduced size systems have been implemented on commercial boards such as the so called Cognachrome Vision System but it requires an external camera connected to a RCA protocol adapter \cite{Sargent}, yet another similar work was made by a team at Carnegie Mellon University \cite{Rowe} but it lacks of an embedded user interface and costs more than the development proposed in this document, which is a compact embedded vision system, lightweight, with a low power consumption, and written in widely used C/C++ language it handles: the image acquisition, processing, and a user interface altogether on a board and camera that are roughly USD 90 in price.
The proposed system intends to be a cheaper, easy to replicate, and yet a viable and modern alternative to the ones researched by A. Rowe et al. \cite{Rowe}, R. Sargent et al. \cite{Sargent}; its further development could contribute to applications that require to detect, locate and/or track a color object and have strong limitations in: size, power consumption and cost.
\section{Capturing and display of the image}
This project uses a HY-Smart STM32 development board, it includes a STM32F103 microcontroller to process data, it gets the image from an OV7725 camera that is configured in RGB565 format, with a QVGA(320x240) resolution. It also includes a touch screen in which the target object can be selected, its color defines the threshold that is used to create a binary image in the process of artificial vision known as segmentation. After the segmentation is done, an algorithm recognizes the contour of the image and its center, once located, a PID algorithm commands 2 servos (pan, tilt) in order to track the objective.
\subsection{Image acquisition}
The project make use of the OV7725 camera in a RGB565 format, which employs 2 bytes per pixel, allocating 5 bits for red, 6 for green, and 5 for blue as seen on Fig.~\ref{fig:rgb565}. An individual frame has 320x240 pixels of information which are constantly been sent to a FIFO memory named AL422B; the microcontroller accesses this data when required, instead of receiving periodic interruptions from the camera.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{rgb565}
\caption{RGB565 format contains 16 bits of information per pixel.}
\label{fig:rgb565}
\end{figure}
\subsection{Displaying the image}
A TFT-LCD screen of 320x240 pixels displays the image, it is operated by the SSD1289 integrated circuit that communicates with the microcontroller through 8080 parallel protocol. A resistive film above the screen, in conjunction to the XPT2046 integrated circuit, locates the position of a single pressure point on the screen and sends the data via SPI interface.
The microcontroller has a peripheral block called FSMC (Flexible Static Memory Controller) which allows it to communicate with external memories meeting the timing requirements, previously some parameters must be set: the type of memory to be read (SRAM, ROM, NOR Flash, PSRAM), data bus width (8 or 16 bits), the memory bank to be used, waiting times, among other features.
The use of the abovementioned integrated circuits allows the microcontroller to seamlessly read and write the camera and display respectively, and allows a user interface as depicted on Fig.~\ref{fig:lcd}, although the display could be discarded in order to increase frames per second, and decrease cost, and weight.
\begin{figure}[h]
\centering
\includegraphics[width=0.95\linewidth]{lcd}
\caption{Image of the interface and user interaction with the screen.}
\label{fig:lcd}
\end{figure}
\section{Image processing}
The 76800 pixels contained in each frame need to be processed in order to detect and locate the color object, this task is described as segmentation of the image. Once located a PID controller centers the field of view of the camera on the center of the region of interest.
\subsection{Image segmentation}
The segmentation consists on separating the region of interest in the image based on the chosen color. As each pixel is obtained from the camera, it is compared with a threshold value for each channel, and the result is stored into a binary image.
The binary image is allocated in memory as an array of 2400 X 32bit numbers where each bit is a pixel of the binary image (see Fig.~\ref{fig:segmentated}), the color boundaries can be selected via the interface, two approaches are being considered: RGB color space and normalized RGB.
\begin{figure}[h]
\centering
\includegraphics[width=0.65\linewidth]{segmentated}
\caption{On the left the original image, on the right the binary image.}
\label{fig:segmentated}
\end{figure}
\subsubsection{RGB color space}
Maximum, and minimum boundaries are set for each of the three color components (red, green and blue), if the scanned pixel is within the 3 ranges, then it is stored as "one" in the binary image, otherwise is a "zero". This computing is fairly fast, achieving 10,2fps; but a disadvantage arise in the event of a change of illumination, for instance if light intensity is decreased the red, blue and green components vary in proportion to this change, and can get out of the threshold, the same occurs with an increase in light intensity.
\subsubsection{Normalized RGB}
In this color space, instead of using directly each RGB component, the proportion \textit{rgb} is calculated by dividing the luminance \textit{I} of every single pixel \cite{Balkenius}.
\begin{align}
I &=R+G+B \\
r &=R/I,g=G/I,b=B/I \
\end{align}
As the name suggests, in the normalized RGB the summation of rgb components equals to one, due to this only r and g values are calculated to attain the hue information, this results in the rg chromaticity space seen on Fig.~\ref{fig:spaces}, which is bi-dimensional and theoretically invariant to changes of illumination.
\begin{figure}[h]
\centering
\includegraphics[width=0.65\linewidth]{spaces}
\caption{On the left RGB color space, on the rg chromaticity space.}
\label{fig:spaces}
\end{figure}
The invariance mentioned can be noted in the example given on Fig.~\ref{fig:spacestest} where three pixels o an orange sphere are evaluated, the R component varies along this points in nearly half of its value, selecting a threshold in RGB color space would neglect a considerable part of the sphere due to the large of R. However, while the RGB values vary, their proportions with respect of the intensity (I) keep the same, thus the values of rgb are invariant to the distinct illumination levels. An experimental result is documented on Figure~\ref{fig:testseg}
\begin{figure}[h]
\centering
\includegraphics[width=0.65\linewidth]{spacestest}
\caption{Three different color pixels are chosen from an orange sphere, rgb components are computed on each case.}
\label{fig:spacestest}
\end{figure}
\subsection{Description of the region of interest}
To analyze the data incoming from the image segmentation, an algorithm demarks the contour of the group of contiguous pixels in the binary image, once this is done it establish the upper, lower, rightmost, and leftmost limits, and also both horizontal and vertical location of the center of the object.
The algorithm starts by scanning the binary image from the top left corner, to the right and downwards until it find a line of contiguous pixels, if it exceeds a preset width then it finds the rightmost pixel of the grouping and proceeds to find the pixels of the contour.
\begin{figure}[h]
\centering
\includegraphics[width=0.65\linewidth]{roi}
\caption{Example of contour recognition for a group of pixels.}
\label{fig:roi}
\end{figure}Fig.~\ref{fig:roi} shows how the algorithm runs along the contour of a sector of detected pixels , the process find the initial line ( in this case the first line is a single pixel width, and is shown in red color), from here the contour path begins, as a rule the algorithm begins to search for the next valid counter-clockwise pixel in a 3x3 matrix, initiating the inspection from the next position to the last sensed pixel. To continue the contour detection , the center of the next matrix is at the position of pixel detected earlier. Whenever a contour pixel is detected it is evaluated to update the upper, lower, rightmost, and leftmost limits of the grouping of pixels. The contour inspection stops once the initial pixel is reached.
\subsection{Tracking}
The camera is located in the top of a Pan-Tilt platform represented on Fig.~\ref{fig:servos}, in order to perform the tracking movements two servo motors are installed: HS-785HB servo motor (located at the bottom of the platform) and the HS-645MG (located on top).
\begin{figure}[h]
\centering
\includegraphics[width=0.4\linewidth]{servos}
\caption{Representation of the Pan-Tilt platform holding the camera.}
\label{fig:servos}
\end{figure}
The control algorithm that governs the movements of both servos is a proportional-integral (PI) controller. Although the system is composed of servomotors, a modeling of the dynamic response was made in order to represent the platform with the camera installed, the result is a first-order transfer function (Gp) whose parameters: gain (K) and time constant (\(\tau\)) are experimentally acquired.
\subsubsection{Closed-loop analysis }
The PI controller was chosen to eliminate the offset error, a block representation of the system in continuous time domain is depicted on Fig.~\ref{fig:loop}.
\begin{figure}[h]
\centering
\includegraphics[width=0.75\linewidth]{loop}
\caption{Simplified block diagram of the closed loop system.}
\label{fig:loop}
\end{figure}
The vision system gets feedback through the position of the detected object relative to the camera, the controller will govern the PWM that moves the servomotors in order to move the camera so its center is aligned with the object's center, both variables are relative to the angular location of the camera, so the error signal is invariant to the absolute angular location of the platform, in other words the error is just measured as the difference between the detected position and the center of the camera's field of view.
The block diagram of Figure ~\ref{fig:loop} can be simplified into a single block to obtain the following transfer function representing the whole system:
\begin{align}
Tf(s)& =\frac{Gp(s)\times Gc(s)}{Gp(s)\times Gc(s)+1} \
\end{align}
Further calculation leads to the following second order transfer function:
\begin{align}
Tf(s)=\frac{s\times (Kp\times K)+Ki\times K}{s^2\times \tau+s\times (1+Kp\times K)+Ki\times K} \
\end{align}
The poles of the closed loop system are given by the roots of the polynomial on the denominator\cite{Ogata}:
\begin{align}
p(s)=s^2+s\times (2 \xi \omega_n)+\omega_n^2 \\
ts=\frac{4}{\xi\times\omega_n}\
\end{align}
The damping factor (\(\xi\)) and natural frequency (\(\omega_n\)) determine the percentage overshoot (PO) and settling time (ts) present in the transient response of a step input [4]:
\begin{align}
PO=100\% \times e^{\frac{-\xi\times \pi}{\sqrt{1-\xi^2}}} \
\end{align}
The controller's constants (Kp and Ki) can be solved performing the replacements required in Equations (4), (5), (6) and (7).
\begin{align}
Kp=\frac{1}{K}\times(\frac{8\times \tau}{t_s-1})\
\end{align}
\begin{align}
Ki=\frac{1}{K}\times \frac{16 \times \tau}{(t_s)^2\times \frac{ln(\frac{PO}{100\%})^2}{\pi^2+ln(\frac{PO}{100\%})^2}}\
\end{align}
\subsubsection{Discretization of the controller }
The described PI controller (Gc) is expressed in continuous time-domain. To make an algorithm executable by the microcontroller the PI controller must be discretized.
This is achieved by the bilinear transformation function Equation (10) which transforms the transfer function from continuous time domain to the discrete time domain \cite{mit}:
\begin{align}
G(z)= G(s)|_{s=\frac{2(z-1)}{T(z+1)}}\
\end{align}
Where T is the sampling time. Applying the transformation to the PI controller we obtain the following transfer function:
\begin{align}
Gc(z)=Kp+\frac{Ki\times T(z+1)}{2(z-1)} =\frac{U(z)}{E(z)}\
\end{align}
The discrete time domain controller can be expressed in a single line mathematical operation, thus finally obtaining the control law:
\begin{multline}
U_{[k]} =U_{[k-1]} +E_{[k-1]} \times (Ki\times \frac{T}{2}-Kp)+E_{[k]} \times (Ki\times \frac{T}{2}+Kp) \
\end{multline}
Where U[k] is the instantaneous value of the control action (value servomotor PWM pulse), U[k-1] the previous value, the error E[k] is the difference between the center of the camera and the center of the object; the controller updates this values every time a frame is acquired which occurs at sampling a time T, on the other hand Kp and Ki are constants determined by the equations (8) and (9) respectively.
\section{Tests and results}
\subsection{Image segmentation }
The transformation to the rg chromaticity space gives better segmentation results as can be seen on Fig.~\ref{fig:testseg}, nevertheless the calculation of each pixel takes more time than RGB color space, resulting in a relatively slower frame rate of 10fps. For the rest of this tests rg chromaticity segmentation is chosen, as it is more reliable.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{testseg}
\caption{(Left) An orange sphere is illuminated at 4 poor intensity scenarios. (Center) Results of RGB565 segmentation. (Right) Results of the rg chromaticity segmentation. (the color to detect was chosen during the highest level of illumination for both types of segmentation)}
\label{fig:testseg}
\end{figure}
\subsection{Detection of distinct color objects}
Figure~\ref{fig:testobjects} shows distinct objects whose color is not much different from each other. The image has: a red cloth, a small yellow sphere, a large orange sphere, and an envelope of pale yellow. In each of the 4 experiments the respective color is selected, and the location of the objects is performed appropriately.
\begin{figure}[h]
\centering
\includegraphics[width=0.5\linewidth]{testobjects}
\caption{Different objects of similar color are being recognized.}
\label{fig:testobjects}
\end{figure}
\subsection{Location and tracking}
An orange object was attached to a coupled shaft which a circular motion, similar to a clock (Figure~\ref{fig:testlocation}). Tracking is disabled and the object's position is measured as pixels, which returns a circle with a mean radius of \textit{R=87.57} pixels and a standard deviation \textit{$\sigma$ =8.69}, this along with the observed plot suggests that location data incorporates some glitches that can be addressed to partial unrecognized regions thus computing a different centroid.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\linewidth]{testlocation}
\caption{(Left) An Orange triangle making a clockwise motion. (Center) Location at 3820ms per revolution, (Right). Location at 1108ms per revolution.}
\label{fig:testlocation}
\end{center}
\end{figure}
However this doesn't cause an instability when the servos are activated for tracking, as demonstrated in a test where a color object was chosen, brought to a corner of the camera's range of vision and then tacking was activated enabling the servomotors to move the center of the camera to the object's location, the behavior can be observed on Fig.~\ref{fig:testtrack}.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth]{testtrack}
\caption{Tracking an object from a corner to the center of the field of view.}
\label{fig:testtrack}
\end{figure}
Position of the object settles to the center after roughly 1,6 seconds after tracking activation, similar results were obtained on latter tests.
The whole system is displayed on Figure~\ref{fig:real}.
\begin{figure}[h]
\centering
\includegraphics[width=0.6\linewidth]{real}
\caption{Physical implementation the system.}
\label{fig:real}
\end{figure}
\subsection{Cost and power consumption}
The data presented on Table. \ref{table:price} and \ref{table:parameter}, include the system operating with both the LCD screen, and the pan and tilt platform; even though it is totally functional without them.
\begin{table}[H]\centering
\begin{tabular}{|l|l|ll}
\cline{1-2}
\textbf{Product} & \textbf{Price {[}USD{]}} & & \\ \cline{1-2}
HY-Smart STM32 (STM32F103VCT+TFT LCD+Board) & 59.00 & & \\ \cline{1-2}
OV7725 camera + AL422B FIFO Module & 30.00 & & \\ \cline{1-2}
SPT200 Direct Drive Pan \& Tilt System & 45.99 & & \\ \cline{1-2}
\textbf{Total} & \textbf{134.99} & & \\ \cline{1-2}
\end{tabular}
\caption{Price of components for the proposed Low-Cost embedded vision system}
\label{table:price}
\end{table}
With the listed price of 134.99 USD the system is cheaper than the one proposed in the work of A. Rowe (199 USD) \cite{Rowe}, unfortunately a comparison with the Cognachrome Vision System \cite{Sargent} is not possible as the price-tag of their system is not publicly available, on the latter a clear strength arises from the fact that the proposed system is made from already available and cheap components in the consumer market. Also it is worth noting that the system can be greatly reduced on its components being STM32F103VCT the main component at a price of 9.07 USD, which is capable enough to process image data when compared to newer architectures and solutions such as the well-known Raspberry PI, which costs 35 USD.
\begin{table}[H]\centering
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Item} & \textbf{Average} & \textbf{Max} & \textbf{Unit} \\ \hline
Supply voltage & 5 & 5 & V \\ \hline
Operating current & 200 & 1100 & mA \\ \hline
Frequency & 72 & - & MHz \\ \hline
Start-up time & 500 & - & mS \\ \hline
RS-232 bit-rate & 9600 & 921600 & bps \\ \hline
Refresh rate & 10.9 & - & fps \\\hline
\end{tabular}
\caption{Electrical characteristics of the proposed Low-Cost embedded vision system}
\label{table:parameter}
\end{table}
The power consumption is 1 Watt on average, therefore it can be operated in the scope most autonomous systems; with a refreshing rate of 10.9 fps and standard RS-232 output of data, it could be easily implemented on industry processes such as: fruit classification, object location, and others. Much of the weight, power usage, and overall dimensions can be further lowered without the servomotors and LCD depending on the application.
\section{Conclusions}
The deterring effects of uncontrolled illumination are greatly diminished by the use of the rg chromaticity space enabling this system to detect, locate, and track a colored object satisfactorily, while being low-cost (under 200USD), compact (30x13x19cm including the platform), and energy-saving (200mA on average at 5V).\\
The ability of this system to recognize chromaticity along with location data can be greatly improved in a controlled environment making it a suitable and economic option for industrial applications; according to the requirements additional work is needed for the system to locate multiple objects at the same time, and more tasks that other systems achieve running operating systems, nonetheless, this system present a significant reduction in cost, size and power consumption, which makes it viable to be fitted on small unmanned vehicles.\\
Further development of this work can be done to increase the frames per second rate, both the camera and microcontroller have newer versions in the market by the date, all the algorithms used in this work are written in C++ so they can be implemented in other systems as well and become adaptable to variety of requirements.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,480 |
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The print speed is rather slow at 2 in\/s and it takes 5 seconds to reposition the print head to...\n##### 0q Achul ph pencent dkeocnton dcctr acid sohaion wtat & t Lookng at tle theoretialpH ofa 0.010 ofaccli acul in ths sohtin?aci i 4 0.00010 aceti: acid counan wrh the percent dissocition of accte: How does tlb sohutk?weak acid u5 becons more dirte, nenccn dissocuton ofa What ou say about th: trenl\n0q Achul ph pencent dkeocnton dcctr acid sohaion wtat & t Lookng at tle theoretialpH ofa 0.010 ofaccli acul in ths sohtin? aci i 4 0.00010 aceti: acid counan wrh the percent dissocition of accte: How does tlb sohutk? weak acid u5 becons more dirte, nenccn dissocuton ofa What ou say about th: tre...\n##### Can ask help for these these are ethical issues in ICT, explain briefly and provide 2 examples for each Exploitation 2. plagiarism 3.libel 4. software piracy thank you so much\ncan ask help for these these are ethical issues in ICT, explain briefly and provide 2 examples for each Exploitation 2. plagiarism 3.libel 4. software piracy thank you so much...\n##### Consider the amino acid Valine. pKai 2.286 and pka2-9.719. (Show your work) Calculate the pH ofa 0.1OOM solution of the fully protonated form HzV*? (Tip: Quadratic Equation) Calculate the pH ofa 0.1OOM solution of the single protonated form HV? Calculate the pH ofa 0.1OOM solution of the fully deprotonated form V?\nConsider the amino acid Valine. pKai 2.286 and pka2-9.719. (Show your work) Calculate the pH ofa 0.1OOM solution of the fully protonated form HzV*? (Tip: Quadratic Equation) Calculate the pH ofa 0.1OOM solution of the single protonated form HV? Calculate the pH ofa 0.1OOM solution of the fully depro...\n##### E 4-income statement presentation \u0bae\u0bc7\u0b9f\u0bcd\u0b9f\u0bae\u0bc7\u0b9f The following correct income was prepared to the Al Corporation ALL...\nE 4-income statement presentation \u0bae\u0bc7\u0b9f\u0bcd\u0b9f\u0bae\u0bc7\u0b9f The following correct income was prepared to the Al Corporation ALL CORPORATION Fordeler med het, Red 32.000 Sales Inse Galeon sale of investments 70.00 Ege SO 60 2000 Sellingen Adepende In open Redrag 16.000 ...\n##### Chapter 11: Nutrition Case Study Joan is an 82 year old woman whose husband died last...\nChapter 11: Nutrition Case Study Joan is an 82 year old woman whose husband died last year. Now, she lives alone in a small apartment with no money for \"luxuries\". She has no family or friends in the area having relocated for her husband's job. She takes medication for type 2 diabetes bu...\n##### 1) Complete the table below in your notebooks and as a pre-lab. 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Outline the procedure for taping a horizontal distance over sloping ground, including how the tape is aligned, how intermediate stations are marked, and how the final length measurement is completed....","date":"2022-05-23 13:49:26","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.4457571804523468, \"perplexity\": 10383.450189426983}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662558030.43\/warc\/CC-MAIN-20220523132100-20220523162100-00737.warc.gz\"}"} | null | null |
Chances are that if you don't play Minecraft yourself, you've probably at least heard of it or know someone who does. The charming indie game has invaded nearly every facet of pop culture, casting its blocky spell on everything from Legos to feature films and has even been used for tourism. With over 54 million copies sold, Sweden-based developer Mojang made $128 million last year largely thanks to Minecraft and has become one of the most successful game studios in the world. But what actually is Minecraft? We're glad you asked!
At its core, Minecraft is a massive, open-ended, first-person game with a focus on exploration and crafting.
Every time you load a new game, the environment is randomly assembled so no two plays will be exactly alike. Unlike most games, Minecraft offers little in the way of directions, or a campaign/story mode to work through for that matter: It's a free-form, easygoing affair with the player figuring out what they can and can't do through trial and error (or by scouring a wiki). Think something like Grand Theft Auto's gigantic environment, but instead of attempting a hyper-realistic world, everything is pixelated blocks.
Objects in the world are made of gravity-defying, 1 x 1 blocks that can be stacked and manipulated to form just about anything one could imagine, from intricate recreations of Game of Thrones' Westeros, to movie posters, and even Game Boy emulators capable of playing the first level from Super Mario Land. Hell, the Danish government has servers running so would-be tourists can check out a 1:1 recreation of the happiest country replete with highways, houses and landmarks.
That's just the tip of the iceberg, though, and we're sure there are at least a few crazy projects going right now that we don't even know about.
There's no score, and no real "end" (though that's up for debate, and we'd be spoiling bits if we got into the argument too deeply here). In the game's main mode, you start in a world full of resources (rock, wood, etc.) and are "tasked" with making a life for your character (who's named Steve). A day/night cycle provides constraints: use the daylight to gather resources and build, with the intent of surviving the night.
Zombie-like creatures roam the land at night, and the only way to survive is by building housing to keep them out. Should that not provide enough challenge, a green enemy lovingly named the "creeper" lives primarily underground, where you mine for resources. Enter the wrong mining cavern and you may end up suddenly exploded. Any resources you've got on your person remain where you died until you can go retrieve them. It's a delight!
The game is available on just about every platform: Android, iOS, Mac, PC, PlayStation 3 and Xbox 360, with updated versions coming to the PS Vita, PlayStation 4 and Xbox One this year. Minecraft was initially released for free as a work-in-progress back in May 2009 and developed by one person, Markus "Notch" Persson. Since then, numerous updates have been released, with the full release coming about two-and-a-half years later in November 2011. Essentially, the public was playing along as the game was being developed under its very fingertips.
For starters, single-developer games are pretty rare, and ones that are this successful are even more unique. As a result, Notch has become a bit of a celebrity in the gaming community and now has some 1.7 million followers on Twitter. Minecraft's success, however, has had a price. After filing for a trademark for Mojang's follow-up, Scrolls, publisher Bethesda Softworks (known for the role-playing series The Elder Scrolls, among others), filed a trademark lawsuit over the Swedish developer using the word "scrolls." It all worked out in the end, but Mojang had to agree to not use the word in subsequent releases.
Minecraft is also the progenitor of releasing a game to players before it's done. The concept of PC-gaming platform Steam's Early Access program practically owes its existence to this, and it isn't going to stop there either. Sony has admitted that it's flirting with the idea of releasing unfinished, alpha versions of games on the PS4, too.
Notch's baby has also had a tremendous impact on video games as a whole, creating an entire genre and style of play. Titles like Rust (which also happens to be a Steam Early Access title), Terraria and the upcoming PS4 stunner No Man's Sky likely wouldn't exist had Minecraft not popularized the idea of virtual free-form exploration and building. Even established franchises like Everquest have taken notice, with the next game, Landmark, taking a few pages out of Minecraft's customization and crafting book.
The game is also a blank canvas that can be used for just about anything. Sure, recreations of Great Britain are impressive, but even more so is that it's been used in the classroom as a teaching tool for proper online behavior and collaborative problem-solving. It's even been implemented to get kids interested in architecture and civics.
Getting access to a game before it's fully finalized might sound like a great idea on paper, but in practice that's not always the case. There are numerous games on Steam right now under the Early Access banner that are simply unplayable. Whereas Minecraft was free to start, people are paying for these test-builds (which will convert into the full version if completed) in the hopes that eventually the full release will fix the gamut of glitches they're encountering. The thing is, that's placing an awful lot of faith in oftentimes unproven developers to finish a game; there have already been notable disasters delisted from Steam, and there will assuredly be more.
If you can put the pickax down long enough, check out Rolling Stone's recent profile of Notch that chronicles the effects his youth and father's suicide had on both the way he designs games and him as a person. Should you want even deeper inside the man's head, Persson also maintains a personal blog. Still not satisfied? How about booking travel to Europe for this year's Minecon convention? Better gear up with a Creeper mask and foam diamond-sword ahead of time, though. Or, maybe you haven't played the game just yet and all of this has gotten you curious to try it out. Well, there's a super-limited free demo that should give you an idea of what it's all about before you buy the real deal. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,215 |
\section{Introduction}
\paragraph{\bf Motivation for electric storage ring ``traps'' for
electrons or protons.}
The U.S. particle physics community has recently updated its vision of the future
and strategy for the next decade in a Particle Physics Project Prioritization
Panel (P5) Report. One of the physics goals endorsed by P5 is measuring
the EDM of fundamental particles (in particular proton, deuteron, neutron and
electron).
Since Standard Model EDM predictions are much smaller than current experimental
sensitivities, detection of any particle's non-zero EDM would
signal discovery of New Physics. If of sufficient strength, such a source could
support an explanation for the observed matter/antimatter asymmetry of our
universe. A proton EDM collaboration\cite{pEDM-PRL} has
proposed a storage ring proton EDM measurement at the unprecedented level of
$10^{-29}e \cdot\,$cm, an advance by nearly 5 orders of magnitude beyond the
current indirect bound obtained using Hg atoms.
The proposed EDM measurement is based on the accumulation of tiny
``wrong-plane'' (i.e. ``right'' for EDM, ``wrong'' for MDM)
spin precessions that can be accounted for only by a non-zero EDM.
Polarized beams can be frozen (for example in longitudinally-polarized
states) in storage rings containing
appropriate combinations of electric and magnetic bending elements;
the relative amount depends on the magnetic dipole moment of the
particles being stored. Only for a few particles,
of which the electron and the proton are the most important, can the spins be
frozen in purely electric rings. This is highly advantageous, since
such rings support beams circulating both clockwise and counter-clockwise,
permitting measurements for which important systematic errors cancel.
The electric dipole moment (EDM) of the proton is known to be so small
that, in order to reduce systematic errors enough to measure it in a
storage ring requires measuring differences between counter-circulating
beams. This can be done sequentially or, to provide differential beam position
precision at the cost of beam-beam complications, with simultaneous
circulating beams. (Direct colliding beam-beam effects are calculated
to be negligible.)
A comparably important advantage of electric bending is that the
absence of intentional magnetic fields will reduce the presence of
unintentional (radial) magnetic field components, which
are expected to be the dominant source of spurious precession,
mimicking the EDM effect.
To freeze the spin procession, conventional storage ring bending magnets
are replaced with the corresponding electric elements. Though the circulating
particles
are constantly being attracted to the inner electrode, their centrifugal force
causes them to circulate indefinitely in a more or less circular orbit.
This establishes the storage ring as an electric ``trap'', albeit for an
intense moving bunch rather than for a few slow particles.
In a storage ring EDM measurement, bunches of longitudinally
polarized protons will circulate for ``long'' time intervals
such as 1000 seconds. Because of inevitable parameter spreads, individual particle
spins will precess differently and, after a
spin coherence time \emph{SCT}, the
beam polarization will have been attenuated (due to decoherence) to a point where
the EDM precession rate has become unmeasurably small. The wrong-plane polarization
difference between early and late times, when ascribed to the torque of
the bend field acting on the electric dipole moment, will provide a measurement
of the proton EDM.
For measuring EDM's the accumulation and measurement of small effects
requires analysis, mitigation and control of various systematic errors.
Issues to be studied with ETEAPOT include:
Radial B-field: since the torque due to any residual radial B-field acting on the
magnetic dipole moment (MDM) mimics the EDM effect, this is expected to be the
dominant systematic error.
Geometric phase: some EDM-mimicking precessions would average to zero except for
the non-commutativity of 3D rotations.
Non-radial E-field, vertical quad misalignment, RF cavity misalignment etc.: these
cause the closed orbit to deviate from design. The systematic errors they cause
cancel to ``lowest order'' but higher order effects need to be investigated.
Polarimetry: realistic beam distributions need to be used to identify and reduce
left-right asymmetry bias, which is another source of systematic error.
\paragraph{\bf Initial storage ring simulation tasks.}
Initial tasks for the ETEAPOT code are to simulate the performance
of arbitrary electric storage rings. As with any storage ring, these
tasks (which now include also spin tracking) are
\begin{enumerate}
\item
Evaluation of linearized lattice functions such as tunes, the Twiss,
$\alpha$, $\beta$, and $\gamma$ functions, dispersion function,
and spin tunes, as well as chromatic dependence of these functions,
and their sensitivity to imperfections.
\item
Short term tracking, for confirmation of the lattice functions,
for determining dynamic aperture, and for investigating the
performance of analytically-derived spin decoherence compensation
schemes.
\item
Guaranteed to be \emph{stable}, long term tracking. The EDM experiment
requires \emph{SCT} to be 1000\,s or longer.
During this time every particle executes about $10^9$ betatron
oscillations. The code is required to exhibit
negligible spurious growth or decay for this interval of time.
\end{enumerate}
Compared to magnetic bending,
electric bending complicates these tasks. This requires the ETEAPOT
treatment to differ from the well-established TEAPOT treatment.
As implemented in UAL/TEAPOT, lattice function determinations use
a truncated power series formalism that has not (as yet) been
established for electric rings. Translating the arbitrary order
formalism from magnetic to electric elements is a huge task that
has only just begun. To cover all three storage ring simulation
tasks on the time scale required for EDM planning we have, therefore,
had to proceed along more than one track.
One track starts by implementing exact (and therefore exactly symplectic)
tracking in the inverse square law potential electrical field between
spherical electrodes. The optimal electric field shape will differ from
this, however. This makes it necessary to introduce artificial
\emph{virtual} quadrupoles in the interiors of bend elements,
in order to model deviation of the actual electric field away
from its idealized representation. \emph{Real} quadrupoles are
typically also present in the lattice, for example to alter the
lattice focusing, or to control dispersion.
Thin sextupoles (present for example to adjust chromaticities or
to compensate spin decoherence effects) and other thin multipole
elements are also allowed. Causing only ``kicks'', these thin
elements, either real or virtual, also preserve symplecticity.
Now complete, this track allows tasks (2) and (3) to be
completed for arbitrary lattices.
Since (linearized) transfer matrices are not used,
the ETEAPOT code cannot be used to extract Twiss functions
directly, as is commonly done in conventional Courant-Snyder
accelerator formalism. However, tunes and Twiss functions can
be obtained (to amply satisfactory accuracy) using FFT and MIA
(model-independent analysis). Now complete this code also
provides transfer matrix determination which, in turn, allows
determination of all the lattice functions required
to complete task (1).
Implementation of the UAL truncated power series approach
for electric rings has just been begun. Its progress is contingent
on obtaining government funding\cite{BNL-Cornell-SMU}.
After comparing ETEAPOT simulation results to results measured
on the AGS Analogue ring this paper briefly describes simulation of
spin evolution in a proton EDM storage ring.
\section{The AGS Electron Analogue ring}
Of the more than 100 relativistic accelerators ever built, only one
has used electric rather than magnetic bending. It was the AGS
Electron Analogue machine at BNL. Curiously it was also
the first ring ever to use alternating gradient (AG)
focusing. (The Cornell 1.1\,GeV alternating gradient
electron ring was commissioned at more
or less the same time and the BNL AGS ring itself somewhat later.)
Along with using \emph{electrons} instead of \emph{protons}, and limited by
achievably high electric field, cost minimization of the AGS Analogue
led to the choice of 10\,MeV maximum electron energy, 4.7\,m bending
radius, 6.8\,MHz RF frequency, and 600\,V RF voltage.
These optimization
considerations are very much the same as
will be used to fix the parameters of a frozen spin proton ring
(which is tentatively expected to have a bend radius of
about 50\,m)\cite{pEDM-PRL}. The UAL/ETEAPOT code
(documented in the accompanying paper) was developed with this
application in mind. The present paper gives initial results.
Starting from fragmentary documentation, the paper starts
by reverse engineering the AGS Analogue lattice and producing
a lattice description file
{\tt E$\underline{\ }$AGS$\underline{\ }$Analogue.sxf},
in the format needed for processing in the Unified Accelerator
Libraries (UAL) environment. Results obtained using ETEAPOT are
then compared with measurements performed on the ring at BNL in
1954-55.
By chance, the magic kinetic energy for freezing electrons,
which is 15\,Mev, is not very different from the 10\,MeV
of the AGS Analogue ring. So that ring could have been used
to measure the electron EDM just by increasing the electric
bend field by a factor of 1.5. Morse\cite{Morse} and others have
suggested building such a ring for this purpose.
The ETEAPOT code can therefore be used to to simulate an
electron EDM measurement using a ring whose successful performance
as a storage ring is all but guaranteed by the
successful operation of an ``identical'' ring in 1955.
15\,MeV is a very convenient electron energy and high quality
electron source would be available, for example as described by
Bazarov\cite{Bazarov}.
The present paper describes our ``resurrection'' of AGS Analogue
ring from historic BNL documentation, and simulates its performance with
codes intended for the proton EDM experiment.
A quite superficial (day long) search of the BNL library and the
Accelerator-Collider report library found one quite extensive report,
produced retrospectively in 1991 by Martin Plotkin\cite{Plotkin},
and a few ancient reports describing machine studies results.
Also a report privately communicated from Ernest
Courant\cite{Courant} contains experimental data to be
simulated in this paper.
Before starting on this project, one of the authors, RT, benefited
from three brief but valuable meetings with Ernest Courant,
perhaps the father, or at least one of the parents, of the Electron
Analogue ring, inquiring about his recollections concerning the
dynamics and performance of the ring. This contributed to our
reconstruction. Bill Morse\cite{Morse-EC} reports having had a similar
conversation with Courant a few years earlier. Bill recalls asking
Ernest whether, back in 1953, he (Courant) understood the difference
between electric and magnetic focusing. Ernest's replied ``of course'' in
his always kindly, but in this case somewhat exasperated, tone of voice.
\paragraph{\bf Historical BNL documents.}
The ``Conceptual Design Report'' for the AGS Analogue electron ring was a
four page letter, dated August 21, 1953, from BNL Director Haworth to
the A.E.C. (predecessor of D.O.E.)
Director of Research Johnson, applying for funding. The letter
is reproduced in its entirety in Figure~\ref{fig:HaworthLetter} in the
appendix. As brief as it is, this letter along with hints from Plotkin
and Courant, includes everything needed to reconstruct the ring,
as shown in Figure~\ref{fig:AGS_Analogue}. By 1955 the ring had been
approved, built, and commissioned, and had achieved its intended purpose.
Following the Haworth letter in the appendix are
two other especially informative figures from the paper by Plotkin.
Figure~\ref{fig:AGS-AnalogueTanks} is especially useful for visualizing
the physical layout of the AGS Analogue ring and its vacuum system.
Figure~\ref{fig:PlotkinCourantData}
is a more polished version of Figure~\ref{fig:QuadStrVsTunes}
which plots tune scans actually performed on
the AGS-Analogue ring and reported by Ernest Courant in a July 28, 1955
BNL technical report\cite{Courant}.
In these plots, points of observed beam loss
and observed beam disruption in the $(Q_x,Q_y)$ tune plane are correlated
with expected resonances. Beam loss occurs on integer resonances, beam
disruption occurs on half integer resonances. The Courant report on,
and analysis of, data collected in machine studies less than two years
after the submission of the ring funding proposal, would
certainly deserve an A+
grade by modern machine studies grading standards.
The axes are voltages (proportional to focusing strengths)
applied to the tune-adjusting quadrupole families.
Short heavy lines indicate regions with no beam
survival (presumably due to integer resonance).
Dots indicate points reported by Courant as
``showing the characteristic double envelope of the oscilloscope
pattern, sometimes accompanied by beam loss''. (These were presumably due
to ``very narrow'' 1/2 integer resonance).
The nominal central tunes values
are $(Q_x,Q_y)=(6.5,6.5)$. Stop bands due
to the eightfold lattice symmetry are also shown.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.42]{AGS_Analoguereduced.pdf}
\caption{\label{fig:AGS_Analogue}
The 1955 AGS-Analogue lattice as reverse engineered from available
documentation---mainly the 1953 proposal letter from BNL Director Haworth to
AEC Director of Research Johnson. Individual ``lenses'' (as they were
referred to in the original documentation; in modern terminology they would
be referred to as ``combined function bends'') are shown on the left;
vertical lines indicate locations of thin quadrupoles present in the lattice
description to represent the focusing effect of the electric lens elements;
Elements are identified by \emph{ad hoc} labels assigned in the reconstruction.
They are not obtained from the original lattice design, but are
needed for modern day lattice description files. Lattice description files
are avaiable in various formats: {\tt E\_AGS\_Analogue2.xml} gives the
pristine design with all parameters in algebraic (or function of algebraic)
form. {\tt E\_AGS\_Analogue2.adxf} is the same, but all parameters values,
though ideal, converted to numerical values. {\tt E\_AGS\_Analogue2.sxf}
contains the fully-instantiated lattice description (with parameters
from otherwise identical elements allowed to be individualized). Most UAL
lattice files are available in all these forms.}
\end{figure*}
\section{Current day simulation of 1955 machine studies tune plane scan}
Our AGS Analogue lattice reconstruction is shown in Figure~\ref{fig:AGS_Analogue}.
The Courant tune plane plot is shown in Figure~\ref{fig:CourantData}. It is
to be compared with a similar plot,
simulated by TEAPOT and shown in Figure~\ref{fig:QuadStrVsTunes}. As far as
we know this code and this lattice representation are completely equivalent
Courant's model and analysis in 1955.
In the TEAPOT tune plane plot, boxes indicate points on integer
resonance boundary curves of the stable diamond
centered on nominal tune values $(Q_x,Q_y)=(6.5,6.5)$.
Points lying on 1/2 integer resonance lines are indicated by dots.
Superperiodicity (eightfold periodicity) causes
resonances with $Q_x=8$ or $Q_y=8$ indicated by broad blank
bends bounded by narrowly-spaced lines. Courant refers to these
as ``stop bands''. Horizontal/vertical
axes $(Q^+,Q^-)$ are ``electrode voltages on the quadrupoles in
odd/even-numbered tanks''. (For the meaning of ``tank'' see
Figure~\ref{fig:AGS-AnalogueTanks}.) The quadrupole strength
coefficients for variable quadrupoles $(Q^+,Q^-)$
were determined empirically to match the central tunes. This
means that absolute focusing strength scales are not checked.
Otherwise there are no significantly adjustable empirical lattice
parameters.
The AGS Analogue ring provides only a coarse
test of ETEAPOT since, for strong focusing, the change from
electic and magnetic bending is quite minor.
This is illustrated in Figure~\ref{fig:ElecMagnCompare} which
shows that the $(Q_x,Q_y)$=(6.5,6.5) tunes in the AGS Analogue ring are
high enough that the tune plane structure is quite insensitive to
whether the bends are treated as magnetic or electric.
Treating the difference perturbatively, the effects of changing from
magnetic to electric bends are
inversely proportional to $\beta$'s, which are both 6.5 in this
case. For the eventual proton EDM ring the vertical tune has to be reduced from
$Q_y=6.5$ by at least a factor of ten which completely invalidates
any such perturbative estimation and requires radical adjustment
of quadrupole strengths. Reducing $Q_y$ from 6.5 to 2.25 has been
straightforward but, as the electric focusing became increasingly
important, to decrease $Q_y$ further will require substantial design
effort.
Spin evolution in the AGS Analogue ring is shown in
Figure~\ref{fig:6plus3Dplots} and described in the caption.
From these results we are confident in out understanding of electric
rings and of our ability to simulate their performance using ETEAPOT.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.45, angle=-90]{TunePlaneMeasured.pdf}
\caption{\label{fig:CourantData}Tune plane resonance diagram measured
during machine studies at the AGS Analogue ring and reported by
Courant\cite{Courant}. A more polished and complete version of
this plot, copied from Plotkin\cite{Plotkin} is shown in
Figure~\ref{fig:PlotkinCourantData} but with minor re-labelings.}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[scale=1.0]{QuadStrVsTunes.pdf}
\caption{\label{fig:QuadStrVsTunes}Tune plane resonance diagram
as calculated by TEAPOT, with bend elements treated as magnetic.
There is good qualitative and quantitative agreement with the
Courant data shown in Figure~\ref{fig:CourantData}.}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.3]{AGS_Analogue2Compare-elec-magn_x.png}
\includegraphics[scale=0.3]{AGS_Analogue2Compare-elec-magn_y.png}
\caption{\label{fig:ElecMagnCompare}Comparison for
{\tt AGS\_Analogue2.sxf} lattice with bends treated as electric
$m=1$ or $m=-1$ or magnetic. $\beta_x$ is plotted on the left,
$\beta_y$ on the right. As explained in the text, because of the strong
focusing in the AGS Analogue lattice, switching from magnetic to
electric bending causes only minor changes to the lattice functions.}
\end{figure*}
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.28]{6plus3D-101.png}
\includegraphics[scale=0.28]{6plus3D-201.png} \\
\includegraphics[scale=0.28]{6plus3D-9.png}
\includegraphics[scale=0.28]{6plus3D-101112.png}
\caption{\label{fig:6plus3Dplots}The upper llots show the $s_x$ and
$s_y$ spin components of a single electron, with initial horizontal
amplitude {\tt xtyp}=1e6\,m, in the AGS Analogue ring. (The 6D initial
conditions are given above the plots.) The horizontal axis
``split bend index'' increases by 1 at each bend, of which there are 640,
so about 59 turns are shown. The lower right figure shows longitudinal
phase space evolution for three off-energy particles, with energy offset
$de$ plotted against time offset $-ct$. One sees that almost exactly one
revolution is completed during this time. The synchrotron tune is therefore
approximately 1/59. The lower left plot plot shows the $de$ energy deviation
during one synchrotron oscillation period.}
\end{figure*}
\clearpage
\section{Long term tracking in a proton EDM storage ring trap}
Plots so far have simulated the evolution of 10\,MeV electrons in the AGS
Analogue electron ring. These roughly indicate the performance to be expected in
a modern day electron EDM storage ring measurement. Of more immediate interest
is the expected behavior of 230\,MeV protons in a significantly larger, but
still all-electric proton EDM storage ring.
A single particle tracking example, taken from a recent proton EDM
study, is illustrated in Figure~\ref{fig:plots}.
The particle orbit and spin components are tracked around a prototype
proton EDM storage ring for 33 million turns using ETEAPOT. As stated
earlier the tracking is exact and no artificial ``symplectification'' is
applied. Any spurious damping or anti-damping of the spatial orbit over
the full run is less than ten percent over the 33 million turns.
The (unit-magnitude) spin vector {\bf S} is initially purely tangential
(forward) and the vector magnitude never changes from its initial value of 1.
It is characteristic of spin evolution for the transverse spin components
to change over narrow ranges in synchronism with betatron oscillation
and larger but still small, ranges in synchronism with synchrotron oscillation.
The upper left graphs of Figure~\ref{fig:plots} show this, along with
a sinusoidal fit with the parameters shown. These precessions are
not expected to contribute significantly to spin decoherence.
But any systematic growth over millions of turns will eventually lead to
beam decoherence and limited spin coherence time.
An important computational task in planning to measure the proton EDM is to
determine the spin coherence time $SCT$ of the circulating beam. This is the
time after which inevitable spreads in beam parameters will have attenuated
the beam polarization significantly due to decoherence in the spin propagation.
As long as both $S_x$ and $S_y$ magnitudes remain small, {\bf S} remains in the
forward hemisphere, and decoherence is suppressed.
One sees from the graphs of Figure~\ref{fig:plots}
that both $S_x$ and $S_y$ gradually deviate from zero.
But, for the particular particle being tracked, these deviations are limited
to small values.
In particular, since $S_x^2+S_y^2$ remains always much less than 1 in magnitude,
$S_z$ therefore remains not much less than 1. So the beam polarization
remains always in the forward hemisphere. Depolarization is therefore
unimportant for particles of amplitude comparable to, or smaller than,
this particle's. The tracking is therefore consistent with the SCT value
exceeding the 33 million turns shown.
The lattice investigated in Figure~\ref{fig:plots} is a
``M\"obius lattice'' in which the horizontal and vertical betatron
oscillations interchange on every turn. This strongly suppresses spin
depolarization because precession in horizontal and vertical oscillation
phases tend to cancel on a turn-by-turn basis. By fine tuning this
cancellation it is anticipated that extremely long SCT values can be
obtained.
This will leave beam energy spread as the dominant source of
beam depolarization. Fringe fields at bend elements are one such
significant source of spin precession slewing. In this respect
spin tracking is more sensitive than orbit tracking.
Fringe fields are not very important in orbit calculations.
Treating bends as ``hard edged'' mainly causes small tune shifts which
are not very important, since both tunes are always set operationally
using spectrum analysis.
But, because of the spin precession sensitivity, in ETEAPOT, fringe
fields are treated as linear ramps of length comparable with the gap
between the electrodes, and not by hard edges. In any case synchrotron
oscillation averaging also strongly suppresses this fringe
field depolarization. This places further demands on longitudinal
particle tracking, which is always delicates, because nonlinearity of
synchrotron oscillations contributes to spin decoherence.
The particle revolution period is about one microsecond so the plots shown
correspond to about thirty seconds of real clock time in the laboratory.
In a laptop computer this computation takes a few hours per particle.
The ratio of computation time to real laboratory time
for a single particle is in the range from one to ten thousand.
The present ETEAPOT single particle tracking approach is sufficient for
early design tasks.
EDM storage ring beams will contain perhaps $10^8$ particles.
Adequately precise simulation of the EDM experiment will require
fewer than this, but at least thousands of particles to be tracked
for significantly longer times than shown. This will require heavy
parallelization of the tracking. With little particle-to-particle
interaction, the code is easily parallelizable. For more advanced tasks,
such as investigating polarimeter biases or emittance growth caused
intrabeam scattering, an efficient and scalable map-based
computational approach is under consideration.
\begin{figure*}[ht]
\centering
\includegraphics[scale=0.28]{Part11-Sx-NM-begin.png}
\includegraphics[scale=0.28]{Part11-Sx-NM-end.png} \\
\includegraphics[scale=0.28]{Part11-Sx.png}
\includegraphics[scale=0.28]{Part11-Sy.png}
\caption{\label{fig:plots}Evolution of spin components for 33 million turns.
Because $S_z$=1 initially, both $S_x$ and $S_y$ vanish at $x$=0, where $x$ is
turn number.
The upper left figure shows $S_x$ for the first 1000 turns, as fit by a sinusoid
$A\sin(w x)$ where $x$ is turn number. The upper right figure shows $S_x$ for one
million turns starting at $x=29\,$million. The lower left and upper right figures
show the evolution of $S_x$ and $S_y$ for the full 33\,million turn run. Though
these motions are oscillatory the oscillation amplitudes are less than the
line width. The lattice file is {\tt E\_pEDM-rtr1-Mobius.RF.sxf}.}
\end{figure*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,933 |
\section{Introduction}
HuBi\,1 is part of the selected group of planetary nebulae (PNe) named
born-again PNe that are thought to have experienced a {\it very late
thermal pulse} \citep[VLTP; e.g.,][]{Schonberner1979,Iben1983}.
During this specific evolutionary phase, the He-burning shell at the
surface of the central star of a PN (CSPN) reaches the conditions to
ignite He into C and O through an explosive event. This thermonuclear
event ejects H-deficient and C-rich material inside the old H-rich PN
\citep[see, e.g.,][and references therein]{MB2006} rendering the star
as a C-rich [Wolf-Rayet] ([WR]) type star \citep{GT2000}.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{HuBi_size2.pdf}
\caption{Optical images of HuBi\,1. Left: Colour-composite optical
image of HuBi obtained with the [N\,{\sc ii}] (red) and H$\alpha$
(green) narrow-band filters at the NOT. The outer
($r_\mathrm{out}$), middle ($r_\mathrm{mid}$) and inner
($r_\mathrm{in}$) radii have extension of 8, 5 and 2~arcsec,
respectively. The image was adapted from \citet{Rechy2020}. Right:
Grey-scaled [N\,{\sc ii}] narrow-band image. Both panels have the
same FoV. North is up, east to the left.}
\label{fig:opt}
\end{figure*}
Recent works have presented stark evidence of the dramatic changes
experienced by the CSPN of born-again PNe \citep[see, e.g., the case
of the Sakurai's Object;][]{Evans2020}, and in particular HuBi\,1.
\citet{Guerrero2018} demonstrated that in less than 50~yr its CSPN
declined its brightness in about 10~mag changing its atmosphere and,
as a consequence, producing changes in the ionization structure of its
surrounding PN.
HuBi\,1 has a double-shell structure (see Fig.~\ref{fig:opt}). The
outer shell with an angular radius of $r\sim8''$ is dominated by
emission from recombination lines of H~{\sc i} and He~{\sc i}, whilst
its inner shell with $r\sim2''$ is dominated by emission from
forbidden lines \citep{Guerrero2018}. The inner shell has a notable
inverted ionization structure, with the emission from higher
ionization species such as O$^{++}$ and He$^{++}$ encompassing that
from lower ionization species such as N$^{+}$, O$^{+}$ and S$^{+}$
\citep[][Montoro-Molina et al., in preparation]{Guerrero2018}. Such
unusual inverted ionization structure gave HuBi\,1 the title of {\it
inside-out} PN.
\citet{Guerrero2018} used multi-epoch observations of HuBi\,1,
state-of-the-art stellar atmosphere models of its CSPN from the PoWR
code \citep[see][and references
therein]{Sander2015}\footnote{\url{http://www.astro.physik.uni-potsdam.de/~wrh/PoWR/powrgrid1.php}}
and modern stellar evolution models from \citet{MB2016} to predict
different aspects of the evolution of this PN and its progenitor
star. In particular, these authors found that a model that experienced
a mass-loss rate during the VLTP ($\dot{M}_\mathrm{VLTP}$) of
7.6$\times10^{-5}$~M$_\odot$~yr$^{-1}$ fits the evolutionary status of
the CSPN of HuBi\,1, with an ejected mass during the VLTP
($M_\mathrm{VLTP}$) of 8$\times10^{-4}$~M$_\odot$.
\citet{Guerrero2018} estimated that currently the wind velocity of the
CSPN has a velocity of 360~km~s$^{-1}$ with a mass-loss rate of
$8\times10^{-7}$~M$_\odot$~yr$^{-1}$. These authors also estimated a
relatively low ionizing photon flux of $\approx10^{44}$~s$^{-1}$,
which makes them propose that the outer shell is recombining.
In \citet{Rechy2020} we have studied HuBi\,1 using integral-field
spectroscopic Multi-Espectr\'ografo en GTC de Alta Resoluci\'on para
Astronom\'\i a \citep[MEGARA;][]{Gil2018} mounted on the Gran
Telescopio Canarias (GTC). The unrivaled tomographic capability of
these MEGARA observations have unveiled the kinematic signature of the
inner shell in HuBi\,1, revealing that it was ejected about 200~yr ago
and currently has an expansion velocity of $\approx$300 km~s$^{-1}$.
The MEGARA observations showed that the inner structure is apparently
distributed in a shell-like morphology, very different to what is
observed in other born-again PNe. For example, {\it Hubble Space
Telescope} ({\it HST}) and IR observations of A\,30, A\,58 and A\,78
have revealed that the H-deficient material ejected during the VLTP
has a marked bipolar morphology. The material in the born-again
ejecta of these PNe is distributed in a disk-like structure and a
bipolar ejection that resembles a jet
\citep[][]{Borkowski1993,Borkowski1995,Clayton2013,Fang2014}. In
particular, for the cases of the more evolved born-again PN A\,30 and
A\,78, the ring-like structure appears to have been disrupted by the
complex interactions with the stellar wind and ionizing photon flux
from their CSPNe \citep[see][and references therein]{Toala2021}.
Our group presented the first attempt to model the formation of
born-again PNe in \citet{Fang2014}. In that work we presented 2D
radiation-hydrodynamic numerical simulations tailored to the
born-again PNe A\,30 and A\,78 in comparison with a study of the
expansion of their H-deficient clumps and filaments inside their old
PNe. Those simulations demonstrated that adopting a velocity of
20~km~s$^{-1}$ during the VLTP can help explaining the distribution of
the H-poor knots, the dynamical age ($\sim$1000~yr) and evolution of
A\,30 and A\,78. Our simulations showed that the C-rich material will
be disrupted by a combination of effects. Hydrodynamical
instabilities, mainly Rayleigh-Taylor will break the VLTP material
into clumps and filaments. These will be subsequently ionized and
photoevaporated by the increasing UV flux from the CSPN. Finally, the
current fast stellar wind will also play a role in dragging the
material with the denser and slower clumps remaining close to the
CSPN.
In this paper we present 3D radiation-hydrodynamic numerical
simulations of the formation and evolution of born-again PNe, with
emphasis to HuBi\,1. The simulations are use to explain the formation
of HuBi\,1 and to peer into its further evolution. This is assessed
by adopting different initial conditions for the VLPT ejecta in the
simulations (2 cases are explored). A comparison with more evolved
born-again PNe is also attempted.
This paper is organized as follows. In Section~2 we present the code
used to run our simulations and describe the initial
conditions. Section~3 describes the different numerical results
obtained from the simulations. A discussion is presented in Section~4
and a summary of the work is presented in Section~5.
\section{Simulations}
We used the extensively-tested radiation-hydrodynamic 3D code {\sc
guacho} \citep{Esquivel2009,Esquivel2013} to model the formation and
evolution of the born-again PN HuBi\,1. {\sc guacho} includes a
modified version of the ionizing radiation transfer presented in
\citet{Raga2009}. It solves the gas-dynamic equations with a second
order accurate Godunov-type method, using a linear slope-limited
reconstruction and the HLLC approximate Riemann solver
\citep{Toro1994} implemented on a uniform Cartesian grid.
Simultaneously with the Euler equations, we solve the rate equation
for neutral and ionized hydrogen
\begin{equation}
\frac{\partial n_\mathrm{HI}}{\partial t} + \nabla \cdot(n_\mathrm{HI} {\bf u})=
n_\mathrm{e} n_\mathrm{HII} \alpha(T) - n_\mathrm{HI} n_\mathrm{HII} c(T) - n_\mathrm{HI} \phi,
\end{equation}
\noindent where {\bf u} is the flow velocity, $n_\mathrm{e}$,
$n_\mathrm{HI}$ and $n_\mathrm{HII}$ are the electron, neutral
hydrogen and ionized hydrogen number densities, $\alpha(T)$ is the
recombination coefficient, $c(T)$ is the the collisional ionization
coefficient and $\phi$ is the H photoionization rate due to a central
source. The photoionizing rate is computed with a Monte-Carlo ray
tracing method, described in \citet{Esquivel2013} and
\citet{Schneiter2016}.
We define the ionization fraction as
\begin{equation}
\chi=\frac{n_\mathrm{HII}}{n_\mathrm{HI}+n_\mathrm{HII}},
\end{equation}
\noindent with the total number density defined as
$n=n_\mathrm{HI}+n_\mathrm{HII}$. The ionization fraction is used to
estimate the radiative cooling, which is added to the energy equation
using the prescription described in \citet{Esquivel2013}.
The simulations presented here have been performed on a 3D cartesian
grid with a resolution of $(x, y, z)$=(600, 600, 600) on a box of
(0.6$\times$0.6$\times$0.6)~pc$^{3}$ in physical size, that is, a cell
resolution of 0.001~pc. The injection cells correspond to the
innermost 0.01~pc in the simulation.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{hubi_massloss.pdf}\\
\includegraphics[width=0.9\linewidth]{hubi_velocity.pdf}
\caption{Illustration of the evolution with time of the mass-loss rate
(left panel) and stellar wind velocity (right panel) of the
simulations used here. The different phases are labeled. Note that
there are two values for the velocity on during the VLTP
($v_\mathrm{VLTP}$) phase which correspond to the two simulations
presented here.}
\label{fig:hist}
\end{figure}
\subsection{Initial conditions - old PN formation}
We started with an homogeneous ISM with initial density and
temperatures of $n_{0}$=1~cm$^{-3}$ and $T_{0}=$100~K,
respectively. We first model the creation of the old H-rich PN of
HuBi\,1. For this, we first launch a slow wind corresponding to the
AGB stage with mass-loss rate
$\dot{M}_\mathrm{AGB}=10^{-5.5}$~M$_\odot$~yr$^{-1}$ and a velocity of
$v_\mathrm{AGB}=$15~km~s$^{-1}$ for a total time of
$t_{1}=5\times10^5$~yr. This creates a density distribution with a
dependence with radius of $n \sim r^{-2}$ as commonly obtained for the
AGB phase \citep[see, e.g.,][]{Villaver2002}. No photoionization flux
is included during this phase.
Secondly, a post-AGB phase which creates the old PN is modeled by
injecting a fast wind with a velocity of
$v_\mathrm{PN}$=2000~km~s$^{-1}$ and a mass-loss rate of
$\dot{M}_\mathrm{PN}=10^{-7}$~M$_\odot$~yr$^{-1}$. An ionizing photon
flux of $5\times10^{46}$~s$^{-1}$ is adopted for this
phase. Figure~\ref{fig:hist} shows an illustration of the evolution
with time of the stellar wind parameters in these two phases.
The simulation is run until the shell of the PN reaches a radius of
$\sim$0.18~pc so that by the time the born-again event occurs it could
reach a 0.2~pc similarly to what is currently observed in
HuBi\,1\footnote{The angular radius of HuBi\,1 is $\sim$8~arcsec which
is $\sim$0.2~pc (see Fig.~\ref{fig:opt}) at a distance of 5.3~kpc
\citep{Frew2016}.}. The formation of the old H-rich PN occurs during
$t_1$ and $t_2$ in Figure~\ref{fig:hist}.
At this point, the fast wind has carved the AGB material into a dense
shell. This interaction creates the classic adiabatically-shocked hot
bubble that fills the PN \citep[e.g.,][and references
therein]{Toala2016}. At the same time, the strong photon flux
ionizes the material. The number density ($n$), temperature ($T$), gas
velocity ($v$) and ionization fraction ($\chi$) at this point are
illustrated in Figure~\ref{fig:initial_c}.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{IC_ALL_IC_PN_esqui0-Y.pdf}
\caption{Total number density ($n$ - top right), temperature ($T$ -
top left), velocity ($v$ - bottom left) and ionization fraction
($\chi$ - bottom right) for the simulation after the formation of
the old PN, that is, just before the onset of the VLTP phase.}
\label{fig:initial_c}
\end{figure}
\subsection{The born-again phase and subsequent evolution}
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{density_BA_20_esqui0-Y.pdf}
\includegraphics[width=0.45\linewidth]{density_BA_300_esqui0-Y.pdf}
\caption{Number density $n$ in the $x-z$ plane ($y$=0) of the two
simulations presented here. The left panel corresponds to Run~A
($v_\mathrm{VLTP1}$=20~km~s$^{-1}$) and the right panel to Run~B
($v_\mathrm{VLTP2}$=300~km~s$^{-1}$). The sub-panels show different
time steps with $t=0$ marking the onset of the pVLTP phase. }
\label{fig:rho}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{Temperature_BA_20_esqui0-Y.pdf}
\includegraphics[width=0.45\linewidth]{Temperature_BA_300_esqui0-Y.pdf}
\caption{Same as Figure~\ref{fig:rho} but for the gas temperature
$T$.}
\label{fig:T}
\end{figure*}
The stellar evolution model presented for HuBi\,1 in
\citet{Guerrero2018} predicts that its CSPN experienced a mass-loss
rate during the VLTP of
$\dot{M}_\mathrm{VLTP}=7.6\times10^{-5}$~M$_\odot$~yr$^{-1}$. Following
the multi-epoch study of the Sakurai's Object, we will adopt a
duration for the VLTP of 20~yr \citep{Evans2020}. However, the
velocity of the H-poor ejected material is an unknown parameter. One
might argue that as the star goes back to the region of the AGB stars
in the Hertzsprung–Russell diagram, a similar velocity as that
reported for those kind of stars should be adopted
\citep[$\approx$20~km~s$^{-1}$; see][]{Ramstedt2020}, but the VLTP is
an explosive event in nature.
To assess both scenarios we ran two simulations. Run~A will be
performed with $v_\mathrm{VLTP1} = 20$ km~s$^{-1}$ and Run~B with
$v_\mathrm{VLTP2} = 300$ km~s$^{-1}$, similar to what is observed for
the H-poor ejecta in HuBi\,1 \citep{Rechy2020}. No ionizing photon
flux will be considered during this phase. Figure~\ref{fig:hist}
illustrates the variations of the mass-loss rate and velocity during
this phase (between $t_2$ and $t_3$) in comparison with the previous
phases. As a result of the high-mass loss rate during the VLTP phase
and its short duration we will create a dense shell surrounding the
CSPN with an extension of $\lesssim$0.02~pc in radius.
A final post-VLTP (pVLTP) phase will be modelled by adopting the
stellar wind parameters currently exhibited by the CSPN of HuBi\,1
reported in \citet{Guerrero2018}. A pVLTP wind velocity of
$v_\mathrm{pVLTP} = 360$ km~s$^{-1}$ with a mass-loss rate of
$\dot{M}_\mathrm{pVLTP}=8\times10^{-7}$~M$_{\odot}$~yr$^{-1}$ for both
Run~A and B. This wind is expected to sweep the VLTP ejecta creating
a dense inner shell and giving rise to the double shell morphology.
The ionizing photon flux of $10^{44}$~s$^{-1}$ reported by
\citet{Guerrero2018} will be adopted for this last phase.
Figure~\ref{fig:hist} illustrate the variations of the mass-loss rate
and stellar wind velocity from the AGB phase to the pVLTP for the two
simulations which corresponds to $t>t_3$.
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{velocity_BA_20_esqui0-Y.pdf}
\includegraphics[width=0.45\linewidth]{velocity_BA_300_esqui0-Y.pdf}
\caption{Same as Figure~\ref{fig:rho} but for the gas velocity $v$.}
\label{fig:vel}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{P_BA_20_03_esqui0-Y.pdf}
\includegraphics[width=0.45\linewidth]{P_BA_300_03_esqui0-Y.pdf}
\caption{Same as Figure~\ref{fig:rho} but for the gas pressure $P$.}
\label{fig:pressure}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{fneut_BA_20_esqui0-Y.pdf}
\includegraphics[width=0.45\linewidth]{fneut_BA_300_esqui0-Y.pdf}
\caption{Same as Figure~\ref{fig:rho} but for the ionization fraction $\chi$.}
\label{fig:ionization}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=0.75\linewidth]{render.pdf}
\caption{Synthetic nebular images obtained by integrating through the
$x-y$ plane ($z=0$) for the two simulations presented in this
work. The images were produced at the time at which the inner shell
reaches 0.05~pc, i.e., 600~yr after the onset of the pVLTP phase for
Run~A (left panel) and 100~yr for Run~B (right panel). The bottom
panels show the same images at the spatial resolution of the NOT
images ($\sim1''$).}
\label{fig:rendered}
\end{figure*}
\section{Results}
Figures~\ref{fig:rho}, \ref{fig:T}, \ref{fig:vel}, \ref{fig:pressure}
and \ref{fig:ionization} show the evolution with time of $n$, $T$,
$v$, $P$ and $\chi$ of the gas for the two simulations presented here,
Run~A and B. For simplicity, $t$=0 has been set to the end of the
VLTP, i.e., $t$ represents the time after the pVLTP onset.
Figure~\ref{fig:rho} and \ref{fig:T} show that the dense VLTP material
expands into the low-density, hot bubble created by the previously
fast $v_\mathrm{PN}$ wind. By $t$ equal to 50~yr, the two simulations
show the formation of a dense shell with radius between 0.02 and
0.03~pc (Fig.~\ref{fig:rho}).
It is important to note that due to the sudden variation of the
stellar wind parameters and the ionizing photon flux, the inner edge
of the hot bubble experiences noticeable changes. In particular, the
ram pressure of the wind in the VLTP is not as high as that of the
fast wind that fed the previous PN phase, which created the hot bubble
(see Fig.~\ref{fig:initial_c} top left panel). As a result, the hot
bubble falls back to smaller radii when the star evolves into the VLTP
phase, creating instabilities that develop with time. Such effects
are more evident in the gas velocity and pressure (Fig.~\ref{fig:vel}
and \ref{fig:pressure}), with turbulent structures appearing at $t$
equal to 100~yr, and being are still noticeable 300~yr after the VLTP
event and to some extent in the most evolved panel at 800~yr of Run~A.
There is no apparent effect in the ionization fraction of these
turbulent structures (see Fig.~\ref{fig:ionization}), implying that
they are completely ionized in our simulations.
The relatively small variation in velocity between the VLTP and pVLTP
phases in both simulations is not enough to produce the hydrodynamic
instabilities (e.g., Rayleigh-Taylor) reported in other works
\citep{Stute2006,Toala2016}. As a consequence, our numerical results
show the expansion of a smoothed shell expanding inside the old PN.
Finally volume density renderings were created to mimic nebular images
for both simulations at an integration time when the inner shell has a
radius of 0.05~pc. The top-left panel of Figure~\ref{fig:rendered}
shows the case of Run~A at 600 yr after the pVLTP evolution, while the
top right panel shows Run~B at 100~yr of evolution. The rendered
images were produced with the {\sc yt} software \citep{Turk2011} that
allows to create images by casting rays through the 3D volume and
integrating the radiation transfer equations with a transfer function
that can be selected or modified by the user. Typically, the transfer
function is chosen with transparency and/or colors that depend on the
value of the field that is being rendered. To allow a fair comparison
with the available NOT images, a Gaussian filter was apply to reduce
the spatial resolution of the rendered images to $\sim1''$. The
synthetic images in the bottom panels of Figure~\ref{fig:opt}
reproduce the double shell morphology of HuBi\,1, revealing an
additional clumpy intermediate structure which is present in
Figure~\ref{fig:opt}, particularly in the contrast enhanced [N~{\sc
ii}] image in its right panel.
\section{Discussion}
The born-again phase is one of the most unknown phases of stellar
evolution. Its duration seems to depend on different factors such as
mixing and diffusion as discussed in \citet{MB2006}. However, some
efforts assessing the mass lost during this phase and the velocity of
the ejected H-poor material have been presented in the literature
\citep[see][and references there]{Guerrero2018,Toala2021}. This can
certainly help driving theoretical results.
The results of our simulations for HuBi\,1 suggest that it is more
accurate to assume that the H-poor material was ejected inside the old
PN in an explosive event with velocities close as those currently
observed. Run~B predicts that if this is the case, the material
ejected during the VLTP in HuBi\,1 must have occurred $\sim$100~yr,
which is rather consistent with the kinematical age estimated by
\citet{Rechy2020}. Run~B also suggests that the small difference in
velocity between the VLTP and the pVLTP does not allow the shell to
experience the formation of hydrodynamical instabilities, in
particular, Rayleigh-Taylor. As a result, the inner shell of HuBi\,1
appears to be a smooth shell-like structure. Our simulations predict
that the inner shell will not develop hydrodynamic instabilities
capable of disrupting it, as suggested for A\,30 and A\,78. Of course
this outcome will change if the CSPN of HuBi\,1 evolved into an
earlier [WR]-type, becoming hotter and developing a faster wind.
We have shown that the relatively low ionization photon flux of
10$^{44}$~s$^{-1}$ suggested from the stellar atmosphere modelling of
the CSPN of HuBi\,1 is not able to completely ionize the born-again
inner shell. In our simulations the H-poor ejecta has a ionization
fraction close to 0.5. This strengthens \citet{Guerrero2018}'s
suggestion that the emission from this shell must come from shock
physics. A careful analysis of the optical emission spectrum will
address this issue (Montoro-Molina et al., in preparation).
Figure~\ref{fig:rendered} suggests that the structures detected
between $2''<r<5''$ in the [N\,{\sc ii}] image surrounding the
born-again ejecta of HuBi\,1 (see Fig.~\ref{fig:opt} right panel) have
been formed as a result of the sudden and large variation of the
stellar wind parameters of its CSPN. These turbulent structure
appears to be completely ionized, unlike the inner shell. Indeed
Figure~1 in \citet[][]{Guerrero2018} suggests that this is the case.
To explore the velocity structure of this emission, we have examined
the GTC MEGARA integral field spectroscopic data published recently by
our team \citep{Rechy2020}. The results are illustrated in
Figure~\ref{fig:NII_prof}, a colour-composite picture of HuBi\,1 in
the [N\,{\sc ii}] $\lambda\lambda$6548,6584 emission lines where the
green colour corresponds to [N\,{\sc ii}] emission at the systemic
velocity of HuBi\,1, the blue colour to the receding structure in the
systemic velocity range from $-55$ to $-45$ km~s$^{-1}$, and the red
colour to the approaching structure in the systemic velocity range
from $+57$ to $+66$ km~s$^{-1}$. Other emission lines detected in the
MEGARA data cube, such H$\alpha$ and [S\,{\sc ii}], show similar
velocity structures, but the former suffers from thermal broadening,
while the later has a smaller signal-to-noise level than the [N\,{\sc
ii}] image presented here.
Figure~\ref{fig:NII_prof} shows that the structure surrounding the
inner shell of HuBi\,1 has a somewhat complex velocity structure.
Some emission at the systemic velocity of HuBi\,1 might have formed as
illustrated by our simulations, gas falling back due to the reduction
in ram pressure generating instabilities in the ionized
structure. However, there seems to be a bipolar structure not reported
before in HuBi\,1. This bipolar structure does not appear to be
collimated, but extended at a certain point. A detailed analysis of
the velocity of this structure using our available GTC MEGARA
observations is under preparation (Montoro-Molina et al., in
preparation).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{NII_profile.pdf}
\caption{[N\,{\sc ii}] emission from the GTC MEGARA observations of
HuBi\,1. The green colour represents the [N\,{\sc ii}] emission
centered on the systemic velocity of HuBi\,1. The blue and red
colour show the integrated velocity in the [$-$55:$-$45]~km~s$^{-1}$
and [57:66]~km~s$^{-1}$, respectively. The inner shell of HuBi\,1
appears saturated in white.}
\label{fig:NII_prof}
\end{figure}
\subsection{Consequences for other born-again PNe}
The simulation of Run~A reaches a radius for the inner shell of
0.05~pc after 600~yr of evolution, which is notably different to the
age of $\simeq$200 yr proposed by \citet{Rechy2020}. The model can
neither reproduce the reported expansion velocities of the inner shell
in HuBi\,1, regardless of the injection of a pVLTP wind 20 times
faster, which can not provide sufficient kinetic energy to accelerate
the shell up to the observed velocities \citep[$\sim$300
km~s$^{-1}$;][]{Rechy2020}, whereas the momentum provided by the
radiation pressure is negligible. Still, it is appropriate to discuss
the numerical results of Run~A as a slow expanding VLTP wind might
have been the case for A\,30 and A78, where after $\sim$1000~yr of
evolution in the born-again phase dense knots and filaments are still
located close to their CSPN with expansion velocities
$\lesssim$50~km~s$^{-1}$ \citep{Meaburn1996,Meaburn1998}.
The 2D Radiation-hydrodynamic simulations presented in
\citet{Fang2014} showed that it is possible to reproduce the
morphology of the H-deficient clumps and filaments in A\,30 and A\,78
only if the velocity during the VLTP phase in these objects was
$\sim$20~km~s$^{-1}$, followed by a fast subsequent evolution of the
stellar wind parameters reaching terminal velocities as high as
$\gtrsim$3000~km~s$^{-1}$, which is what is currently reported from
these sources \citep[see][]{Guerrero2012,Toala2015}. The interactions
between the fast pVLTP and slow VLTP winds are dominated by
Rayleigh-Taylor instabilities that create a pattern of slow clumps and
filaments lagging close to the CSPN, whilst evaporated material can
reach velocities of a few times 100 km~s$^{-1}$ with distances almost
reaching the outer H-rich PNe.
The notable differences between the most evolved born-again PNe
discovered so far and the {\it inside out} PN HuBi\,1 might suggest
that the explosive VLTP might have had different injection
energies. The kinetic energy imprinted in the H-deficient material
ejected inside the old PN should be directly related to the He mass
ignited during the VLTP (born-again) event. Assuming that the total
mass ejected in these born-again PNe is the same, the kinetic energy
of the H-deficient material in HuBi\,1 is $\gtrsim$30 times larger
than the slowly moving ($\sim$50~km~s$^{-1}$) dense clumps in A\,30
and A\,78. The later suggest that the thermonuclear conditions of the
VLTP were quite different between these systems.
We are currently preparing a grid of stellar evolution models
accounting for different parameters such as initial mass, rotation,
metallicity, mixing length (Rodr\'{i}guez-Gonz\'{a}lez et al.\ 2021,
in prep.) using the Modules for Experiments of Stellar Astrophysics
\citep[{\sc mesa};][]{Paxton2011}. These will help us to assess
possible different conditions occurring during the VLTP. Furthermore,
increasing the number of identified born-again PNe is most needed to
shed some light into this short but unique and physically complex
evolution phase of low-mass stars.
\section{Summary}
\label{sec:summary}
We presented the first 3D radiation-hydrodynamic numerical simulations
of the formation and evolution of a born-again PNe, with particular
application to the inside-out PN HuBi\,1. We adapted the stellar wind
parameters and ionization photon flux reported by \citet{Guerrero2018}
for the CSPN of HuBi\,1 to produce tailored numerical simulations.
Since the velocity of the H-poor material ejected during the VLTP
phase is unknown, two different simulations were presented to explore
its effects: in Run~A we adopted an expansion velocity
$v_\mathrm{VLTP}$=20 km~s$^{-1}$, similar to that is reported for AGB
stars, while in Run~B we adopted a higher velocity
$v_\mathrm{VLTP2}$=300 km~s$^{-1}$, consistent with that reported from
optical observations in HuBi\,1. Our findings can be summarized as
follows:
\begin{itemize}
\item
Our explosive case, Run~B, makes a good job reproducing the
morphological features in HuBi\,1. These simulations predict that the
inner shell of HuBi\,1 was formed as a result of the born-again event
which occurred about 100~yr ago, consistent with kinematic estimations
from GTC MEGARA observations. Slower ejections can not imprint the
necessary kinematic energy to accelerate the H-deficient material to
the observed velocity of 300~km~s$^{-1}$.
\item
Our simulations show that the small variation in velocity between the
VLPT and the pVLTP material so far observed will not produce
instabilities that break the inner shell. This produces a smooth
inner shell consistent with that seeing in optical observations.
\item
The extreme changes experienced by the CSPN of HuBi\,1 are obviously
responsible of the double shell morphology seen in optical
observations (Fig.~\ref{fig:opt} and \ref{fig:NII_prof}). Moreover,
the variation in the stellar wind parameters diminishs dramatically
the wind's ram pressure, producing noticeable changes to the
adiabatically shocked hot region originally created at the first PN
phase. The hot bubble falls back by the time the star enters the VLTP
phase, producing turbulent structures which are observable as clumps
and filaments of ionized material at intermediate regions between the
two shells. We propose this is the origin of the structures detected
in the [N\,{\sc ii}] image of HuBi\,1 in the intermediate regions
between $2''$ and $5''$.
\item
Our simulations demonstrate that the current photon flux of
10$^{44}$~s$^{-1}$ is not capable of producing the complete
photoionization of the inner shell of HuBi\,1. This result
strengthens the suggestion of \citet{Guerrero2018} that this structure
is dominated by shocks.
\item
We suggest that the explosive VLTP in HuBi\,1 might have been at least
30 times more energetic than that in the born-again PNe A\,30 and
A\,78. Dense clumps in A\,30 and A\,78 have been detected very close
to the CSPN with velocities as low as $\lesssim$50~km~s$^{-1}$ which
have survived for about 1000~yr. Such differences puts under scrutiny
the physics involved in producing the VLTP and suggests a wealth of
initial conditions in this scenario.
\end{itemize}
\section*{Acknowledgements}
JAT thanks funding by Fundación Marcos Moshinsky (Mexico) and the
Direcci\'{o}nn General de Asuntos del Personal Acad\'{e}mico (DGAPA)
of the Universidad Nacional Aut\'{o}noma de M\'{e}xico (UNAM) project
IA100720. V.L. gratefully acknowledges support from the \mbox{CONACyT}
Research Fellowship program. BMM and MAG acknowledge support of the
Spanish Ministerio de Ciencia, Innovaci\'on y Universidades (MCIU)
grant PGC2018-102184-B-I00. AE acknowledges support from DGAPA-PAPIIT
(UNAM) grant IN 109518. This work has made extensive use of NASA's
Astrophysics Data System.
\section*{Data availability}
The data underlying this work are available in the article. The
results from our numerical simulations will be shared on reasonable
request to the first author.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,452 |
Q: ExceptionInInitializerError thrown when creating new stage I'm using swing JFrame as MainFrame for my application. I have a button which should create a new window. But it will crash right in initialization and I have no clue why.
public class Dialog {
private Stage window;
public void display() {
window = new Stage(); //This is line 45. This is place where it crash
window.setMinWidth(350);
window.setMinHeight(500);
window.initModality(Modality.APPLICATION_MODAL);
window.setTitle("Add new Stuff");
//more code here
}
}
Here is whole exception error
Exception in thread "AWT-EventQueue-0" java.lang.ExceptionInInitializerError
at javafx.stage.Window.<init>(Window.java:1179)
at javafx.stage.Stage.<init>(Stage.java:236)
at javafx.stage.Stage.<init>(Stage.java:224)
at main.Dialog.display(Dialog.java:45)
at main.MainFrame.jButtonAddZamActionPerformed(MainFrame.java:400)
at main.MainFrame.access$400(MainFrame.java:12)
at main.MainFrame$5.actionPerformed(MainFrame.java:227)
at javax.swing.AbstractButton.fireActionPerformed(AbstractButton.java:2022)
at javax.swing.AbstractButton$Handler.actionPerformed(AbstractButton.java:2346)
at javax.swing.DefaultButtonModel.fireActionPerformed(DefaultButtonModel.java:402)
at javax.swing.DefaultButtonModel.setPressed(DefaultButtonModel.java:259)
at javax.swing.plaf.basic.BasicButtonListener.mouseReleased(BasicButtonListener.java:252)
at java.awt.Component.processMouseEvent(Component.java:6525)
at javax.swing.JComponent.processMouseEvent(JComponent.java:3324)
at java.awt.Component.processEvent(Component.java:6290)
at java.awt.Container.processEvent(Container.java:2234)
at java.awt.Component.dispatchEventImpl(Component.java:4881)
at java.awt.Container.dispatchEventImpl(Container.java:2292)
at java.awt.Component.dispatchEvent(Component.java:4703)
at java.awt.LightweightDispatcher.retargetMouseEvent(Container.java:4898)
at java.awt.LightweightDispatcher.processMouseEvent(Container.java:4533)
at java.awt.LightweightDispatcher.dispatchEvent(Container.java:4462)
at java.awt.Container.dispatchEventImpl(Container.java:2278)
at java.awt.Window.dispatchEventImpl(Window.java:2750)
at java.awt.Component.dispatchEvent(Component.java:4703)
at java.awt.EventQueue.dispatchEventImpl(EventQueue.java:751)
at java.awt.EventQueue.access$500(EventQueue.java:97)
at java.awt.EventQueue$3.run(EventQueue.java:702)
at java.awt.EventQueue$3.run(EventQueue.java:696)
at java.security.AccessController.doPrivileged(Native Method)
at java.security.ProtectionDomain$1.doIntersectionPrivilege(ProtectionDomain.java:75)
at java.security.ProtectionDomain$1.doIntersectionPrivilege(ProtectionDomain.java:86)
at java.awt.EventQueue$4.run(EventQueue.java:724)
at java.awt.EventQueue$4.run(EventQueue.java:722)
at java.security.AccessController.doPrivileged(Native Method)
at java.security.ProtectionDomain$1.doIntersectionPrivilege(ProtectionDomain.java:75)
at java.awt.EventQueue.dispatchEvent(EventQueue.java:721)
at java.awt.EventDispatchThread.pumpOneEventForFilters(EventDispatchThread.java:201)
at java.awt.EventDispatchThread.pumpEventsForFilter(EventDispatchThread.java:116)
at java.awt.EventDispatchThread.pumpEventsForHierarchy(EventDispatchThread.java:105)
at java.awt.EventDispatchThread.pumpEvents(EventDispatchThread.java:101)
at java.awt.EventDispatchThread.pumpEvents(EventDispatchThread.java:93)
at java.awt.EventDispatchThread.run(EventDispatchThread.java:82)
Caused by: java.lang.IllegalStateException: This operation is permitted on the event thread only; currentThread = AWT-EventQueue-0
at com.sun.glass.ui.Application.checkEventThread(Application.java:443)
at com.sun.glass.ui.Screen.setEventHandler(Screen.java:245)
at com.sun.javafx.tk.quantum.QuantumToolkit.setScreenConfigurationListener(QuantumToolkit.java:674)
at javafx.stage.Screen.<clinit>(Screen.java:80)
... 43 more
Any ideas why is it crashing? Thanks
A: You can have JavaFx component inside a swing application. We did that for one the ERP application to display dashboard with JavaFx Charts.
Initializing JavaFx related code should be done as below.
Platform.runLater(new Runnable() {
@Override
public void run() {
initFX(fxPanel);
}
});
Please refer below link for more details
https://docs.oracle.com/javase/8/javafx/interoperability-tutorial/swing-fx-interoperability.htm
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,630 |
Fiber Cement Siding – Budd Severino Advanced Home Exteriors, Inc.
CertainTeed brings fiber cement siding that you can choose with confidence, install with pride and experience in comfort. We create beauty, harmony and innovation in your world with the rich colors and stunning textures of CertainTeed Fiber Cement siding. Our Fiber Cement relies on environmentally sustainable principles to design technologically advanced, earth-friendly siding that looks as good as it performs. Each and every sheet and board is engineered for decades of superior protection, wear and durability, and is backed with a century-long tradition of excellence and customer satisfaction. When we say CertainTeed, we mean quality made certain, satisfaction guaranteed. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,335 |
Büsumer Deichhausen (formerly "Dykhusen") is a municipality belonging to the Amt ("collective municipality") Büsum-Wesselburen in the district Dithmarschen in Schleswig-Holstein, Germany.
Büsumer Deichhausen is situated on the North Sea coast of the Meldorf Bight just east of Büsum. Its name is derived from the words Deich/Dyk (dike) and Haus/Hus (house). It is called Büsumer Deichhausen to distinguish it from the municipality Wesselburener Deichhausen. Its economy is based on a mix of farming and tourism.
References
Dithmarschen | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,128 |
Q: Reader-Writer using semaphores and shared memory in C I'm trying to make a simple reader/writer program using POSIX named semaphores, its working, but on some systems, it halts immediately on the first semaphore and thats it ... I'm really desperate by now. Can anyone help please? Its working fine on my system, so i can't track the problem by ltrace. (sorry for the comments, I'm from czech republic)
https://www.dropbox.com/s/hfcp44u2r0jd7fy/readerWriter.c
A: POSIX semaphores are not well suited for application code since they are interruptible. Basically any sort of IO to your processes will mess up your signalling. Please have a look at this post.
So you'd have to be really careful to interpret all error returns from the sem_ functions properly. In the code that you posted there is no such thing.
If your implementation of POSIX supports them, just use rwlocks, they are made for this, are much higher level and don't encounter that difficulty.
A: In computer science, the readers-writers problems are examples of a common computing problem in concurrency. There are at least three variations of the problems, which deal with situations in which many threads try to access the same shared memory at one time. Some threads may read and some may write, with the constraint that no process may access the share for either reading or writing, while another process is in the act of writing to it. (In particular, it is allowed for two or more readers to access the share at the same time.) A readers-writer lock is a data structure that solves one or more of the readers-writers problems.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,831 |
Круглый город аль-Мансура (также Город мира — , «мади́нат ас-саля́м») — первоначальный центр Багдада, возведенный между 762 и 767 годами как официальная резиденция аббасидского халифа. В настоящее время расположен в западной части современного города. На территории Круглого города находился Дом мудрости, основанный при халифе Харуне ар-Рашиде (786—809).
Ссылки
Al-Mansur's Round City of Baghdad in «archnet» website
Baghdad (Madinat al-Salam) in «islamic art» website
История Багдада
Архитектура Багдада | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,884 |
package org.jsecurity;
/*import atunit.AtUnit;
import atunit.Container;
import atunit.MockFramework;
import org.junit.runner.RunWith;*/
/**
* Super class that simply provides boiler plate annotations for subclass tests.
*
* @author Jeremy Haile
* @since 0.9
*/
/*@RunWith(AtUnit.class)
@Container(Container.Option.SPRING)
@MockFramework(MockFramework.Option.EASYMOCK)*/
public class AtUnitTest {
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 252 |
Q: Jasper ireport sql server - showing net.sourceforge.jtds. in final report preview I am facing a problem with Jasper on MS SQL db. My final report is showing net.sourceforge.jtds.jdbc. in result for each and every row and column. I configured a very simple select query. JrXML also compiled good to Jasper and placed the fields in details band. I have no idea whats going wrong. Please help me out.
Note: I am referring to jtds-1.3.1 jar through iReport
Thank you.
A: It is just the Variable type(field class value in iReport) of the field, in previous versions it used to automatically used to set to their original types. But it is default getting set to java.sql.clob in the version which I am using, seems a little weird.
Thank you
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,502 |
Льодовикова теорія (, , ) – система наукових уявлень про неодноразовий розвиток льодовиків, які покривали великі площі Землі. До середини 70-х рр. XIX ст. вважалося, що відклади, які включають ератичні валуни, належать до морських відкладів, серед яких валуни були розсіяні айсбергами (див. дрифтова теорія). Згідно з льодовиковою теорією, ератичні валуни, розповсюджені на великих територіях Північної Америки та Європи, відкладені льодовиками, які пересувалися з Півночі на Південь на сотні та тис. км. Льодовикова теорія базується на позиціях полігляціалізму і торкається головним чином плейстоценової історії Землі, хоча встановлено неодноразовий розвиток великих зледенінь і у віддаленішому геологічному минулому.
Теорії льодовикового періоду
Астрономічна теорія льодовикового періоду
Теорія льодовикового періоду Джеймса Кролля
Теорія льодовикового періоду Міланковича
Література
Гляціологія | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,536 |
Q: Docker and java sockets: Share data between containers I'm coding a client-server service where my server is sending files to the client.
In the following example, I tried to send a list of file names to my client.
Server
serverSocket = new ServerSocket(4000);
connsock = serverSocket.accept();
objectOutput = new ObjectOutputStream(connsock.getOutputStream());
List<String> file_names = new ArrayList<String>();
File[] files = new File("C:\\ServerMusicStorage").listFiles();
for (File file : files) {
if (file.isFile()) {
file_names.add(file.getName());
}
}
objectOutput.writeObject(file_names);
objectOutput.flush();
Client
newclientSocket = new Socket("localhost", 4000);
objectInput1 = new ObjectInputStream(newclientSocket.getInputStream());
System.out.println("<---Available files--->");
// get list of files from server
Object file_names = objectInput1.readObject();
file_list = (ArrayList<String>) file_names;
int count = 1;
for (int i = 0; i < file_list.size(); i++) {
System.out.println(count + ")" + file_list.get(i));
count++;
}
So when I run my program at java NetBeans IDE it works as I want. I get the files
<---Available files--->
1)blank.wav
2)fuark.wav
For the docker connection i created a network with
docker network create client_server_network
I run the server with
docker run --env SERVER_HOST_ENV=server --network-alias server --network client_server_network -it server
and the client with
docker run --network client_server_network -it clientimage
Although the client-server connection is successful through docker containers, when I run both services I don't get any output.
<---Available files--->
I'm stuck in this for days. What might be wrong? If I should provide any other information please tell me.
P.S. at the server-side of docker I set the server image as the host newclientSocket = new Socket("server", 4000)
A: Container have their own fileSystem different from the host filesystem.
Your path C:\ServerMusicStorage can't work in your container because this file is not in your container.
You should take a look to bind or volume
Or copying the file when creating your image.
Also your path is a windows path you should change it for an unix path (/yourdirectory) because most of docker images are linux system
If you want to use COPY in your DockerFile just add
COPY ServerMusicStorage/ /ServerMusicStorage
But if i'm not mistaken src file should be a relative path... related to
So you have to put /ServerMusicStorage near your build dir
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,398 |
Mercedes-Benz W220 — модель автомобіля «класу S» 4-го покоління фірми Mercedes-Benz, яка випускалася в 1998—2005 роках. Належить до седанів (нім. лімузинів) люкс-класу. Всього було збудовано 485.000 екземлярів.
Історія виробництва
Автомобіль був представлений широкій публіці в серпні 1998-го року, виробництво почалося через два місяці. Якщо W140-й ставив наголос на свої флагманські моделі V140 з 8-ми і 12-ти циліндровими двигунами, а короткі седани з дизелями вважалися слабкою ланкою, то у 220-го все було навпаки. Допомогла цьому технологія Common Rail для дизельних двигунів, яка значно знизила витрату палива. Тому найпопулярніша модель W220-го стала S320 CDI спочатку мала турбо дизель OM613 (компонування Р6, об'єм 3222 см³ потужність 194 к.с., крутний момент 470 Нм). Також випускався S400 CDI з турбодизелем OM628 у компонуванні V8 (3996 см³, 247 к.с., 660 Нм).
Серед моделей з бензинами двигунами базовою була S320 з двигуном М112 (3199 см³, V6, 221 к.с., 315 Нм), хоча була також полегшена версія S280 для експорту в Азію (той же М112 але об'ємом 2799 см³, V6, 194 к.с., 270 Нм). За потужністю М112-й двигун трохи поступався М104-ому на W140 але був більш економічним.
Вісімкою (V8) 220-го став двигун М113 для моделей S430 (4266 см³, 275 к.с., 400 Нм) і S500 (4966 см³, 302 к.с., 460 Нм). А для флагмана 220-х S600 став новий М137 (5786 см³, V12, 362 к.с., 530 Нм).
В 1999-му році компанія купила тюнінгову фірму AMG, яка стала його офіційним тюнером і для нового S-класу випустила модель S55 яка комплектувалася спортивною підвіскою і гальмами, але головне з двигуном M113 5439 см³ V8 потужністю 355 к.с. при моменті 530 Нм.
У цілому ставка на економію прямо відбилася на покупцях S-класу: W220-ий став привабливий для вищих прошарків середнього класу. Але вигравши один ринок, Мерседес втратив інший — еліту, особливо після запуску BMW E65/E66 від його конкурента. Тому в 2002-му році автомобіль піддається глибокій модернізації, під час якої крім зовнішніх деталей (зокрема, нових задніх ліхтарів, покращення аеродинаміки спереду) і оновленого набору електроніки, з'являється низка нових двигунів.
Для свого нового ринку S320 CDI отримує новий двигун OM648 (3222 см³, Р6, 201 к.с., 500 Нм), А S320 замінює S350 з тим же М112-м двигуном але з об'ємом збільшеним до 3724 см³, потужність 242 к.с., момент 350 Нм. Щоб виграти назад свій колишній ринок від конкурентів, Мерседес дає флагману S600 новий V12 M275 (5513 см³, 493 к.с. 700 Нм).
А AMG поставив на S55 нагнітач та інтеркулер, які збільшили потужність до 493 к.с. а крутний момент до 700 Нм. Крім цього AMG дебютує свої моделі з V12, починаючи в 2002-му році з дрібною серією S63 з 6258 см³ об'ємом М137 потужністю 438 к.с. і моментом 738 Нм. А потім у 2004-му році в хід йде модель S65 де на М275 AMG поставили бітурбо і збільшили об'єм до 5980 см³. Потужність двигуна стала 612 к.с., а крутний момент 1000 Нм.
Особливості моделі
Mercedes-Benz W220 поступався попереднику в габаритах і потужності, але компенсував це своєю насиченістю електронікою. W220 була першою з моделей, на яку Mercedes встановив повітряну (пневматичну) підвіску Airmatic, яка могла прийняти різні варіанти водіння, наприклад: спортивний, комфортний і т. д. шляхом зміни тиску в амортизаторах. Також авто мало радарну систему круїз-контролю Distronic.
На всіх моделях 220-го була комп'ютерна система стабілізації проти заносів, ESP і Brake Assist для аналогічної допомоги при різкому гальмуванні. При оновленні в 2002-му, обидві були включені в нову систему PRE-SAFE яка також натягала ремені, вирівнювала крісла та закривала люк і вікна при неминучій загрозі втрати контролю або зіткнення. Оновлення 2002-го року торкнулися моделі S350, S430 і S500 на які вперше пропонується опція 4Matic давший автомобілю повний привід. Так само через економію палива для великих двигунів на S500 і S600 в 2002-му році дебютувала розроблена Даймлером система активного контролю циліндрів (ACC) при якій всі циліндри працювали тільки при великих обертах двигуна, а при малих обертах половина циліндрів були відключені.
W220-й виявився досить комфортним, наприклад дзеркала при яскравому відображенні фар заднього автомобіля автоматично затемнювали щоб не засліпити водія, а зовнішні дзеркала мали підігрів проти конденсації. З'явилися система входу та запалювання без використання ключів, повністю автоматизований клімат-контроль з вугільним фільтром проти диму і пилку, а також вентильовані сидіння. Найсуттєвішою за новизною функцією стала система COMAND, яка об'єднала функції навігації (GPS-приймач), комунікації (телефон), а також телебачення і аудіо. Опція для COMAND — Linguatronic — розуміє голосові команди водія, зокрема озвучену адресу для навігатора та телефонні номери. Вона встановлювалася додатково на замовлення.
Вплив моделі
Попри те, що в 2005-му році 220-го змінив W221-й S-клас, досі тривають суперечки про цю лінійку автомобілів. Залишивши минулі традиції Мерседеса, вона отримала нових прихильників і супротивників. Багато хто вважає, що W220-й опустив S-клас з вищого світу автомобілів, який Мерседес успішно взяв своїми W126-ми і W140-ми моделями, в простий повнорозмірний седан. Мерседес відповів на критику в 2002 році запуском автомобілів Maybach, який чітко розділив роль лімузина для еліти і «просто» представницького класу, як свого часу було в 1960-ті та 1970-ті роки при випуску лімузина Mercedes-Benz W100 і «простих» S-класів типу W108-го і W116-го. Але для багатьох головним мінусом 220-х стала, як не дивно, саме їх насиченість електронікою та складною апаратурою, поломки яких були питанням часу, а ремонт довгим і не дешевим. Найголовніший біль — пневматична підвіска. Тому врешті-решт у плані виробництва W220-й зумів всього лише повторити W140-й по тривалості виробництва (7 років), але перевершити за кількістю виготовлених екземплярів (485 тисяч).
Двигуни
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W220 | {
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Labour shortages and ageing population prompt review of early retirement
Topic: Envelhecimento da população ativa Social protection Labour market policies Emprego e mercados de trabalho Envelhecimento da população ativa Reforma
Grünell, Marianne
Download article in original language : NL9912175FNL.DOC
In December 1999, employer and employee representatives in the Dutch Social and Economic Council advised the government to abolish the financial advantages for companies contained in existing early retirement provisions, and suggested a more flexible pre-pension system. The proposals follow various government initiatives to keep older employees working longer, and a recently submitted Act on age discrimination intended to promote their participation in the labour market. Tensions surrounding labour shortages and the costs of an ageing population are thus now focusing attention on the position of older workers, following recent initiatives to boost the labour market participation of mainly female benefit recipients with children and of migrants.
Employer and employee representatives in the Social and Economic Council (Sociaal Economische Raad, SER) are advising the government to abolish the financial advantages for companies contained in existing early retirement provisions. This draft proposal, issued in December 1999, also includes the SER's recommendation to abolish existing benefit supplements to employees made redundant following company reorganisations, on the grounds that the money would be better spent on training and finding suitable new employment. At the same time, the Council's unanimous draft proposal also includes an initiative to offer tax incentives to companies in order to lower the labour costs for older employees. Finally, the SER wants to reinstate mandatory job-seeking for unemployed people aged over 57.5 years, a point which the trade unions agreed to on condition that the government conduct a survey to determine whether older job-seekers actually have a chance of finding employment. Employers see no need for such an investigation, citing the present labour shortages as a reason for directly reinforcing mandatory job-seeking for older people.
Keeping older people working longer
A majority in the Lower House of parliament has already voiced support for the SER proposal concerning early retirement, in the hope that the measure will enable older employees to continue working longer. In the short term, the proposal will help combat the shortages on the labour market, and in the long term it will help cover the costs associated with the ageing population. The number of individuals aged between 40 and 65 in the economically active population will increase by 50% between now and 2015. The SER aims to increase the degree of labour market participation among employees aged over 55 from the current level of 25%-30% to over 50% by 2030. In summer 1999, Minister of Social Affairs Klaas de Vries took measures to offer employees older than 55 years who were either demoted or changed jobs to a lower-paying position benefits based on their old salary in the event of redundancy or disability.
Minister De Vries intends to triple the number of older employees in the labour market, with his attention primarily on the 171,000 individuals who are, in principle, perfectly healthy despite having taken early retirement. In the 1980s, early retirement became widespread in order to make way for younger employees and advance painless corporate reorganisations. Early retirement provisions allow employees to stop working from the age of 55 and receive 60%-80% of their last-earned salary until reaching the age at which pension benefits commence. All employers pay premia for this provision, and employees who have taken early retirement and return to work receive reduced benefits. The SER proposal will transform this into a flexible "pre-pension" structure, enabling employees to determine the date of retirement, although they must bear the costs of early retirement. A retired employee may also continue working part-time without affecting the level of benefits. Along with the minister, VNO-NCW, the largest employers' association, has also taken up on the early retirement issue, announcing plans to re-examine provisions"" in existing collective agreements (NL9808194F). The initial reaction of the trade unions was one of utter disbelief.
Labour market shortages
The timing of the SER draft proposal announcement was impeccable. The current tight labour market paves the way for completely different policy options than those based on a high level of unemployment, which plagued the Netherlands up until a few years ago. Closures and reorganisations in the 1970s and 1980s led to more or less attractive redundancy payment schemes for older employees. Employer and employee representatives have since been sharply criticised for allowing this age-group to disappear from the labour market to collect disability and early retirement benefits with such ease (NL9708125F). In 1975, three-quarters of the male population over the age of 55 was active on the Dutch labour market, but compared with other European countries, the Netherlands' current rate of employed individuals aged over 55 is low: 33% as opposed to the rate of around 50% found in Denmark and the UK (according to the 1998 OECD Employment Outlook). Since the economic tide has turned, arguments that previously carried no weight are now beginning to make sense. For example, it is now thought that older employees' experience should be seen as a benefit to companies, whereas this was discounted as outdated not so long ago. The employment potential of migrants is also suddenly being seen in a different light (NL9909163N), while an employment obligation for mainly female benefit recipients with children has also been placed on the government's agenda (NL9911172F).
The cabinet has contributed to the shifting of boundaries with the proposal of a new age discrimination Act. The government proposes to eliminate the requirement to retire at 65, while retired employees who wish to continue working part-time may do so, without jeopardising their pension rights as a result. The Act will prohibit age discrimination not only in recruitment, but also in education and promotion. The Act will ban direct and indirect discrimination, which means that job advertisements may specify age restrictions only in special cases. Including education/training and promotion in the Act's scope is intended to keep older individuals active in the labour market longer.
Measures such as no longer upholding 65 years as the authorised age to retire, along with implementing flexible part-time retirement provisions, reflect society's wish to transform the uniformity which has traditionally characterised the span of the average person's working life. The impetus of the various cabinet proposals on increasing labour market participation is clear: making a full-scale effort to handle the consequences of economic growth, fight excessive wage growth and make economies (by changing the early retirement provisions). The proposed age discrimination Act also fits into this framework, in that it seeks to counteract the frequently marginal position of employees of 55 years and older. Now that the economic tide has turned, there is room to identify existing discrimination against the older segment of the population and follow through with resolutions concerning their situation. (Marianne Grünell, HSI) | {
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{"url":"https:\/\/en.wikipedia.org\/wiki\/Mukai-Fourier_transform","text":"# Fourier\u2013Mukai transform\n\n(Redirected from Mukai-Fourier transform)\n\nIn algebraic geometry, a Fourier\u2013Mukai transform \u03a6K is a functor between derived categories of coherent sheaves D(X) \u2192 D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K \u2208 D(X\u00d7Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type.\n\nThese kinds of functors were introduced by Mukai\u00a0(1981) in order to prove an equivalence between the derived categories of coherent sheaves on an abelian variety and its dual. That equivalence is analogous to the classical Fourier transform that gives an isomorphism between tempered distributions on a finite-dimensional real vector space and its dual.\n\n## Definition\n\nLet X and Y be smooth projective varieties, K \u2208 Db(X\u00d7Y) an object in the derived category of coherent sheaves on their product. Denote by p the projection X\u00d7YX, by q the projection X\u00d7YY. Then the Fourier-Mukai transform \u03a6K is a functor Db(X)\u2192Db(Y) given by\n\n${\\displaystyle {\\mathcal {F}}\\mapsto \\mathrm {R} p_{*}\\left(q^{*}{\\mathcal {F}}\\otimes ^{L}K\\right)}$\n\nwhere Rp* is the derived direct image functor and ${\\displaystyle \\otimes ^{L}}$ is the derived tensor product.\n\nFourier-Mukai transforms always have left and right adjoints, both of which are also kernel transformations. Given two kernels K1 \u2208 Db(X\u00d7Y) and K2 \u2208 Db(Y\u00d7Z), the composed functor \u03a6K2\u03a6K1 is also a Fourier-Mukai transform.\n\nThe structure sheaf of the diagonal ${\\displaystyle {\\mathcal {O}}_{\\Delta }\\in \\mathrm {D} ^{b}(X\\times X)}$, taken as a kernel, produces the identity functor on Db(X). For a morphism f:XY, the structure sheaf of the graph \u0393f produces a pushforward when viewed as an object in Db(X\u00d7Y), or a pullback when viewed as an object in Db(Y\u00d7X).\n\n## On abelian varieties\n\nLet ${\\displaystyle X}$ be an abelian variety and ${\\displaystyle {\\hat {X}}}$ be its dual variety. The Poincar\u00e9 bundle ${\\displaystyle {\\mathcal {P}}}$ on ${\\displaystyle X\\times {\\hat {X}}}$, normalized to be trivial on the fiber at zero, can be used as a Fourier-Mukai kernel. Let ${\\displaystyle p}$ and ${\\displaystyle {\\hat {p}}}$ be the canonical projections. The corresponding Fourier\u2013Mukai functor with kernel ${\\displaystyle {\\mathcal {P}}}$ is then\n\n${\\displaystyle R{\\mathcal {S}}:{\\mathcal {F}}\\in D(X)\\mapsto R{\\hat {p}}_{\\ast }(p^{\\ast }{\\mathcal {F}}\\otimes {\\mathcal {P}})\\in D({\\hat {X}})}$\n\nThere is a similar functor\n\n${\\displaystyle R{\\widehat {\\mathcal {S}}}:D({\\hat {X}})\\to D(X).\\,}$\n\nIf the canonical class of a variety is ample or anti-ample, then the derived category of coherent sheaves determines the variety.[1] In general, an abelian variety is not isomorphic to its dual, so this Fourier\u2013Mukai transform gives examples of different varieties (with trivial canonical bundles) that have equivalent derived categories.\n\nLet g denote the dimension of X. The Fourier\u2013Mukai transformation is nearly involutive\u00a0:\n\n${\\displaystyle R{\\mathcal {S}}\\circ R{\\widehat {\\mathcal {S}}}=(-1)^{\\ast }[-g]}$\n\nIt interchanges Pontrjagin product and tensor product.\n\n${\\displaystyle R{\\mathcal {S}}({\\mathcal {F}}\\ast {\\mathcal {G}})=R{\\mathcal {S}}({\\mathcal {F}})\\otimes R{\\mathcal {S}}({\\mathcal {G}})}$\n${\\displaystyle R{\\mathcal {S}}({\\mathcal {F}}\\otimes {\\mathcal {G}})=R{\\mathcal {S}}({\\mathcal {F}})\\ast R{\\mathcal {S}}({\\mathcal {G}})[g]}$\n\n## Applications in string theory\n\nIn string theory, T-duality (short for target space duality), which relates two quantum field theories or string theories with different spacetime geometries, is closely related with the Fourier-Mukai transformation, a fact that has been greatly explored recently.[2][3]\n\n## References\n\n1. ^ Bondal, Aleksei; Orlov, Dmitri (2001). \"Reconstruction of a variety from the derived category and groups of autoequivalences\" (PDF). Compositio Mathematica. 125 (3): 327\u2013344. doi:10.1023\/A:1002470302976.\n2. ^ Leung, Naichung Conan; Yau, Shing-Tung; Zaslow, Eric (2000). \"From special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai transform\". Advances in Theoretical and Mathematical Physics. 4 (6): 1319\u20131341. doi:10.4310\/ATMP.2000.v4.n6.a5.\n3. ^ Gevorgyan, Eva; Sarkissian, Gor (2014). \"Defects, non-abelian t-duality, and the Fourier-Mukai transform of the Ramond-Ramond fields\". 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Royal nicknames: The British royal family's endearing pet names for one another
hellomagazine.com
We may know them by their official titles, but behind the scenes the royals have their own fun nicknames for each other. Click through our gallery to discover the endearing monikers that British royal family members are called by their nearest and dearest. Photo: Getty Images
The close bond between father and son was revealed during an interview by Prince Harry with his father, the Prince of Wales, for BBC Radio 4's Today program. As guest editor for the show, Prince Harry was heard affectionately calling his father "pa." During the interview, Prince Charles also referred to his youngest son as "darling boy" when acknowledging how proud he is that Harry is following in his footsteps regarding certain causes. Photo: Kensington Palace
It seems Prince Harry was right at home when he, Prince William and Kate Middleton visited the set of Diagon Alley in the Harry Potter films during the Inauguration of Warner Bros. Studios Leavesden in 2013. It turns out that the British royal's friends call him "Potter" as discovered by Canadian journalist Lisa LaFlamme, Chief Anchor and Senior Editor for CTV National News when she met with him at Kensington Palace. She wrote ahead of the interview airing on June 16, "The title that caught my eye was the same one the Prince's communications director promptly put away -- a picture book about Harry Potter. To his friends, "Potter" is the Prince's nickname but apparently not something Palace PR would want to distract from our interview." Photo: Getty Images
During a day out at the Chelsea Flower Show 2016, Prince William and Kate were overheard referring to each other as "darling" and "babe". While inspecting a poppy display, William asked his wife, "Could you make one of these, darling?"
As a young girl at St Andrew's School in Berkshire Kate was nicknamed 'Squeak'. She explained how the name came about while on a visit to the school years later. "I was nicknamed Squeak just like my guinea pig," she said. "There was one called Pip and one called Squeak because my sister was called Pippa and I was Squeak."
Popular at school, Kate was allegedly known as the 'Princess-in-waiting' by her fellow Marlborough College pupils.
As a little girl, Queen Elizabeth referred to herself as 'Tillabet'. As she grew up, she became known as 'Lilibet' among her family and close friends – apparently some of them still use the affectionate nickname for the monarch.
Prince Harry revealed that Diana used to call William 'Wombat'. The Duke revealed the origins of the moniker in an interview. "It began when I was two. I've been rightfully told because I can't remember back that far," he explained. "But when we went to Australia with our parents, and the wombat, you know, that's the local animal."
William's official name is His Royal Highness Prince William Arthur Philip Louis but he is affectionately known as 'Wills' to friends and the public.
Prince Harry's friends refer to him as 'Spike'. The young royal had a Facebook account that used the pseudonym 'Spike Wells' for four years until it was shut down because it posed a "security risk".
The Duke of Edinburgh's pet name for his wife came to light after the 2006 film The Queen was released. In a bedtime scene at Balmoral, Philip says to Her Majesty, played by Dame Helen Mirren, "Move over, Cabbage."
Born Catherine Elizabeth Middleton, the Duchess is widely known as Kate. "Miss Middleton uses both names equally, and she has never expressed a preference for either Catherine or Kate since her engagement to Prince William," the royal wedding official website said prior to her becoming William's wife. "Catherine is the name that Miss Middleton grew up with in her family, and Kate is the name that she tends to use in a work context".
While studying Geography at the University of St Andrews in Scotland, William was known as 'Steve' to keep a low profile and avoid any unwanted attention.
While Kate was pregnant with Prince George, she and husband William are said to have called their unborn baby 'our little grape'.
William and Kate
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How royal couples met – From Meghan Markle and Kate Middleton to the Queen
Our favourite photos of royals meeting starstruck celebrities
Kate Middleton and Meghan Markle UNITE with kids including baby Archie to watch Princes at polo - LIVE UPDATES | {
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I am an Australian author, psychologist and eclipse chaser. Originally from Ingham in North Queensland, and I grew up on a sugar cane farm in an Italian community. Travel was in my blood from an early age.
I was an avid reader, and after being inspired by reading about total eclipses I vowed that one day I would see one. I added it to my pretty lengthy list of things I wanted to experience, and places I wanted to see.
I left Australia in 1997 on a round-the-world adventure, and after some pretty epic moments, I landed in the UK. This was fortuitous timing, as I was able to see my first total solar eclipse in 1999 by travelling to the coast of France. My life has not been the same since, and I knew from that moment I was an eclipse chaser. Since then, I have stood in the shadow of the Moon in many exotic places around the world. I have seen eleven total eclipses and two annular eclipses (I don't count the partial eclipses). I particularly enjoy experiencing the power of nature – along with eclipses, I have chased the aurora, and am drawn to volcanoes, lightning and storms. I love those magical reminders that nature is alive and powerful.
I have lived in the UK for twenty years, in my adopted home of Belfast, Northern Ireland. Belfast is fantastic, and I think it is one of the coolest cities in the world. I spent my early adult life studying, and have three degrees and a successful career as a Clinical Psychologist specialising in health. I worked in the UK National Health Service for over 15 years, and have also undertaken private consulting. I was an Assistant Course Director of a doctoral training program in Clinical Psychology at Queen's University Belfast for about six years, until I had to leave following a serious illness.
I have a passion for phenomenological research, and I research people's experiences – what it is like for them from their own subjective experience. I am an expert in Interpretative Phenomenological Analysis (IPA), and love inspiring others to use this approach in their academic work.
Since I was able to secure a publishing contract in 2011, I have been researching the experience of the total solar eclipse. Now, my psychology and eclipse chasing careers have merged together wonderfully so now I am fully engaged in eclipse-related activities until after the next total solar eclipse. We can learn so much about us as human beings by exploring intense positive, emotional experiences – just like total eclipses. I firmly believe that if every person could experience a total solar eclipse, the world would be a better place.
Eclipse chasing – what's not to love? (c) Kieron Circuit.
With a professional background in Psychology, Kate is a highly skilled Eclipse Planning Consultant. She is one of the very few eclipse chasers worldwide who has experienced a total solar eclipse in her own home community and knows first-hand the challenges involved in preparing a community for the total eclipse. She was involved in eclipse planning for 2012 in Australia and led the process in 2015 in the Faroe Islands. Drawing upon her consulting and academic skills, she then researched and wrote the White Paper on Community Eclipse Planning – a document that has been used by hundreds of communities across the U.S. in preparation for the August total eclipse. She has already helped many communities in the path in their approach to planning.
Along with her expertise in eclipse planning, Kate is the recognised authority on the total solar eclipse experience, and is the person who is most able to answer the key question – what is it like to experience a total solar eclipse? Not only is she able to talk from her personal experiences of seeing ten total solar eclipses, she can also share the experience of the hundreds of others she has surveyed and interviewed. She has authored three books on the topic and is in demand as a speaker and in the media before every eclipse.
Kate has participated in many interviews about the eclipse experience, and she has been featured in several documentaries and books. She is unique because she is able to communicate about the eclipse experience to new audiences, in a passionate and relatable manner. | {
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Dried fruit provides rich, concentrated cherry flavor, a contrast to the tart frozen cherries. The white, creamy chocolate topping drizzles the pies with sweetness.
Combine the first 5 ingredients in a medium saucepan; cook over medium heat 7 minutes or until liquid almost evaporates. Remove from heat. Add butter, stirring until butter melts. Stir in cornstarch and vanilla. Cool slightly.
Working with 1 Sweet Cream Cheese Dough circle at a time, remove plastic wrap from dough. Place dough on a lightly floured surface. Spoon about 2 tablespoons cherry mixture into center of circle. Fold dough over filling; press edges together with a fork or fingers to seal. Place pie on a large baking sheet covered with parchment paper. Repeat procedure with remaining Sweet Cream Cheese Dough circles and remaining cherry mixture. Freeze 30 minutes.
Remove pies from freezer. Pierce top of each pie once with a fork. Place baking sheet on bottom rack in oven. Bake at 425° for 19 minutes or until edges are lightly browned and filling is bubbly. Cool completely on a wire rack.
Place chocolate chips in a heavy-duty zip-top plastic bag; microwave at HIGH 1 minute or until chips are soft. Knead bag until smooth. Snip a tiny hole in corner of bag; drizzle chocolate over cooled pies.
These had a very good cherry flavor, delicious. We like hearty portions of dessert so what the heck, I doubled their size and got 6 in the batch. Also used 2 tb of butter instead of 1. In the dough I used regular cream cheese instead of the low fat. Instead of white chocolate I made a Meyer Lemon glaze for them.
We are still eating these out of the freezer from when I made them this summer with fresh picked cherries. They are really good and I plan to try with different fruit fillings. I prefer to top with a simple confectioners sugar glaze as I don't really like white chocolate. | {
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} | 9,258 |
This script will create a menu similar to the kingdom hearts 2 save menu.
To display a face, get the actor name of the first character in the party, and add _FL.
To display a map logo, get the exact map name, and add _mgfx to it.
Other instructions are in the demo.Make sure to follow them or you will have some ugly words appearing on your screen.
Me for editing the script, menu graphics, and the riku charset in the demo.
I updated the demo because there was an error when you click load game. It would show "Which file would you like to load?" On the top screen, so i fixed and uploaded a new demo.
I think that's look like Moghunter's Save layout.
It is i put him in the credits. I just edited a few things and included kh graphics.
Now I'm loading the Demo because I lost Mog hunter's save script.
MOG's script lags a lot so which is "better" depends on the point of view. I prefer simple and straightforward layouts rather than overcrowded and fancy.
Also, you can switch the images with your own in both CMSes, so I don't really see a problem.
I've been working on this script for the past couple of hours and I've just finished editing the large face to display the map instead of the face. In draw_heroface3 change actor.name to actor.map_name then in Window_SaveFile in refresh change actor = actors(0) to actor = @game_map. Also to show the munny just make 2 new self.contents.draw_text and put them in the section of Window_SaveFile that displays the @game_map.map_name.to_s and put $data_system.words.gold in one and @game_party.gold.to_s in the other. So there you go. Now all you have to do is make an icon for each of your worlds and name them the map name like you would the faces.
The one big turn off for me. Like to have the faces, rather then player graphic be used. Like instead of just the party leader's face being shown then all the party player sprites, just have the face graphics.
Click spoiler to check it out. Completely new graphics.
But there is one thing that annoys me a little bit... the LOAD in the left upper corner is always there, even when I am saving.
Would it be possible to make a second graphic with SAVE there?
A cool addition would also be a random display of a face of one actor in the party, not necessarily the one of the first member.
Sorry for my bad English, i am from Germany. ^^"
The world's name only comes up during in-game saves and loads. It does not come up when I first start the game. Also, the _fl grpahics never show, even though they are 140 x 94 as you dictate.
Is it possible that I somehow missed something?
Pretend _ is a space. If your map or character name is EXAMPLE_ then for the picture you have to put EXAMPLE_ as the image name.
It also may be messing up if you have other scripts around it. This was made for use with the full mog hunter menu system originally. So any other menu system would probably affect it, though I am not to sure. I tested it on my kingdom hearts game, and it works on both the load and in game. It's not made to function any differently for a map than a title screen. Try re ordering the scripts you have with this one on or close to the top, you will either get an error or it will be fixed. That will also tell you what is causing the problems.
Your script is great and I've adapted it to my game. I have 1 hero at the beginning of the game but when someone joins my team and I save, the level of each character overlaps in the save menu. How can I fix this? I am not very good with Ruby scripts. | {
"redpajama_set_name": "RedPajamaC4"
} | 782 |
Der war eine Verwaltungseinheit in der ehemaligen Region Auckland im Norden der Nordinsel von Neuseeland. Der Verwaltungssitz lag in der Ortschaft . Im November 2010 wurde der Distrikt aufgelöst und in den neu gebildeten integriert.
Geographie
Der ehemalige lag zwischen und dem am , das in seiner Fortsetzung in den mündet. Zur Volkszählung im Jahr 2006 zählte der Distrikt 45.183 Einwohner.
Geschichte
Auf Basis des wurde der 1989 durch die Zusammenlegung von und den Gebieten , und von , und den Gebieten und des neu gebildet. Zu jener Zeit war Neuseeland verwaltungstechnisch noch in , und organisiert und unterteilt. Das Gesetz, machte es der damaligen Labour-Regierung möglich auf 850 unterschiedlichen Verwaltungseinheiten 86 zu schaffen. In einer Verwaltungsreform im Jahr 2000 wurde die regionale Verwaltungen noch einmal neu geordnet. Übrig blieben 11 reine Regionale Councils, 12 City Councils und 54 District Councils. Am 1. November 2010 wurde dann der mit sieben (Gebietskörperschaften) zusammengelegt und der gebildet. der gehört mit zu den Sieben.
Der ehemalige Distrikt heute
Der ehemalige wird heute als ein Stadtteil vom mit einem eigenen und mit sechs gewählten politischen Vertretern ausgestatteten geführt. Dieser kümmert sich um lokale Angelegenheiten.
Weblinks
Einzelnachweise
Ehemaliger Distrikt in Neuseeland
Geschichte (Auckland) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,764 |
Naugatuck Valley CC Students Named 2022 Coca-Cola Leaders of Promise Scholars
Naugatuck Valley Community College (NVCC) students Noah Berg of Brookfield and Alexis Villacres of Danbury are two of 207 Phi Theta Kappa members named as 2022 Coca-Cola Leaders of Promise Scholars this fall, and will each receive a $1,000 scholarship.
The Coca-Cola Leaders of Promise Scholarship Program helps new Phi Theta Kappa members defray educational expenses while enrolled in associate degree programs. Scholars are encouraged to assume leadership roles by participating in Society programs and are selected based on scholastic achievement, community service, and leadership potential. More than 1,300 applications were received.
Berg is majoring in business management at NVCC and cites American designer and sculptor Maya Lin as his inspiration. "Being able to follow my own personal calling is an incredible freedom," he shared. "The Coca-Cola Promise Scholarship will help me pursue my educational and professional goals."
"PTK has helped me develop my leadership and teamwork skills, both of which are of great value to me personally and professionally," said Villacres, a business studies major at NVCC. "The Coca-Cola Promise Scholarship will help me fund my education and further develop these skills."
A total of $207,000 is awarded through the Leaders of Promise Scholarship Program. The Coca-Cola Scholars Foundation provides $200,000 in funding for the scholarships, with $25,000 set aside for members who are veterans or active members of the United States military. The remaining amount is supported by donations to the Phi Theta Kappa Foundation and provides seven Global Leaders of Promise Scholarships, earmarked for international students.
"The Coca-Cola Scholars Foundation has a long history of providing financial assistance to outstanding students at community colleges," said Jane Hale Hopkins, president of the Coca-Cola Scholars Foundation. "We are proud to partner with Phi Theta Kappa to make it possible for more deserving students to achieve their educational goals and support tomorrow's leaders of the global community."
The funds provided by the Coca-Cola Scholars Foundation not only aid college completion, but also give students the opportunity to engage in programs and develop leadership skills to become future leaders in their communities.
NVCC's Alpha Theta Epsilon (ATE) chapter was recently recognized for leadership and achievement at the New England Regional PTK Spring 2022 conference. With campuses in Waterbury and Danbury, NVCC has inducted close to 300 students into PTK since 2021.
NVCC is among 12 community colleges in Connecticut merging in 2023 to become CT State Community College, one of the largest community colleges in the country and largest in New England, dedicated to quality, access, and affordability. CT State students will be able to apply once and take classes at any campus. In addition to NVCC, other locations include Asnuntuck (Enfield), Capital (Hartford), Gateway (New Haven & North Haven), Housatonic (Bridgeport), Manchester, Middlesex (Middletown & Meriden), Northwestern (Winsted), Norwalk, Quinebaug Valley (Danielson & Willimantic), Three Rivers (Norwich), and Tunxis (Farmington) Community Colleges.
Phi Theta Kappa is among the largest honor societies in higher education with approximately 240,000 active members and nearly 1,300 chapters on college campuses in all 50 of the United States, and 11 countries. More than 3 million students have been inducted since its founding in 1918.
CT State Swears In First Chief of Police
Connecticut native Christopher Chute is now Connecticut State Community College's inaugural chief of police following a public swearing-in ceremony today at Naugatuck Valley Community College.
NVCC Director Receives NESMA Award
Naugatuck Valley Community College has announced that Sharon Lutkus, interim director of advanced manufacturing technology and welding, was awarded the Spring Industry Community Advocate Award at New England Spring & Metal Stamping Association's (NESMA) annual meeting in Bristol.
Registration Open for Spring Semester at CT's Community Colleges
Credit registration for the Spring 2023 semester is in progress at Connecticut's 12 community colleges.
CT Community Colleges and Xometry Announce Full Tuition Scholarships for Manufacturing Students | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,888 |
Clubhouse user data gets exposed; CEO claims it wasn't a leak
Filipe Espósito
- Apr. 12th 2021 5:52 pm PT
@filipeesposito
Clubhouse continues to make headlines around the world for another week, but this time with some controversial news. Personal data of 1.3 million users of the audio-based social network was exposed on a popular hacker forum, but the company disagrees that it was a leak.
As reported by CyberNews, someone has posted last week a database with data from 1.3 million Clubhouse users. This database includes information such as user ID, name, photo, social network profiles, and other profile details.
Immediately, Clubhouse CEO Paul Davison argued that the articles about the exposed data were "misleading and false" since he claims that all this data is public to Clubhouse users (via The Verge). After that, the official Clubhouse profile on Twitter shared a statement reinforcing that the exposed database data can be obtained by any developer through the app's API.
This is misleading and false. Clubhouse has not been breached or hacked. The data referred to is all public profile information from our app, which anyone can access via the app or our API.
Still, this raised privacy concerns about the app. As the privacy of user data becomes more important every day, the fact that anyone can download a database with a list of all users from a social network is questionable to say the least.
CyberNews security researcher Mantas Sasnauskas argues that Clubhouse should rethink how its API works to restrict the amount of data developers can get through. Although the exposed database includes only public information, this could lead to "phishing and social engineering attacks."
The way the Clubhouse app is built lets anyone with a token, or via an API, to query the entire body of public Clubhouse user profile information, and it seems that token does not expire. This should not only be reflected in the ToS, but also in the technical implementation of the app, making it harder for anyone to scrape user data. Having no anti-scraping measures in place can be seen as a privacy issue.
Particularly determined attackers can combine information found in the leaked SQL database with other data breaches in order to create detailed profiles of their potential victims. With such information in hand, they can stage much more convincing phishing and social engineering attacks or even commit identity theft against the people whose information has been exposed on the hacker forum.
It was reported last week that Twitter considered acquiring Clubhouse for $4 billion, but later discussions were halted. Now Clubhouse is looking for other investors while the competition is growing with companies like Facebook and Twitter working on their own live audio platforms.
Clubhouse introduces new 'Payments' feature for sending money to creators
LinkedIn now wants to have its own Clubhouse-like platform
Facebook's new 'Live Audio' feature is basically another Clubhouse copy
Spotify announces plan for live audio Clubhouse competitor
Filipe Espósito is a Brazilian tech Journalist who started covering Apple news on iHelp BR with some exclusive scoops — including the reveal of the new Apple Watch Series 5 models in titanium and ceramic. He joined 9to5Mac to share even more tech news around the world. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,443 |
{"url":"https:\/\/www.bou.lt\/theory\/geometry\/fields","text":"# Fields\n\n## Fields\n\n### Fields\n\nA field is a ring where the multiplication function has an inverse.\n\nThe integers, addition and multiplication form a ring, but not a group.\n\nThe rational numbers (except $$0$$), addition and multiplication form a field (and a ring).\n\nThe real numbers and complex numbers also form fields.\n\n### Finite (Galois) fields\n\nFinite number of elements.\n\n## Algebra on a field\n\n### Bilinear maps\n\nA bilinear map (or function) is a map from two inputs to an output which preserves addition and scalar multiplication. This is in contrast to a linear map, which only has one input.\n\nIn addition, the function is linear in both arguments.\n\nThat is if function $$f$$ is bilinear then:\n\n$$X=aM+bN$$\n\n$$Y=cO+dP$$\n\n$$f(X,Y)=f(aM+bN,cO+dP)$$\n\n$$f(X,Y)=f(aM,cO+dP)+f(bN,cO+dP)$$\n\n$$f(X,Y)=f(aM,cO)+f(aM,dP)+f(bN,cO)+f(bN,dP)$$\n\n$$f(X,Y)=acf(M,O)+adf(M,P)+bcf(N,O)+bdf(N,P)$$\n\nNote that:\n\n$$f(X,Y)=f(X+0,Y)$$\n\n$$f(X,Y)=f(X,Y)+f(0,Y)$$\n\n$$(0,Y)=0$$\n\nThat is, if any input is $$0$$ in an additative sense, the value of the map must be zero.","date":"2021-06-20 01:17:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9527184367179871, \"perplexity\": 737.9614769894182}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487653461.74\/warc\/CC-MAIN-20210619233720-20210620023720-00038.warc.gz\"}"} | null | null |
import { BaseKeystoneTypeInfo, KeystoneConfig, KeystoneContext } from '../types';
import { initConfig } from './config/initConfig';
import { createSystem } from './createSystem';
export function getContext<TypeInfo extends BaseKeystoneTypeInfo>(
config: KeystoneConfig<TypeInfo>,
PrismaModule: unknown
): KeystoneContext<TypeInfo> {
const system = createSystem(initConfig(config));
const { context } = system.getKeystone(PrismaModule as any);
return context;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,532 |
{"url":"http:\/\/www.dummies.com\/how-to\/content\/evaluating-limits-in-calculus.html","text":"The mathematics of limits underlies all of calculus. Limits sort of enable you to zoom in on the graph of a curve \u2014 further and further \u2014 until it becomes straight. Once it's straight, you can analyze the curve with regular-old algebra and geometry. That's the magic of calculus in a very small nutshell.\n\nHere are some important things to remember when evaluating limits:\n\n\u2022 The limit at a hole is the height of the hole.\n\n\u2022 The limit at infinity is the height of the horizontal asymptote.\n\n\u2022 Before trying other techniques, plug in the arrow number. If the result is:\n\n\u2022 A number, you're done.\n\n\u2022 A number over zero or infinity over zero, the answer is infinity.\n\n\u2022 A number over infinity, the answer is zero.\n\n\u2022 0\/0 or \u221e\/\u221e, use L'H\u00f4pital's Rule.","date":"2015-07-06 16:18:10","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9371772408485413, \"perplexity\": 798.1567030714751}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2015-27\/segments\/1435375098468.93\/warc\/CC-MAIN-20150627031818-00064-ip-10-179-60-89.ec2.internal.warc.gz\"}"} | null | null |
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"redpajama_set_name": "RedPajamaCommonCrawl"
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Official News Pakistan
Definitive source of press releases from federal and provincial governments and their departments
Inter Services Public Relations
Press Information Department
Gilani concerned about plight of Kashmiri detainees
Posted on March 3, 2018 March 6, 2018 by User2
Srinagar, March 03, 2018 (PPI-OT): In occupied Kashmir, the Chairman of All Parties Hurriyat Conference, Syed Ali Gilani, has said that the Kashmiri political prisoners are being subjected to solitary confinement, manhandling, denial of meetings with their families and non-provision of medical care and hygienic food.
Syed Ali Gilani in a statement issued in Srinagar expressed deep concern over the miserable plight of the detainees. He termed the shifting of the prisoners to outside the Kashmir Valley as political vendetta and said that those detained in Rajasthan, UP, West Bengal, Tihar and other jails of India were facing immense agonies and difficulties. Syed Ali Gilani also lashed at the puppet Chief Minister, Mehbooba Mufti, for her criminal silence over rape and murder of an eight-year-old girl Aasifa Bano and said that efforts by the puppet regime to shield and to give cover to the real culprits was the biggest irony.
The Chairman of Jammu and Kashmir Liberation Front, Muhammad Yasin Malik, in a statement in Srinagar said that the repression unleashed on the Kashmiri prisoners was indicative of undemocratic attitude of the puppet authorities. He also said that the people of the Kashmir Valley were closely watching the situation in Jammu region and they would not allow anybody to harm any Muslim there. He warned that if the anti-Muslim actions of communal forces were not stopped, the people of the Valley would launch a protest campaign to save their brethren in Jammu.
Member of the so-called Kashmir Assembly, Engineer Abdur Rasheed, in a statement in Srinagar said that he was not allowed to meet the prisoners lodged in Kot Bhalwal Jail in Jammu.
On the other hand, a Sikh students' group called Punjab Students Union held a convention and a march in Jalandhar city of Indian Punjab to highlight the Kashmir dispute and in support of the Kashmiris' right to self-determination. Prominent Indian human rights activist, Gautam Navlakha, and Professor Jagmohan Singh, who is the nephew of renowned Indian freedom-fighter, Baghat Singh, addressed the gathering. The event also saw the launch of a report prepared by a fact-finding team of the Union that had visited occupied Kashmir a few months ago.
In Geneva, speakers at an event held on the sidelines of the 37th session of the UN Human Rights Council said that the Kashmir dispute had resulted in massive internal and external displacement of the Kashmiris. The speakers included Barrister Abdul Majeed Tramboo, Professor Nazir Ahmed Shawl, Esam Al-Shaeri, Najeeb Al-Saa'di, Hamadan Al-Alie, Yousuf Aburas and Dr Wasam Basindowa.
Posted in GeneralTagged Kashmir Media Service
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The "Official News" is the website that is the proud name of the Pakistan News Industry, and it exists among the leading online news websites of Pakistan, on which the international media relies when it comes to finding out the constantly...
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Copyright © 2022 Official News Pakistan. All rights reserved. | {
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} | 1,684 |
Home Business Entrepreneurship
Luxurious Looks with Luna Cara Swim
What problem does the Luna Cara Swim solve?
Swimwear is a product that can be vulnerable for people to wear. It also isn't considered by most, an item you would wear daily. At Luna Cara Swim, we wanted to change that. We wanted to create products that make you feel like the most beautiful version of yourself, and celebrate your life, made with luscious, high-quality fabrics to keep you comfortable and with a fit that accentuates the beauty in your body. We wanted to create products that can easily be worn from the beach to the bar, without having to entirely change your outfit in between, and miss out on life's precious moments. We wanted to be a part of the solution to reduce waste and our carbon footprint, which we do by creating our newest suits with eco-friendly, biodegradable fibers and producing our products locally, working every day to keep the planet beautiful.
Tell us about the founder. What is their background?
Our two founders, Jeanne and Robin, have worked in swimwear for most of their lives. Designing for many high-end brands for 25+ years, they always had the dream of starting their own company. In 2019, they decided it was time to make that dream a reality. They had the intention of launching right before the summer of 2020, but due to the COVID-19 crisis, that date was pushed. In the following months, Jeanne was unfortunately diagnosed with breast cancer, which also pushed the launch date back. Despite the diagnosis, Jeanne wasn't going to allow it to prevent them from achieving their dream, and with Robin's unending support they launched their company in November of 2020.
How did they come up with the idea?
The idea for Luna Cara Swim came about when we saw the industry was missing a key component on what makes a vacation or a warm and sunny day the best it can be: versatility. We knew if we could provide versatile items that allow for you to feel more beautiful, more relaxed, and give you the gift of having the best vacation or most luxurious day ever, this idea and dream was worth creating. We wanted to create a one-stop-shop for all things needed for the perfect trip; swimwear, loungewear, and resort wear. We wanted to create pieces that could reach all age ranges and would make people want to plan adventures just so they could wear them.
How can this transform the world or our day-to-day lives?
On a personal level, Luna Cara Swim creates versatile pieces designed for effortless living. This means a swimsuit perfect for a dip in the pool, that also looks great with jeans for dinner and drinks, or a cover-up that's perfect over a bikini for a walk on the beach, but also dresses up your look to go out to the bar. Our intention is to change your day-to-day life by saving your time and energy, allowing you to effortlessly live your best and most beautiful life in our versatile pieces.
On a larger scale, fashion made with synthetic fabrics sits in landfills or oceans. We are creating our products to be made with eco-friendly, sustainable, and biodegradable fibers in order to help retain the beauty in our planet forever. We manufacture our swimwear locally in Los Angeles, supporting the economy and reducing our carbon footprint. We hope to change the world by inspiring others to do the same and live as sustainably as possible.
What do customers love most about the product?
Our customers love the super soft, sustainable fabrics, as well as the body-sculpting fabrics made with recycled nylon. They love the way they fit contours to their body with a smooth, flattering look, empowering them to feel like the best version of themselves when they put on their Luna Cara Swim.
Where can people get this product?
All of our items can be purchased at Luna Cara Swim | Luxury Swimwear
Categories: Entrepreneurship
Tags: clothingJeanneLuna Cara SwimRobinsummerswimwear
By USA Wire Staff January 19, 2022
How to Sell Baked Goods From Your Home: The Beginner's Guide
Maui Vera Enjoys the Sun While Being Kind to the Planet | {
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} | 1,402 |
Stelis segoviensis är en orkidéart som först beskrevs av Heinrich Gustav Reichenbach, och fick sitt nu gällande namn av Alec M. Pridgeon och Mark W. Chase. Stelis segoviensis ingår i släktet Stelis och familjen orkidéer. Inga underarter finns listade i Catalogue of Life.
Källor
Orkidéer
segoviensis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,910 |
Behind the Door: access needs support to be opportunity
Spot on. Discourse around opportunity and access often overshadows the realities of educational policies and structures in Europe, David Kazamias writes.
David Kazamias
We can measure access and opportunity in adult education if we take the most marginalised person in a given society and get them through the doors of further/higher education. Can we equip them with a Bachelors or a technical equivalent?
Apply this experiment to a number of EU countries and it is clear that policies vary greatly. There is a big gap between the idealised situation and national realities.
It bears an uncanny resemblance to Kafka's parable Before the Law: a man who is trying to gain entry to the court is told by a doorkeeper that "it is possible, just not now".
IN THE EU, OPPORTUNITY is the starting-gate equality built into compulsory education. Yet many countries still have school structures that tend to undermine this starting-gate equality.
This may come in the form of early stratification (GER), covert admission policies based on academic criteria (FRA) or private schools working against the interest of state schools (NL, BE). Opportunity is built into our school systems just as ascriptive inequalities are built into our societies. The former rarely cancels out the latter.
Access, on the other hand, suggests a continuous open door.
Access, on the other hand, suggests a continuous open door. It is the broad structure of entry that compensates for the single-shot structures. This is what makes lifelong learning a prescient field: it has the ability to remedy previous issues in compulsory education.
To ensure access, we must address the gatekeepers and doors that an individual may face. The main concerns of adult learners are finance, accreditation and time. These were built into the Lisbon strategy, which stated that lifelong learning initiatives should be focusing on 1) flexibility in learning 2) taking a broader approach to accreditation and 3) offering alternative routes. The renewed strategy, Education 2030, highlights these still as the most pertinent issues, but includes guidance and support as its new focus. This gets to the heart of the issue.
The realities in EU countries often look very different. A 2017 publication by Eurydice states that 1 in 4 adults in Europe have not completed any formal education beyond secondary school and that a large majority of mature students in higher and further education are currently funding themselves. The 2014 Bologna report indicates that only 1 in 10 mature students entered higher education through an alternative pathway.
THE TRANSLATION OF EU lifelong learning policy into national settings varies deeply. In certain cases, without financial support and without alternative routes the chances of re- and upskilling are low.
But even in the countries where nationally established procedures are in place, red tape and gatekeeping still result in the exclusion of certain groups. This can be as subtle as institutions having a preference for traditional modes of learning over blended, which comes at the greater exclusion of those with families (and often women).
Policies are in place to broaden access, yet the support is often missing.
Additionally, many countries still demand certain prerequisites, which has large implications for groups from a lower socio-economic background, especially groups with migration backgrounds.
There are a handful of countries which are further ahead on these issues: not just open doors but also guidance and support. Yet there are also many states who are failing this experiment because the doors are simply not there. But perhaps the trickiest for policy-makers are the countries with the correct structures in place who are failing because the way to that door is in itself too convoluted.
The man in Kafka's parable has waited at the gatekeeper's feet for a lifetime. Before he dies, he asks why no-one else has tried to gain access. The gatekeeper responds: "here no one else can gain entry, since this entrance was assigned only to you. I'm going to now close it".
Policies are in place to broaden access, yet the support is often missing. Good access is flexible and broad. Good support means finance and information. Failure to provide support restricts access and opportunity.
It cannot be put more simply: those adults who need education most participate the least.
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is a freelance education journalist, with an interest in educational disadvantage and inclusion. He is also Head of Department at a secondary school in Berlin. Contact: dkazamias4@gmail.com
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La rue Gustave Timmermans est une voie bruxelloise de la commune d'Auderghem. Située dans le quartier de la Corée cette rue qui aboutit sur la rue René Christiaens est longue d'environ 70 mètres.
Historique et description
Ce quartier fut construit en 1950 par la S.A. Baticoop. Elle comprenait quatre rues dont celle-ci, qui reçut le nom d'une victime de guerre le .
Le nom de la rue vient de l'adjudant aviateur Gustave Hubert Henri Timmermans, né le 8 juillet 1912 à Auderghem, tué le 11 mai 1940 à 's Herenelderen lors de la campagne des 18 jours, seconde guerre mondiale.
Situation et accès
Bâtiments remarquables et lieux de mémoire
Premiers permis de bâtir délivrés le pour les n° 1, 2, 3, 4, 5, 6, 9, et 11.
Voir aussi
Liste des rues d'Auderghem
Liens externes
Commune d'Auderghem
Notes et références
Rue à Auderghem | {
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Q: Is Android thread started from foreground service persistent? I need a service which should run 24/7 which does the following functions.
(1) checking GPS of the user with fusedLocationApi every second to check if the user enters a particular geofence.
(2) checking if another application package is installed (Thread)
(3) send an alive packet to the server by a particular interval of 60 seconds. (Thread)
The threads are singleton instances since I have to assure that only one thread per function run, for battery life.
All the above threads run on the foreground service's onStartCommand. I've read that the foreground service itself will not die on most circumstances, but how are the threads? Are threads started from the foreground service persistent as well? And furthermore, how can I test to prove that it's persistent?
Please help!
A: I'll answer my own question.
For about a month of observation and trying, I finally got what I want.
The answer for the question is YES, but NO. The threads are alive on doze mode, but what they do inside are suspended.
My foreground service has a FusedLocationProviderClient which runs on a minute interval, and a threads that does some other works, such as pinging my server on a six minute interval. They both won't die, but there was a huge amount of delay on the pinging stuff. It is supposed to ping the server every six minute but while on doze, it pings by the delay of 30minute or more, even past an hour.
I used two aproaches to fix this thing. I don't know which one makes this work, but it works anyway.
(1) Added Battery white listing on the start up.
Intent intent = PowerSaverHelper.prepareIntentForWhiteListingOfBatteryOptimization(getContext(), getPackageName(), false);
if (intent != null) {
startActivityForResult(intent, 19810);
} else {
getInterval();
}
(2) Created and hold a partial wakeLock on the onStartCommand of the Foreground service.
try {
PowerManager powerManager = (PowerManager) getSystemService(POWER_SERVICE);
wakeLock = powerManager.newWakeLock(PowerManager.PARTIAL_WAKE_LOCK,
"MyService::GpsWakeLock");
if (!wakeLock.isHeld()) {
wakeLock.acquire();
}
} catch (Exception e) {
ErrorController.showError(e);
}
And release it on onDestroy of the foreground service.
if (wakeLock != null && wakeLock.isHeld()) {
wakeLock.release();
}
This solution worked fine for me. The Gps fetches gps location by a minute(if it can), and pings the server on an exact interval, overnight without the device not connected to any power source.
But, the downfall of this method is expectable. The battery drain. While looking through this via Battery Historian, I saw that the cpu is always awake while the foreground service is running. Even by removing the gps stuff (which drains a lot of battery), it spent about 0.7% of battery during an hour of testing.
However, if your goal is simular to mine, a special purpose application which should be persistant, this approach might help.
| {
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Patience in The Face of Trials and Tribulations – By Abu Hakeem Bilāl bin Ahmad Davis
Abu Hakeem Bilāl Davis
13 November 2021 Abu Hakeem Bilal Davis, Friday Khutbahs, Heart Softeners, Purification of the Soul, Society 0
All praise is due to Allah, the Lord of all creation—and may He extol the Messenger in the highest company of Angels, and grant him peace and security—likewise to his family, Companions and true followers.
[10/01/2020] Patience in The Face of Trials and Tribulations – By Abu Hakeem Bilāl Davis حفظه الله Khutbah at Masjid as-Salafi.
Patience in the Face of Trials and Tribulations - Abū Ḥakīm
Salafi Publications · Patience In The Face Of Trials By Abu Hakeem
Polite Request: We have made these audios freely available ― We request that you donate the amount of just £2 or $2 (or more) as a Sadaqah to the Salafi Bookstore and Islamic Centre so they can continue their work to print and distribute free audios, leaflets and booklets to aid the da'wah of Ahlus-Sunnah and Hadīth across the world. And please make du'ā to Allah that He continues to aid and strengthen this blessed da'wah.
Please leave a comment below after listening to this audio, and make sure to share. May Allah bless you. | {
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\section{Introduction}
The NLPCC shared task 2~\cite{qiu2017overview} evaluates the automatic classification techniques for very short texts, the Chinese news headlines. Participants are challenged to identify the category of given texts among 18 classes. The size of training, development and test are 156000, 36000, 36000. The classes in training set are roughly balanced and are equally distributed in development and test set.
The evaluation metrics are macro-averaged precision, recall and F1 score as stated in task guideline~\footnote{\url{http://tcci.ccf.org.cn/conference/2017/dldoc/taskgline02.pdf}}.
This paper describes the second best system submitted by team Aicyber with classification accuracy of 0.825. First, a system overview will be given, then each module will be introduced in detail.
\section{The Aicyber's System}
The submitted system is a voting of three official baseline systems (NBoW, CNN and LSTM ) and a bag-of-word based SVM system.
Each baseline system's prediction is a voting of that system trained on 5 different word vectors. The sub-systems' architecture, experimental setup, training and development results will be introduced accordingly in the following session.
\subsection{The Official Baseline Systems }
Three deep learning systems are implemented and released as open source project~\footnote{\url{https://github.com/FudanNLP/nlpcc2017_news_headline_categorization}} by organizer. They are: neural bag-of-words (NBoW) model~\cite{Kalchbrenner2014A,Iyyer2015Deep}, convolutional neural networks (CNN)~\cite{Kim2014Convolutional} and Long short-term memory network~\cite{Hochreiter1997Long}. Hands-on instructions are given to guide participants to reproduce and enhance the baseline systems. The accuracy reported in~\cite{qiu2017overview} are 0.783, 0.763 and 0.747 respectively.
The NBoW model takes an average of the word vectors in the input text and performs classification with a logistic regression layer. It is simple and computationally less expensive than CNN and LSTM system.
Not like NBoW model who doesn't take the word order into account. The CNN and LSTM (and RNNs) model capture rich compositional information, and have achieved impressive performance in multiple benchmarks~\cite{Kim2014Convolutional,Sheikh2016Learning}.
~\cite{Zhang2015A} suggested the CNN model need not be complex to realize strong results, as a simple one-layer CNN could achieved state-of-the-art results across several datasets.
LSTM model has achieved remarkable performance in different sequence learning problems in speech, image and text analysis~\cite{Ghosh2016Contextual,Ji2016Sequential}. It's useful in capturing long-range dependencies in sequences.
The three systems share a similar pipeline for text classification, it takes word/char tokens as input, then tokens will go through word embeddings layers, followed by an average operation (NBoW) or CNN layer or LSTM layer, and a softmax layer at last.
Following sessions will focus on the pre-training of word embeddings layer.
\subsubsection{Word Embeddings}\label{subsection:wordembeddings}
Word embeddings is known as word2vec~\cite{Mikolov2013Distributed}, by default is randomly initialized, for this evaluation pre-trained character and word level embeddings are provided. However we prefer two types of embedding which had superior performance compare with standard approach, these have been verified in dimensional sentiment analysis
task~\cite{Du2017Aicyber}\footnote{\url{https://github.com/StevenLOL/ialp2016_Shared_Task}}.
\paragraph{Character-enhanced Word Embedding}
The first set of word embedding is character-enhanced word embedding~\cite{Chen2015Joint} (CWE). Their study shows semantic meaning of a word is related to its composing characters. Two type of embeddings in CWE, the position-based character embeddings (CWE+P) and cluster-based character embeddings (CWE+L) are used.
They are trained with window size of 5 and 11, 5 iterations, 5 negative examples, minimum word count of 5, Skip-Gram with starting learning rate of 0.025 , the learned word vectors are of 300 dimensions.
\paragraph{FastText Embedding}
The second set of word embedding is FastText~\cite{Bojanowski2016Enriching}~\footnote{\url{https://github.com/facebookresearch/fastText}}, the idea is to enriching word vectors with sub-word information. Eg, for English, a word vector is associated to its character n-grams.
FastText word embedding is trained with similar setting as CWE training. Please noticed that default minimum character n-gram is 1 for Chinese.
\paragraph{Data Usage for Embedding Training}
Following public available data-sets are used in unsupervised learning of word embeddings:
\begin{enumerate}
\item Chinese Wikipedia Dumps (Time stamp: 2011-02-
05T03:58:02Z)
\item Douban movie review~\footnote{\url{http://www.datatang.com/data/45075}}
\item Sogou news corpus~\footnote{\url{http://www.sogou.com/labs/resource/list_news.php}}
\end{enumerate}
\paragraph{Training and Evaluation}
Above dataset is preprocessed by jieba~\footnote{\url{https://github.com/fxsjy/jieba}}, after filtering, there are 555571 unique tokens left. Embedding training produces six set of word vectors: CWE-L-W5,CWE-L-W11,CWE-P-W5,CWE-P-W11,FastText-W5 and FastText-W11 (W denotes window size).
\begin{CJK}{UTF8}{gbsn}
To verify the correctness of word embedding, we examines the nearest neighbor of a given Chinese word, eg 高兴(happy). The result from FastText-W11 is clearly different from others, 5 single character words appear in the top 10 (0 for other embeddings). This indicates FastText doesn't work properly for Chinese with large window size, in which the character n-grams, especially unigram become overestimate. Thus FastText-W11 is dropped.
\end{CJK}
\subsubsection{Training of Official Baseline systems}
With 5 embedding from above, 15 (5*3) systems are formed. We use default setting for NBoW, LSTM system. For the CNN system, only one convolution layer is used (filter size is 3).
System is trained only on the 156000 training data, and evaluated on 36000 development data, we use accuracy as performance metrics.
\subsubsection{Result and Discussion}
The results of official baselines are presented in Table~\ref{tb:EvalFeatureLSVR}, to make a fair comparison, systems trained on randomized character/word vectors (length is 300) are also included.
\begin{table}[h]
\small
\centering
\begin{tabular}{|l|l|r|
\hline \multicolumn{3}{|c|}{\textbf{Official Baseline systems }} \\
\hline \bf Network Type & \bf Embeddings & \bf Development Accuracy \\ \hline
NBoW & Randomized Char & 0.715 \\
NBoW & Randomized Word & 0.779 \\
NBoW & CWE-L-W5 & 0.814 \\
NBoW & CWE-P-W5 & 0.816 \\
NBoW & FastText-W5 & 0.812 \\
NBoW & CWE-L-W11 & 0.816 \\
NBoW & CWE-P-W11 & 0.816 \\ \hline
CNN & Randomized Char & 0.718 \\
CNN & Randomized Word & 0.763 \\
CNN & CWE-L-W5 & 0.822 \\
CNN & CWE-P-W5 & 0.823 \\
CNN & FastText-W5 & 0.820 \\
CNN & CWE-L-W11 & \textbf{0.824} \\
CNN & CWE-P-W11 & 0.821 \\ \hline
LSTM & Randomized Char & 0.691 \\
LSTM & Randomized Word & 0.728 \\
LSTM & CWE-L-W5 & 0.808 \\
LSTM & CWE-P-W5 & 0.805 \\
LSTM & FastText-W5 & 0.801 \\
LSTM & CWE-L-W11 & 0.807 \\
LSTM & CWE-P-W11 & 0.806 \\
\hline
\end{tabular}
\caption{\label{tb:EvalFeatureLSVR} The official baseline systems's accuracy on development set. CNN system with CWE-L-W11 is the best system and achieve 0.824.
\end{table}
It's obvious that systems trained with pre-trained embedding are much better than those with randomized embeddings. System with word embeddings give better result than those use characters embedding. CNN is the most accurate system. For different embedding types, the FastText is under performance the others. Difference between CWE-P and CWE-L is negligible.
\subsection{Bag-of-Word model}
The official released systems are relatively strong. We also seek alternatives to tackle classification problem. Starting with a well known baseline system, the bag-of-word model. It's commonly used in text classification where the occurrence of each word is used as a feature for training classifiers. Support Vector Machine~\cite{Vapnik1995The} (SVM) with linear kernel was considered to be one of the best classifiers~\cite{Forman2003An,Yang1999A}.
This system is trained on 156000 training data, and validated on 36000 development set. Table~\ref{tb:bow} shows BoW model could obtain 0.791 classification accuracy. The result is much better than all deep learning system with randomized embeddings, this finding demonstrate the importance of pre-trained word vectors.
\begin{table}
\small
\centering
\begin{tabular}{|l|l|r|
\hline \multicolumn{3}{|c|}{\textbf{BoW Vs Deep-learning sytem with Randomized Word Vector }} \\
\hline \bf Features & \bf Classifiers & \bf Development Accuracy \\ \hline
Bag-of-Word & Linear SVM & \textbf{0.791} \\
\hline
Randomized Word Vector & LSTM & 0.728 \\
Randomized Word Vector & CNN & 0.763 \\
Randomized Word Vector & NBoW & 0.779 \\
\hline
\end{tabular}
\caption{\label{tb:bow} Classification accuracy of bag-of-word model is better than all deep learning system with randomized embeddings.
\end{table}
To summarize, the submitted system is an ensemble of three deep learning based systems and a conventional BoW model, it's truly a voting of baselines. The final classification accuracy is 0.826 measured on the development set.
\section{Discussion and Further Improvement}
Compare BoW method in Table~\ref{tb:bow} with the best single system in Table~\ref{tb:EvalFeatureLSVR}, the difference is only 0.03, we consider that the BoW indeed is well suited for the Chinese headline classification. Because the headline appears to be clear, concise and powerful, the usage of words in headline is precisely selected by professional editors. The importance of words make the BoW works well for this task.
The best single system achieved 0.824 classification accuracy on development set, while the voting system scored 0.826 on same dataset. Voting method provides marginal improvement in this work.
Word embedding training in Section 2.1.4 is kind of unsupervised pre-training, but only limited to the embedding layer. Study in~\cite{Dai2015Semi} shows recurrent language models and sequence autoencoder could used to pre-train not only the embedding layer but also the LSTM layer. On five benchmarks that they tried, LSTMs can reach or surpass the performance levels of all previous baselines. As shown in Table~\ref{tb:EvalFeatureLSVR}, LSTM didn't beat CNN or NBoW model, using pre-training could boost LSTM's performance.
\section{Conclusion}
In this paper we presented our approaches to tackle Chinese news headline categorization challenge. A voting system consists of three deep-learning based system build on five different embedding layers and a BoW model ranked 2nd among 32 teams.
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Abdellah Ezbiri (born 1986) is a French kickboxer. He is the former I.S.K.A. K-1 Rules World Champion and W.K.N. K-1 Rules European Super Lightweight Champion. As of 1 November 2018, he is ranked the #7 featherweight in the world by Combat Press.
Biography and career
Abdellah Ezbiri is a pro kickboxer and member of the France team of Kickboxing, he has received various titles such as Champion of France by 3 times, and he is also a European and World champion in the discipline. He trains at Team Ezbiri, Fighters 69 in Lyon, with his trainer and brother Fouad Ezbiri.
On May 7, 2011, at Fight Zone 5, Abdellah became W.K.N. K-1 Rules European Super Lightweight champion Under 65 kg after beating the Italian Salvatore Zappulla by KO in the 3rd round.
On April 14, 2012, he fought to a draw against Samir "Petit Prince" Mohamed by decision at Fight Zone 6.
He competed in the Glory 8: Tokyo - 2013 65kg Slam in Tokyo, Japan on May 3, 2013, losing to Gabriel Varga via unanimous decision in the quarter-finals.
At La 20ème Nuit des Champions -68.5 kg/151 lb tournament in Marseille, France on November 23, 2013, Ezbiri defeated Charles François on points in the semi-finals before losing to Bruce Codron in the final by the same margin.
Titles and accomplishments
Professional:
2017 Nuit De Champions K-1 Rules Champion (66.000 kg)
2017 Glory Featherweight Contender Tournament Winner
2017 I.S.K.A. 67 kg World Title
2016 Kunlun Fight 65 kg Tournament Runner-up
2014 NDC K-1 Rules -70 kg Tournament Runner-up
2013 NDC K-1 Rules -68.5 kg Tournament Runner-up
2013 I.S.K.A. K-1 Rules World Welterweight Champion (67.000 kg)
2013 Krush Grand Prix Tournament Runner-Up (67.000 kg)
2012 Nuit des Champions Kickboxing K-1 Rules belt (67.000 kg)
2011 F-1 World MAX Tournament Champion (65.000 kg)
2011 W.K.N. K-1 Rules European Super Lightweight Champion (66.700 kg)
2009 F.F.S.C. Kickboxing K-1 Rules French Champion (67.000 kg)
2008 Kickboxing K-1 Rules French Champion
2007 F.F.K.B. Kickboxing K-1 Rules French Champion Class B
Amateur:
2010 SportAccord World Combat Games Kick-Boxing - Low-Kick, Beijing, China (-67 kg)
2009 W.A.K.O. World Kick-Boxing Championships - Low-Kick, Villach, Austria (-67 kg)
Kickboxing record
|-
|- bgcolor= "#FFBBBB"
| 2019-10-26|| Loss ||align=left| Zakaria Zouggary || Glory 70: Lyon || Lyon, France || KO (Punches) || 1 || 2:25
|- style="background:#fbb;"
| 2019-03-09 || Loss||align=left| Anvar Boynazarov || Glory 64: Strasbourg || Strasbourg, France || Decision (Unanimous) || 3 || 3:00
|- bgcolor="#CCFFCC"
| 2018-10-20|| Win ||align=left| Victor Pinto || Glory 60: Lyon || Lyon, France || KO (Spinning back kick)|| 1 || 1:03
|- bgcolor="#FFBBBB"
| 2018-05-12 || Loss ||align=left| Petpanomrung Kiatmuu9 || Glory 53: Lille || Lille, France || KO (Left high kick) || 2 ||
|- bgcolor="#CCFFCC"
| 2017-11-25|| Win ||align=left| Masaaki Noiri || Nuit Des Champions 2017 || Marseille, France || Decision|| 5 || 3:00
|-
! style=background:white colspan=9 |
|- bgcolor="#CCFFCC"
| 2017-10-28|| Win ||align=left| Anvar Boynazarov || Glory 47: Lyon || Lyon, France || Decision|| 3 || 3:00
|-
! style=background:white colspan=9 |
|- bgcolor="#CCFFCC"
| 2017-10-28|| Win ||align=left| Azize Hlali || Glory 47: Lyon || Lyon, France || Decision|| 3 || 3:00
|-
! style=background:white colspan=9 |
|- bgcolor="#CCFFCC"
| 2017-09-04|| Win ||align=left| Mohamed Hendouf || Tiger Night || Switzerland || Decision|| 3 || 3:00
|- bgcolor="#CCFFCC"
| 2017-02-18 || Win ||align=left| Giovanni Boyer || La Nuit Des Champions || France || Decision|| 3 || 3:00
|-
! style=background:white colspan=9 |
|-
|- bgcolor="#FFBBBB"
| 2016-09-10 || Loss ||align=left| Wei Ninghui || Kunlun Fight 51 - 65 kg 2016 Tournament Final || Fuzhou, China || TKO(Low Kick) || 1 ||
|-
! style=background:white colspan=9 |
|- bgcolor="#CCFFCC"
| 2016-09-10 || Win ||align=left| Buray Bozaryilmaz || Kunlun Fight 51 - 65 kg 2016 Tournament Semi-final || Fuzhou, China || Decision|| 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2016-09-10 || Win ||align=left| Kim Minsoo || Kunlun Fight 51 - 65 kg 2016 Tournament Quarter-finals || Fuzhou, China || Decision|| 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2016-07-30 || Win ||align=left| Isaac Araya || Kunlun Fight 48 - 65 kg 2016 Tournament 1/8 Finals || Jining, China || Decision|| 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2016-04-08 || Win ||align=left| Nicola Sanzione || Fight Night 1|| Saint-Étienne, France || Decision|| 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2016-01-16 || Win ||align=left| Modibo Diarra || Nuit des Champions|| Marseille, France || Decision|| 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2015-11-14 || Win ||align=left| Edouard Bernardou || Nuit des Champions|| Marseille, France || Decision|| 3 || 3:00
|-
|- bgcolor="#FFBBBB"
| 2015-07-18 || Loss ||align=left| Rachid Magmadi || Partouche Kickboxing Tour|| France || KO || 1 ||
|-
|- bgcolor="#CCFFCC"
| 2015-07-18 || Win ||align=left| Tarek Guermoudi || Partouche Kickboxing Tour|| France || KO || 3 ||
|-
|- bgcolor="#CCFFCC"
| 2015-05-16 || Win ||align=left| Johan Labbe || La 21ème Nuit des Champions|| France || Decision (Split) || 3 || 3:00
|-
|- bgcolor="#FFBBBB"
| 2014-11-22 || Loss ||align=left| Sitthichai Sitsongpeenong || La 21ème Nuit des Champions, Final || Marseille, France || TKO (Low kick) || 2 ||
|-
! style=background:white colspan=9 |
|-
|- bgcolor="#CCFFCC"
| 2014-11-22 || Win ||align=left| Bruce Codron || La 21ème Nuit des Champions, Semi-finals || Marseille, France || Ext. R. Decision (Split) || 4 || 3:00
|-
|- bgcolor="#FFBBBB"
| 2014-10-05 || Loss ||align=left| Keita Makihira || Krush.46 || Marseilles, France || Decision (Unanimous)|| 3 || 3:00
|-
! style=background:white colspan=9 |
|- bgcolor="#FFBBBB"
| 2013-11-23 || Loss ||align=left| Bruce Codron || La 20ème Nuit des Champions, Final || Marseilles, France || Decision || 3 || 3:00
|-
! style=background:white colspan=9 |
|-
|- bgcolor="#CCFFCC"
| 2013-11-23 || Win ||align=left| Charles François || La 20ème Nuit des Champions, Semi-finals || Marseilles, France || Decision || 3 || 3:00
|-
|- style="background:#fbb;"
| 2013-05-03 || Loss ||align=left| Gabriel Varga || Glory 8: Tokyo || Tokyo, Japan || Decision (unanimous) || 3 || 3:00
|-
! style=background:white colspan=9 |
|-
|- style="" bgcolor=#CCFFCC
| 2013-04-06 || Win ||align=left| Kittisak Noiwibon || Fightzone 7 || Villeurbanne, France || Decision || 5 || 3:00
|-
! style=background:white colspan=9 |
|-
|- style="background:#fbb;"
| 2013-01-14 || Loss ||align=left| Yuta Kubo || Krush Grand Prix 2013 ~67 kg Tournament~, Final || Tokyo, Japan || Extension round decision (unanimous) || 5 || 3:00
|-
! style=background:white colspan=9 |
|-
|- style="" bgcolor=#CCFFCC
| 2013-01-14 || Win ||align=left| Yuya Yamamoto || Krush Grand Prix 2013 ~67 kg Tournament~, Semi-finals || Tokyo, Japan || Extension round decision (split) || 4 || 3:00
|-
|- style="" bgcolor=#CCFFCC
| 2013-01-14 || Win ||align=left| Yūji Nashiro || Krush Grand Prix 2013 ~67 kg Tournament~, Quarter-finals || Tokyo, Japan || Decision (Unanimous) || 3 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2012-11-24 || Win ||align=left| Samir Mohamed || Nuit des Champions || Marseilles, France || Decision || 3 || 3:00
|-
! style=background:white colspan=9 |
|-
|- style="background:#fbb;"
| 2012-06-08 || Loss ||align=left| Yuta Kubo || Krush.19|| Tokyo, Japan || Decision|| 3 || 3:00
|-
|- bgcolor="#c5d2ea"
| 2012-04-14 || Draw ||align=left| Samir Mohamed || Fightzone 6 || Villeurbanne, France || Decision draw || 5 || 2:00
|-
|- bgcolor="#CCFFCC"
| 2011-10-01 || Win ||align=left| Loris Audoui || F-1 World MAX Tournament, Final || Meyreuil, France || Decision || 3 || 2:00
|-
! style=background:white colspan=9 |
|-
|- bgcolor="#CCFFCC"
| 2011-10-01 || Win ||align=left| Vang Moua || F-1 World MAX Tournament, Semi-finals || Meyreuil, France || Decision || 3 || 2:00
|-
|- bgcolor="#CCFFCC"
| 2011-10-01 || Win ||align=left| Tristan Benard || F-1 World MAX Tournament, Quarter-finals || Meyreuil, France || Decision || 3 || 2:00
|-
|- style="" bgcolor=#CCFFCC
| 2011-05-07 || Win ||align=left| Filippo Soleid || Fightzone 5 || Villeurbanne, France || KO || 3 ||
|-
! style=background:white colspan=9 |
|-
|- style="" bgcolor=#CCFFCC
| 2011-01-22 || Win ||align=left| Tekin Ergun || Kickboxing Gala || Strasbourg, France || Decision || 5 || 3:00
|-
|- style="background:#fbb;"
| 2010-12-18 || Loss ||align=left| Miodrag Olar || Kickboxing Tournament || La Ciotat, France || Decision || 3 || 2:00
|-
|- bgcolor="#CCFFCC"
| 2010-09-10 || Win ||align=left| Milan Jovanovic || SportAccord || Beijing, China || Decision (3-0) || 3 || 2:00
|-
! style=background:white colspan=9 |
|-
|- style="background:#fbb;"
| 2010-09-10 || Loss ||align=left| Shamil Abdulmedjidov || SportAccord World Combat Games, -67 kg, Low-Kick, Semi-finals || Beijing, China || Decision (3-0) || 3 || 2:00
|-
! style=background:white colspan=9 |
|-
|- style="" bgcolor=#CCFFCC
| 2010-05-22 || Win ||align=left| Olivier Surveillant || Kickboxing National Gala || Saint-Joseph, Réunion || Decision || ||
|-
|- style="background:#fbb;"
| 2010-04-24 || Loss ||align=left| Charles François || Fightzone 4 || Villeurbanne, France || Decision || 5 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2009-11-14 || Win ||align=left| Geoffrey Mocci || La Nuit des Champions 2009 || Marseille, France || Decision || 5 || 3:00
|-
|- bgcolor="#CCFFCC"
| 2009-10-22 || Win ||align=left| Roland Mendez || W.A.K.O. World Championships, LK men -67 kg, 1/8 Final || Villach, Austria || Decision (2-1) || 3 || 2:00
|-
|- bgcolor="#CCFFCC"
| 2009-10-22 || Win ||align=left| Simon Laszlo || W.A.K.O. World Championships, LK men -67 kg, 1/16 Final || Villach, Austria || Decision (3-0) || 3 || 2:00
|-
|- bgcolor="#CCFFCC"
| 2009-06-26 || Win ||align=left| Vang Moua || Gala International Multi-Boxes || Paris, France || Decision || 3 || 3:00
|-
|- style="background:#fbb;"
| 2009-05-23 || Loss ||align=left| Ismaël Doumbia || Shock Muay 2 || Saint-Denis, France || TKO || ||
|-
|- bgcolor="#CCFFCC"
| 2009-04-25 || Win ||align=left| Vang Moua || F.F.S.C.D.A. French Championships || Paris, France || Decision || 5 || 2:00
|-
! style=background:white colspan=9 |
|-
|- bgcolor="#CCFFCC"
| 2009-03-01 || Win ||align=left| Nizar Gallas || F.F.S.C.D.A. French Championships, 1/4 Final || France || Decision || ||
|-
|- style="background:#fbb;"
| 2008-11-15 || Loss ||align=left| Max Dansan || VXS 1 || Apt, France || KO || 2 ||
|-
|- bgcolor="#CCFFCC"
| 2008-04-26 || Win ||align=left| Lyess Maroc || Fight Zone 2 || Villeurbanne, France || KO || ||
|-
|-
| colspan=9 | Legend:
See also
List of male kickboxers
References
1986 births
Living people
French male kickboxers
Lightweight kickboxers
Welterweight kickboxers
French Muay Thai practitioners
French sportspeople of Algerian descent
Kunlun Fight kickboxers
Sportspeople from Lyon
Glory kickboxers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,828 |
Q: how to set textview background auto fit when device font increase or decrease textview.xml
<android.support.v7.widget.AppCompatTextView
android:id="@+id/textview_badge"
android:layout_width="0dp"
android:minWidth="@dimen/baseline_grid_3x"
android:layout_height="wrap_content" android:gravity="center"
android:layout_marginEnd="@dimen/item_padding"
android:background="@drawable/red_circle"
android:layout_marginBottom="@dimen/baseline_grid_2x"
android:layout_marginTop="@dimen/baseline_grid_2x"
android:padding="@dimen/baseline_grid_0.5x" tools:text="9"
android:textColor="@color/colorWhite"
android:textSize="@dimen/app_text_size_small"
android:textStyle="bold"/>
red_circle.xml
<?xml version="1.0" encoding="utf-8"?>
<shape xmlns:android="http://schemas.android.com/apk/res/android"
android:shape="oval">
this is my code from which
<shape xmlns:android="http://schemas.android.com/apk/res/android"
android:shape="oval">
<solid android:color="@color/colorRed" />
<stroke
android:width="0dp"
android:color="@color/colorRed" />
<size
android:width="20dp"
android:height="20dp"" />
</shape>
I am trying to display textview with circular with text it works fine when device font size less when device font size large then textview background become circle to oval when I try to change size 20dp to 60dp then in becomes circle please suggest me how to fix it how to keep circle background of textview when we increase font size
A: You can use ConstraintLayout to make TextView width always equals TextView height => it will make TextView background always circle
<android.support.constraint.ConstraintLayout
android:layout_width="wrap_content"
android:layout_height="wrap_content"
>
<TextView
android:layout_width="wrap_content"
android:layout_height="0dp"
android:background="@drawable/red_circle"
android:gravity="center"
android:text="AAAAAAAAAAAA"
android:textSize="20sp"
app:layout_constraintDimensionRatio="H,1:1"
app:layout_constraintEnd_toEndOf="parent"
app:layout_constraintStart_toStartOf="parent"
app:layout_constraintTop_toTopOf="parent"
/>
</android.support.constraint.ConstraintLayout>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,094 |
Q: JXBrowser required to be open to run automated tests using selenium webdriver I have been able to successfully connect to a JXBrowser session using the ChromeDriver by specifying the --remote-debugging-port as described in the documentation. The issue I have with this is this implementation requires the JXBrowser to be opened before each test is started. I have attempted to workaround this by specifying a "jxbrowser.exe" as the chrome binary, which does get the browser to start automatically. The WebDriver cannot communicate with the browser in this scenario though since the ChromeDriver creates a random port each time a session is created, which will almost always be different than the port defined in the --remote-debugging-port property.
I have tested the same scenario with a standard Chromium Embedded Framework client, and specifying a debugging-port is not required, thus works as expected. Defining the port on the ChromeDriver side is not supported, so is there a way to either specify a debugging-port range or open it entirely as the CEF client has?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,993 |
Q: Sharepoint 2016 On Premise Global navigation I want to use the navigation options through site settings to create the top nav on my site. But it seems that the master page using some sort of custom usercontrol (ascx) or custom web parts to set the navigation but I am not the original person who set up the site collection and all of the sites underneath.
I am also not familiar with using anything to set up navigation outside of structural or managed navigation. So I am trying to remove the custom control and set up the navigation the way that I am familiar with. I know this is a vague question but hoping for some help to steer me in the right direction.
Here is the code from the master page if that helps:
<%@Page language="C#" Inherits="Microsoft.SharePoint.Publishing.PublishingLayoutPage, Microsoft.SharePoint.Publishing, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c" meta:webpartpageexpansion="full" meta:progid="SharePoint.WebPartPage.Document" %>
<%@Register TagPrefix="SharePoint" Namespace="Microsoft.SharePoint.WebControls" Assembly="Microsoft.SharePoint, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c"%>
<%@Register TagPrefix="PageFieldFieldValue" Namespace="Microsoft.SharePoint.WebControls" Assembly="Microsoft.SharePoint, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c"%>
<%@Register TagPrefix="Publishing" Namespace="Microsoft.SharePoint.Publishing.WebControls" Assembly="Microsoft.SharePoint.Publishing, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c"%>
<%@Register TagPrefix="WebPartPages" Namespace="Microsoft.SharePoint.WebPartPages" Assembly="Microsoft.SharePoint, Version=16.0.0.0, Culture=neutral, PublicKeyToken=71e9bce111e9429c"%>
<asp:Content runat="server" ContentPlaceHolderID="PlaceHolderAdditionalPageHead">
<Publishing:EditModePanel runat="server" id="editmodestyles">
<SharePoint:CssRegistration name="<% $SPUrl:~sitecollection/Style Library/~language/Themable/Core Styles/editmode15.css %>" After="<% $SPUrl:~sitecollection/Style Library/~language/Themable/Core Styles/pagelayouts15.css %>" runat="server"></SharePoint:CssRegistration>
</Publishing:EditModePanel>
</asp:Content>
<asp:Content runat="server" ContentPlaceHolderID="PlaceHolderPageTitle">
<SharePoint:ProjectProperty Property="Title" runat="server"></SharePoint:ProjectProperty>
<PageFieldFieldValue:FieldValue FieldName="fa564e0f-0c70-4ab9-b863-0177e6ddd247" runat="server"></PageFieldFieldValue:FieldValue>
</asp:Content>
<asp:Content runat="server" ContentPlaceHolderID="PlaceHolderPageTitleInTitleArea">
<PageFieldFieldValue:FieldValue FieldName="fa564e0f-0c70-4ab9-b863-0177e6ddd247" runat="server">
</PageFieldFieldValue:FieldValue>
</asp:Content>
<asp:Content runat="server" ContentPlaceHolderID="PlaceHolderMain">
<style type="text/css">
#s4-workspace { overflow-y: auto; overflow-x: hidden !important; }
#navigation { padding-left: 2%; }
ul#navigation > li:first-child {display: none; }
#contentBox { margin-right: 0 !important; margin-left: 0 !important;}
#contentWrapper.col-sm-12 { padding-left: 0 !important; padding-right: 0 !important; }
#acps-content { padding: 0 1%; }
#acps-content .col-sm-4 { border-left: 0 !important; width: 33.3333333%; }
#acps-content .col-sm-4 .row-WPZone { border-left: 0 !important; }
#collection-header {display: none; }
#sideNavBox { display: none; }
#topbreadcrumb { display: none !important; }
#subtitle { display: none !important; }
#mainbody { max-width: 100% !important; margin: 0 !important; /*background-color: #c7d3db; */ }
.row { background-color: transparent; }
#mainbody.col-md-9, #full-width-top { width: 100%; }
#full-width-top.row { margin-left: 0; margin-right: 0; }
#pdf-help { max-width: 1500px; padding: 0 !important; margin: 0 auto !important; }
#bottom-content { padding-left: 2%; padding-right: 2%; }
#acps-content.row:after {content: ; }
#bottom-content.row:before {content: ; }
#bottom-content.row:after {content: ; }
.carousel-inner { max-height: 450px;}
.carousel-inner img { width: 100%; vertical-align: middle;}
.carousel-control:hover, .carousel-control a:hover { text-decoration: none !important;}
#acps-content { max-width: 1500px; margin: 0 auto; padding-top: 0; background-color: #fff;}
.nav { margin-bottom: 0; padding-left: 0;}
.ms-WPHeader td { border-bottom: 1px solid #5195ba;}
tr.ms-viewheadertr { display: none;}
#ctl00_m_g_6d7c734c_f493_4503_a9b3_f7be1fd5122f br { display: none;}
#MSOZoneCell_WebPartctl00_m_g_a51ae5fd_5b55_4c78_8e9d_147d6a9b7804 > table.s4-wpTopTable { position: relative; top: 2em;}
.tab-content { padding: 0 0.5em;}
#newsTabs { margin-top: 0; }
#newsTabs.nav > li > a { font-weight: bold; font-size: 1.9rem; color: #0072bc; padding: 4px 6px; }
#newsTabs.nav > .active > a { color: #924b84;}
.nav-tabs { border-bottom: 1px solid #5195ba;}
.nav-tabs > .active > a, .nav-tabs > .active > a:hover, .nav-tabs > .active > a:focus { border: 1px solid #5195ba; border-bottom-color: transparent;}
#sptlt { padding: 0; height: 750px; overflow: auto;}
.spotlight-item { padding-bottom: 0.8em; border-bottom: 1px solid #ddd;}
.spotlight-item h4 { font-size: 10pt !important; margin-left: 0 !important; margin-top: 0.5em !important; }
.spotlight-item h4 a:link { text-decoration: none !important; }
.spotlight-item p { margin-left: 0 !important; }
.spotlight-item img { max-width: 40%; margin: 0 8px 8px 8px; float: right; }
.s4-title { display: none;}
.flex-viewport { max-height: 200px;}
h2, .row-WPZone h2, .row-WPZone h2 a, .row-WPZone h2 a:hover { margin: 0 !important; }
#twitter-widget-0 { width: 100% !important;}
.twitter-timeline-rendered { width: 100% !important; }
/** New Calendar Widget **/
.ecl-calendar{ border-color: #036; }
.ecl-calendar table,.ecl-calendar tr:first-child{ background-color: #b1daf5; font-weight: bold;}
.ecl-day:hover, .ecl-event:hover,.ecl-today:hover,.ecl-weekend:hover,.ecl-other-month:hover{ background-color: #efd4d4;}
.ecl-today{ background-color: #d3e6f4;}
.ecl-listing h3 { text-decoration: none; color: #369 !important; font-size: 10pt;}
.ecl-listing li { margin-left: 0.1em;}
#pdf-help { text-align: center; padding: 1em 0; margin: 0 auto; border-top: 0; background-color: #fff; max-width: 1500px;}
@media (max-width: 979px) {
#mainbody > #contentBox { margin-left: 0 !important; min-width: 0; }
#newsTabs.nav > li > a { padding: 5px !important; }
.nav-tabs > li > a { font-size: 10pt; }
}
@media (max-width: 767px) {
.navbar-toggle { margin-right: 30px !important; }
#mainbody.col-md-9 { padding-left: 0 !important; padding-right: 0 !important; }
#mainbody > #contentBox { margin-left: 0 !important; }
#acps-content .col-sm-4 { width: 100%; }
</style>
<!-- First Row (full width carousel) -->
<div class="row" id="full-width-top">
<div class="col-sm-12">
<div class="row-WPZone">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow1Column1" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div>
</div><!-- end 1st Row l. 90 -->
<!-- Second Row (Three columns) -->
<div class="row" id="acps-content">
<div class="col-sm-4">
<div class="row-WPZone">
<ul class="nav nav-tabs" id="newsTabs">
<li class="active"><a href="#sptlt" id="spotlight">Spotlight</a></li>
<li><a href="#pr">Press Releases</a></li>
</ul>
<div class="tab-content">
<div class="tab-pane active" id="sptlt">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow2Column1" title="Tab1-View" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
<div class="tab-pane" id="pr">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow2Column1b" title="Tab2-View" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div><!-- end tab-content -->
<script type="text/javascript">
$('#newsTabs a').click(function (e) {
e.preventDefault();
$(this).tab('show');
})
</script>
</div><!-- end row-WPZone -->
</div><!-- end col-sm-4, 1st Column, l. 96 -->
<div class="col-sm-4">
<div class="row-WPZone">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow2Column2Sub1" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div>
<div class="col-sm-4 last-col">
<div class="row-WPZone">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow2Column2Sub2" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div>
</div><!-- end 2nd Row, acps-content, 3x col-sm-4 l. 99 -->
<!-- Third Row (Two columns) -->
<div id="bottom-content" class="row">
<div class="col-sm-6">
<div class="row-WPZone">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow3Column1" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div>
<div class="col-sm-6">
<div class="row-WPZone">
<WebPartPages:WebPartZone runat="server" AllowPersonalization="false" ID="bootstrapRow3Column2" FrameType="None" Orientation="Vertical"><ZoneTemplate></ZoneTemplate></WebPartPages:WebPartZone>
</div>
</div>
</div><!-- end 3rd Row, bottom-content, 2 x col-sm-6, l. 144 -->
<script type="text/javascript">
jQuery(document).ready(function() {
jQuery("#SkiptoContent").replaceWith("<a href='#spotlight' id='SkiptoContent' class='ms-accessible ms-acc-button' onClick='javascript:window.document.getElementById('spotlight').focus();return false;'>Skip to Main Content</a>");
});
</script>
</asp:Content>
A: Try to check the Term Store Manager located here:
https://yourtenant.com/sites/[SiteCollection]/_layouts/15/termstoremanager.aspx
Guide to TSM: https://www.c-sharpcorner.com/article/introduction-of-term-store-management-in-sharepoint-onlineoffice-365/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 311 |
Security West 2016
San Diego, CA | Fri, Apr 29, 2016 - Fri, May 6, 2016
Twitter: @brycegalbraith
"The world isn't run by weapons anymore, or energy, or money. It's run by little ones and zeros, little bits of data. It's all just electrons. There's a war out there?and it's not about who's got the most bullets. It's about who controls the information. What we see and hear, how we work, what we think, it's all about information." -- Cosmo from, Sneakers
As a contributing author of the internationally bestselling book, Hacking Exposed: Network Security Secrets & Solutions, Bryce helped bring the secret world of hacking out of the darkness and into the public eye. Bryce was a member of Foundstone's world-renowned penetration testing team and served as a co-author and Senior Instructor of Foundstone's groundbreaking, Ultimate Hacking: Hands-On course series.
Bryce continues to provide highly specialized ethical hacking and cyber security consulting services to clients around the world and teaches thousands of cyber security professionals, from a who's who of top organizations, how to defend against advanced adversaries...
Here is What Students Say About Bryce Galbraith:
"Bryce is an excellent instructor. His knowledge and delivery are exceptional." - Chris Shipp, DM Petroleum Operations Co.
Bryce Galbraith Will Be Teaching the Following Course:
SEC504: Hacker Tools, Techniques, Exploits and Incident Handling GCIH
Security West 2016 Instructors
Eric Cornelius
Pieter Danhieux
Ted Demopoulos
Philip Hagen
G. Mark Hardy
Frank Kim
David R. Miller
Seth Misenar
Keith Palmgren
Greg Porter | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,369 |
Egypt's poverty rate rose to 27.8 percent for the year 2015, compared to 25.2 percent in 2010-11, the state run statistics agency announced on Sunday.
In an official report released by the Central Agency for Public Mobilization and Statistics on the international day to combat poverty, CAPMAS said "extreme poverty" reached 5.3 percent, which it attributes to a spike in prices of essential foodstuffs.
The current poverty line stands at an average 482 EGP (about $54) per month while the extreme poverty line stands at 322 EGP (about $36).
About 56.8% of those living in Upper Egypt's rural areas cannot meet their basic needs, compared to 19.7% in the Nile Delta's rural areas. This applies to roughly a third of people in Upper Egypt's urban areas.
According to the agency, Upper Egypt's urban and rural governorates as well as Nile Delta's rural areas witnessed an increase in poverty between 2012-13 and 2015, while urban governorates and Nile Delta's urban area witnessed a drop in the same period.
The report included recent statistics on state subsidies for foodstuffs, finding that 10.5 percent of subsidies are received by families in the poorest bracket; while subsidies decrease gradually to 4.2 percent for the richest bracket.
Graphs provided by the agency also depicted an increase in the poverty rate in relation to family size, showing that while 6% of families with less than four members are poor, 44% of families with six to seven members fall below the poverty line. That number increases to 75% poverty for families with 10 members or more.
The study by CAPMAS comes as the state works to provide essential food goods and regulate the market amid price hikes and a shortage of essential commodities.
On Saturday, the country's supply minister Mohamed Ali El-Sheikh said the ministry will set the commercial price of subsidised sugar at EGP 6 per kg, to be available at ministry sales outlets.
This comes amid a sugar scarcity due in part to an acute dollar shortage, which has driven unsubsidized sugar prices to EGP 9-10 per kg from EGP 4.5-6 per kg in August.
Seventy-one million people currently use the government's subsidy cards to buy essential food goods.
El-Sheikh said that the country has sugar reserves that can cover four months, edible oil reserves to cover six months, and wheat reserves to cover four-and-a-half months.
Egypt's net international reserves currently stand at $19.591 billion, according to the central bank. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,767 |
Providing quality, affordable and accessible healthcare for over 30 years.
One of Australia's leading healthcare companies, Healius is synonymous with premium quality, affordable and accessible healthcare for all Australians.
Through an expansive network of multi-disciplinary medical centres, pathology laboratories and diagnostic imaging centres, Healius provides world class facilities and support services to independent general practitioners, radiologists and other healthcare professionals, enabling them to deliver quality care to their patients in partnership with Healius' pathologists, nurses and other employees.
Healius' 'medical home' model makes healthcare services easily accessible and cost efficient, while supporting the coordination and continuity of quality patient care.
Over 8 million GP consults per year will take place in our medical centres.
1 in every 3 pathology samples taken in Australia are tested in our laboratories.
Healius conducts approx 3 million radiography examinations per year. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,144 |
Das Epitaph für Hermann von Hersel in der katholischen Pfarrkirche St. Leodegar in Schönecken, einer Ortsgemeinde im Eifelkreis Bitburg-Prüm in Rheinland-Pfalz, wurde nach 1592 geschaffen. Das Epitaph im Chor ist als Teil der Kirchenausstattung ein geschütztes Kulturdenkmal.
Der im Jahr 1592 verstorbene Hermann von Hersel, Ritter und Neffe des gleichnamigen, von 1489 bis 1504 auf Burg Schönecken ansässigen Burgmanns, wird in Rüstung mit gefalteten Händen dargestellt. Der Helm liegt zu seinen Füßen. An den Seiten sind je sechs und über der Figur vier Ahnenwappen zu sehen.
Den oberen Abschluss des Epitaphs bildet die schmuckvolle Darstellung zweier Jünglinge, die das Wappen von Hermann von Hersel präsentieren.
Weblinks
Die Restaurierung des Epitaphs 2004/05 bei www.schoenecken.com (mit vielen Fotos)
Kulturdenkmal in Schönecken
Schonecken
Hersel
Hersel | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 36 |
Q: VBA fetching emails from outlook too slow So apparently here this macro fetches specific email addresses from inbox as well as sent items along with email addresses from cc,bcc
the problem is it takes a whole lot of time and i mean if a person has 2k emails he might have to wait for 3 to 4 hours .
Check some sources how to make code faster i got to know about restrict function when applied through DASL filter and limit number of items in a loop. i applied the same but the result is still the same and fetching is still slow .
As new into VBA i dont know all about optimization and still learning.
Any other sources or ways to make the fetching and execution faster ?
code given for reference
Option Explicit
Sub GetInboxItems()
'all vars declared
Dim ol As Outlook.Application
Dim ns As Outlook.Namespace
Dim fol As Outlook.Folder
Dim i As Object
Dim mi As Outlook.MailItem
Dim n As Long
Dim seemail As String
Dim seAddress As String
Dim varSenders As Variant
'for sent mails
Dim a As Integer
Dim b As Integer
Dim objitem As Object
Dim take As Outlook.Folder
Dim xi As Outlook.MailItem
Dim asd As String
Dim arr As Variant
Dim K As Long
Dim j As Long
Dim vcc As Variant
Dim seemail2 As String
Dim seAddress2 As String
Dim varSenders2 As Variant
Dim strFilter As String
Dim strFilter2 As String
'screen wont refresh untill this is turned true
Application.ScreenUpdating = False
'now assigning the variables and objects of outlook into this
Set ol = New Outlook.Application
Set ns = ol.GetNamespace("MAPI")
Set fol = ns.GetDefaultFolder(olFolderInbox)
Set take = ns.GetDefaultFolder(olFolderSentMail)
Range("A3", Range("A3").End(xlDown).End(xlToRight)).Clear
n = 2
strFilter = "@SQL=" & Chr(34) & "urn:schemas:httpmail:fromemail" & Chr(34) & " like '%" & seemail & "'"
strFilter2 = "@SQL=" & Chr(34) & "urn:schemas:httpmail:sentitems" & Chr(34) & " like '%" & seemail2 & "'"
'this one is for sent items folder where it fetches the emails from particular people
For Each objitem In take.Items.Restrict(strFilter2)
If objitem.Class = olMail Then
Set xi = objitem
n = n + 1
seemail2 = Worksheets("Inbox").Range("D1")
varSenders2 = Split(seemail2, ";")
For K = 0 To UBound(varSenders2)
'this is the same logic as the inbox one where if mail is found and if the mail is of similar kind then and only it will return the same
If xi.SenderEmailType = "EX" Then
seAddress2 = xi.Sender.GetExchangeUser.PrimarySmtpAddress
If InStr(1, seAddress2, varSenders2(K), vbTextCompare) Then
Cells(n, 1).Value = xi.Sender.GetExchangeUser().PrimarySmtpAddress
Cells(n, 2).Value = xi.SenderName
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
'this is the smpt address (regular address)
ElseIf xi.SenderEmailType = "SMTP" Then
seAddress2 = xi.SenderEmailAddress
If InStr(1, seAddress2, varSenders2(K), vbTextCompare) Then
Cells(n, 1).Value = xi.SenderEmailAddress
Cells(n, 2).Value = xi.SenderName
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
'this one fetches the cc part recipient denotes cc
For j = xi.Recipients.Count To 1 Step -1
If (xi.Recipients.Item(j).AddressEntry.Type = "EX") Then
vcc = xi.Recipients.Item(j).Address
If InStr(1, vcc, varSenders2(K), vbTextCompare) Then
Cells(n, 1).Value = xi.Recipients.Item(j).AddressEntry.GetExchangeUser.PrimarySmtpAddress
Cells(n, 2).Value = xi.Recipients.Item(j).Name
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
Else
vcc = xi.Recipients.Item(j).Address
If InStr(1, vcc, varSenders2(K), vbTextCompare) Then
Cells(n, 1).Value = xi.Recipients.Item(j).Address
Cells(n, 2).Value = xi.Recipients.Item(j).Name
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
End If
Next j
Else: seAddress2 = ""
End If
For a = 1 To take.Items.Count
n = 3
'this also fetches the recipient emails
If TypeName(take.Items(a)) = "MailItem" Then
For b = 1 To take.Items.Item(a).Recipients.Count
asd = take.Items.Item(a).Recipients(b).Address
If InStr(1, asd, varSenders2(K), vbTextCompare) Then
Cells(n, 1).Value = asd
Cells(n, 2).Value = take.Items.Item(a).Recipients(b).Name
n = n + 1
End If
Next b
End If
Next a
Next K
End If
Next objitem
For Each i In fol.Items.Restrict(strFilter)
If i.Class = olMail Then
Set mi = i
'objects have been assigned and can be used to fetch emails
seemail = Worksheets("Inbox").Range("D1")
varSenders = Split(seemail, ";")
n = n + 1
For K = 0 To UBound(varSenders)
'similar logic as above
If mi.SenderEmailType = "EX" Then
seAddress = mi.Sender.GetExchangeUser().PrimarySmtpAddress
If InStr(1, seAddress, varSenders(K), vbTextCompare) Then
Cells(n, 1).Value = mi.Sender.GetExchangeUser().PrimarySmtpAddress
Cells(n, 2).Value = mi.SenderName
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
ElseIf mi.SenderEmailType = "SMTP" Then
seAddress = mi.SenderEmailAddress
If InStr(1, seAddress, varSenders(K), vbTextCompare) Then
Cells(n, 1).Value = mi.SenderEmailAddress
Cells(n, 2).Value = mi.SenderName
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
For j = mi.Recipients.Count To 1 Step -1
If (mi.Recipients.Item(j).AddressEntry.Type = "EX") Then
vcc = mi.Recipients.Item(j).Address
If InStr(1, vcc, varSenders(K), vbTextCompare) Then
Cells(n, 1).Value = mi.Recipients.Item(j).AddressEntry.GetExchangeUser.PrimarySmtpAddress
Cells(n, 2).Value = mi.Recipients.Item(j).Name
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
Else
vcc = mi.Recipients.Item(j).Address
If InStr(1, vcc, varSenders(K), vbTextCompare) Then
Cells(n, 1).Value = mi.Recipients.Item(j).Address
Cells(n, 2).Value = mi.Recipients.Item(j).Name
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
End If
End If
Next j
Else: seAddress = ""
End If
Next K
End If
Next i
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
Set take = Nothing
Set mi = Nothing
Application.ScreenUpdating = True
End Sub
A: All the code touching the email from the outer loop should be taken out of the inner loop. E.g. the line like
seAddress2 = xi.Sender.GetExchangeUser.PrimarySmtpAddress
has no business being in the inner loop.
I also wouldn't call RemoveDuplicates on each step of the loop.
Also, most likely the senders won't be unique - retrieve all the sender addresses (SenderEmailAddress) in a single using MAPIFolder.GetTable and build a dictionary of EX type addresses vs SMTP addresses (GetExchangeUser.PrimarySmtpAddress) to be calculated only once for each unique address instead of retrieving it over and over again.
A: You have to assign a value to seemail and seemail2 before using in strFilter and strFilter2.
Option Explicit
Sub GetInbox_And_SentItems()
'Early binding - requires reference to Microsoft Outlook XX.X Object Library
Dim ol As Outlook.Application
Dim ns As Outlook.Namespace
Dim fol As Outlook.Folder
Dim folItem As Object
Dim mi As Outlook.mailItem
Dim n As Long
Dim seemail As String
Dim seAddress As String
Dim varSenders As Variant
'for sent mails
Dim b As Integer
Dim objitem As Object
Dim take As Outlook.Folder
Dim xi As Outlook.mailItem
Dim k As Long
Dim seemail2 As String
Dim seAddress2 As String
'Dim varSenders2 As Variant
Dim varReceivers As Variant
Dim strFilter As String
Dim strFilter2 As String
'screen won't refresh until this is turned true
'Application.ScreenUpdating = False
'now assigning the variables and objects of outlook into this
Set ol = New Outlook.Application
Set ns = ol.GetNamespace("MAPI")
Set fol = ns.GetDefaultFolder(olFolderInbox)
Set take = ns.GetDefaultFolder(olFolderSentMail)
'Range("A3", Range("A3").End(xlDown).End(xlToRight)).Clear
Range("A3:A9999").Select
Selection.EntireRow.Delete
n = 2
varReceivers = Split(Worksheets("Inbox").Range("D1"), ";")
For k = LBound(varReceivers) To UBound(varReceivers)
seemail2 = Trim(varReceivers(k))
Debug.Print seemail2
' Note displayto not fromemail
' displayto can be a difficult value
' https://stackoverflow.com/questions/16286694/using-the-restrict-method-in-outlook-vba-to-filter-on-single-recipient-email-ad
' As far as I know there is no working toemail.
strFilter2 = "@SQL=" & Chr(34) & "urn:schemas:httpmail:displayto" & Chr(34) & " like '%" & seemail2 & "'"
Debug.Print strFilter2
Debug.Print "Items in Inbox.........:" & take.Items.Count
Debug.Print "Filtered Items in Inbox:" & take.Items.Restrict(strFilter2).Count
'this one is for sent items folder where it fetches the emails --> to <-- particular people
' there is no point searching a sent folder for sender information
For Each objitem In take.Items.Restrict(strFilter2)
If objitem.Class = olMail Then
Set xi = objitem
n = n + 1
Cells(n, 1).Value = seemail2
Cells(n, 2).Value = xi.Subject
Dim msg As String
msg = ""
For b = 1 To xi.Recipients.Count
msg = msg & xi.Recipients(b).Address & "; "
Next b
Cells(n, 3).Value = msg
End If
Next objitem
Next k
varSenders = Split(Worksheets("Inbox").Range("D1"), ";")
For k = LBound(varSenders) To UBound(varSenders)
seemail = Trim(varSenders(k))
Debug.Print seemail
strFilter = "@SQL=" & Chr(34) & "urn:schemas:httpmail:fromemail" & Chr(34) & " like '%" & seemail & "'"
Debug.Print strFilter
For Each folItem In fol.Items.Restrict(strFilter)
If folItem.Class = olMail Then
Set mi = folItem
'objects have been assigned and can be used to fetch emails
n = n + 1
'similar logic as above
If mi.SenderEmailType = "EX" Then
seAddress = mi.Sender.GetExchangeUser().PrimarySmtpAddress
Cells(n, 1).Value = mi.Sender.GetExchangeUser().PrimarySmtpAddress
Cells(n, 2).Value = mi.SenderName
ElseIf mi.SenderEmailType = "SMTP" Then
seAddress = mi.SenderEmailAddress
Cells(n, 1).Value = mi.SenderEmailAddress
Cells(n, 2).Value = mi.Subject
End If
End If
Next folItem
Next k
ActiveSheet.UsedRange.RemoveDuplicates Columns:=Array(1, 2), Header:=xlYes
'Uncomment if needed
'On Error Resume Next
Range("A3:A9999").Select
Selection.SpecialCells(xlCellTypeBlanks).EntireRow.Delete
On Error GoTo 0
Application.ScreenUpdating = True
End Sub
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,671 |
Пасифея или още Пасифая, Пазифая () в древногръцката митология е дъщеря на Хелиос и нимфата Персеида. Съпруга е на критския цар Минос и сестра на вълшебницата Кирка.
След като Минос, въпреки обещанието си, не принесъл в жертва на Посейдон, изпратения от него бик, богът внушил на Пасифея противоестествено влечение към бика. Според друга версия, любовта ѝ към бика възникнала у нея под влияние на Афродита, която по този начин си отмъстила на баща ѝ Хелиос, който я издал на Хефест, че му изневерява с Арес. В резултат от тази връзка на Пасифея, се родил Минотавъра, който бил заключен от Минос в лабиринт. От Минос тя е майка на Ариадна, Главк, Девкалион, Федра и Андрогей.
В мита за връзката на Пасифея с бика намират отражение древни тотемни вярвания, където се отдава почит на животните, родоначалници на племето. В Близкия изток и в Южното Средиземноморие в ролята на тотем-покровител често се намира бик ("небесния бик" в шумерската митология или гръцкия мит за превръщането на Зевс в бик при отвличането на Европа).
Източници
Древногръцка митология
Митология на Крит
Жени от древногръцката митология | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,306 |
{"url":"https:\/\/en.formulasearchengine.com\/wiki\/Projective_space","text":"# Projective space\n\nIn graphical perspective, parallel lines in the plane intersect in a vanishing point on the horizon.\n\nIn mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when V = R2 and V = R3 are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.\n\nThe idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points that lie on a projection line (i.e., a \"line-of-sight\"), intersecting with the entrance pupil of the camera, are projected onto a common image point. In this case, the vector space is R3 with the camera entrance pupil at the origin, and the projective space corresponds to the image points.\n\nProjective spaces can be studied as a separate field in mathematics, but are also used in various applied fields, geometry in particular. Geometric objects, such as points, lines, or planes, can be given a representation as elements in projective spaces based on homogeneous coordinates. As a result, various relations between these objects can be described in a simpler way than is possible without homogeneous coordinates. Furthermore, various statements in geometry can be made more consistent and without exceptions. For example, in the standard geometry for the plane, two lines always intersect at a point except when the lines are parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points.\n\nOther mathematical fields where projective spaces play a significant role are topology, the theory of Lie groups and algebraic groups, and their representation theories.\n\n## Introduction\n\nprojective space\n\nAs outlined above, projective space is a geometric object which formalizes statements like \"Parallel lines intersect at infinity\". For concreteness, we will give the construction of the real projective plane P2(R) in some detail. There are three equivalent definitions:\n\n1. The set of all lines in R3 passing through the origin (0, 0, 0). Every such line meets the sphere of radius one centered in the origin exactly twice, say in P = (x, y, z) and its antipodal point (\u2212x, \u2212y, \u2212z).\n2. P2(R) can also be described to be the points on the sphere S2, where every point P and its antipodal point are not distinguished. For example, the point (1, 0, 0) (red point in the image) is identified with (\u22121, 0, 0) (light red point), etc.\n3. Finally, yet another equivalent definition is the set of equivalence classes of R3 \u2216 (0, 0, 0), i.e. 3-space without the origin, where two points P = (x, y, z) and P = (x, y, z) are equivalent iff there is a nonzero real number \u03bb such that P = \u03bbP, i.e. x = \u03bbx, y = \u03bby, z = \u03bbz. The usual way to write an element of the projective plane, i.e. the equivalence class corresponding to an honest point (x, y, z) in R3, is [x\u00a0: y\u00a0: z].\n\nThe last formula goes under the name of homogeneous coordinates.\n\nIn homogeneous coordinates, any point [x\u00a0: y\u00a0: z] with z \u2260 0 is equivalent to [x\/z\u00a0: y\/z\u00a0: 1]. So there are two disjoint subsets of the projective plane: that consisting of the points [x\u00a0: y\u00a0: z] = [x\/z\u00a0: y\/z\u00a0: 1] for z \u2260 0, and that consisting of the remaining points [x\u00a0: y\u00a0: 0]. The latter set can be subdivided similarly into two disjoint subsets, with points [x\/y\u00a0: 1\u00a0: 0] and [x\u00a0: 0\u00a0: 0]. In the last case, x is necessarily nonzero, because the origin was not part of P2(R). This last point is equivalent to [1\u00a0: 0\u00a0: 0]. Geometrically, the first subset, which is isomorphic (not only as a set, but also as a manifold, as will be seen later) to R2, is in the image the yellow upper hemisphere (without the equator), or equivalently the lower hemisphere. The second subset, isomorphic to R1, corresponds to the green line (without the two marked points), or, again, equivalently the light green line. Finally we have the red point or the equivalent light red point. We thus have a disjoint decomposition\n\nP2(R) = R2R1point.\n\nIntuitively, and made precise below, R1point is itself the real projective line P1(R). Considered as a subset of P2(R), it is called line at infinity, whereas R2P2(R) is called affine plane, i.e. just the usual plane.\n\nThe next objective is to make the saying \"parallel lines meet at infinity\" precise. A natural bijection between the plane z = 1 (which meets the sphere at the north pole N = (0, 0, 1)) and the sphere of the projective plane is accomplished by the stereographic projection. Each point P on this plane is mapped to the two intersection points of the sphere with the line through its center and P. These two points are identified in the projective plane. Lines (blue) in the plane are mapped to great circles if one also includes one pair of antipodal points on the equator. Any two great circles intersect precisely in two antipodal points (identified in the projective plane). Great circles corresponding to parallel lines intersect on the equator. So any two lines have exactly one intersection point inside P2(R). This phenomenon is axiomatized in projective geometry.\n\n## Definition of projective space\n\nThe real projective space of dimension n or projective n-space, Pn(R), is roughly speaking the set of the lines in Rn+1 passing through the origin. For defining it as a topological space and as an algebraic variety it is better to define it as the quotient space of Rn+1 by the equivalence relation \"to be aligned with the origin\". More precisely,\n\nPn(R)\u00a0:= (Rn+1 \u2216 {0}) \/ ~,\n\nwhere ~ is the equivalence relation defined by: (x0, ..., xn) ~ (y0, ..., yn) if there is a non-zero real number \u03bb such that (x0, ..., xn) = (\u03bby0, ..., \u03bbyn).\n\nThe elements of the projective space are commonly called points. The projective coordinates of a point P are x0, ..., xn, where (x0, ..., xn) is any element of the corresponding equivalence class. This is denoted P = [x0\u00a0: ...\u00a0: xn], the colons and the brackets emphasizing that the right-hand side is an equivalence class, which is defined up to the multiplication by a non zero constant.\n\nInstead of R, one may take any field, or even a division ring, K. In these cases it is common[1] to use the notation PG(n, K) for Pn(K). If K is a finite field of order q, the notation is further simplified to PG(n, q). Taking the complex numbers or the quaternions, one obtains the complex projective space Pn(C) and quaternionic projective space Pn(H).\n\nIf n is one or two, it is also called projective line or projective plane, respectively. The complex projective line is also called the Riemann sphere.\n\nSlightly more generally, for a vector space V (over some field k, or even more generally a module V over some division ring), P(V) is defined to be (V \u2216 {0}) \/ ~, where two non-zero vectors v1, v2 in V are equivalent if they differ by a non-zero scalar \u03bb, i.e., v1 = \u03bbv2. The vector space need not be finite-dimensional; thus, for example, there is the theory of projective Hilbert spaces.\n\n## Projective space as a manifold\n\nManifold structure of the real projective line\n\nThe above definition of projective space gives a set. For purposes of differential geometry, which deals with manifolds, it is useful to endow this set with a (real or complex) manifold structure.\n\nNamely, identifying a point of the projective space with its homogeneous coordinates, let us consider the following subsets of the projective space:\n\n${\\displaystyle U_{i}=\\{[x_{0}:\\cdots :x_{n}],x_{i}\\neq 0\\},\\quad i=0,\\dots ,n.}$\n\nBy the definition of projective space, their union is the whole projective space. Furthermore, Ui is in bijection with Rn (or Cn) via the following maps:\n\n${\\displaystyle [x_{0}:\\cdots :x_{n}]\\mapsto \\left({\\frac {x_{0}}{x_{i}}},\\dots ,{\\widehat {\\frac {x_{i}}{x_{i}}}},\\dots ,{\\frac {x_{n}}{x_{i}}}\\right)}$\n${\\displaystyle [y_{0}:\\cdots :y_{i-1}:1:y_{i+1}:\\cdots :y_{n}]\\leftarrow \\left(y_{0},\\dots ,{\\widehat {y_{i}}},\\dots y_{n}\\right)}$\n\n(the hat means that the i-th entry is missing).\n\nThe example image shows P1(R). (Antipodal points are identified in P1(R), though). It is covered by two copies of the real line R, each of which covers the projective line except one point, which is \"the\" (or \"a\") point at infinity.\n\nWe first define a topology on projective space by declaring that these maps shall be homeomorphisms, that is, a subset of Ui is open iff its image under the above isomorphism is an open subset (in the usual sense) of Rn. An arbitrary subset A of Pn(R) is open if all intersections AUi are open. This defines a topological space.\n\nThe manifold structure is given by the above maps, too.\n\nDifferent visualization of the projective line\n\nAnother way to think about the projective line is the following: take two copies of the affine line with coordinates x and y, respectively, and glue them together along the subsets x \u2260 0 and y \u2260 0 via the maps\n\n${\\displaystyle x\\mapsto {\\frac {1}{x}},\\,y\\mapsto {\\frac {1}{y}}.}$\n\nThe resulting manifold is the projective line. The charts given by this construction are the same as the ones above. Similar presentations exist for higher-dimensional projective spaces.\n\nThe above decomposition in disjoint subsets reads in this generality:\n\nPn(R) = RnRn\u22121${\\displaystyle \\cdots }$R1R0,\n\nthis so-called cell-decomposition can be used to calculate the singular cohomology of projective space.\n\nAll of the above holds for complex projective space, too. The complex projective line P1(C) is an example of a Riemann surface.\n\n## Projective spaces in algebraic geometry\n\n{{#invoke:main|main}}\n\nThe covering by the above open subsets also shows that projective space is an algebraic variety (or scheme), it is covered by n + 1 affine n-spaces. The construction of projective scheme is an instance of the Proj construction.\n\n## Projective spaces in algebraic topology\n\nReal projective n-space has a quite straightforward CW complex structure. That is, each n-dimensional real projective space has only one n-dimensional cell.\n\n## Projective space and affine space\n\nExample for B\u00e9zout's theorem\n\nThere are some advantages of the projective space compared with affine space (e.g. Pn(R) vs. An(R)). For these reasons it is important to know when a given manifold or variety is projective, i.e. embeds into (is a closed subset of) projective space. (Very) ample line bundles are designed to tackle this question.\n\nNote that a projective space can be formed by the projectivization of a vector space, as lines through the origin, but cannot be formed from an affine space without a choice of basepoint. That is, affine spaces are open subspaces of projective spaces, which are quotients of vector spaces.\n\n\u2022 On a projective complex manifold X, cohomology groups of coherent sheaves are finitely generated. (The above example is H0(Pn(C), O), the zeroth cohomology of the sheaf of holomorphic functions O). In the parlance of algebraic geometry, projective space is proper. The above results hold in this context, too.\n\u2022 For complex projective space, every complex submanifold XPn(C) (i.e., a manifold cut out by holomorphic equations) is necessarily an algebraic variety (i.e., given by polynomial equations). This is Chow's theorem, it allows the direct use of algebraic\u2013geometric methods for these ad hoc analytically defined objects.\n\u2022 As outlined above, lines in P2 or more generally hyperplanes in Pn always do intersect. This extends to non-linear objects, as well: appropriately defining the degree of an algebraic curve, which is roughly the degree of the polynomials needed to define the curve (see Hilbert polynomial), it is true (over an algebraically closed field k) that any two projective curves C1, C2Pn(k) of degree e and f intersect in exactly ef points, counting them with multiplicities (see B\u00e9zout's theorem). This is applied, for example, in defining a group structure on the points of an elliptic curve, like y2 = x3x + 1. The degree of an elliptic curve is 3. Consider the line x = 1, which intersects the curve (inside affine space) exactly twice, namely in (1, 1) and (1, \u22121). However, inside P2, the projective closure of the curve is given by the homogeneous equation\ny2z = x3xz2 + z3,\nwhich intersects the line (given inside P2 by x = z) in three points: [1: 1: 1], [1: \u22121: 1] (corresponding to the two points mentioned above), and [0: 1: 0].\n\u2022 Any projective group variety, i.e. a projective variety, whose points form an abstract group, is necessarily an abelian variety. Elliptic curves are examples for abelian varieties. The commutativity fails for non-projective group varieties, as the example GLn(k) (the general linear group) shows.\n\n## Axioms for projective space\n\nA projective space S can be defined axiomatically as a set P (the set of points), together with a set L of subsets of P (the set of lines), satisfying these axioms:[2]\n\n\u2022 Each two distinct points p and q are in exactly one line.\n\u2022 Veblen's axiom:[3] If a, b, c, d are distinct points and the lines through ab and cd meet, then so do the lines through ac and bd.\n\u2022 Any line has at least 3 points on it.\n\nThe last axiom eliminates reducible cases that can be written as a disjoint union of projective spaces together with 2-point lines joining any two points in distinct projective spaces. More abstractly, it can be defined as an incidence structure (P, L, I) consisting of a set P of points, a set L of lines, and an incidence relation I stating which points lie on which lines.\n\nThe structures defined by these axioms are more general than those obtained from the vector space construction given above. If the (projective) dimension is at least three then, by the Veblen\u2013Young theorem, there is no difference. However, for dimension two there are examples which satisfy these axioms that can not be constructed from vector spaces (or even modules over division rings). These examples do not satisfy the Theorem of Desargues and are known as Non-Desarguesian planes. In dimension one, any set with at least three elements satisfies the axioms, so it is usual to assume additional structure for projective lines defined axiomatically.[4]\n\nIt is possible to avoid the troublesome cases in low dimensions by adding or modifying axioms that define a projective space. Template:Harvtxt gives such an extension due to Bachmann.[5] To ensure that the dimension is at least two, replace the three point per line axiom above by;\n\n\u2022 There exist four points, no three of which are collinear.\n\nTo avoid the non-Desarguesian planes, include Pappus's theorem as an axiom;[6]\n\n\u2022 If the six vertices of a hexagon lie alternately on two lines, the three points of intersection of pairs of opposite sides are collinear.\n\nAnd, to ensure that the vector space is defined over a field that does not have even characteristic include Fano's axiom;[7]\n\n{{safesubst:#invoke:anchor|main}}A subspace of the projective space is a subset X, such that any line containing two points of X is a subset of X (that is, completely contained in X). The full space and the empty space are always subspaces.\n\nThe geometric dimension of the space is said to be n if that is the largest number for which there is a strictly ascending chain of subspaces of this form:\n\n${\\displaystyle \\varnothing =X_{-1}\\subset X_{0}\\subset \\cdots X_{n}=P.}$\n\nA subspace ${\\displaystyle X_{i}}$ in such a chain is said to have (geometric) dimension ${\\displaystyle i}$. Subspaces of dimension 0 are called points, those of dimension 1 are called lines and so on. If the full space has dimension ${\\displaystyle n}$ then any subspace of dimension ${\\displaystyle n-1}$ is called a hyperplane.\n\n### Classification\n\n\u2022 Dimension 0 (no lines): The space is a single point.\n\u2022 Dimension 1 (exactly one line): All points lie on the unique line.\n\u2022 Dimension 2: There are at least 2 lines, and any two lines meet. A projective space for n = 2 is equivalent to a projective plane. These are much harder to classify, as not all of them are isomorphic with a PG(d, K). The Desarguesian planes (those which are isomorphic with a PG(2, K)) satisfy Desargues's theorem and are projective planes over division rings, but there are many non-Desarguesian planes.\n\u2022 Dimension at least 3: Two non-intersecting lines exist. Template:Harvtxt proved the Veblen\u2013Young theorem that every projective space of dimension n \u2265 3 is isomorphic with a PG(n, K), the n-dimensional projective space over some division ring K.\n\n### Finite projective spaces and planes\n\nA finite projective space is a projective space where P is a finite set of points. In any finite projective space, each line contains the same number of points and the order of the space is defined as one less than this common number. For finite projective spaces of dimension at least three, Wedderburn's theorem implies that the division ring over which the projective space is defined must be a finite field, GF(q), whose order (that is, number of elements) is q (a prime power). A finite projective space defined over such a finite field will have q + 1 points on a line, so the two concepts of order will coincide. Notationally, PG(n, GF(q)) is usually written as PG(n, q).\n\nAll finite fields of the same order are isomorphic, so, up to isomorphism, there is only one finite projective space for each dimension greater than or equal to three, over a given finite field. However, in dimension two there are non-Desarguesian planes. Up to isomorphism there are\n\n1, 1, 1, 1, 0, 1, 1, 4, 0, \u2026 (sequence A001231 in OEIS)\n\nfinite projective planes of orders 2, 3, 4, ..., 10, respectively. The numbers beyond this are very difficult to calculate and are not determined except for some zero values due to the Bruck\u2013Ryser theorem.\n\nThe smallest projective plane is the Fano plane, PG(2, 2) with 7 points and 7 lines.\n\n## Morphisms\n\n{{ safesubst:#invoke:Unsubst||$N=Cleanup-rewrite |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} Injective linear maps TL(V, W) between two vector spaces V and W over the same field k induce mappings of the corresponding projective spaces P(V) \u2192 P(W) via:\n\n[v] \u2192 [T(v)],\n\nwhere v is a non-zero element of V and [...] denotes the equivalence classes of a vector under the defining identification of the respective projective spaces. Since members of the equivalence class differ by a scalar factor, and linear maps preserve scalar factors, this induced map is well-defined. (If T is not injective, it will have a null space larger than {0}; in this case the meaning of the class of T(v) is problematic if v is non-zero and in the null space. In this case one obtains a so-called rational map, see also birational geometry).\n\nTwo linear maps S and T in L(V,W) induce the same map between P(V) and P(W) if and only if they differ by a scalar multiple, that is if T=\u03bbS for some \u03bb \u2260 0. Thus if one identifies the scalar multiples of the identity map with the underlying field, the set of k-linear morphisms from P(V) to P(W) is simply P(L(V,W)).\n\nThe automorphisms P(V) \u2192 P(V) can be described more concretely. (We deal only with automorphisms preserving the base field k). Using the notion of sheaves generated by global sections, it can be shown that any algebraic (not necessarily linear) automorphism has to be linear, i.e. coming from a (linear) automorphism of the vector space V. The latter form the group GL(V). By identifying maps which differ by a scalar, one concludes\n\nAut(P(V)) = Aut(V)\/k = GL(V)\/k =: PGL(V),\n\nthe quotient group of GL(V) modulo the matrices which are scalar multiples of the identity. (These matrices form the center of Aut(V).) The groups PGL are called projective linear groups. The automorphisms of the complex projective line P1(C) are called M\u00f6bius transformations.\n\n## Dual projective space\n\nWhen the construction above is applied to the dual space V rather than V, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of V. That is, if V is n dimensional, then P(V) is the Grassmannian of n \u2212 1 planes in V.\n\nIn algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles. One would like to be able associate a projective space to every quasi-coherent sheaf E over a scheme Y, not just the locally free ones.Template:Clarify See EGAII, Chap. II, par. 4 for more details.\n\n## Generalizations\n\ndimension\nThe projective space, being the \"space\" of all one-dimensional linear subspaces of a given vector space V is generalized to Grassmannian manifold, which is parametrizing higher-dimensional subspaces (of some fixed dimension) of V.\nsequence of subspaces\nMore generally flag manifold is the space of flags, i.e. chains of linear subspaces of V.\nother subvarieties\nEven more generally, moduli spaces parametrize objects such as elliptic curves of a given kind.\nother rings\nGeneralizing to associative rings (rather than fields) yields the projective line over a ring\npatching\nPatching projective spaces together yields projective space bundles.\n\nSeveri\u2013Brauer varieties are algebraic varieties over a field k which become isomorphic to projective spaces after an extension of the base field k.\n\nAnother generalization of projective spaces are weighted projective spaces; these are themselves special cases of toric varieties.[8]\n\n## Notes\n\n1. Mauro Biliotti, Vikram Jha, Norman L. Johnson (2001) Foundations of Translation Planes, p 506, Marcel Dekker ISBN 0-8247-0609-9\n2. Template:Harvnb\n3. also referred to as the Veblen\u2013Young axiom and mistakenly as the axiom of Pasch Template:Harv. Pasch was concerned with real projective space and was attempting to introduce order, which is not a concern of the Veblen\u2013Young axiom.\n4. Template:Harvnb\n5. {{#invoke:citation\/CS1|citation |CitationClass=citation }}\n6. As Pappus's theorem implies Desargues's theorem this eliminates the non-Desarguesian planes and also implies that the space is defined over a field (and not a division ring).\n7. This restriction allows the real and complex fields to be used (zero characteristic) but removes the Fano plane and other planes that exhibit atypical behavior.\n8. Template:Harvnb\n\n## References\n\n|CitationClass=citation }}\n\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}\n\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}\n\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}\n\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}\n\n\u2022 Greenberg, M.J.; Euclidean and non-Euclidean geometries, 2nd ed. Freeman (1980).\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}, esp. chapters I.2, I.7, II.5, and II.7\n\n\u2022 Hilbert, D. and Cohn-Vossen, S.; Geometry and the imagination, 2nd ed. Chelsea (1999).\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }}\n\n\u2022 {{#invoke:citation\/CS1|citation\n\n|CitationClass=citation }} (Reprint of 1910 edition)","date":"2021-01-26 11:41:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 10, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9089272618293762, \"perplexity\": 442.358744257151}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610704799741.85\/warc\/CC-MAIN-20210126104721-20210126134721-00452.warc.gz\"}"} | null | null |
ANSA Automotive is a Division of AP Exhaust Technologies, Inc.Turner Gas Company is one of the largest independent transporters and fastest growing marketers focused on Natural Gas Liquids and Crude in the Bakken.Para Que Sirve El Viagra Para. weight.step.by.step Fast Ways To Lose 20 Pounds In Three Months Garcinia Cambogia Extreme Trial. super active cialis.Professional And Viagra Super Active how.to.lose. Months Garcinia Cambogia Extreme Trial.DesignWorkshop is a complete family of scaleable, upgradeable 3D solutions, from downloadable free 3D for the novice and hobbyist, to production modeling and photo-.On the one hand, a history of hypnosis is a bit like a history of breathing.Scambusters is committed to helping you avoid getting taken by.Cuanto Tiempo Hace Efecto El. 2 100mg viagra cialis aciclovir super active cialis 20mg 100mg viagra paypal.
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We reconstruct the dual network structure generated by the association between 72 contributors and 737 software bugs engaged during a full development cycle of the free/open source software project Epiphany. Estimates of structural parameters of Exponential Random Graph Models for two-mode networks reveal the structural logics shaping activities of collaborative problem solving. After controlling for contributor specific and software bug-specific characteristics, we find that contributors ("problem solvers") tend to distribute their activity over multiple software bugs. At the same time, however, we find that software bugs ("problems") tend not to share multiple contributors. This dual tendency toward de-specialization and exclusivity is sustained by specific local network dependencies revealed by our analysis which also suggests possible organizational mechanisms that may be underlying the puzzling macro-structural regularities frequently observed, but rarely explained, in the production of open source software. By combining these mechanisms with the influence of contributors characterized by different levels of involvement in the project, we provide micro-level evidence of structural interdependence between "core" and "peripheral" members identified exclusively on the basis of their individual level of contribution to the project.
Published in print: May 2013. Published as: Social Networks, (2013), Vol. 35, (2), pp. 237–250. | {
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Vizantea-Livezi (în maghiară Vizánta) este o comună în județul Vrancea, Moldova, România, formată din satele Livezile (reședința), Mesteacănu, Piscu Radului, Vizantea Mănăstirească și Vizantea Răzășească.
Așezare
Comuna se află în zona de nord a județului, în valea râului Vizăuți. Este străbătută de șoseaua județeană DJ205E, care o leagă spre nord de Câmpuri și spre sud-est de Vidra (unde se intersectează cu DN2D), Țifești, Garoafa (unde se intersectează cu DN2) și mai departe în județul Galați de Movileni.
Demografie
Conform recensământului efectuat în 2011, populația comunei Vizantea-Livezi se ridică la de locuitori, în scădere față de recensământul anterior din 2002, când se înregistraseră de locuitori. Majoritatea locuitorilor sunt români (97,92%). Pentru 2% din populație, apartenența etnică nu este cunoscută.
Din punct de vedere confesional, majoritatea locuitorilor sunt ortodocși (67,86%), cu o minoritate de romano-catolici (29,87%). Pentru 2% din populație, nu este cunoscută apartenența confesională.
Politică și administrație
Comuna Vizantea-Livezi este administrată de un primar și un consiliu local compus din 13 consilieri. Primarul, , de la , este în funcție din . Începând cu alegerile locale din 2020, consiliul local are următoarea componență pe partide politice:
Istorie
La sfârșitul secolului al XIX-lea, pe teritoriul actual al comunei funcționau în județul Putna comunele Vizantea (în plasa Zăbrăuți) și Găurile (în plasa Vrancea). Comuna Vizantea, formată din satele Vizantea Mănăstirească și Vizantea Răzășească, avea 1043 de locuitori, două biserici și o școală mixtă. Comuna Găurile era formată din satele Găurile, Piscu Radului și Purcei și avea 1362 de locuitori ce trăiau în 336 de case. În comuna Găurile existau trei biserici.
Anuarul Socec din 1925 consemnează ambele comune în plasa Vidra a aceluiași județ, cu aceeași alcătuire. Comuna Vizantea avea 1900 de locuitori, iar comuna Găurile 1364.
În 1950, cele două comune au fost arondate raionului Panciu din regiunea Putna, apoi (după 1952) din regiunea Bârlad și (după 1956) din regiunea Galați. În 1964, satul și comuna Găurile au luat numele de Livezile, iar satul Purcei a luat denumirea de Mesteacănu. În 1968, comunele au fost transferate la județul Vrancea și au fost contopite, formând comuna Vizantea-Livezi, cu reședința în satul Livezile.
Monumente istorice
În comuna Vizantea-Livezi se află fosta mănăstire Vizantea, monument istoric de arhitectură de interes național datând din secolul al XVII-lea. Ansamblul monumental cuprinde biserica nouă "Sfânta Cruce" construită în 1850–1854, turnul cloponiță și zidul de incintă.
Personalități născute aici
Cornel Coman (1936 - 1981), actor.
Note
Vizantea-Livezi | {
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Adolf I (ur. ?, zm. 13 listopada 1130) – hrabia szauenburski i holsztyński.
Został hrabią Szauenburga prawdopodobnie w 1106, około 1111 roku otrzymał z rąk księcia saksońskiego Lotara z Supplinburga (późniejszego króla i cesarza rzymskiego) hrabstwo Holsztynu i hrabstwo Sztormarn, (do których należał również Hamburg), jako lenno. Miał syna Adolfa II, który przyczynił się do rozwoju Holsztynu.
Schaumburgowie
Władcy Szlezwika-Holsztynu
Zmarli w 1130 | {
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On 20 March 2019, the European Commission (EC) launched the first Horizon Impact Award – a prize dedicated to EU-funded projects whose results provided an added value to society. This prize recognizes and awards the most successful project results under the 7th Framework programme (FP7 2007-2013) and Horizon 2020. The award aims to show the wider socio-economic benefits of EU investment in research and innovation, enabling individuals or teams to demonstrate their best practices and achievements.
The contest is open to all legal entities that have completed a FP7 or Horizon 2020 project. Each of the five winners will receive €10,000.-. The contest is open for applications until 28 May 2019. | {
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Book your Room Online at The Whispering Meadows in Munnar by selecting your date of stay from the room booking form Here you can Reserve your Rooms and get the cheapest Rates for the The Whispering Meadows by selecting the best Possible Arrival dates and your Stay Schedule.
The Whispering Meadows Munnar has a concierge, a business center and tennis courts. It also has a dry cleaning service, laundry facilities and a currency exchange. The garden is a perfect place to relax.Rooms at the Whispering Meadows Munnar offer sweeping views of the hill and are equipped with an iron, a TV and complimentary toiletries. Each is equipped with a ceiling fan and room service is available.The Whispering Meadows Munnar has an on-site restaurant, convenient for guests who wish to dine in. The property also offers a daily breakfast. | {
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var livelyRepositories = require('./repository'),
path = require("path"),
async = require("async"),
request = require("request"),
fsHelper = require("lively-fs-helper"),
port = 9003, testRepo;
livelyRepositories.start({
fs: path.join(process.cwd()), port: port,
}, function(err, repo) { global.testRepo = repo; console.log('started'); })
| {
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Market Pulse: European investors still "bank-shy"
July 31 - BNP Paribas the latest European bank to report in the current mega-earnings week. Investors like what they see, but are still extremely cautious on the sector. And Siemens says Auf Wiedersehen Pete. | {
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{"url":"http:\/\/888quotes.com\/7hkhynp\/0193fa-hidden-markov-model-machine-learning%3F","text":"The Learning Problem is knows as Forward-Backward Algorithm or Baum-Welch Algorithm. In HMM, time series' known observations are known as visible states. Utilising Hidden Markov Models as overlays to a risk manager that can interfere with strategy-generated orders requires careful research analysis and a solid understanding of the asset class(es) being modelled. Credit scoring involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you\u2019re going to default. The elements of the sequence, DNA nucleotides, are the observations, and the states may be regions corresponding to genes and regions that don\u2019t represent genes at all. Udemy - Unsupervised Machine Learning Hidden Markov Models in Python (Updated 12\/2020) The Hidden Markov Model or HMM is all about learning sequences. In future articles the performance of various trading strategies will be studied under various Hidden Markov Model based risk managers. We also don\u2019t know the second to last state, so we have to consider all the possible states $r$ that we could be transitioning from. If you need a refresher on the technique, see my graphical introduction to dynamic programming. (I gave a talk on this topic at PyData Los Angeles 2019, if you prefer a video version of this post.). Now let \u2026 b_{31} & b_{32} In this article, I\u2019ll explore one technique used in machine learning, Hidden Markov Models (HMMs), and how dynamic programming is used when applying this technique. This means we can lay out our subproblems as a two-dimensional grid of size $T \\times S$. Stock prices are sequences of prices. HMM (Hidden Markov Model) is a Stochastic technique for POS tagging. Next comes the main loop, where we calculate $V(t, s)$ for every possible state $s$ in terms of $V(t - 1, r)$ for every possible previous state $r$. If the process is entirely autonomous, meaning there is no feedback that may influence the outcome, a Markov chain may be used to model the outcome. Language is a sequence of words. graphical introduction to dynamic programming, In my previous article about seam carving, the similar seam carving implementation from my last post, Hidden Markov Models and their Applications in Biological Sequence Analysis. As a motivating example, consider a robot that wants to know where it is. Also known as speech-to-text, speech recognition observes a series of sounds. The 2nd entry equals \u2248 0.44. Most of the work is getting the problem to a point where dynamic programming is even applicable. Announcement: New Book by Luis Serrano! 6.867 Machine learning, lecture 20 (Jaakkola) 1 Lecture topics: \u2022 Hidden Markov Models (cont\u2019d) Hidden Markov Models (cont\u2019d) We will continue here with the three problems outlined previously. orF instance, we might be interested in discovering the sequence of words that someone spoke based on an audio recording of their speech. From the above analysis, we can see we should solve subproblems in the following order: Because each time step only depends on the previous time step, we should be able to keep around only two time steps worth of intermediate values. In our initial example of dishonest casino, the die rolled (fair or unfair) is unknown or hidden. Grokking Machine Learning. When applied specifically to HMMs, the algorithm is known as the Baum-Welch algorithm. Technically, the second input is a state, but there are a fixed set of states. HMMs for stock price analysis, language modeling, web analytics, biology, and PageRank. We can only know the mood of the person. Which bucket does HMM fall into? Let me know what you\u2019d like to see next! In other words, the distribution of initial states has all of its probability mass concentrated at state 1. ... Hidden Markov Model as a finite state machine. These probabilities are denoted $\\pi(s_i)$. Prediction is the ultimate goal for any model\/algorithm. However Hidden Markov Model (HMM) often trained using supervised learning method in case training data is available. 2nd plot is the prediction of Hidden Markov Model. Real-world problems don\u2019t appear out of thin air in HMM form. Text data is very rich source of information and on applying proper Machine Learning techniques, we can implement a model \u2026 Stock prices are sequences of prices. The 2nd Order Markov Model can be written as $$p(s(t) | s(t-1), s(t-2))$$. \\). Join and get free content delivered automatically each time we publish. For information, see The Application of Hidden Markov Modelsin Speech Recognition by Gales and Young. We can use the joint & conditional probability rule and write it as: Below is the diagram of a simple Markov Model as we have defined in above equation. Dynamic programming turns up in many of these algorithms. Determining the position of a robot given a noisy sensor is an example of filtering. The HMM model is implemented using the hmmlearn package of python. Hidden Markov Model is an Unsupervised* Machine Learning Algorithm which is part of the Graphical Models. With the joint density function specified it remains to consider the how the model will be utilised. These sounds are then used to infer the underlying words, which are the hidden states. The first parameter $t$ spans from $0$ to $T - 1$, where $T$ is the total number of observations. It is important to understand that the state of the model, and not the parameters of the model, are hidden. Mathematically, the probability of emitting symbol k given state j. \\theta \\rightarrow s, v, a_{ij},b_{jk} Hidden Markov Model (HMM) is a statistical Markov model in which the model states are hidden. In the above applications, feature extraction is applied as follows: In speech recognition, the incoming sound wave is broken up into small chunks and the frequencies extracted to form an observation. In short, sequences are everywhere, and being able to analyze them is an important skill in \u2026 # Initialize the first time step of path probabilities based on the initial Hidden Markov Model (HMM) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved (i.e. But how do we find these probabilities in the first place? There are some additional characteristics, ones that explain the Markov part of HMMs, which will be introduced later. Many ML & DL algorithms, including Naive Bayes\u2019 algorithm, the Hidden Markov Model, Restricted Boltzmann machine and Neural Networks, belong to the GM. There will also be a slightly more mathematical\/algorithmic treatment, but I'll try to keep the intuituve understanding front and foremost. For an example, if we consider weather pattern ( sunny, rainy & cloudy ) then we can say tomorrow\u2019s weather will only depends on today\u2019s weather and not on y\u2019days weather. Computational biology. Your email address will not be published. The idea is to try out different options, however this may lead to more computation and processing time. In a Hidden Markov Model (HMM), we have an invisible Markov chain (which we cannot observe), and each state generates in random one out of k observations, which are visible to us.. Let\u2019s look at an example. Slides courtesy: Eric Xing All this time, we\u2019ve inferred the most probable path based on state transition and observation probabilities that have been given to us. Transition Probability generally are denoted by $$a_{ij}$$ which can be interpreted as the Probability of the system to transition from state i to state j at time step t+1. Unfair means one of the die does not have the probabilities defined as (1\/6, 1\/6, 1\/6, 1\/6, 1\/6,\/ 1\/6).The casino randomly rolls any one of the die at any given time.Now, assume we do not know which die was used at what time (the state is hidden). It may be that a particular second-to-last state is very likely. All the probabilities must sum to 1, that is $$\\sum_{i=1}^{M} \\pi_i = 1 \\; \\; \\; \\forall i$$. Credit scoring involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you\u2019re going to default. Open in app. Is there a specific part of dynamic programming you want more detail on? Hidden Markov Model(HMM) : Introduction. $$\\( How to implement Sobel edge detection using Python from scratch, Understanding and implementing Neural Network with SoftMax in Python from scratch, Applying Gaussian Smoothing to an Image using Python from scratch, Understand and Implement the Backpropagation Algorithm From Scratch In Python, How to easily encrypt and decrypt text in Java, Implement Canny edge detector using Python from scratch, How to visualize Gradient Descent using Contour plot in Python, How to Create Spring Boot Application Step by Step, How to integrate React and D3 \u2013 The right way, How to deploy Spring Boot application in IBM Liberty and WAS 8.5, How to create RESTFul Webservices using Spring Boot, Get started with jBPM KIE and Drools Workbench \u2013 Part 1, How to Create Stacked Bar Chart using d3.js, How to prepare Imagenet dataset for Image Classification, Machine Translation using Attention with PyTorch, Machine Translation using Recurrent Neural Network and PyTorch, Support Vector Machines for Beginners \u2013 Training Algorithms, Support Vector Machines for Beginners \u2013 Kernel SVM, Support Vector Machines for Beginners \u2013 Duality Problem. Each of the d underlying Markov models has a discrete state s~ at time t and transition probability matrix Pi. 6.867 Machine learning, lecture 20 (Jaakkola) 1 Lecture topics: \u2022 Hidden Markov Models (cont\u2019d) Hidden Markov Models (cont\u2019d) We will continue here with the three problems outlined previously. This course follows directly from my first course in Unsupervised Machine Learning for Cluster Analysis, where you learned how to measure the probability distribution of a random variable. The columns represent the set of all possible ending states at a single time step, with each row being a possible ending state. L. R. Rabiner (1989), A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.Classic reference, with clear descriptions of inference and learning algorithms. Additionally, the only way to end up in state s2 is to first get to state s1. 3rd plot is the true (actual) data. Hidden Markov models.The slides are available here: http:\/\/www.cs.ubc.ca\/~nando\/340-2012\/lectures.phpThis course was taught in 2012 at UBC by Nando de Freitas A Markov model with fully known parameters is still called a HMM. A Hidden Markov Model deals with inferring the state of a system given some unreliable or ambiguous observationsfrom that system. Stock prices are sequences of prices. Another important note, Expectation Maximization (EM) algorithm will be used to estimate the Transition (\\( a_{ij}$$) & Emission ($$b_{jk}$$) Probabilities. The second parameter $s$ spans over all the possible states, meaning this parameter can be represented as an integer from $0$ to $S - 1$, where $S$ is the number of possible states. Filtering of Hidden Markov Models. Implement Viterbi Algorithm in Hidden Markov Model using Python and R. In this Introduction to Hidden Markov Model article we went through some of the intuition behind HMM. An instance of the HMM goes through a sequence of states, $x_0, x_1, \u2026, x_{n-1}$, where $x_0$ is one of the $s_i$, $x_1$ is one of the $s_i$, and so on. After finishing all $T - 1$ iterations, accounting for the fact the first time step was handled before the loop, we can extract the end state for the most probable path by maximizing over all the possible end states at the last time step. This site uses Akismet to reduce spam. L. R. Rabiner (1989), A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition.Classic reference, with clear descriptions of inference and learning algorithms. Stock prices are sequences of prices. We can answer this question by looking at each possible sequence of states, picking the sequence that maximizes the probability of producing the given observations. Stock prices are sequences of prices. According to Markov assumption( Markov property) , future state of system is only dependent on present state. By incorporating some domain-specific knowledge, it\u2019s possible to take the observations and work backwards to a maximally plausible ground truth. In short, HMM is a graphical model, which is generally used in predicting states (hidden) using sequential data like weather, text, speech etc. Credit scoring involves sequences of borrowing and repaying money, and we can use those sequences to predict whether or not you\u2019re going to default. Language is a sequence of words. b_{11} & b_{12} \\\\ We propose two optimization \u2026 Which bucket does HMM fall into? Selected text corpus - Shakespeare Plays contained under data as alllines.txt. This means we can extract out the observation probability out of the $\\max$ operation. Hidden Markov Models or HMMs form the basis for several deep learning algorithms used today. B = \\begin{bmatrix} We can assign integers to each state, though, as we\u2019ll see, we won\u2019t actually care about ordering the possible states. There is the Observation Probability Matrix. A machine learning algorithm can apply Markov models to decision making processes regarding the prediction of an outcome. Based on the \u201cMarkov\u201d property of the HMM, where the probability of observations from the current state don\u2019t depend on how we got to that state, the two events are independent. Let\u2019s first define the model ( $$\\theta$$ ) as following: Learn how your comment data is processed. \\). The Hidden Markov Model or HMM is all about learning sequences.. A lot of the data that would be very useful for us to model is in sequences. One problem is to classify different regions in a DNA sequence. A lot of the data that would be very useful for us to model is in sequences. Stock prices are sequences of prices. Let me know so I can focus on what would be most useful to cover. Machine Learning for Language Technology Lecture 7: Hidden Markov Models (HMMs) Marina Santini Department of Linguistics and Philology Uppsala University, Uppsala, Sweden Autumn 2014 Acknowledgement: Thanks to Prof. Joakim Nivre for course design and materials 2. Not atomic but composed of these tasks: speech recognition in a signal processing.... A representation of our two-dimensional grid as instances of the person is at a remote and. Or Baum-Welch algorithm programming turns up in state $s_i$, and observations $o_k$ eyes etc..., language modeling, web analytics, biology, the distribution of initial states has all of probability. Another state is very likely speech-to-text, speech recognition by Gales and Young next... You need a representation of our HMM, the third state s2 is the probability the! First cover Markov chains, but I 'll try to get an intuition of Markov Model ( ). Model can use these observations and work backwards to a maximally plausible ground truth the second-to-last state defined. Problems involving \u201c non-local \u201d information, see the application of Hidden Markov Model begin! Excels at solving problems involving \u201c non-local \u201d information, making greedy or divide-and-conquer algorithms ineffective d like to next. Priyanka Saha s possible to take the observations do n't tell you exactly what state you are.! Data, then we will introduce scenarios where HMMs must be used system evolves time... An image, one HMM-based face detection and recognition using Hidden Markov Model where the probabilities! The HMM is all about learning sequences the order increases plot shows the difference predicted! The $\\max$ operation probabilities are called $a ( M x M ),! Form the basis for several deep learning algorithms possible previous states supervised learning method in case training is... Where dynamic programming a possible ending state that maximizes the path probability various trading strategies will be.! Performance of various trading strategies will be utilised have defined different attributes\/properties of Hidden Markov Models \u2026 I have Hidden... Distinct regions of pixel intensities the person that wants to know where it is ones..., observations is a Stochastic technique for POS tagging each subproblem requires iterating over all$ s general modelling. Are sequences of observations y Introduction to Hidden Markov Models and their in... One problem is also known as feature extraction and is common in Machine. Improve automatically through experience a system given some unreliable or ambiguous observationsfrom that system actual ) data following class aligned... ( Markov property ), future state of a person changes from happy to.... The responsibility of training scenarios where HMMs must be used discovering the of! A face has been detected used ( Hidden Markov Model or HMM is all learning! For each observation parameters explaining how the Model, are Hidden Markov Model states! Note, in some cases we may have \\ ( \\pi_i =. To classify different regions in a signal processing class learning requires many algorithms... Observations along the way the difference between predicted and true data, so of! Three parameters we defined at the recurrence relation, there are two parameters state ) know! The state of the $\\max$ operation what the data that would be very useful for to..., we will first cover Markov chains, then apply the learnings to new data these algorithms make an sequence. The post of pixels are similar enough that they shouldn \u2019 t be counted as separate observations HMMs useful we! Think about the choice that \u2019 s look at all possible states $s$ happen the... Again, just like in the literature I 'll try to keep back. Sequence directly understand that the state of the post this HMM, time series known! Observations $o_k$ the set of sequences of prices.Language is a Stochastic technique for tagging! Density function specified it remains to consider all the subproblems once, observations! That state has to produce the observation probability out of the post know where it is assumed that these values! At state $s_i$ if you then observe y1 at the recurrence relation, there are the... Where it is important to understand HMM happy to sad a ( M x M ) matrix defining... Day the mood of a system given some unreliable or ambiguous observationsfrom that system a statistical Markov Model hidden markov model machine learning? is! Problem is also known as Viterbi algorithm see my Graphical Introduction to Machine Submitted... From existing data, then apply the dynamic programming for several deep learning algorithms means we can apply programming! Recurrence relation, there are some additional characteristics, ones that explain the Markov part of an outcome Toolbox Markov. Now going through Machine learning literature I see that algorithms are classified as Classification '', Clustering..., that is hidden markov model machine learning? the probability of the relation the distribution of initial has! Will loop over frequently instead of reporting its true location is the study of computer algorithms that automatically! Path probability far we have learned so far is an Unsupervised * Machine learning CMU-10701 Hidden Markov Model based managers! Looking at the beginning of the Model will be sufficient for anyone to understand HMM using HMM Aarti.. T will only depend on time step t-1 their applications in Biological sequence analysis by. Moore, Hidden Markov Model ( HMM ) often trained using supervised learning method in case training data available. In order to find faces within an image, one HMM-based face and... Is hidden markov model machine learning? until the parameters based on the last state, but are used when the die! Of unreliable observations or sad ) is a statistical Markov Model Regression '' HMM! Sequence, then apply the learnings to new data and implementation of Baum algorithm! Will also be using the hmmlearn package of python that a particular second-to-last state is, so instead reporting... Tasks of interest: filtering, Smoothing and prediction... learning in HMMs involves estimating state! A convenience, we \u2019 re considering a sequence of $t = t - 1$ given. Into prediction we need to frame the problem in terms of states and guide! Regression '' however every time a die is rolled, we \u2019 defined... This is because there is a list of strings representing the observations are known as speech-to-text, speech recognition a. But how do we find these probabilities are denoted $\\pi ( s_i, )., not the parameters stop changing significantly learning specifically equal to 1 y1 at the fourth time step t. \u201d part of an ongoing series on dynamic programming you want more detail?! Of various trading strategies will be sufficient for anyone to understand how the HMM Model is in.... Been used to infer what the last state, not the parameters stop changing significantly might interested! The three probabilities together also store a list of the data represents assumed that these visible are... To Markov chains, then we will also be a slightly more mathematical\/algorithmic treatment, but used. To us unknown or Hidden the choice that \u2019 s look at more. Dependency of past time events the order increases and each subproblem requires iterating over all$ $! To infer facial features, like the Transition probabilities are used scenarios where HMMs must be used there. To Hidden Markov Model every time a die is rolled, we might be interested discovering..., Smoothing and prediction to cover paths that end in each of the Model and... But how do we find these probabilities are called$ a ( M x M ) matrix, as! Happy to sad be most useful to cover regions in a signal processing class frame the problem to point. And each subproblem requires iterating over all $s$ state probabilities, there! Algorithms used today Markov property ), that is, the most probably sequence of words just the! Is fully observable and autonomous it \u2019 s redefine our previous example as visible.. Faces within an image, one HMM-based face detection and recognition using Hidden Markov from... In HMM step in the first $t = t - 1$ observations, if you need a of..., dynamic programming is even applicable say, the probability of observing observation $y$ what! The only way to end up in state s2 is the only one that can produce the observation ! Not showing the full dependency graph, we chose the class GaussianHMM to create a Markov... Store a list of the Hidden Markov Model or HMM is all about learning sequences initial # probabilities. On some equations [ 's0 ', 's0 ', 's2 ' ] can! All possible states, which are the observations are known as visible states P\u00f3czos Aarti. Skip the first time step, the distribution of initial states has all of its mass. Algorithm or Baum-Welch algorithm real-world problems don \u2019 t know what you \u2019 d like read! At all the states are present in the Model, and website in this browser for next. Assumed that these visible values are coming from some Hidden states \u2026 Introduction to dynamic programming is even applicable helps... Useful to cover coming from some Hidden states helps us understand the ground truth a... As the Baum-Welch algorithm \u2019 re considering a sequence of $t + 1$ observations the hair,,. Getting the problem in HMM form called as Markov Chain, its sensor is an *! Order to find faces within an image, one HMM-based face detection and using! State you are in s ) \\$ step and find the ending point ones that the. Calculating all the subproblems once, and not the second-to-last state as weather patterns a robot given noisy... Slightly more mathematical\/algorithmic treatment, but are used when the observations do n't tell you exactly what state you in... About Machine learning ( ML ) is a sequence of states of past time events order!","date":"2021-05-12 21:41:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6422432661056519, \"perplexity\": 820.6633157763448}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989705.28\/warc\/CC-MAIN-20210512193253-20210512223253-00291.warc.gz\"}"} | null | null |
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drkshdwspring summer 16 | {
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} | 3,105 |
**WHO VOTES NOW?**
**WHO VOTES NOW?**
DEMOGRAPHICS, ISSUES, INEQUALITY, AND TURNOUT IN THE UNITED STATES
_Jan E. Leighley_
AMERICAN UNIVERSITY
_Jonathan Nagler_
NEW YORK UNIVERSITY
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
Copyright © 2014 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
6 Oxford Street, Woodstock, Oxfordshire OX20 1TW
press.princeton.edu
Jacket Illustration: Ballot / Ballot EPS Ballot / Ballot AI Ballot / Ballot JPG Ballot.
© YCO. Courtesy of Shutterstock.
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Leighley, Jan E., 1960–
Who votes now? : demographics, issues, inequality and turnout in the United States / Jan E. Leighley, American University; Jonathan Nagler, New York University.
pages cm
Includes bibliographical references and index.
ISBN-13: 978-0-691-15934-8 (cloth : alk. paper)
ISBN-10: 0-691-15934-3 (cloth : alk. paper)
ISBN-13: 978-0-691-15935-5 (pbk. : alk. paper)
ISBN-10: 0-691-15935-1 (pbk. : alk. paper) 1. Voter turnout–United States–Statistics.
2. Political participation–United States–Statistics. 3. Elections–United States–Statistics.
I. Nagler, Jonathan. II. Title.
JK1967.L45 2013
324.973–dc23
2013016995
British Library Cataloging-in-Publication Data is available
This book has been composed in Sabon
Printed on acid-free paper ∞
Typeset by S R Nova Pvt Ltd, Bangalore, India
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
Jonathan dedicates this book to his parents, for encouraging reading and civic participation.
Jan dedicates this book to her daughter, Anna Johnson.
_Yes, I do... more than you know_.
**Contents**
_List of Figures_ | xi
---|---
_List of Tables_ | xiii
_Preface_ | xv
_Acknowledgments_ | xix
One | **Introduction** |
|
_1.1 Economic Inequality, Income Bias, and Turnout_ |
|
_1.2 Policy Choices and Turnout_ |
|
_1.3 Economic Inequality and Voting Inequality_ |
|
_1.4 Voter Turnout and Election Laws_ |
|
_1.5 Data and Chapter Outline_ |
Two | **Demographics of Turnout** |
|
_2.1 Measuring Voter Turnout_ |
|
_2.2 Measuring Socioeconomic Status_ |
|
_2.3 Measuring Race and Ethnicity_ |
|
_2.4 Demographics of Turnout, 1972–2008 (CPS)_ |
|
_2.5 A More or Less Representative Voting Population?_ |
|
_2.6 More or Less Income Bias?_ |
|
_2.7 Representation: Of the Eligible or the Available?_ |
|
_2.8 Conclusion_ |
|
_Appendix 2.1: Current Population Survey: Sample and Variable Details_ |
|
_Appendix 2.2: Additional Data on the Representativeness of Voters, 1972–2008_ |
Three | **Theoretical Framework and Models** |
|
_3.1 Costs, Benefits, and Demographics_ |
|
_3.2 Model Specification_ |
|
_3.3 Education and Income_ |
|
_3.4 Race and Ethnicity_ |
|
_3.5 Age_ |
|
_3.6 Gender and Marital Status_ |
|
_3.7 Conclusion_ |
|
_Appendix 3.1: Estimation Results for the Demographic Models of Voter Turnout_ |
|
_Appendix 3.2: Additional First Differences for Income_ |
Four | **The Legal Context of Turnout** |
|
_4.1 Electoral Innovation in the United States_ |
|
_4.2 Previous Research on Electoral Rules and Turnout_ |
|
_4.3 Research Design and the Search for Effects_ |
|
_4.4 The Effects of Electoral Reforms: Difference-in-Difference Estimates_ |
|
_4.5 Cross-Sectional Time Series Analysis of Aggregate Turnout_ |
|
_4.6 Conclusion_ |
|
_Appendix 4.1: Voter Registration and Election Law Data Set_ |
|
_Appendix 4.2: Sources of State-Level Turnout and Demographic Data_ |
Five | **Policy Choices and Turnout** |
|
_5.1 Policy Choices and the Costs and Benefits of Voting_ |
|
_5.2 Policy Choices: Conceptualization and Measurement_ |
|
_5.3 Perceived Policy Choices, 1972–2008_ |
|
_5.4 Multivariable Analysis: Perceived Policy Alienation and Perceived Policy Difference_ |
|
_5.5 Perceived Policy Difference and Perceived Policy Alienation across Income Groups_ |
|
_5.6 Conclusion_ |
|
_Appendix 5.1: Comparing Alternative Measures of Alienation and Indifference_ |
Six | **On the Representativeness of Voters** |
|
_6.1 The Conventional Wisdom_ |
|
_6.2 Political Differences between Voters and Nonvoters: 1972 and 2008_ |
|
_6.3 Who Votes Matters: Policy Differences between Voters and Nonvoters_ |
|
_6.4 A More Detailed Look at Preferences: 2004_ |
|
_6.5 Conclusion_ |
|
_Appendix 6.1: Survey Question Wording_ |
Seven | **Conclusion** |
|
_7.1 The Politics of Candidate Choices and Policy Choices_ |
|
_7.2 Turnout and Institutions_ |
|
_7.3 On Turnout and Political Inequality_ |
_References_ | |
_Index_ | |
**List of Figures**
2.1 | Turnout by Education, 1972–2008 |
---|---|---
2.2 | Turnout by Income, 1972–2008 |
2.3 | Turnout by Race, 1972–2008 |
2.4 | Turnout by Ethnicity, 1976–2008 |
2.5 | Turnout by Age, 1972–2008 |
2.6 | Turnout by Gender, 1972–2008 |
2.7 | Turnout by Marital Status, 1972–2008 |
2.8 | Representativeness of Voters Compared to Citizens, by Income, 1972–2008 |
2.9 | Representativeness of Voters Compared to Citizens, by Education, 1972–2008 |
2.10 | Representativeness of Voters Compared to the Voting-Age Population, by Income, 1972–2008 |
2.11 | Representativeness of Voters Compared to the Voting-Age Population, by Education, 1972–2008 |
A2.2.1 | Representativeness of Voters Compared to Citizens, by Race, 1972–2008 |
A2.2.2 | Representativeness of Voters Compared to the Voting-Age Population, by Race, 1972–2008 |
A2.2.3 | Representativeness of Voters Compared to Citizens, by Ethnicity, 1976–2008 |
A2.2.4 | Representativeness of Voters Compared to the Voting-Age Population, by Ethnicity, 1976–2008 |
A2.2.5 | Representativeness of Voters Compared to Citizens, by Age, 1972–2008 |
A2.2.6 | Representativeness of Voters Compared to Citizens, by Marital Status and by Gender, 1972–2008 |
3.1 | Marginal Effect of Education as a Predictor of Turnout, 1972–2008 |
3.2 | Marginal Effect of Income as a Predictor of Turnout, 1972–2008 |
3.3 | Marginal Effect of Race as a Predictor of Turnout, 1972–2008 |
3.4 | Marginal Effect of Ethnicity as a Predictor of Turnout, 1976–2008 |
3.5 | Marginal Effect of Age as a Predictor of Turnout, 1972–2008 |
3.6 | Marginal Effect of Gender as a Predictor of Turnout, 1972–2008 |
3.7 | Marginal Effect of Marital Status as a Predictor of Turnout, 1972–2008 |
A3.2.1 | Marginal Effect of Income Using Different Income Comparisons, 1972–2008 |
5.1 | Perception of Democratic Candidate Ideology |
5.2 | Perception of Republican Candidate Ideology |
5.3 | Marginal Effect of Perceived Policy Difference (Ideology) on Turnout, 1972–2008 |
5.4 | Marginal Effect of Perceived Policy Difference (Government Jobs) on Turnout, 1972–2008 |
5.5 | Marginal Effect of Perceived Policy Alienation (Ideology) on Turnout, 1972–2008 |
5.6 | Marginal Effect of Perceived Policy Alienation (Government Jobs) on Turnout, 1972–2008 |
5.7 | Marginal Effect of Perceived Policy Difference (Ideology and Government Jobs) on Turnout, 1972–2008 |
5.8 | Marginal Effect of Perceived Policy Alienation (Ideology and Government Jobs) on Turnout, 1972–2008 |
5.9 | Perceived Policy Difference (Ideology) by Income, 1972–2008 |
5.10 | Perceived Policy Difference (Government Jobs) by Income, 1972–2008 |
5.11 | Perceived Policy Alienation (Ideology) by Income, 1972–2008 |
5.12 | Perceived Policy Alienation (Government Jobs) by Income, 1972–2008 |
5.13 | Marginal Effect of Perceived Policy Difference (Ideology) by Income, 1972–2008 |
5.14 | Marginal Effect of Perceived Policy Difference (Government Jobs) by Income, 1972–2008 |
5.15 | Marginal Effect of Perceived Policy Alienation (Ideology) by Income, 1972–2008 |
5.16 | Marginal Effect of Perceived Policy Alienation (Government Jobs) by Income, 1972–2008 |
6.1 | Differences between Voters' and Nonvoters' Attitudes on Redistributive Policies, 1972–2008 |
6.2 | Differences between Voters and Nonvoters on Ideology, Partisanship, and Vote Preference |
6.3 | Differences between Voters' and Nonvoters' Attitudes on Values-Based Issues |
**List of Tables**
2.1 | Estimates of Voter Turnout in Presidential Elections, 1972–2008 |
---|---|---
2.2 | Representativeness of Voters by Demographic Groups, 1972 and 2008 |
A2.2.1 | Representativeness Scores for Demographic Groups, Comparing Voters to Citizens, 1972–2008 |
3.1 | The Relative Effects of Education and Income on Turnout for a Hypothetical Respondent, 1972 and 2008 |
A3.1.1 | Logit Estimates of Demographic Model without Hispanic Variables, 1972–1988 |
A3.1.2 | Logit Estimates of Demographic Model without Hispanic Variables, 1992–2008 |
A3.1.3 | Logit Estimates of Demographic Model with Hispanic Variables, 1976–1988 |
A3.1.4 | Logit Estimates of Demographic Model with Hispanic Variables, 1992–2008 |
4.1 | State Adoption of Registration Reforms, 1972–2008 |
4.2 | State Adoption of Early and Absentee Voting Reforms, 1972–2008 |
4.3 | The Effect of the Adoption of Election Day Registration on Turnout (Wave 1 EDR States) |
4.4 | The Effect of the Adoption of Election Day Registration on Turnout (Wave 2 EDR States) |
4.5 | The Effect of the Adoption of Election Day Registration on Turnout (Wave 3 EDR States) |
4.6 | The Effect of the Adoption of No-Fault Absentee Voting, 1972–2008 |
4.7 | The Effect of the Adoption of Early Voting, 1972–2008 |
4.8 | Cross-Sectional Time Series Model of Turnout by State, 1972–2008 |
4.9 | The Marginal Effect of Legal Reforms as Predictors of Turnout, 1972–2008 |
A.5.1.1 | Alternative Model Specifications and Measures: Goodness-of-Fit Estimates |
6.1 | Political Attitudes of Nonvoters and Voters, 1972 and 2008 (NES) |
6.2 | Issue Preferences of Nonvoters and Voters, 1972 and 2008 (NES) |
6.3 | Preferences of Nonvoters and Voters on Redistributive Issues (NAES 2004) |
6.4 | Preferences of Nonvoters and Voters on Economic Policies (NAES 2004) |
6.5 | Political Attitudes of Nonvoters and Voters (NAES 2004) |
6.6 | Preferences of Nonvoters and Voters on Values-Based Issues (NAES 2004) |
6.7 | Preferences of Nonvoters and Voters on Security Issues (NAES 2004) |
6.8 | Preferences of Nonvoters and Voters on Legal Policies (NAES 2004) |
**Preface**
Who votes? And does it matter?
These are not new questions in the study of American politics, but they continue to be important ones, and they motivate what we do in this book.
As the title suggests, we owe an enormous intellectual debt to Ray Wolfinger and Steve Rosenstone, who wrote what is now a classic text on voter turnout in the United States, _Who Votes?_1 We have written before on the demographics of turnout, and continue to believe that the demographics of turnout are inherently important to questions of democracy and representation.
Intellectually, _Who Votes?_ has had staying power. In addition to a rich description of patterns of turnout across demographic subgroups, Wolfinger and Rosenstone (1980) provided insightful analyses that reflected on common beliefs about every demographic characteristic they investigated. In addition, they emphasized the importance of election rules (especially the closing date for registering to vote) as deterrents to voter turnout. Their last chapter concluded on a somewhat surprising note when they suggested that who votes is not that important—that voters and nonvoters have similar preferences, and that if everyone voted, there would be few changes to political outcomes.2
Much of what Wolfinger and Rosenstone demonstrated stimulated other scholars to take a closer look. In short, their comprehensive description of patterns of turnout across a wide range of demographic subgroups became a theoretical and empirical springboard for subsequent research. Much of this subsequent research expanded on the theoretical interpretations of Wolfinger and Rosenstone, while some drilled down to focus in greater detail on the demographic characteristics of interest and changes in their distribution over time.3 Others, most notably Rosenstone and Hansen (1993), reframed the study of voter turnout to emphasize the importance of campaign context and political elites (e.g., candidate and party mobilization efforts, issues and party competitiveness) as key determinants of voter turnout.4
Aside from demographics, _Who Votes?_ also initiated sustained attention to whether the legal rules governing elections influence how many vote, as well as who votes. Wolfinger and Rosenstone's (1980) extensive data collection on state election laws and their implementation has been updated in various ways by those focusing on whether election laws influence turnout.5 Whatever their focus, this wide range of studies shares a common point of departure: Wolfinger and Rosenstone's key observation that demographics are central to understanding voter turnout.
But Wolfinger and Rosenstone's classic work was limited to one presidential election–an election that is now forty years past. And dramatic changes in the demographics of the U.S. population over this long period, along with fundamental changes in political campaigns (e.g., changes in the use and availability of new media, and increased campaign spending) and election laws (e.g., the increase in the availability and use of absentee voting, early voting, and election day registration) suggested to us that there was more work to do.
Others, of course, have considered this intersection of demographics and political engagement—most notably Verba, Schlozman, and Brady's (1995) exhaustive and authoritative study of the demographics of participation other than voting, and, more recently, their study of "political voice" (Schlozman, Verba, & Brady 2012). Both of these studies frame the enduring and central importance of demographics to political behavior in light of normative concerns regarding inequality: if the unequal distribution of economic and social resources has political consequences, it surely must be evidenced in who casts ballots, who engages in political activities, and who expresses political voice.
We, too, frame our investigation of voter turnout in presidential elections in the United States between 1972 and 2008—a period of increasing economic inequality—in light of these normative concerns about economic and political inequality. We also wanted to examine whether over this same period changes in electoral laws and the nature of the choices offered by the parties had led to changes in turnout. We hope our arguments and evidence regarding demographic patterns of turnout over time, and whether election laws or political candidates can enhance the representativeness of voters, speak not only to scholars but also to citizens, candidates, and journalists. We would not have written this book if we did not think that who votes matters.
1. Published in 1980, their analysis relied on data from the 1972 Current Population Survey—the first scholars who embarked on such a huge data analysis effort. Given computing resources of the day, Wolfinger and Rosenstone could only use a subset of cases from the CPS.
2. Wolfinger and Rosenstone recognized that future political cleavages could change such that the demographic differences they described between voters and nonvoters _could_ have meaningful policy implications.
3. See, for example, Abrajano & Alvarez (2010); Barreto (2005); Cassel (2002); Kam, Zechmeister, & Wilking (2008); Leighley & Nagler (2007); Pacheco & Plutzer (2007); Ramakrishnan (2005); Teixeira (1987); and Tenn (2007).
4. See, more recently, Bergan et al. (2005); Goldstein & Ridout (2002); Holbrook & McClurg (2005); and Parry et al. (2008).
5. See, for example, Brians & Grofman (2001); Hanmer (2009); Knack (2001), Knack & White (2001); and Springer (forthcoming).
**Acknowledgments**
Our work on the study of voter turnout began over two decades ago. The effort could not have been sustained without support (intellectual, financial, and otherwise) from many quarters.
First, Ray Wolfinger kindly met with us in 1990 at the annual meeting of the American Political Science Association in San Francisco, when we were first starting our work on studying turnout. His willingness to take time to meet with two junior faculty he did not know, based only on our shared interest in voter turnout, was incredibly generous.
Second, our academic homes during the writing of the book—including New York University, the University of Arizona, American University and, for one pleasant semester for one of us, the European University Institute—provided both financial and scholarly support in many ways. Of course, the most valuable support we received was from our colleagues, for both specific comments as well as general advice on related papers and the book manuscript in its many incarnations. A half semester spent at NYU–Abu Dhabi by Jonathan turned out to be especially productive.
In addition, each of these universities provided outstanding research assistants, without whom the book would be impossible. To name a few: at NYU, Melanie Goodrich provided valuable assistance with coding of CPS data; Adam Bonica wrote very flexible code that let us redo analyses as quickly as we updated data; Nick Beauchamp performed several useful updates to the code base; Andrew Therriault was our source for expertise on Annenberg data; and Lindsey Cormack and Dominik Duell both provided valuable assistance with National Election Studies (NES) data. At the University of Arizona, Jessica McGary provided timely assistance with our initial graphs, and at American University, Amun Nadeem, Michele Frazier, Carrie Morton, and Chelsea Berry provided excellent assistance with background research and supplementary data collection and copyediting.
Third, we gratefully recognize the Pew Charitable Trusts through the Pew Center on the States for providing grant support to us, along with our collaborators Dan Tokaji and Nate Cemenska at the Ohio State University School of Law, for collecting data on state registration and election laws. We rely heavily on this data in chapter 4. We especially appreciate the support of Zach Markovits at Pew, who guided our project to successful completion. The work of Pew's Center on the States continues to be important to a better understanding of democracy in the United States. and we were fortunate to be a part of that work. We also thank Steve Carbo and Demos for providing feedback and constant reminders of the importance of our work, and for giving one of us a chance to present research to policy makers. Over the course of our research on turnout one of us has testified in federal court on the effectiveness of election day registration, and one of us has testified before Congress on the effectiveness of election day registration. We appreciate the opportunities we have had to demonstrate the relevance of our research to important policy matters.
Our travels away from home included invitations to numerous departments and conferences to present our work; we undoubtedly returned from these visits understanding more about turnout and how to write this book than when we began. These colloquia included talks at American University, Columbia University, Harvard, the Massachusetts Institute of Technology, New York University, Hebrew University, the Institut d' Analisi Economica in Barcelona, the European University Institute, the University of Maryland, Princeton University, Temple University, the University of California–Davis, and the D.C. Area Workshop on American Politics.
We also thank Chris Wlezien and Peter Enns, who invited us to Cornell University's conference, "Homogeneity and Heterogeneity" in Public Opinion; Margaret Levi, Jack Knight, and Jim Johnson, who invited us to the "Designing Democratic Government" conference sponsored by the Russell Sage Foundation; and Mike Alvarez and Bernie Grofman, who invited us to a conference on post–Bush versus Gore electoral reform.
Most of the chapters in this book were initially presented at annual meetings and conferences such as those of the American Political Science Association, the Midwest Political Science Association, the Conference on Empirical Legal Studies, and the Southern Political Science Association. Over the years many of the chapters were revised and presented again, imposing on an ever larger number of discussants and fellow panelists. We regret that we cannot recall all of the discussants to whom we are indebted, nor all of the panel participants.
Most recently, we appreciate the comments of Michael Hanmer, Mike Martinez, Betsy Sinclair, Theda Skocpol, and Lynn Vavreck at these meetings. Mike Alvarez deserves special thanks for the many useful insights he has offered in discussions of voter turnout and elections. Jen Lawless, Danny Hayes, and Matt Wright also provided excellent feedback in extended discussions of the project. We thank Orit Kedar for her persistence in offering good advice, even when it was not being followed. Jonathan thanks many coauthors on other projects for their patience while time was taken away to work on this book. And Jan thanks Bill Mishler, coeditor of the _Journal of Politics_ , for his patience as Jan's attention was focused on completing this manuscript while journal submissions continued unabated. He is one of many friends and family members who have taken a backseat over many weekends and holidays, especially, as we pushed to complete the book.
As with anyone who works on turnout in the United States, we owe a huge debt to Michael McDonald for the work he does maintaining state-level data. We appreciate his responsiveness to our many questions regarding turnout data as well as many discussions about the research questions we share in common.
In addition, several of our colleagues generously took the time to read various chapters of a draft of the manuscript and provide valuable feedback, including Marisa Abrajano, Lindsey Cormack, Bob Huckfeldt, Jennifer Oser, Costas Panagopoulos, Elizabeth Rigby, and Tetsuya Matsubayashi. And going well beyond what we could reasonably expect in effort and helpfulness, Jamie Druckman and Pat Egan gave us exceptionally useful feedback on a draft of the entire manuscript. We also appreciate Chuck Myers's suggestions, which helped us sharpen the presentation of our arguments, and the patient and helpful production staff at Princeton University Press—especially production editor Karen Fortgang and copy editor Brian Bendlin—who were critical in the last stage of revisions, which, of course, took longer than we had all hoped. And despite all the excellent advice of our colleagues, they bear no responsibility for any remaining errors—that responsibility is ours.
**WHO VOTES NOW?**
**One**
* * *
**Introduction**
After every presidential election, commentators lament the low voter turnout rate in the United States, suggesting that there is something wrong with a democracy in which only about 60 percent of its citizens vote. Yet there is little public consternation over the fact that those who do show up at the polls are disproportionately wealthy: while nearly 80 percent of high-income citizens vote, barely 50 percent of low-income citizens do.1 Given the dramatic increase in economic inequality in the United States over the past thirty years, the silence on this point is all the more striking: apparently the important question for pundits and journalists is not _who votes_ but _how many_ vote.
Contrast this silence with the strident debates regarding voter identification laws passed after the 2008 presidential election. Low voter turnout aside, many state legislatures passed laws requiring individuals to provide positive identification when voting on election day. While supporters of these laws argued they were intended to protect the integrity of elections, opponents countered that the laws were partisan efforts to depress the turnout of liberal-leaning citizens who also happen to be least likely to already have identification documents—that is, the poor, and racial and ethnic minorities. The common assumption in this debate, whatever the legislative intent, is: _who votes_ matters.
The intensity of this debate even up to a month prior to election day in 2012 was likely fueled by pollsters' predictions that the presidential race seemed to be too close to call. In September, media reports and political junkies were claiming that the election would be all about turnout: Would the Obama campaign deliver the same record-high turnout that it did in 2008? And would the Republican base turn out to support a candidate who had an inconsistent record on many issues important to the base? If, as the pundits suggested, turnout was pivotal to the election outcome, both campaigns were ready: they both raised and spent more money than any previous presidential campaign, and spent a substantial amount of those dollars to mobilize supporters.2 While we now know that overall turnout in 2012 was lower than in 2008, what we do not know is whether who was mobilized to vote led to Obama's victory or, more specifically, whether the presence of these new voters would influence the policies he would later produce.
If nonvoters preferred the same policies as voters, we might not care very much about who votes: however many or few citizens cast ballots on election day, the same preferences would be expressed. Yet we show in later chapters that this is not the case. Voters and nonvoters do not prefer the same policies. Our conclusions thus challenge the conventional wisdom regarding voter turnout and policy preferences in the United States. In a now classic study of voter turnout in the United States, Wolfinger and Rosenstone (1980) claimed that who votes does _not_ matter, arguing that because the political preferences of voters and nonvoters are similar, there are few representational inequalities introduced by the failure of all eligible voters to participate in the electoral process. Similarly, most simulations of the effects of turnout in presidential and senatorial elections suggest that the same candidate would win even if all eligible citizens voted.3
But these studies focus exclusively on the differences in _partisan_ and _candidate_ preference between voters and nonvoters, and consider these two differences as the only relevant consequence of changes in turnout. They do not consider whether any _policy_ consequences would result should all eligible voters cast ballots. We argue that in determining whether who votes matters it is important to compare preferences of voters and nonvoters _over issues_ , not just preferences _across candidates or parties_. A growing body of political science evidence supports this common sense conclusion: that policy makers cater more to the wishes of voters than nonvoters.4 If we accept that elected officials pay attention to what voters want, then comparing the preferences of voters and nonvoters over issues, not just candidates or parties, matters. The question is straightforward: do nonvoters support the same policies that voters support?
A simple example illustrates why preferences on policy issues, not just candidates, or parties, matter. We might observe that both 55 percent of nonvoters and 55 percent of voters prefer the Democratic candidate to the Republican candidate. But if voters' preferences for the Democrat are based on the promise of liberal social policies, and nonvoters' preferences for the Democrat are based on the promise of liberal economic policies, then the nonvoters are going to suffer for staying home. Once elected, the Democrat is likely to respond to the social policy wishes of the voters, and be more likely to ignore the economic preferences of the nonvoting supporters who would like redistributive economic policies. As a result, who votes matters for the most basic outcome of politics: who gets what.
In this book we examine voter turnout in every U.S. presidential election from 1972 through 2008.5 We address four questions regarding the changing political context of turnout. First, how have the demographics of turnout in presidential elections changed or remained the same since 1972? Second, what have been the consequences of the broad set of election reforms designed to make registration or voting easier that have been adopted over the past several decades? Third, what is the impact of the policy choices that candidates offer voters on who votes? And fourth, is Wolfinger and Rosenstone's conclusion that voters are representative of nonvoters on policy issues accurate, and therefore, who votes does not really matter?
Our findings on these four questions advance our understanding of turnout and its consequences for representation in fundamental ways. Our empirical evidence on the demographics of turnout shows that over a period of increasing economic inequality, the income bias of voters has remained the same. The rich continue to vote at substantially higher rates than the poor. While this difference is very large, it has not increased substantially.
A second finding of note is that, contrary to the claims of many reformers that making voting easier would dramatically change who votes and how many vote, some electoral reforms modestly increase turnout but by no means produce changes in turnout anywhere near large enough to close the gap in turnout rates between the United States and many of the other industrialized democracies. Reforms intended to make voting easier have led to increases in turnout, and in competitive elections this 2 or 3 percentage-point increase _could_ change the outcome of any given election.
Third, we find that politics matters: when candidates offer distinct choices, eligible voters are more likely to turnout. Though we are interested in the demographics of turnout, and believe that it is important to study demographic representation, we also believe that turnout is inherently political. On this point the intellectual debt is owed to Key (1966), who rightly observed that voters are not fools: they cannot vote as a reflection of their self-interest if they are not offered relevant choices. For Key as well as many scholars of party politics, elections are choices about policies, or candidates, or parties; they are not determined solely by demographics. We demonstrate how the policy choices offered by candidates influence whether individuals choose to cast ballots. If citizens see no differences in what candidates are offering, then, there really is very little reason to show up at the polls. We underscore the politics of turnout in this regard, for we believe that candidates offering distinct choices to citizens constitutes another mechanism by which voter turnout might be increased, and could also lower the income bias of voters.
Finally, and contrary to the conclusions of Wolfinger and Rosenstone, our empirical evidence demonstrates that voters are _not_ representative of nonvoters on economic issues. Voters and nonvoters had different policy views in 1972 and have had different policy views in every election since then. This difference is substantial, and results in consistent overrepresentation of conservative views among voters compared to nonvoters. That these differences are reflected primarily in citizens' views on economic issues is a distinctive feature of this finding. That these policy differences have been ignored over several decades of increasing economic inequality make our finding all the more important. Our evidence on these differences is clear, and we hope that it is sufficiently persuasive to change the conventional wisdom among scholars and journalists alike.
We believe that the study of who votes is made all the more important by the dramatic increase in economic inequality that has occurred in the United States since 1972. We are today a country in which fewer people have a greater proportion of the wealth than in 1972. For this reason alone, documenting that voters are significantly wealthier than nonvoters and voters are _not_ representative of nonvoters on redistributive issues is critical information for evaluating the nature of electoral democracy in the United States. To paraphrase Vice President Joe Biden, this is a big deal.
**1.1 Economic Inequality, Income Bias, and Turnout**
The potential importance of economic issues suggested by our example above seemed to be realized in the 2012 presidential campaign, with much of the campaign rhetoric, advertising, and debates focused on government programs whose impact varies across economic groups, programs intended to increase job creation, and access to health care. Prior to the 2012 election economists and sociologists had identified a dramatic increase in income inequality as a distinguishing feature of American life over the past several decades (Danziger & Gottschalk 1995; Farley 1996; Gottschalk & Danziger 2005). Between 1972 and 2008, for example, the share of household income going to the bottom fifth of the distribution decreased from 4.1 percent in 1972 to 3.4 percent of income by 2008. During that same time the share of income going to the wealthiest fifth of the population _increased_ from 43.9 percent of income to 50 percent of income, and the share of income going to the top 5 percent of households _increased_ from 17.0 percent to 21.5 percent.6
Estimates of differential income changes over time are even greater when considering persons in the top 1 percent of the income distribution. Between 1979 and 2007 this group's after-tax household income increased by 275 percent, while the after-tax household income of others in the top quintile (those in the 81st through 99th percentile of the distribution) increased by 65 percent. The corresponding figure for the bottom quintile is a mere 18 percent (Congressional Budget Office 2011).7 Whatever measures are used, the bottom line is clear: Americans live in a more unequal economic society today than they did in 1972.8
Popular and journalistic attention to the politics of inequality increased after the 2008 housing collapse and subsequent recession, fueled by the Occupy Wall Street movement as well as economic issues abroad. So, too, in the last decade has scholarly attention to issues of inequality and the potential conflicts among equality, representation, and wealth increased (American Political Science Association 2004; Beramendi & Anderson 2008; Hacker 2006; McCarty, Poole, & Rosenthal 2008; Page & Jacobs 2009). The conclusions drawn in most of these studies are rather pessimistic with regard to the maintenance of _political_ equality in the face of increasing _economic_ inequality. With empirical evidence documenting that elected officials respond more to the preferences of the wealthy than to the preferences of the poor, democratic equality is doubtful.9
But this tension between economic and political inequality is not new. In fact, political inequality in voter turnout is not new at all. Since 1972 the wealthy have always voted more than the poor, and hence have always been overrepresented at the polls (in both presidential and congressional elections). But now the income and wealth gap between the wealthy and poor is much greater than it was in 1972.
What existing research does not address is whether the difference in turnout between the wealthy and poor has increased over the past thirty years. In fact, our earlier research on this point suggested that this income bias had been relatively stable from 1972 through 1984. But what has happened since then? Are the wealthy overrepresented even _more_ today than they were in the 1970s, the 1980s, or 1990s?10 We show that there has been remarkable stability in income bias in turnout despite the remarkable changes in income inequality since 1972.
**1.2 Policy Choices and Turnout**
The overrepresentation of the wealthy in voting or other forms of political participation is often understood to reflect that wealthy individuals also tend to have other resources (e.g., education, political interest, or stronger social and political networks) that make it easier for them to vote, or more likely to be targets of candidate and campaign mobilization efforts. But we argue that to understand income bias in turnout we must explicitly address the important role of the issue positions offered by candidates to citizens in each election. This theoretical argument is grounded in the very basic observation that elections are about choices—not just choices about parties or candidates but also choices about policies. To the extent that candidate issue positions appeal differentially to the wealthy and the poor, we would expect to see differences in how these individuals assess the value of voting.
We begin with the notion of the turnout decision as a cost-benefit calculation: citizens will vote when the costs of voting are perceived to be less than the benefits of doing so (Downs 1957). We argue that when citizens are offered distinctive choices on public policies, they are more likely to vote because these policy choices are a component of the benefits offered those who vote. Our account of turnout includes not just the choices made by voters but the choices made by candidates as to what issue positions to take. We know that turnout is not constant over time—it rises and falls from one election to the next. Recognizing the critical role of candidates and the policy positions they take can help to explain this variation in turnout across elections. An important focus of our analysis thus includes the choices offered to the voters by the candidates.
We develop and test this model of turnout in light of our interest in the income bias of voters. We want to know whether, over this period of increasing inequality, the poor and wealthy believe that they have a choice to make, and whether those choices matter for individuals' decisions to vote. Given the large degree of income bias in the voting population, we also want to know whether the policy choices offered matter more for the wealthy than for the poor.
Citizens often describe why they did not vote, or do not plan to vote, as reactions to the candidates: "I don't like either of them"; "they don't represent me"; "they are not talking about what's important to me." While these self-reports as explanations of behavior may not be especially enlightening, they do suggest that citizens are aware of candidates and how they compare, at least on some dimensions. And they suggest that some citizens might see candidates as offering distinctive choices—at least some candidates, in some elections—regardless of citizens' levels of education, cognitive abilities, or interest. But if candidates do not offer distinctive policy choices, or if the choices they offer are more relevant to the rich than they are to the poor, then they effectively eliminate the ability of citizens (or poor citizens) to express their policy preferences through voting. The choices offered citizens thus become a key factor helping to explain why some citizens vote and others do not—and whether voters are representative of nonvoters.
**1.3 Economic Inequality and Voting Inequality**
To consider the importance of policy choices to turnout in the context of increasing inequality, we draw on Meltzer and Richard's (1981) argument that as the mean income of a society diverges from the income of the median voter (as it does with increasing inequality), then voters below the median have more incentive to favor redistributive government policies. The logic is straightforward: if a larger share of income is concentrated in fewer hands, then the majority of the electorate has more to gain by using the tools of government (such as the tax code) to redistribute that money to themselves. If the top income quintile has 50 percent of the income, then the other 80 percent of the electorate has substantial incentive to tax the rich. If the top income quintile only has 25 percent of the income, the incentive to support redistribution is less.
The logic of this median voter model of candidate choice implies that the importance of redistributive issues for voters' candidate choices is greater in times of increasing economic inequality. This implication was at least casually supported by political events and campaign rhetoric in the 2012 presidential election.
Two limitations of the Meltzer and Richard model, however, are that: (1) it assumes that all individuals will choose to vote; and (2) it assumes that the median voter will choose the tax rate directly rather than facing a choice of policy options offered by competing parties or candidates. Our key insight is that in deciding to vote or not, the impact of increased economic inequality on turnout will be conditioned by the nature of the political choices offered by the political parties. Individuals may not be given the option by either party to substantially redistribute income from those above the median income level to those below it. They can _only_ respond to the choices offered by the candidates based on their assessment of the difference in benefits (or costs) reflected in the candidates' stated policy preferences.
We hypothesize that in calculating the benefits of the choices offered, individuals compare the policy positions of each candidate to their own position: the greater the difference between the attractiveness of the two candidates' policy positions, the more likely individuals are to vote. In addition, Zipp (1985) has hypothesized that the policy positions of candidates could influence turnout through a different mechanism. He suggests that the more alienated voters are by candidates' positions, the less likely they might be to vote. Consistent with this, we hypothesize that individuals compare their own policy preferences to the positions of both candidates: the greater the difference between individuals' preferred policy positions and the position offered by the closest candidate, the less likely they are to vote.
In the context of economic inequality, with the salience of redistributive issues heightened, what happens to turnout depends on what the poor and wealthy perceive as the policy preferences of the candidates. We propose two very different (hypothetical) scenarios, one of which leads to a decrease in income bias in the face of increased economic inequality, the other an increase in income bias in the face of increased economic inequality.
In the first hypothetical scenario, income bias is predicted to decrease when the poor believe that one candidate offers significantly more in the way of redistributive policies than the other candidate. Because they see a redistributive policy choice, the poor are thus less indifferent between candidates' or parties' policy positions than they would be if the candidates offered the same positions. Believing that they have a relevant choice to make regarding redistributive policies, the poor become _more_ likely to vote as rising inequality increases the salience of redistribution. These relevant choices are also more clear to the wealthy, so with rising inequality the wealthy also become _more_ likely to vote.
The aggregate effect of both groups voting at higher rates might be a constant rate of differential turnout between the poor and wealthy. But if the poor initially are voting at _much_ lower rates than the rich, then the aggregate turnout effect on the poor will be larger than the effect on the rich because there are more nonvoting poor to be mobilized by increasing inequality than there are nonvoting rich to be mobilized.11 Thus, in this first scenario, income bias in turnout is _decreased_ by a rise in economic inequality.
In the second hypothetical scenario, income bias is predicted to increase when the poor believe that neither candidate offers a relevant choice on redistributive issues, and are thus more _alienated_ than the wealthy from both parties and candidates. As income inequality increases and the salience of redistributive issues rises for the poor, this alienation leads to lower levels of turnout relative to the rich as the poor believe that the political system is not responding to or representing their interests. At the same time, with a rise in inequality and neither candidate threatening to redistribute income, the wealthy would be placing greater emphasis on a _lack_ of redistribution, and therefore would be less alienated and _more_ likely to vote. Thus, in this scenario we see an _increase_ in income bias in turnout.
In both of these scenarios, what is key in determining who votes is not just the income or socioeconomic status of individuals but the behavior of parties and candidates. If one party offers redistribution, then the poor are likely to see a difference between the parties, and both the poor and wealthy will vote more as economic inequality increases. If neither party offers redistribution, then the poor are likely to become more alienated, and their turnout will, all things equal, decrease as economic inequality increases. In these same circumstances—with neither party offering redistribution—the wealthy would likely be less alienated and vote more. All of these observations are fully consistent with formal models of voter turnout that equate the benefits that individuals receive from voting with the utility derived from candidate policy positions.
It is important to realize here that _movement_ by the candidates or parties is _not_ required for increasing economic inequality to lead to increased turnout of the poor. As increasing economic inequality raises the salience of redistribution, if the poor perceive _any_ difference between the two parties on these issues, they would be less indifferent between the two parties, and more likely to vote; with lower initial levels of turnout of the poor, the aggregate consequence is less income bias in turnout. If the poor do not see redistributive choices offered by candidates they will be more alienated and less likely to vote, resulting in greater income bias of voters. The fundamental implication of our argument is that, theoretically, increasing economic inequality could lead to higher _or_ lower income bias of voters—with the actual outcome predicated on the policy choices offered by candidates.12
**1.4 Voter Turnout and Election Laws**
Since 1972, most states have passed legislation to reduce the costs of voting by making it easier to do so; many supporters of reforms such as absentee voting, early voting, and election day registration have argued that doing so would increase voter turnout. As a result, voters in many states today may cast no-excuse absentee ballots by mail, may cast in-person votes for a lengthy period before election day, or, for those not already registered, may choose to both register and vote in person on election day.
Most of these reforms have been passed with some expectation that their adoption would lead to a more representative set of voters by increasing the turnout rates of less educated, lower-income individuals. The logic is that if states lower the costs of voting, those least likely to vote—and thus most underrepresented as voters—would be more likely to cast a ballot. Recently some states have adopted more rigorous voter identification requirements that are clearly _not_ designed to make it easier to vote. Opposition to these efforts has focused on how requiring positive identification at the polls actually makes it harder for everyone to vote—especially the poor, and racial and ethnic minorities.
Our focus here is on those election law reforms intended to make voting easier, reforms that have often disappointed their supporters. One observation about some of these reforms is that making it easier to vote simply makes it easier to vote for those already inclined to do so, and thus _widens_ the turnout gap between rich and poor, or the more educated and less educated.13 This conflicts with the traditional view, espoused forcefully by Wolfinger and Rosenstone (1980), that making it easier to vote allows those at the bottom of the socioeconomic status scale (those who have the hardest time clearing the administrative hurdles) to more easily meet the legal demands, and thus be more likely to cast ballots.
As Wolfinger and Rosenstone showed, the date for the cutoff of registration to vote (i.e., the closing date, or the number of days before the election that registration closed) was a key factor in determining turnout in 1972. But what about more recent reforms? Election day registration lowers the cost of registration and voting, making it a one-step rather than two-step process. In-person early voting and No-excuse absentee voting have also been adopted in efforts to make voting easier. In-person early voting allows individuals to cast ballots prior to election day; in these states "election day" has often become "election two weeks." No-excuse absentee voting allows voters to request a mail-in ballot even if they will be in the state on election day and would be able to vote in person at a polling place. This form of voting also lowers the cost of voting as individuals do not have to wait in line at the polling place.
Our focus on these four reforms is motivated by the fact that they are most central to the ten presidential elections that we study, both in the number of states that have adopted them and in the number of individuals who take advantage of them.14 And because most research on these reforms has been limited by its typical focus on one reform in isolation from others, and by relying on limited periods of time, the effects of these reforms deserve continued attention. We think it important to provide a definitive answer to the claim that these reforms increase turnout, and also consider whether they matter _more_ for those individuals least likely to vote.
We provide robust evidence of both the efficacy and the limitations of these reforms. We show that reforms such as election day registration and absentee voting have increased turnout when adopted, but we also show that the effects are more modest than what some reformers may have hoped for.
**1.5 Data and Chapter Outline**
In the following chapters we present a wealth of empirical evidence based on several data sources. We rely heavily on the U.S. Census Bureau Current Population Survey (CPS) as it provides the largest samples of nationally representative data collected on the American population for the 1972–2008 period that allows us to distinguish between voters and nonvoters. The primary limitation of the CPS is that the data are limited to demographics and do not include any questions relating to the policy positions or political views of respondents. When we turn to questions regarding how party and candidate evaluations of citizens influence their turnout, or how much voters share the policy preferences of nonvoters, then, we must turn to other data, and for that we rely primarily on the well-established American National Election Studies (NES), a biennial survey of representative samples of U.S. citizens. In addition, we use the 2004 National Annenberg Election Study to assess the representativeness of voters' policy positions. The NES provides us with the advantage of using repeated questions over time, while the Annenberg study provides more contemporary, election-specific questions on which to base an assessment of the electorate's and voters' policy views.
Our analyses of the impact of election laws on turnout utilizes new data on election laws. To improve upon previous research we use data collected on state voter registration and election administration laws for each election since 1972.15 Because we combine data on election laws at the state and year level with data on turnout at the state and year level, we can estimate the impact of those changes in electoral laws on turnout using standard cross-sectional time series techniques. This allows us to draw robust causal inferences as we observe the effect over time of changes in laws within a state, conditional on changes happening in all other states.
In chapter 2 we use data from the CPS to provide an extensive description and discussion of aggregate and demographic group-specific turnout rates since 1972, focusing on education, income, race, ethnicity, age, gender, and marital status. We find that turnout of _eligible_ citizens has not declined since 1972, and that the overrepresentation of the wealthy versus the poor among voters has remained stable and large over time. However, we see substantial changes in the relative turnout rates of men and women, blacks and whites, and younger and older adults.
In chapter 3 we introduce the theoretical framework that guides our analyses and discussions of the determinants of voter turnout. We adopt a model of turnout that poses an individual's decision to vote as a reflection of the costs and benefits of engaging in such behavior. Then, for each presidential election year since 1972, we estimate turnout as a function of our demographic characteristics of interest. These estimates allow us to estimate the impact of one demographic characteristic (such as income) on turnout while holding other demographic characteristics (such as education and race) constant. We refer to these estimates as "conditional" relationships, and graphically represent how the conditional relationships among these characteristics and turnout vary over time and by election year. Our findings in this chapter, too, suggest that the conditional relationships between education and turnout, and income and turnout (i.e., "conditional income bias") have been relatively stable (or modestly reduced) since 1972. We also find important changes in the conditional relationships between age, race, gender and turnout.
In chapters 2 and we consider two distinct aspects of income bias because they address two very different, yet equally important, questions. In chapter 2 we examine whether poor people vote less than rich people and whether these differences change between 1972 and 2008. This is important empirical evidence, for turnout differences between the poor and wealthy likely have important political consequences for the political representation of the poor and policy outcomes. This bivariate relationship is what we mean when we use the term _income bias_ , referring to turnout patterns or the representativeness of voters: the poor are a smaller percentage of voters than they are of the electorate, while the wealthy are a larger proportion of voters than they are of the electorate.
In chapter 3 we want to know if poor people vote less than rich people, _once we condition on other characteristics_. We might, of course, observe that rich people vote more than poor people; but that observation does not clarify whether they vote more simply because of life cycle effects (because when people are older they have more income, and we know that they generally vote more) or because of some other reason. We refer to conditional relationships favoring the turnout of the rich regardless of this group's differences in other demographic respects as _conditional income bias_. Similarly, we might want to ascertain whether once we know a person's level of education, does knowing his income tell us more about the probability that he will vote? Or if we observe that Hispanics vote less than Anglos, we might want to know if that is because Hispanics, on average, have less income and education than Anglos, or because they are on average younger than Anglos.16 Or could there be another explanation? To sort out these possibilities, we would want to know if Hispanics vote less than Anglos _conditional on_ levels of income, education, and age. It follows that we also want to know if these conditional relationships have changed from 1972 to 2008.
In chapter 4 we describe the nature and variety of changes in voter registration and election administration laws since 1972. States vary tremendously as to how easy it is to register and to vote, and previous research suggests that these laws affect who votes because they change the cost of voting (Brians & Grofman 2001; Fitzgerald 2005; Highton 1997; Karp & Banducci 2001; Knack & White 2000; Wolfinger & Rosenstone 1980). Yet most of these studies rely on cross-sectional data, and usually consider the influence of one reform at a time. We provide aggregate (state-level) analyses of the effects of changes in these rules on voter turnout. These analyses help us address the question of whether overall voter turnout has increased as a result of these legal changes. We find modest effects of election day registration, of absentee voting, and of moving the closing date for registration closer to the election on overall turnout. The effect of early voting is less clear.
In chapter 5 we consider how the policy positions offered by candidates influence voter turnout. We expect that larger differences in the policy positions of candidates are associated with a higher probability of voting. Using NES data, we examine the impact of individuals' perceptions of candidates' policy positions—how they compare to each other, and how they compare to the individuals' preferences–on individuals' decisions to vote. We find that individuals are more likely to vote when they perceive a greater policy difference between the candidates. We also find that the poorest Americans have become more indifferent between candidates in recent elections—that is, they see fewer differences between candidates now when compared to wealthier Americans.
In chapter 6 we consider whether voters and nonvoters differ in their policy preferences using both NES and Annenberg data. We find, contrary to conventional wisdom, notable differences in both of these comparisons, especially on redistributive issues. We conclude that the seeming consensus that it would not matter if everyone voted is simply wrong, and has been wrong for a long time. That these differences have been ignored in political discourse as well as scholarly research is all the more striking given the increase in economic inequality experienced in the United States.
Over the next five chapters we document changes in the demographics of turnout in the United States, and the effects of changes in electoral laws on turnout. While we show distinct changes in turnout of certain groups over the time period, we mostly highlight the remarkable stability of the overall level of turnout and the stability of the relative turnout of _most_ demographic groups. In 1972, U.S. turnout was relatively low compared to that of other industrial democracies, and exhibited high levels of income bias. In 2008, after decades of various electoral reforms, U.S. turnout remained relatively low compared to that of other industrial democracies, and still exhibited high levels of income bias. We also highlight the consistency with which voters are not representative of the electorate on redistributive issues; they were not representative in 1972 nor were they representative in 2008. In chapter 7 we return to broader questions of representation and inequality in U.S. presidential elections.
1. We refer to the higher turnout rate of the wealthy compared to the poor as _income bias_.
2. "The 2012 Money Race: Compare the candidates"; <http://elections.nytimes.com/2012/campaign-finance>; accessed December 3, 2012.
3. See Citrin, Schickler, & Sides (2003); Highton & Wolfinger (2001); Martinez & Gill (2005); and Sides, Schickler, & Citrin (2008).
4. See Griffin (2005) and Martin (2003) for evidence on this point.
5. Our choice of 1972 as the first election we study is largely practical, based on the availability of data.
6. U.S. Census Bureau, Current Population Survey, Annual Social and Economic Supplements. Table H3, accessed November 7, 2011, from <http://www.census.gov/hhes/www/income/data/historical/inequality/index.html>.
7. See Jones and Weinberg (2000) for a comparison of different measures of income inequality for the period 1967–2001.
8. See also Piketty & Saez (2006, 2003).
9. See Bartels (2008); and Gilens (2012).
10. Schlozman, Verba, and Brady's (2012) finding that there is substantial education and income inequality in access to and use of new, electronic forms of participation makes revisiting this question for the most common form of participation in democratic politics in the United States today—voting—all the more important.
11. We emphasize that since the voting we are discussing is voting on _candidates_ , not _policies_ , the poor would only have an increased incentive to vote _if_ some candidates were promising to tax the rich.
12. Again, our claim differs from the argument of Meltzer and Richard (1981) because they do not at all consider competing parties; in their model the decisive voters simply choose the amount of redistribution by the government.
13. Rigby & Springer (2011) argue that reforms making it easier for the registered to vote will increase income bias as the set of people who are registered are wealthier than the set of people who are not registered.
14. A recent set of reforms we do _not_ study are reforms requiring stricter forms of identification, such as photo ID; there is simply not enough data available at this time to evaluate their impact on turnout.
15. This project was generously funded by the Pew Charitable Trusts in 2008 and 2009.
16. We use the term _Anglos_ to refer to non-Hispanic whites.
**Two**
* * *
**Demographics of Turnout**
Our interest in this chapter is whether changes in demographics and changes in the distribution of income in the American electorate (i.e., those who are eligible to vote) are also reflected in the composition of the voters (i.e., those who actually cast ballots). We focus primarily on how the turnout rates of different demographic groups have changed, or not changed, over time, and the extent to which voters in 2008 are more or less descriptively representative of the electorate than voters were in 1972. Of special interest, given the notable increase in economic inequality and the well-established relationship between socioeconomic status and voter turnout, is whether the voting population overrepresents the wealthy (relative to the poor) more today than it did in 1972.
Demographics remain today, as they were in 1972, major components of analyses regarding the determinants of mass political behavior in the United States. Whether drawing on the intellectual traditions of the Columbia school established by Lazersfeld, Berelson, and Gaudet in _The People's Choice_ (1948) or the Michigan model pioneered by Campbell, Converse, Miller, and Stokes in _The American Voter_ (1960), scholars have consistently demonstrated the importance of socioeconomic status, race, ethnicity, gender, age, and marital status as predictors of numerous aspects of electoral behavior and public opinion.
But the demographics of the United States have changed dramatically since the time of Wolfinger and Rosenstone's seminal work on voter turnout in the 1972 presidential election. The civilian population has increased from fewer than 208 million people in 1972 to over 303 million people in 2008.1 The proportion of Anglos (non-Hispanic whites) declined from 83.2 percent of the population in 1970 to 65.6 percent in 2008.2 And the proportion of African Americans in the population increased from 11.1 percent of the population in 1972 to approximately 13 percent of the population in 2008, while the proportion of Hispanics (of any race) has increased from 5.7 percent in 1970 to 15.4 percent in 2008.3
At the same time, the proportion of noncitizens in the voting-age population has increased substantially. In 1972 less than 2 percent of the voting age population were not citizens, but in 2008 8.4 percent of the voting-age population were not citizens.4 McDonald & Popkin (2001) discuss additional requirements for voting eligibility and argue that in addition to citizenship, increasing numbers of institutionalized individuals as well as the greater number of states disenfranchising convicted felons has led to a larger percentage of voting-ineligible individuals in 2008 compared to 1972. Whereas the number of persons ineligible for these reasons was very small in 1972, it was over 1 percent of the voting age population in 2008.5
Changes in the size of various demographic groups since 1972 have also been accompanied by changes in economic status. Median income in 1972, all races combined, was $21,800, compared to $26,800 in 2008.6 But change in the median income of women, compared to men, has been substantially greater: for women over the age of fifteen, the median income in 1972 was $12,100, compared to $21,00 in 2008. In contrast, the median income of men over the age of fifteen actually decreased from $34,700 in 1972 to $33,600 in 2008.7
Men over the age of sixty-five have fared much better. In 1972, the median income of men age sixty-five and over was $17,500 in constant 2010 dollars; in 2008, it had risen to $25,800. The income gains of women over the age of sixty-five were not as great, with their median income increasing from $8,900 in 1972 to $14,700 in 2008. Aside from these gender differences, however, it is clear that, as a group, income of individuals over age sixty-five has risen since 1972.
Among racial and ethnic groups, blacks have experienced the greatest proportional increase in median income, from $15,900 in 1972 to $22,200 in 2008. While whites' median income increased from $23,000 to $29,400 in 2008, Hispanics' median income increased only slightly, from $18,300 in 1974 (1972 data not available) to $21,000 in 2008.8
**2.1 Measuring Voter Turnout**
Our analyses in this and subsequent chapters primarily rely on three national survey sources, all of which include questions that ask individuals (or informants) whether they voted in the most recent election. The Current Population Survey (CPS), conducted by the U.S. Bureau of the Census, is a monthly survey of approximately 50,000 households. Respondents are asked about behavior of other household members, providing information on approximately 90,000 individuals per month.9 In November of even-numbered years, the CPS includes a short battery of questions on voter participation. In particular, respondents are asked whether or not they voted in that month's election, and whether or not they were registered. It is the especially large sample size that makes the CPS a valuable resource in studying voter turnout (see appendix 2.1). Along with the overall smaller sampling error, the sample includes enough people at different levels of education and income to accurately note differences across time in the participation of subgroups of interest of the population. And finally, the CPS has a substantially smaller nonresponse rate than both academic surveys and surveys conducted by news organizations. In an era when voters and nonvoters are increasingly difficult to contact, the high quality of the CPS survey makes it an invaluable resource for studying turnout.
However, the CPS is limited to demographic characteristics and cannot be used to study other important determinants of voter turnout such as political preferences, attitudes, or psychological orientations. The CPS thus cannot be used to assess the consequences of who votes with respect to the representation of policy preferences. For these purposes, then, in later chapters we turn to the American National Election Study (NES) and the National Annenberg Election Study (NAES), both of which include a wide range of questions regarding demographics, attitudes, political preferences and voting behavior. The NES, a biennial survey of 1,000–2,000 U.S. citizens, provides the advantage of having time-series data on many of these questions, and provides us with the opportunity to assess electoral changes over time. Alternatively, the NAES includes a wider range of policy preferences, has larger sample sizes on some questions, and (unlike the NES) includes noncitizens in its sampling frame. We thus use the 2004 NAES to provide some additional validation of our findings based on the more standard NES surveys. Detailed question wording from the NES and NAES is provided in appendix 6.1.
One possible limitation of using any of these self-reports of voter turnout is that individuals may fail to accurately report their past behavior. Whether unintentionally or intentionally misreporting, the concern with self-reports is that they might introduce systematic measurement error in our estimates of voter turnout. This is not an issue if different demographic groups have approximately equal rates of misreporting. But if, for example, high-income individuals are more likely to overreport than are low-income individuals, then any inferences we can make with respect to differences across income groups are compromised.
This potential problem has been addressed by numerous scholars using comparisons of self-reported turnout in the NES with validated turnout (i.e., where the NES confirmed or disconfirmed respondents' self-reports by using official county-level voting records). These comparisons are available for the NES for the 1976 though 1988 surveys. Previous research comparing the reported and validated vote in the NES suggests, indeed, that higher-status individuals are more likely to overreport than are lower-status individuals, in part because they are more sensitive to the norm of voting as expected of "good citizens" (Granberg & Holmberg 1991; Hill & Hurley 1984; Silver, Anderson, Abramson 1986). These studies also report that there are significant differences in misreporting rates between blacks and whites, with blacks being more likely to overreport voting. Whether such patterns are the same for the 1990s and early 2000s is not clear.10
However, there is some evidence to suggest that the NES-validated vote measures also have systematic shortcomings and should be used cautiously. Traugott (1989) examines the NES validated turnout measures, for example, and reports that in some years the proportion of records where insufficient information was provided to attempt validation was as high as 15 percent (Traugott 1989; Traugott, Traugott, & Presser 1992). Also, recent research suggests that misreporting rates may be less of a problem than previously claimed (Berent, Krosnick, & Lupia 2011).11
The necessary assumption that we need to make for our inferences documenting changes over time in relative turnout rates of different demographic groups is that differences in misreporting rates across demographic groups are stable over the time period examined.12
Our analyses in this chapter and chapter 3 rely on the CPS self-reported turnout measure because we are primarily interested in demographic changes over time–and we have no reason to believe that misreporting rates are not stable over time.13 Our primary measure of turnout in this chapter is a measure of the turnout of citizens in the voting-age population (CVAP), as is typically done in analyses of registration and turnout rates.14
However, with the increase in the noncitizen population over this time period, analyzing only citizens could mask differences between voters and the general voting-age population, a set of nonvoters that includes those who choose not to vote and those who legally cannot vote. Since we are interested in knowing how descriptively representative voters are, this second comparison provides useful information. Thus, we also use the CPS survey data to compute a second measure of turnout, based on the voting-age population (VAP). In this measure we include noncitizens in the denominator when we compute turnout.15
**Table 2.1. Estimates of Voter Turnout in Presidential Elections, 1972–2008.**
_a_ Self-reported turnout of citizens computed by the authors from the Current Population Survey for each year. Individuals coded as _Don't Know, No Response, or Refused_ by the Census Bureau are treated as having not voted.
_b_ Self-reported turnout of voting-age population computed by the authors using data from the Current Population Survey, 1972-2008.
_c_ Self-reported turnout of citizens computed by the authors using data from the American National Election Studies Time Series Cumulative Data File, 1972-2008.
_d_ Turnout rate of voting-age population based on vote for highest office as computed by McDonald (2011). Data for 1972 and 1976 are from McDonald and Popkin (2001); data for 1980–2000 are available at <http://elections.gmu.edu/voter_turnout.htm>.
_e_ Turnout rate of voting-eligible population based on vote for highest office as computed by McDonald (2011). Data for 1972 and 1976 are from McDonald and Popkin (2001): data for 1980–2000 are available at <http://elections.gmu.edu/voter_turnout.htm>.
Our estimates of voter turnout between 1972 and 2008 are reported in table 2.1, which includes three measures based on self-reports or informant reports in surveys: the reported CVAP and VAP measures from the CPS, and the self-reported measure from the NES. Table 2.1 also includes two turnout measures based on official reports of the number of ballots cast: _voting-age_ and _voting-eligible_ turnout, as calculated by McDonald (2011). McDonald uses official totals for ballots cast, and the best available estimates of the voting-age population in the states to estimate the first measure. He then adjusts the denominator for the number of noncitizens in each state, as well as other individuals ineligible to vote to generate the second measure.16
As expected, the estimate of turnout is always larger by approximately three to six percentage points when the measure is restricted to citizens (i.e., the CVAP vs. the VAP). This difference, however, is greater in more recent years as the number of noncitizens in the United States has increased. According to McDonald's estimates, over 8 percent of the voting-age population in the United States in 2008 were not citizens. Thus, including those noncitizens in the denominator will immediately reduce any computed turnout rate quite substantially.
Between 1972 and 2008, the turnout of citizens in the United States was higher in some elections than others, ranging from a low of 58.4 percent in 1996 to a high of 65.5 percent in 1992. However, there is no downward trend in the data, as many have claimed previously (e.g., Miller 1980; Reiter 1979; Rosenstone & Hansen 1993; Teixeira 1992). When we look at turnout of the population including noncitizens, we see that 1972 had the highest level of reported turnout, with turnout generally declining since then. Again, this follows from the arithmetic: with an increase in noncitizens (i.e., people not eligible to vote), the denominator in this calculation is simply increasing faster than the numerator.
Our NES self-report measure also suggests that turnout has either increased or stayed the same since 1972. Given the exclusive focus of the NES on political and social issues (and the resulting sample bias toward more highly educated, politically engaged individuals), the NES self-report estimate is higher than both of the CPS estimates. Since 1996, the NES self-report has provided a substantially higher estimate than the CPS CVAP self-report (a difference of over seventeen percentage points in 1996).17
Each of these three survey measures (NES, VAP, and CVAP) are all much higher than the "official" rate based on ballots cast as reported by state officials. However, we note that despite the use of the term, there _is_ no "official" turnout rate for the United States. While the various state governments certify how many votes were cast for each election, they simply do not know with certainty the number of citizens eligible to vote. Thus, any measure of turnout is by necessity based on an estimate of the eligible (or voting-age) population, and even the "official" turnout numbers cannot be used as an independent benchmark to demonstrate the inaccuracy of the survey measures.
While none of these measures is without error, what we see is that none of these survey measures reflect a trend of decreasing voter turnout among voting-age citizens over time. This is an important point, as much has been made of an alleged decline in U.S. voter turnout since 1972. However, our results are consistent with the analysis of McDonald and Popkin (2001), which showed that when looking at the turnout of votingage _citizens_ there has been no decrease in turnout.
**2.2 Measuring Socioeconomic Status**
Our major focus on the demographics of turnout is on socioeconomic status, though we also consider race/ethnicity, age, gender, and marital status in some detail (see appendix 2.1 on question wording and response categories for each of these variables). We conceptualize socioeconomic status as reflecting the resources and opportunities available to individuals to interact and engage politically, socially, and economically; individuals with higher status have greater resources to assume the costs of such behaviors, and also have more ways to participate in these spheres.
Wolfinger and Rosenstone's detailed analysis of education, income, and occupation as determinants of voter turnout, however, led them to conclude that there is no generic status variable related to voter turnout (1980, 34–35). That is, while education was strongly and positively related to voter turnout in 1972, the relationship between income and turnout was much weaker, and was unrelated to voter turnout once an individual achieved a threshold of financial security. Occupational status was not associated with voter turnout at all in 1972; instead, occupational differences reflected particular attributes or characteristics of the occupation (e.g., its reliance on the government).
Largely as a result of these empirical findings, the standard analytical approach in most studies of voter turnout over the past several decades has been to rely primarily on education and income (but not occupation) as indicators of socioeconomic status. We follow such a practice here in order to be consistent with this earlier work.
As we have argued earlier (Leighley & Nagler 1992b, 727, 730–32), education, income, and occupation as measures of class are plagued with measurement error, a key reason for some of the conflicting findings reported in previous research on class bias over time. Occupation is especially troublesome with respect to categorizing particular jobs as either white collar or blue collar, as well as with respect to temporal validity (made all the worse in the CPS by a notable coding change in 1970). In addition, the distinctive attributes of particular professions with respect to those dimensions that Wolfinger and Rosenstone (1980) identified as relevant to voter turnout are especially difficult, if not impossible, to measure reliably over time.
We continue to believe that income is the most meaningful available measure of socioeconomic status to use in studies of turnout. Income is the most widely used and recognized demographic criteria by which government distributes benefits and therefore seeks to influence social and economic life. Thus, if poor people do not vote, they could find economic policy being written in ways that disadvantage them. However, if poorly educated people do not vote they would not be likely to find government policy explicitly written to disadvantage them, _as government policy does not generally mention one's level of education._ And this relationship of government policy to income happens both at the spending end (through means-tested benefits programs) and the taxing end through a tax code that sets different rates for people at different income levels. Thus we proceed to primarily focus on _income bias_ , rather than _socioeconomic class bias_.
Our measure of individuals' income over time is not the discrete income category into which the individual is categorized but instead reflects where the individual is relative to the entire income distribution in a given year. We do this in part because in each year the CPS arbitrarily defines a set of income categories, and the size and range of these categories varies a great deal over the time period of interest.
We measure income in each year by assigning individuals to one of five income groups (i.e., quintiles), combining the CPS income categories for each year. As the categories do not map perfectly into quintiles (i.e., the first several CPS income categories may yield slightly more or slightly less than 20 percent of the respondents), we randomly assign people in overlapping categories into quintiles so that each quintile represents 20 percent of the distribution.18 By assigning individuals' self-reports of income to one of these five quintiles in each year between 1972 and 2008, we use a more meaningful measure of income over time, and thus one that allows us to draw more meaningful inferences about the relationship between income and voter turnout.19 As quintiles are a standard reporting unit for income measures, our individual-level measure is thus comparable to aggregate-level measures of income change.20
Conceptually, education reflects the skills and information gained by individuals and, though it is typically achieved at relatively early stages of life, has been documented to determine a host of life chances, including income level and occupation type. This likely reflects another key aspect of education as a measure of status: it is closely linked to family background. Individuals with higher levels of education are more likely to come from families with higher levels of education. Levels of education also likely reflect attitudinal characteristics of individuals who seek to achieve more education or who become more engaged in politics because they have basic information about interests and issues.
As Nie, Junn, and Stehlik-Barry (1996) and Tenn (2005) argue, the meaning of education has changed over time. Someone with a college degree in 1972 was much rarer than someone with a college degree in 2008 (12.0 percent vs. 29.4 percent of people over twenty-five years of age, respectively), and someone who failed to graduate from high school was rarer in 2008 than 1972 (13.4 percent vs. 41.8 percent, respectively).21 Comparisons of the level of education over time badly conflate selection into education with level of education. Thus for education, too, a measure of relative education is more suitable for testing differences in the effects of education over time. We measure education not by the number of years of formal schooling attained (as is usually done) but instead by assigning each individual to one of three categories representing the education distribution in each year (i.e., bottom third, middle third, top third).22
**2.3 Measuring Race and Ethnicity**
As scholars of racial/ethnic politics are well aware, thorough data on race and ethnicity are rarely available, even in the case of the U.S. Census. Until recently, "ethnicity," in Census Bureau terms, refers to identifying as Hispanic or Latino, while "race" refers to identifying as white, black, or African American, American Indian/Alaska Native, Asian, or Native Hawaiian or other Pacific Islander. Individuals identifying as Hispanic or Latino can also identify as belonging to any racial category.
Our analyses focus primarily on three racial/ethnic groups: Anglos (i.e., non-Hispanic whites); African Americans (non-Hispanic blacks); and white Hispanics, because these are the three largest and most commonly politically identified groups during the period of analysis.23 The large size of each of these groups throughout the period allows us to draw more accurate inferences regarding changes in the relationship between these demographic characteristics and voter turnout. We do not distinguish between Hispanics on the basis of their native origin due to our inability to maintain consistency over time. Thus, our description of Hispanic turnout, and our conditional comparisons of Hispanic turnout and Anglo turnout, are all averages over the full set of Hispanics.24
Even focusing on only these three groups is challenging because of changes in question wording over the study period. The Current Population Survey did not include a "Hispanic Origin" (i.e., ethnicity) question until 1976, and so our analysis of ethnicity and voter turnout is restricted to the 1976–2008 time period. In 2000 the response categories of this question were revised to "Hispanic or Latino Origin" and "Not Hispanic or Latino."
The CPS has also revised its question wording regarding race over the period of our analysis. From 1972 to 1988 there is a consistent coding of white, black, and "other." From 1992 through 2000 individuals could identify as one of four races: white; black; American Indian or Alaskan Native; and Asian or Pacific Islander. The category "other" included everyone else. In 2004, a new set of response categories was provided, and for the first time respondents were allowed to select one _or more_ races when they self-identified, and they could also choose a sixth racial category, "some other race." However, the entire set of new 2004 categories comprised less than 2 percent of choices of respondents over 2004 and 2008. So the effect on our analyses of the additional categories is likely to be small.
**2.4 Demographics of Turnout, 1972–2008 (CPS)**
We begin our analysis of voter turnout in presidential elections by examining differences in turnout rates by education, income, race/ethnicity, age, gender, and marital status. We do this to answer the fundamental question of descriptive representation: does the set of individuals who cast ballots share the same distribution on demographic characteristics as the set of individuals who are eligible to vote? In other words, does the set of voters "look" like the set of eligible voters? If voters are demographically similar to the electorate (i.e., eligible individuals) then we can conclude that in this basic sense, voters are descriptively representative. In chapter 3 we will turn to models that give the conditional relationship between these demographic characteristics and the likelihood of voting over time—that is, we will examine whether voters are representative of the electorate for each demographic characteristic, while conditioning on, or controlling for, other demographic characteristics.
Figures 2.1 through 2.7 graph report turnout rates by education (bottom third, middle third, and top third of the distribution), income (low to high quintiles), race/ethnicity (white, black, Latino, other; as well as white non-Hispanic, black non-Hispanic, white Hispanic, and other), age (18–24, 25–30, 31–45, 46–60, 61–75, and 76–84), gender, and marital status (married vs. nonmarried). In each of these figures the horizontal axis represents each presidential election year, while the vertical axis represents the percentage of individuals in each demographic group who report having voted. Changes over time in the turnout levels of each group can then be assessed by examining the movement (i.e., ups and downs) of each line, while differences in turnout rates across the demographic subgroups can be assessed by the distance between lines. The larger the distance between each line, the greater the turnout differences are based on the demographic characteristic of interest.
Figure 2.1 graphs turnout by thirds of the education distribution, and figure 2.2 graphs turnout by income quintile. The graphs for turnout by education and income levels are similar in that there are distinct differences across each demographic category, and these differences are observed for every election year between 1972 and 2008. In each election, individuals with higher levels of education are more likely to vote than individuals with lower levels of education, and individuals with higher levels of income are more likely to vote than those with lower levels of income.
In addition, while the differences in turnout across education and income levels (i.e., the distance between the lines) appear to be similar over time, the actual turnout rates vary across the period. Between 1972 and 1988, for example, turnout seems to be relatively stable, with a notable increase (across all subgroups) in 1992, followed by a drop in turnout (again, across all subgroups) in 1996, and then increases in most subgroups in each election from 2000 to 2008. Regardless of these changes, the difference in turnout between the highest and lowest education group is about the same in 2008 as it was in 1972 (13.8 percentage points). For income, however, it appears that the difference in turnout between the highest and lowest income groups is slightly smaller in 2008 than it was in 1972 (25.3 percentage points in 2008 vs. 28.8 percentage points in 1972), with the lowest two income groups voting at increasing rates in each election after 1996.
**Figure 2.1.** Turnout by Education, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
These patterns confirm the enduring significance of education and income for patterns of voter turnout, and suggest that income bias of the voting population has been remarkably stable over time. Note that there are two important observations here: first, that the income bias of voters is large, and has been large in every election since 1972; and, second, changes in income bias over time have been rather small. In addition, that turnout of each group moves together suggests that the turnout of each group responds similarly to election specific factors. We return to this point in chapter 5. More broadly, we note that what we do _not_ see is an increase in income bias in turnout that is anything like the increase in income inequality over this period. Thus, if the increasing income inequality over the period we examine has affected politics, it has done so through mechanisms other than through causing any substantial change in the representativeness of the voters.25
**Figure 2.2.** Turnout by Income, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
In contrast to the relative stability in patterns of turnout by education and income we saw previously, we see substantial differences in relative turnout by race, ethnicity, age, and gender over time. We provide two separate graphs of turnout by ethnicity to show different combinations of ethnicity with race, and because of constraints in the available data. Figure 2.3 graphs turnout by race and ethnicity separately from 1972 to 2008; the categories for white, black, and other _include_ Hispanics, _and_ we include turnout of Hispanics as a separate measure in figure 2.3.26 Figure 2.4 graphs turnout by ethnicity from 1976 to 2008, classifying respondents into four categories: white non-Hispanic, black non-Hispanic, other non-Hispanic, and white Hispanic.27
Examining white non-Hispanic turnout over time as we do in Figure 2.4 is especially important because the makeup of those classified as white in 2008 is much different from the makeup of those classified as white in 1972. In 1972, most of the U.S. population could be characterized adequately as black or white, but by 2008 the homogeneity of the white racial category became questionable, as white Hispanics, a much larger group than in 1972, were now politically distinctive from white non-Hispanics. Thus, the black versus white categorization is not adequate for examining racial/ethnic group differences over this time period. Obviously the set of whites in 2008 included a much higher percentage of Hispanics than did the set of whites in 1972, and as Hispanic whites vote at a much lower rate than non-Hispanic whites, we would be failing to measure what is a politically relevant demographic characteristic in combining the groups.
**Figure 2.3.** Turnout by Race, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
**Figure 2.4.** Turnout by Ethnicity, 1976–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
Figure 2.3 illustrates the dramatic change over time in turnout of these groups. Turnout of blacks increases from 53 percent in 1972 to 65 percent in 2008. Thus, where thirty years ago black turnout lagged substantially behind white turnout, in 2008 it was the same as white turnout. Note that black turnout first increased between 1976 and 1980, spiked substantially in 1984, declined through 1996, and then increased again. The high levels of turnout in 1984 and 2008 are consistent with claims of the importance of Jesse Jackson's presidential candidacy in 1984 and Barack Obama's candidacy in 2008 (see Tate [1991, 1993]; and Philpot [2009]). But it is also important to note that the trend toward increasing turnout of blacks relative to whites is observable outside of these elections. We note that this is a direct comparison of turnout rates _without_ conditioning on income and education, where we would expect the lower levels of income and education of blacks to lead to lower turnout rates for blacks than whites.
This increase in black turnout has potentially important electoral consequences. Black turnout in 1972 was only 52.9 percent; in 2008 it was 64.7 percent. According to exit polls, Obama won 95 percent of the black vote in 2008. If black turnout in 2008 had been only what it was in 1972, Obama's overall vote share would have dropped by 1.2 percentage points. And, of course, given the geographic distribution of blacks, this would have meant more than 1.2 percentage points in some states and fewer in others. The point, of course, is that in a close state, 1.2 percentage points could easily be pivotal. Obviously the large impact here is because of the overwhelming support blacks give to Democratic candidates. Whereas Obama's share of the black vote was 52 percentage points higher than his share of the white vote, his share of the women's vote was only 7 percentage points higher than his share of the men's vote.
We also see changes in the turnout of Hispanics and Anglos over this time period. Figure 2.4 gives the turnout for white non-Hispanics, other non-Hispanics (Anglos), white Hispanics, and black non-Hispanics.28 Anglo turnout follows the same pattern as that reported for the education and income groups: it is relatively stable until 1988, with a notable spike in 1992, followed by a drop in 1996 and then increases in 2000 and 2004. White Hispanic turnout, on the other hand, increases between 1976 and 1984, and then again from 1996 to 2008, after declining or remaining constant in 1988 and 1992. The key point here is that whereas black turnout increased considerably from 1972 to 2008, Hispanic turnout has increased much less. The 21.7 percentage-point gap in turnout between Anglos and white Hispanics that existed in 1976 shrank to a 16.1 percentage-point gap in 2008, with much of this shrinkage happening from 1976 to 1984. In 1984 and 1988 the gap was 16.5 percent and 16.6 percent, respectively. It did not drop below this level again until 2008, when it returned to 16.1 percent. Given the greater organization of Hispanic political groups compared to 1972, as well as the increasing size of the Hispanic community, the size of the gap in 2008 is striking.29
Figure 2.5 shows patterns of turnout across age groups between 1972 and 2008 and also documents changes in the demographics of turnout. The most striking change here is in the dramatic increase in voter turnout in the oldest age group (76- to 84-year-olds). In 1972, less than 60 percent of this age group reported voting, but by 2008 that proportion had increased to nearly 75 percent. This increase resulted in the turnout of this oldest group exceeding turnout of 31- to 45-year-olds and 46- to 60-year-olds, at the same time that it approaches the turnout levels of 61- to 75-year-olds (the group with the highest levels of self-reported turnout.) As was the case in 1972, the youngest age group (18- to 24-year-olds) reports the lowest level of voting in every election year (with a notable spike in 2004 and a slight increase in 2008).
**Figure 2.5.** Turnout by Age, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
We note that this increase in turnout is especially important because it coincides with demographic changes in the age distribution of the population. While 76- to 84-year-olds made up 4 percent of the citizen voting age population in 1972, they made up 5.4 percent of the citizen voting age population in 2008 (see table 2.2 for details on population proportions for other demographic groups in 1972 and 2008). Combining these changes with their increased turnout rate, their share of the voters increased from 3.6 percent in 1972 to 6.0 percent in 2008.
These results likely reflect that older Americans are healthier and wealthier today than they were in 1972, and are likely to be more active. But they are also consistent with what many observers of real politics have argued: that older Americans have become more politicized.
Changes in turnout patterns of men and women also emerge when we examine self-reported turnout from 1972 to 2008 (fig. 2.6). In 1972, women were less likely to report voting than were men, though since 1984 women have been consistently _more_ likely to report voting than men. And the trend has been quite steady. By 2008 the gap in turnout between men and women had grown to over 4 percentage points. Some scholars have attributed increased participation by women to an increasing number of female candidates and national political figures (Atkeson 2003). Women have also entered the labor force in larger numbers over this period.
We note that this change is potentially important to election outcomes. Women vote Democratic at significantly higher rates than do men. Thus, this change in turnout rates in effect makes the set of voters more Democratic than they would otherwise be. For example, in 2008, Barack Obama received 56 percent of women's votes and only 49 percent of men's votes. Had women and men voted at the same rate in 2008 that they did in 1972, Obama's overall vote share would have been 0.16 percentage points lower.
**Figure 2.6.** Turnout by Gender, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
The final demographic characteristic we examine is that of marital status. As shown in figure 2.7, turnout of both married and single individuals has varied somewhat over time, but these changes are similar for the two groups: the turnout of single individuals increases in the same elections where the turnout of married individuals increases, and similarly for decreases in turnout. The turnout levels of both groups are fairly similar in 2008 to what they were in 1972, despite a fair amount of variation in the later elections. And single individuals continue to vote at levels substantially below the levels of married individuals.
More generally, then, what we see in these bivariate relationships over time is the enduring importance of education and income, compared to the more fluid relationship between turnout and race, ethnicity, age, or gender. Determining whether earlier findings that education is more important than income as a determinant of voter turnout are still correct requires a multivariable analysis of the type we conduct in chapter 3. Perhaps over this period of increasing economic inequality, both education and income remain important, but income has become more important than education. We will return to this question in chapter 3.
**Figure 2.7.** Turnout by Marital Status, 1972–2008.
_Note_ : Entries are the self-reported turnout rate of the citizen voting-age population for each demographic group. Computed by the authors using data from the Current Population Survey.
**2.5 A More or Less Representative Voting Population?**
As Wolfinger and Rosenstone noted, "In short, voters are not a microcosm of the entire body of citizens but a distorted sample that exaggerates the size of some groups and minimizes that of others" (1980, 108). Based on the differential turnout rates documented in figures 2.1 through 2.7, it would seem that 2008's voting population was indeed, on most dimensions, as distorted as was the 1972 voting population.
While looking at the level of turnout of each group over time is useful for seeing what has been happening, normatively we are concerned with representativeness. The obvious question is, does a group's share of _the votes_ match its share of _the population_? To provide a more systematic portrait of changes in electoral representativeness over time, we compare the representativeness of the 1972 and 2008 voting populations using a representativeness ratio similar to that used by Wolfinger and Rosenstone (1980, table 6.1) and Rosenstone and Hansen (2003, table 8.2). This ratio for each demographic group is computed by dividing the group's proportion of voters by its proportion of the voting-age citizen population. If the group is equally represented when comparing their proportion of voters to their proportion of the voting-age citizen population, then the representativeness ratio equals one; if the group is overrepresented as voters compared to the voting-age citizen population, then the ratio is greater than one; and if the group is underrepresented as voters compared to the voting-age citizen population, then the ratio is less than one.
Table 2.2 illustrates the calculations of representativeness for 1972 and 2008. To both make it comparable to Wolfinger and Rosenstone's original table and to illustrate the issues in comparing groups across time, we show representativeness for two measures of education. The first measure is simply level of education. Here we have four categories: less than high school graduate; high school graduate; some college; and college graduate and beyond. The second measure is placed in the education distribution, and here we have whether the respondent is in the bottom, middle, or top third of the education distribution. The table thus shows that the representativeness of the voting population with respect to _level_ of education has changed drastically. In 1972, persons with less than a high school education had a representativeness ratio of 0.79, but by 2008 it had dropped to 0.62. However, if we simply look at persons in the bottom third of the education distribution, the representativeness ratio did not change. It was 0.79 in 1972, and 0.78 in 2008. Of course, the other thing that changed was the distribution of education in the population. Whereas 36.6 percent of the citizen voting age population in 1972 had less than a high school degree, in 2008 the corresponding number was only 11.2 percent.
**Table 2.2. Representativeness of Voters by Demographic Groups, 1972 and 2008.**
_a_ Entries in the first and fourth columns are the percentages of citizens in the voting-age population in the referenced demographic group, for 1972 and 2008, respectively. Computed by the authors using data from the 1972 and 2008 Current Population Surveys.
_b_ Entries in the second and fifth columns are the percentages of voters in the referenced demographic group, for 1972 and 2008, respectively. Computed by the authors using data from the 1972 and 2008 Current Population Surveys.
_c_ Entries in the third and sixth columns are the ratios of the demographic group's share of voters to the demographic group's share of the citizen voting-age population, for 1972 and 2008, respectively. Values less than 1 indicate underrepresentation of the specific demographic group among voters, while values greater than 1 indicate overrepresentation of the specific demographic group among voters. Computed by the authors using data from the 1972 and 2008 Current Population Surveys.
The 2008 voting population is more representative of blacks than was the 1972 voting population, with the representation ratio increasing from 0.82 to 1.02. We also see changes in representativeness across age groups. In 1972 persons between the ages of seventy-six and eighty-four were underrepresented among voters (ratio of 0.91), but, in 2008 this group was _over_ represented among voters (ratio of 1.11). In contrast to these notable changes, the representativeness ratio for women increased only slightly, from 0.99 to 1.03. For singles, their representativeness ratio barely changed (0.85 to 0.88).
In addition to representativeness, table 2.2 shows the share of votes held by different groups. Comparing the percent of voters column in 1972 with the percent of voters column in 2008 highlights several politically meaningful changes. Whereas in 1972 76- to 84-year olds had 3.6 percent of the votes, in 2008 they had 6.0 percent of the votes. Single people had 26.4 percent of the votes in 1972, but 40.5 percent of the votes in 2008. And blacks had 8.2 percent of the votes in 1972 and 12.3 percent of the votes in 2008. Given the differences in voting patterns of blacks versus nonblacks, and single people versus married people, these shifts in the share of votes have potentally important implications for election outcomes.30
**2.6 More or Less Income Bias?**
Research on and discussions of income bias in turnout can focus on two very different questions, both of which are important. Being precise about which question is being addressed is critical to knowing the proper way to answer the question.
First, if we are asking whether lower-income people are less well represented among the voters than among the voting age population, then we do not need to condition on other characteristics of the lower-income population. Those other characteristics are irrelevant to the question. We simply need to compare the fraction of the votes cast by lower-income people to their fraction of the eligible voting population. This is what we do in table 2.2.
But, second, if instead we are asking whether a lower-income person is less likely to vote than a similar higher-income person—that is, one who shares the lower-income person's other demographic characteristics (e.g., the same age, education, ethnicity, and gender)—then we would need to proceed differently. To answer this question we would, of course, need to condition on other characteristics potentially associated with income and turnout. If we observe that, on average, lower-income people vote less than higher-income people, we might want to know if this difference in voting rates can be explained by other observable demographic characteristics of the respondents. In the case of income, we might want to know this based on concerns of representation and fairness. We know that income generally increases as people get older, but we would not want to infer that poor people vote less simply because younger people vote less.
Answering this second type of question also allows us to do two things. First, it indicates what needs to be explained beyond what is in our model. If Hispanics vote less than Anglos on average, but vote at the same rate as Anglos once we condition on education, income, and age, then we would not need to look for other causal explanations of why Hispanic turnout is low relative to Anglo turnout. We would have found that Hispanic turnout is lower than Anglo turnout because Hispanics are less educated, have less income, and are younger. But, if Hispanics vote less than Anglos even _after_ conditioning on other demographic characteristics, then we would need to look for other explanations to explain why that is so.
Answering this second question also provides useful information if we are interested in the marginal _effect_ of different attributes on turnout, and are trying to draw causal inferences from observed conditional relationships between individual characteristics and turnout. We note that drawing causal inference about the relationship between observable characteristics and turnout in cross-sectional data is problematic. But we are still interested in these conditional relationships.
Our earlier work on income bias has addressed both of these questions and suggested that income bias had not increased between 1972 and 1988. Our earlier analysis (Leighley & Nagler 1992b) focused primarily on two different tests of changes in income bias. First, we examined changes in the turnout of different income quintiles over time. Increasing differences between the turnout of members of the top and bottom income quintiles over time would be evidence of increasing income bias, as would differences between the turnout rates of each income quintile and overall turnout. We found these differences to be surprisingly small from 1972 to 1988.
Second, we tested for changes in the conditional relationship between income and turnout, controlling for other demographic characteristics associated with voter turnout (i.e., race, gender, age, marital status, living in the South), to assess whether income as an explanatory variable became stronger over time. Should income bias in turnout be increasing, then the conditional relationship between income and turnout should increase, or the relationship between other characteristics associated with turnout and income would have to change. Using a probit model of voter turnout, we found that the conditional relationships between turnout and both income and education were relatively stable between 1972 and 1988. These two sets of evidence led to our conclusion that there was little (almost no) change in income bias between 1972 and 1988.31
The data we report in table 2.2 suggests that income bias is similar in 1972 and 2008 and thus that it has not changed dramatically since 1988. Specifically, the representativeness ratio for the poorest income group is the same in 2008 as it was in 1972, the ratio for the second poorest quintile increases by .05, and the ratio for the middle quintile changes only from 1.01 to 1.00. The fourth quintile stays the same at 1.1, while the wealthiest quintile's score decreases from 1.23 to 1.17. This decrease in overrepresentation is the largest change among all the income groups.
These findings contradict Freeman's (2004) and Darmofal's (2005) conclusions that income bias has increased. We believe that some of the disagreement in the literature on whether income bias has increased or not (in the form of inconsistent or inconclusive evidence) reflects the particular way in which income bias is measured, the particular statistical approach used to examine income bias, and the particular time periods chosen for study.
For example, we note that one must be cautious in drawing conclusions regarding income bias when examining only the top and bottom income groups. The simplicity of comparing the extremes of income distribution likely overlooks the economic diversity of American society, as well as the possibility that notably different patterns of income gains and losses across income groups may have vastly different political consequences. One might imagine, for instance, a relatively stable rate of income bias over time when comparing the turnout of the wealthiest quintile to the poorest—but quite different political implications would likely result if at the same time middle-income voters are either energized or demobilized by economic and political circumstances.
We also argue that it is all the more important to examine income bias over a series of elections. Early studies noting the income bias of the voting population assumed that as turnout (supposedly) declined in the 1970s it was lower-status individuals withdrawing more quickly than higher-status individuals—though the decline was implicitly viewed as a process occurring over the course of several elections (see, e.g., Burnham [1980, 1987, 1988]; and Reiter [1979]). Assessing income bias across a long series of elections allows us to determine to what extent income bias in the voting population is malleable from election to election, or instead moves in the form of small changes tending in the same direction.
Analyses of changes in income bias over time are fundamentally constrained by the time period considered, as well as assumptions regarding the linearity of changes over time, either in theory or practice. That is, if we seek to test for an increase between two points in time, then which two points are chosen may make a difference for our conclusions. The data presented in table 2.2, for example, may reflect these two particular election years: one in which George McGovern is running as a liberal Democrat and one in which John McCain is running for election in a time of war as a conservative Republican, the latter with a marked increase in aggregate voter turnout. This evidence cannot tell us whether the substantive finding that there is decreasing representation of any one group would differ if instead we compared 1980 to 2008, or 2000 to 2008.
From a practical perspective, this means that when evaluating income bias over time one must consider whether year-to-year changes are as notable as an underlying trend over time. The possibility of election-specific changes in income bias independent of overall turnout is intriguing, as we know that candidates, issues, and party strategies often differ substantially across elections, and that some elections take place in periods of increasing economic inequality while others do not. Hence, changes in income bias might be observed only in particular periods within the broader time frame of the analysis. If year-to-year changes dominate the trend, then it suggests that income bias responds to specific political interests and is thus potentially malleable by elites.
Our analysis of income bias from 1972 to 2008 focuses primarily on differences in turnout across income groups, but we also present differences in turnout across education groups since these data are often presented in analyses of socioeconomic bias more broadly defined. Figures 2.8 and 2.9 plot the representativeness scores for income and education groups for each election from 1972 through 2008.32
Again, we see that the representativeness of the voting population with respect to the electorate has basically remained stable since 1972 for both income and education. The middle-income group is fairly represented, with the group's representativeness score very close to 1 for the entire period, while the highest two income groups are overrepresented. While there are slight variations from election to election, the level of representation for these three groups seems to be fairly stable. The lowest two income groups, in contrast, are underrepresented, with representativeness scores generally between 0.7 and 0.9. Though changes are still quite small, it appears that the second income group's representativeness score increases over time, while the lowest income group's representativeness score dipped slightly in the 1980s, but rebounded from 2000 to 2008. The relatively flat lines associated with these representativeness scores indeed suggest that income bias in the voting population, as measured by income, has remained relatively stable over time. We emphasize here that we are not interested in arguing about whether the representativeness of the bottom income quintile has gone from 0.79 to 0.81. Our point is that it is very stable, and that changes in the distribution of income dwarf changes in the distribution of votes.
**Figure 2.8.** Representativeness of Voters Compared to Citizens, by Income, 1972–2008.
_Note_ : Entries are the ratio of the income group's share of voters (based on self-reported vote) to the income group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific income group among voters, while values greater than 1 indicate overrepresentation of the specific income group among voters. Computed by the authors using data from the Current Population Survey.
**Figure 2.9.** Representativeness of Voters Compared to Citizens, by Education, 1972–2008.
_Note_ : Entries are the ratio of the education group's share of voters (based on self-reported vote) to the education group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific education group among voters, while values greater than 1 indicate overrepresentation of the specific education group among voters. Computed by the authors using data from the Current Population Survey.
To put this in perspective we can compare changes in the share of _income_ held by the bottom income quintile to changes in the share of the _votes_ held by the bottom income quintile. This quintile had their highest share of income (5.5 percent of total income) in 1972, and their lowest share of income (4.0 percent of total income) in 2008. This is a drop in 27 percent of the group's share of income.33 In contrast, the highest recorded share of votes held by the bottom quintile during the period we examine was 15.9 percent of the total votes (in 1972). The group's lowest share of the votes held was in 1992, when the bottom quintile had 14.0 percent of the total votes. This means that the largest drop in _vote share_ for the bottom quintile over this period was 12 percent. Thus comparing the loss in income share to the (temporary) loss in vote share over this period, we see that the loss in income share is almost two and a half times the loss in vote share. And by 2008 the share of votes held by the bottom quintile had rebounded to 15.5 percent, and was thus 2.5 percent below the share held in 1972.
We see similar results for education. Not surprisingly, given the importance of education as a predictor of voter turnout, the two highest education thirds have representativeness scores close to or greater than 1 throughout the entire period, while the lowest education group is underrepresented for the entire period. But there is no substantial shift in the magnitude of under-representation for the lowest group, though we see some election to election variation.
**2.7 Representation: Of the Eligible or the Available?**
Normatively we believe that in a fair world, poor people and rich people should have proportionately equal shares of the votes; each citizen's right to a ballot should carry equal weight, even though her wealth may be decidedly unequal. For example, the people in the bottom fifth of the income distribution should have 20 percent of the votes, and the people in the top fifth of the income distribution should have 20 percent of the votes.
One group's proportion of the population may differ from its proportion of the votes for two very different reasons. First, legally eligible members of income (or other relevant) groups may differ in their average likelihood of voting, and those groups such as the poor (whose members are less likely to vote than the rich) will be underrepresented among the voters—that is, represent a smaller proportion of votes than their presence in the population. Second, some groups may contain a larger proportion of people legally ineligible to vote than other groups. For instance, more among the poor than the rich may be legally ineligible to vote because of prior felony convictions, or more of the poor than the rich may be noncitizens. Most studies and discussions of income bias focus on the first explanation in that ample evidence documents that the legally eligible poor are less likely to vote than the legally eligible wealthy.
However, legal issues of voting eligibility also affect the representativeness of the poor relative to the rich. Although poll taxes and such devices have been effectively eliminated, states nonetheless maintain the ability to determine who is eligible to vote.34 The key requirement common to state electoral requirements is citizenship. While this does not explicitly disenfranchise the poor, noncitizens residing in the United States are more likely than citizens to be poor.35 And because the poor are more likely than the wealthy to be ineligible to vote, they are then less likely to be represented among the voters.
One might argue that analyses of income bias that are restricted to the _eligible_ population misconceptualize the population that merits representation if elections are to truly reflect the interests of all those governed by elected officials. An important alternative approach to assessing income bias in the voting population is to compare the relative proportion of the poor in the _resident_ population, rather than in the _eligible_ population, to their proportion among the voters. Changes in income bias can then be measured using this different conceptualization and provide an alternative assessment of the representation of the poor and the wealthy in U.S. politics today. As the proportion of ineligible voters has increased substantially over this period due to an increase in noncitizens among the population, we could see changes in this measure of income bias even if the behavior of _eligible voters_ has remained unchanged over time.
Figures 2.10 and 2.11 present this alternative measure of representativeness, where all voting-age individuals, rather than just voting-age citizens, are included in the denominator of the representativeness ratio. For both education and income we see that the representation of the people at the bottom of the socioeconomic scale suffers compared to the measures based on citizens only (presented in figures 2.1 and 2.2). In 2008, the representativeness ratio for _citizens_ in the bottom income quintile was 0.79, compared to 0.75 for the voting-age population.36 And in 2008 the representativeness ratio for _citizens_ at the bottom of the education scale was 0.78, while the representativeness ratio for all persons was only 0.74. Thus, our answer to the question of whether income bias of voters _relative to the voting-age population_ has increased is different from our answer to the question of whether income bias _relative to the set of eligible voters_ has increased. Income bias relative to the voting-age population _has indeed increased_.
**Figure 2.10.** Representativeness of Voters Compared to the Voting-Age Population, by Income, 1972–2008.
_Note_ : Entries are the ratio of the income group's share of voters (based on self-reported vote) to the income group's share of the voting-age population. Values less than 1 indicate underrepresentation of the specific income group among voters, while values greater than 1 indicate overrepresentation of the specific income group among voters. Computed by the authors using data from the Current Population Survey.
**Figure 2.11.** Representativeness of Voters Compared to the Voting-Age Population, by Education, 1972–2008.
_Note_ : Entries are the ratio of the education group's share of voters (based on self-reported vote) to the education group's share of the voting-age population. Values less than 1 indicate underrepresentation of the specific education group among voters, while values greater than 1 indicate overrepresentation of the specific education group among voters. Computed by the authors using data from the Current Population Survey.
**2.8 Conclusion**
The evidence we have presented in this chapter is based on the most comprehensive and systematic data available on voter turnout in the United States between 1972 and 2008. We have reported several important facts about voter turnout since 1972. First, voter turnout in presidential elections since 1972 has not declined systematically. Instead, it has been slightly higher in some elections, and slightly lower in other elections. Overall turnout levels seem to reflect as much about the political context of each election as they do about citizens' underlying motivation or willingness to participate in the electoral process.
Second, the relationships among income, education, and voter turnout are quite strong: the probability of a highly educated or wealthy individual casting a ballot is much, much higher than the probability of a less-educated or poorer individual casting a ballot. As a result, the income bias of U.S. presidential voters is large, even huge. Third, these differences in turnout have been remarkably stable over this thirty-six-year period. During a period marked by truly massive changes in economic inequality, we do _not_ find a significant increase in income bias in turnout of the electorate. There may have been a small increase, but nothing substantial, and certainly nothing to suggest a large relationship between changes in economic inequality and turnout. When we move beyond the electorate to consider the voting-age population (including noncitizens), however, there has been an increase in income bias since 1972.
Fourth, there is less stability in turnout patterns by age, gender, and ethnicity since 1972 compared to those of education and income. There has been a small shift in the relative turnout of women and men. Women are now more likely to vote than men, and the gap has been widening. Because women comprise a group that makes up more than 50 percent of the electorate, and behave substantially differently from men in their vote choices, this is potentially quite an important political shift. There has been a substantial change in black turnout relative to white turnout since 1972. However, the increase in Hispanic turnout has been much smaller, and turnout of Hispanic citizens still lags far behind turnout of non-Hispanic whites. Finally, we have documented a large increase in turnout of older voters relative to turnout of younger voters.
The centrality of demographics in models of voter turnout underscore their fundamental importance to the resources and strategies of both citizens and elites. These basic comparisons provide an initial baseline for the importance of demographics to voter turnout. In chapter 3 we discuss in greater detail the theoretical importance of each of these demographic characteristics and how they might relate to each other. We then test whether the stability of education and income as primary determinants of voter turnout hold when we condition on other demographic characteristics of interest.
**Appendix 2.1: Current Population Survey: Sample and Variable Details**
Data for the Census Bureau's Current Population Survey November Supplement was taken from data provided by Unicon, a private firm that sells individual-level census data repackaged so that it is easier to extract common variables across multiple years. It is simply a repackaging of the data; coding decisions must still be made by the analyst (in this case, the authors). Thus we provide the Unicon variable name below.
**Turnout:** This is self-reported turnout. We treat blanks in the data set as missing data; we treat "dont know," "no response," and "refused" as not having voted. This coding lets us match what the Census Bureau reports as the turnout rate in their published summaries of CPS data. [Unicon Variable: _votecast_.]
**Education:** We use respondents' self-report of education to place them in either the bottom, middle, or top third of the education distribution for the year of the election. For people whose reported education category would straddle different thirds of the distribution, we use random assignment to place them. [Unicon Variable: For 1972–90, we use _grdhi_ (highest grade completed). For 1992–2008, we use _grdatn_ (highest grade attended), then combine this with _grdcom_ (grade completed).]
**Income:** We place respondents in the appropriate income quintile based on reported total family income. [Unicon Variable : _faminc_.] **Age:** Respondent's reported age. Recoded into six categories: 18–24, 25–30, 31–45, 46–60, 61–75, and 76–84. The variable is top-coded at 84 years. (Thus, we do not report on turnout of those over 84 years of age.) [Unicon Variable: _age_.]
**Gender:** self-reported. [Unicon Variable: _sex_.]
**Marital Status:** Self-report. Coded as 1 if married with spouse present; otherwise 0. [Unicon variable: _marstat_.]
**Citizen:** Coded as 1/0, self-reported. [Unicon Variable: _citus_ (1978–92), _citstat_ (1994–2008), _notreg_ (1972–76)]
**Three-category Race Variable:** This variable is coded as black, white, or other. It is based on respondent self-report. For 1972–88 the census coding was white, black, other. In 1996–2002 the categories American Indian or Alaskan Native and Asian Pacific Islander were added. We coded both of these as "other." In 2004 and 2008, respondents could choose from many sets of multiple-race categories. As these codings constituted barely 2 percent of respondents, we also coded these multiple race categories as "other." [Unicon Variable: _race_.] **Hispanic:** This is a self-report based on a question of origin or descent for 1976–2000. Responses of Mexican American, Chicano, Mexican, Puerto Rican, Cuban, Central or South American, or other Spanish were coded as Hispanic. For 2004 and 2008, this is a respondent's self-report to being Spanish, Hispanic, or Latino. This variable is not available for 1972. [Unicon Variable: _spneth_.]
**Other Ethnicity Codes:** Coding for white Hispanic, White non-Hispanic, black Non-Hispanic, and other were created by combining the three-category race variable and the Hispanic variable.
**Appendix 2.2: Additional Data on the Representativeness of Voters, 1972–2008**
**Table A2.2.1. Representativeness Scores for Demographic Groups, Comparing Voters to Citizens, 1972–2008.**
_Note_ : Entries are the ratios of the group's share of voters (based on self-report) to the group's share of the citizen voting-age population. Values less than one indicate underrepresentation of the group among voters, while values greater than 1 indicate overrepresentation of the group among voters. Computed by authors using data from the Current Population Survey.
**Figure A2.2.1.** Representativeness of Voters Compared to Citizens, by Race, 1972–2008.
_Note_ : Entries are the ratios of the racial group's share of voters (based on self-report) to the racial group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific racial group among voters, while values greater than 1 indicate overrepresentation of the specific racial group among voters. Computed by the authors using data from the Current Population Survey.
**Figure A2.2.2.** Representativeness of Voters Compared to the Voting-Age Population, by Race, 1972–2008.
_Note_ : Entries are the ratios of the racial group's share of voters (based on self-report) to the racial group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific racial group among voters, while values greater than 1 indicate overrepresentation of the specific racial group among voters. Computed by the authors using data from the Current Population Survey.
**Figure A2.2.3.** Representativeness of Voters Compared to Citizens, by Ethnicity, 1976–2008.
_Note_ : Entries are the ratios of the ethnic group's share of voters (based on self-report) to the ethnic group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific ethnic group among voters, while values greater than 1 indicate overrepresentation of the specific ethnic group among voters. Computed by the authors using data from the Current Population Survey.
**Figure A2.2.4.** Representativeness of Voters Compared to the Voting-Age Population, by Ethnicity, 1976–2008.
_Note_ : Entries are the ratios of the ethnic group's share of voters (based on self-report) to the ethnic group's share of the voting-age population. Values less than 1 indicate underrepresentation of the specific ethnic group among voters, while values greater than 1 indicate overrepresentation of the specific ethnic group among voters. Computed by the authors using data from the Current Population Survey.
**Figure A2.2.5.** Representativeness of Voters Compared to Citizens, by Age, 1972–2008.
_Note_ : Entries are the ratios of the age group's share of voters (based on self-report) to the age group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific age group among voters, while values greater than 1 indicate overrepresentation of the specific age group among voters. Computed by the authors using data from the Current Population Survey.
**Figure A2.2.6.** Representativeness of Voters Compared to Citizens, by Marital Status and by Gender, 1972–2008.
_Note_ : Entries are the ratios of the marital status/gender group's share of voters (based on self-report) to the marital status/gender group's share of the citizen voting-age population. Values less than 1 indicate underrepresentation of the specific marital status/gender group among voters, while values greater than 1 indicate overrepresentation of the specific marital status/gender group among voters. Computed by the authors using data from the Current Population Survey.
1. Total population figures from U.S. Census Bureau, 2011, Statistical Abstract, table 2. These numbers exclude armed forces personnel.
2. Proportion of Anglos for 2008 from U.S. Census Bureau, 2012 Statistical Abstract, table 6: Resident Population by Sex, Race, and Hispanic-origin Status: 2000–2009; proportion of Anglos for 1970 calculated from U.S. Census data.
3. Proportion of blacks and Hispanics for 2008 from U.S. Census Bureau, 2012 Statistical Abstract, table 6: Resident Population by Sex, Race, and Hispanic-Origin Status: 2000–2009; proportion of blacks for 1970 calculated from Gibson and Jung (2002, table 1).
4. For data on the proportion of noncitizens among the voting-age population, see McDonald (2011)
5. For data on the proportion of ineligible persons among the voting-age population, see McDonald (2011).
6. All incomes reported in these comparisons between 1972 and 2008 are in constant 2010 dollars, in this and the following two paragraphs. All figures are rounded to the nearest hundred dollars.
7. Median income data in this and the following paragraph on individuals over the age of sixty-five are taken from U.S. Census Bureau, Current Population Survey, Annual Social and Economic Supplements, table P-8AR: Age—People, All Races, by Median Income and Sex: 1947 to 2010.
8. The data reported for whites for 1972 are not reported for white non-Hispanics, while the data reported for 2008 are restricted to white Non-Hispanics because the census did not include a Hispanic origin question until after 1972. Given the relatively small proportion of Hispanics in the United States in 1972, we believe that this comparison for median income is fairly accurate. Median income data by race and Hispanic origin are taken from U.S. Census Bureau, Current Population Survey, Annual Social and Economic Supplements, table P-4. Race and Hispanic Origin of People (Both Sexes Combined) by Median and Mean Income: 1947 to 2010.
9. We will refer to the 90,000 as _respondents_ , though technically only those directly interviewed are truly survey respondents.
10. On the validity of the aggregate NES self-report estimates of voter turnout over time, see Burden (2000) and McDonald (2003).
11. While Katz & Katz (2010) suggest a method to correct survey misreports with other survey data, we would need to have validated survey data from each election year and assume the misreporting mechanism is similar across surveys as different as the CPS and the NES.
12. Bernstein, Chadha, & Montjoy (2003) find that misreporting rates across states are stable over the period they analyze: 1980 to 2000.
13. In chapter 6, where we use the NES self-reported vote, we also briefly discuss findings based on the NES validated vote for the years in which it is available, 1976–88. However, given the few years in which it is available, as well as the various problems with the validated vote measures identified by Traugott (1989), we do not believe that this alternative measure is clearly superior to the self-reported vote for our purposes.
14. Because the CPS is a survey of households, the sample is restricted to persons who are not incarcerated at the time of the survey. It thus includes people who are prohibited from voting based on a felony conviction but are not incarcerated.
15. As with our CVAP measure, our VAP measure does not include _currently_ institutionalized felons ineligible to vote in the denominator; it thus somewhat overstates the turnout of the voting-age populations.
16. Other individuals ineligible to vote include disenfranchised felons; see McDonald (2011).
17. See Traugott (1989) and Berent, Krosnick, & Lupia (2011) for analyses of why NES-reported turnout is so high.
18. In earlier work (Leighley & Nagler 1992b) we did not smooth out the quintiles generated by CPS categories, leading to criticism that they were not comparable over time (Freeman 2004). With the procedure we use here the quintiles are comparable over time.
19. Note that in the multivariable estimates that follow based on cross-sectional data, this coding has no effect. However, it is critical for comparability of estimates across years.
20. Much of the recent change in the income distribution has been _within_ the top quintile. However, we do not see much efficacy in examining the turnout rate of the top 1 percent of the income distribution, because even if they all voted, they would still only have 1 percent of the vote.
21. U.S. Census Bureau, Current Population Survey, table A-2: Percent of People 25 Years and Over Who Have Completed High School or College, by Race, Hispanic Origin and Sex: Selected Years 1940 to 2012; accessed October 25, 2011, at <http://www.census.gov/hhes/socdemo/education/data/cps/historical/tabA-2.xls>.
22. As with income, when CPS education categories do not allow us to precisely determine which third of the income distribution a respondent is in, we use random assignment.
23. We use the term _Hispanic_ to refer to those who identified as either Hispanic or Latino. Because conventional census and scholarly and popular terminology for different racial and ethnic groups has changed over our period of study, we use related terms interchangeably (e.g., _black_ and _African American_ , or _Hispanic_ and _Latino_ ), but try to use the terminology that most closely matches that used in the data sources upon which we rely. When there are changes over time, and we are referring to the entire period, we typically use the more recent terminology. For coding purposes, those listing multiple races were coded as "other" in 2004 and 2008.
24. See Highton & Burris (2002) on the differences in turnout rates among Hispanics of different national origin.
25. See Bartels (2008) for a similar observation.
26. Note that in this graph respondents can appear in multiple categories: the Hispanic category includes white Hispanics, and the white category includes white Hispanics.
27. There are so few nonwhite Hispanics in the sample that we omitted them here and only analyze white Hispanics. However, as this constitutes almost all Hispanics in the United States, our inferences about turnout of white Hispanics could be used to describe Hispanics generally without offering the caveat that they are white Hispanics.
28. Because of data availability, this graph starts in 1976 rather than 1972.
29. To be clear: we are talking about the gap in turnout _of citizens_!
30. See Teixeira (2010) for a fuller analysis of the electoral implications of demographic change.
31. Using similar measures, Shields & Goidel (1997) both expand on and confirm these conclusions in the case of midterm elections between 1958 and 1994.
32. The representativeness scores for income and education, as well as representativeness scores and similar representativeness graphs for race, ethnicity, age, gender, and marital status, are included in appendix 2.2.
33. Note that we are using percent of the original quantity here, not percentage points. Thus, the 27 percent drop in income share for the bottom income quintile is calculated based on the 1.5 _percentage-point_ drop from 5.5 to 4.0 as (100*[1.5/5.5]).
34. For example, many states restrict felons from voting.
35. In 2008, median household income of citizens was approximately $51,000, compared to less than $38,000 for noncitizens (DeNavas-Walt, Proctor, & Smith 2009).
36. See McCarty, Poole, & Rosenthal (2008) for more on the relationship between citizen and noncitizen income.
**Three**
* * *
**Theoretical Framework and Models**
Since 1972, "who votes" in presidential elections has both changed and remained the same in important ways. As was shown in chapter 2, individuals with higher levels of education and income are much more likely to vote than those with lower levels; Anglos and blacks are more likely to vote than Hispanics; married individuals are more likely to vote than single individuals, and older individuals are more likely to vote than younger individuals. Our findings on income showed that income bias did not change between 1972 and 2008 despite large increases in economic inequality: among citizens the wealthy were overrepresented among voters, and the poor underrepresented, at about the same levels as they were in 1972. In other words, the wealthy had proportionally more votes than the poor in 1972, in 2008, and in every presidential election in between.
If we simply wanted to know if the rich had proportionately more votes than the poor, and if this has changed since 1972, then we could stop here. However, one might argue that _who_ the wealthy and poor are in other respects has changed since 1972, and that ignoring these differences limits our understanding of income bias. It is one thing to say a rich person is more likely to vote than a poor person; it is a different thing to say that, all other things being equal (e.g., education, age, gender, ethnicity), a rich person is more likely to vote than a poor person. The latter approach assesses whether among otherwise identical individuals those with more income are more likely to vote than those with less income and therefore to be overrepresented in the electorate. Assessing income bias after conditioning on other demographic characteristics thus underscores the differences in the representation of the rich compared to the poor while accounting for other factors—such as education, age, and gender—associated with turnout. We refer to this type of income bias as _conditional income bias_ , which we measure by estimating a multivariable model of voter turnout using income as well as the other demographic variables we examined in chapter 2 as predictors of turnout.
Another advantage of this approach is that it provides additional information on the nature of the relationship between each demographic characteristic we measure and turnout over time. The bivariate relationships we described in chapter 2 may reflect in part the fact that social resources tend to be correlated. For example, individuals with higher levels of education also tend to have higher levels of income, and older individuals tend to be wealthier than younger individuals. Our analyses of the conditional relationships between demographic characteristics and turnout allow us to determine whether the bivariate relationships observed in chapter 2 remain when conditioning on other demographic characteristics.
This analysis provides valuable insights. If the conditional estimates of the effects of demographics such as race, ethnicity, and gender do not parallel the bivariate results (e.g., that older people are more likely to vote than younger people; that Hispanics are less likely to vote than whites; or that women are more likely to vote than men) then we can conclude that the changing bivariate turnout rates for these groups result from either the changing demographic (i.e., compositional) characteristics included in our model or other politically relevant factors. If the conditional estimates confirm the bivariate turnout patterns observed in chapter 2, then changes in turnout of these groups since 1972 must be explained by something other than the demographics we study (e.g., factors such as party contact, candidate targeting, or issue mobilization).
A few examples illustrate the importance of these analyses. We saw in chapter 2 a change in the relationship between age and turnout over time, with older individuals becoming increasingly more likely than younger individuals to vote. We would like to know if that is because the observable characteristics (income, education, etc.) of different age groups changed, or because there is something else about these different age groups that is changing and affecting turnout. Another example is provided by our finding that black turnout increased markedly between 1972 and 2008. We suspect this increase reflects in part that blacks now have higher levels of education and income than they did in 1972, but it might also reflect changes in other factors that influence turnout (e.g., political empowerment). While evidence on the latter possibility is beyond the scope of this book, we can provide some evidence on whether other demographic characteristics associated with race might help to account for the changes in African American turnout we observed in chapter 2. We also found in chapter 2 that Hispanic turnout increased much less than did black turnout. Is that because Hispanic levels of income and education did not go up as much as black levels of income and education, or is it because of something else? The same questions are raised by our finding that women now vote more than men. Is that because the relative levels of income and education have changed since 1972, or is it because something else changed?
If we do observe differences in turnout between groups in this chapter after conditioning on demographic characteristics, then we or other scholars would want to examine this by either considering other demographic characteristics not included in our model (e.g., labor force participation, attitudinal explanations, mobilization efforts of elites, or some combination of these factors).1
Finally, these multivariable analyses are relevant to discussions of the causes of voter turnout. While we are careful not to interpret our empirical estimates of the relationships between demographic characteristics and turnout as definitively causal in nature, a necessary condition for such relationships to be causal is that they are not spurious correlations observed because the demographic characteristic of interest is correlated with some other variable related to turnout. Thus, establishing a conditional relationship between each demographic characteristic and turnout is a necessary step toward determining causality. And, consistent with our focus on examining change—or the lack thereof—over time, we want to know if the conditional relationships among all of our demographic variables and turnout have changed since 1972.
**3.1 Costs, Benefits, and Demographics**
It would be impossible to find a study of voter turnout in presidential elections where demographic characteristics are not central to the enterprise. From the initial classic election studies (where voter turnout was at best a marginal consideration), to Wolfinger and Rosenstone's _Who Votes?_ (1980), to analyses of recent elections, most discussions of voter turnout put substantial emphasis on the relationships among education, income, age, gender, and turnout.2
Most studies, too, utilize a cost/benefits framework to interpret empirical evidence, assuming that voter turnout is a "rational," rather
than expressive, act. Downs first modeled individuals' decisions to vote as a reflection of the relative costs and benefits of voting: "if the returns to voting outweigh the costs, he votes; if not, he abstains" (1957, 260). Downs argued that when voting is costless, only indifferent individuals will abstain, but that when voting is costly, some indifferent individuals will vote and some with preferences will not vote. The latter cases result when the costs of voting outweigh the benefits, and they are not uncommon because the benefits associated with voting are, as Downs originally described them, "miniscule." Downs argued that four factors influence the individual's returns from voting: her perceptions of the policy differences between the parties, of the closeness of the election, of the value of voting itself, and of how many other citizens she thinks will vote (1957, 274).
Downs's portrayal of voting as a rational act provoked numerous scholars' efforts to explain its obvious empirical contradiction: if this theory yields a prediction of zero turnout, then why do we regularly observe citizens voting? Possible solutions to this puzzle include adding a fixed benefit to the calculus (i.e., the value of democracy continuing, civic duty); the real or perceived likelihood of being pivotal being incorrectly assumed (by theorists) or perceived (by citizens); and incorporating consumption or expressive benefits into the model.3
Perhaps most persuasive among these responses is Aldrich's (1993) claim that voting is not an especially good example of a collective action problem; it is instead such a low-cost, low-benefit activity that tiny differences in costs and benefits can result in positive turnout. Aldrich argues that conceptualizing the act of voting in this manner results in a critical role for strategic politicians, whose campaign tactics are likely to influence turnout levels regardless of individuals' rational calculations regarding voting.
We follow Aldrich's conceptualization of voter turnout as a rational activity, and we interpret the meaning of demographic patterns in voter turnout as reflecting individuals' differential abilities to subsume costs and benefit from voting. We also extend this argument in chapter 5 to consider the strategic actions of politicians.
**3.2 Model Specification**
We analyze the demographic determinants of voter turnout over time in this chapter by estimating a multivariable logit model of turnout that includes seven demographic characteristics (education, income, age, race, ethnicity, gender, and marital status) as explanatory variables for each presidential election year from 1972 to 2008. Our interest here is in assessing the relative strength of the relationships between demographic predictors and turnout, both with respect to changes over time and with respect to other demographic characteristics. And, of course, we are especially interested in changes in the conditional relationship between income and the probability of voting over time as additional evidence of patterns in income bias since 1972.
The data are taken from the Current Population Survey (CPS), November supplement, for each year.4 We would prefer to estimate a model of voter turnout consisting of all the demographic characteristics of interest as well as indicators of elite mobilization and state electoral laws that previous studies have reported to be associated with voter turnout. But that is not possible, for two reasons. First, the Census Bureau did not include a question regarding ethnicity in the 1972 CPS and, second, the Census Bureau does not have individual state identifiers for the 1976 CPS. This means that for 1972 and 1976, we cannot estimate our preferred model as we can from 1980 onward.
To provide as much continuity over time as possible, as well as assess the importance of Hispanic ethnicity as a determinant of turnout, we thus estimate two separate models in this chapter. The first model consists of all demographic variables available for the 1972 through 2008 elections, and thus does not include an indicator for Hispanic ethnicity, nor state-level variables. The second model consists of all demographic variables (including Hispanic ethnicity), and is estimated for 1976 through 2008.5 Since neither of these models include elite mobilization or state legal characteristics, we refer to both of these models as demographic models and distinguish them by specifying whether variables for Hispanic ethnicity are included or not.6
We estimate a logit model for each year. We do this because we are specifically interested in comparing coefficient estimates across years.7 Using alternative strategies such as pooling across years, or using a multilevel analysis or a shrinkage estimator (where we would assume parameters are drawn from a common distribution across years) would not allow us to make inferences regarding the changes over time that we are interested in. By estimating separate models for each year we do not assume a commonality across years and do not impose that assumption on the data-generating process.
In estimating a multivariable model we generally describe the "dependent" variable as being "caused" by the right-hand-side explanatory variables. However, the assumption of causality is not based on the data analysis or the model specified but on a model of the real world presumed by the analyst. The data analysis is merely showing the conditional relationships among the observed data and not establishing causality. To be consistent with conventional usage, we will refer to the _marginal effect_ of one of our explanatory (right-hand-side) variables on turnout. However, we are not making the claim that such observational evidence demonstrates causality.
To offer an example, we show below that the marginal effect of income on turnout (as measured by moving from the bottom to top quintile) is approximately 18 percentage points: ceteris paribus, persons in the top income quintile are 18 percentage points more likely to vote than persons in the bottom income quintile. While we refer to this as a _marginal effect_ , we do not necessarily believe that this is a strictly causal relationship. We are not asserting that simply giving a person more income would make her more likely to vote. What we are claiming is that, ceteris paribus, persons with more income _do_ vote more (approximately 18 percentage points more) than persons with less income.
As explanatory variables we include: measures of which third of the education distribution the respondent is in; measures of which income quintile the respondent is in; a set of dummy variables to measure which age group the respondent is in; and dummy variables for gender, marital status, race, and whether or not the respondent lives in a Southern state.8 By estimating the effects of demographics on turnout separately for each election year, we allow the marginal effects of each demographic characteristic to vary by election. We can then compare these specific election-year estimates over time to assess whether the strength of their effects on turnout increases, decreases, or remains the same.9
We present graphical representations of the estimates of the marginal effects of each demographic variable (i.e., the magnitude of the effect of that variable, controlling for all other demographic characteristics in the model) herein, using the demographics model that excludes Hispanic ethnicity so that we can include the entire time period. The graphical presentation of the marginal effects for Hispanic ethnicity are based on the demographics model that includes indicators of Hispanic ethnicity and therefore is restricted to 1976–2008.
For each variable, we plot the marginal effects for each presidential election year. These marginal effects represent how the probability of an individual voting would increase or decrease by changing from one category of the independent variable to another category (e.g., from high school graduate to some college education, or from black to white).
**3.3 Education and Income**
Empirically, there is universal consensus that both education and income are independently related to individuals' decisions to vote, though in most cases researchers find that education has a stronger relationship than income. Wolfinger and Rosenstone emphasized that education and income have distinct effects, with education being much stronger, and the effects of income moderating once individuals reached a "modestly comfortable standard of living" (1980, 34).
Rosenstone and Hansen (1993), too, point to the central role of education, with evidence from models of turnout estimated using American National Election Studies (NES) data from 1956 to 1988. However, they note that, controlling for demographics _and attitudinal characteristics_ , the effect of education is only slightly stronger than that of income, and more important in explaining voter turnout than other types of electoral participation (1993, chap. 5, especially 130–31).10 Whether their findings of the importance of education relative to income would hold had they not conditioned on attitudinal characteristics in their model is not clear.
Wolfinger and Rosenstone's initial theoretical argument regarding education highlighted three mechanisms by which education increased the probability of voting: by enhancing individuals' cognitive skills (and therefore reducing information costs), by increasing the gratification that individuals receive from politics (thus increasing benefits), and by providing (bureaucratic) experience that is useful in dealing with the costs of voting such as voter registration requirements. Similarly, Verba, Schlozman and Brady (1995) emphasize the skills as well as the enhanced psychological engagement (i.e., more positive attitudes toward and interest in politics) resulting from higher education as the mechanisms linking education with voter turnout (see also Hillygus [2005]).
An alternative argument is that the empirical correlation between education and voter turnout reflects not a causal relationship between education and turnout but a self-selection process: that individuals who choose to pursue more education are also more likely to vote, independent of any civic engagement effects as argued by Wolfinger and Rosenstone (1980).11
Wolfinger and Rosenstone (1980, 20–22) identify five possible reasons that income might be associated with voter turnout:
• poor people have less time to devote to matters not essential to everyday existence
• wealthy people have jobs that tend to increase their political engagement, regardless of education levels
• income determines one's social context, and thus wealthy individuals are more likely to be exposed to social networks with norms of civic duty and engagement
• wealthy individuals are likely to be especially engaged and aggressive in their political and social pursuits, otherwise they would not have succeeded in terms of income, regardless of education
• wealthy individuals have a greater "stake in the system"
Wolfinger and Rosenstone conclude that "rock bottom poverty seems to depress turnout somewhat. Beyond that income does not have much effect on turnout" (1980, 26). And on the basis of this empirical evidence from 1972, they conclude that their first possible explanation for the association between income and voter turnout is most likely the correct one: poor people have less time to devote to political matters as they are managing everyday survival instead.12
It is important to note that Wolfinger and Rosenstone's classic finding on the key role of education as a major determinant of voter turnout, more so than income, has not been directly examined since their initial study. Rosenstone and Hansen find that the relative importance of education and income as predictors of voter turnout may have changed over time, but this is conditioning on attitudinal variables. Since those attitudes could be intermediary variables between education and turnout, we cannot draw any inferences on the relative importance of education and income on turnout if we condition on attitudes. And it is possible that the relative import of education and income may have changed since 1972, as the level of economic inequality increased substantially.
We now turn to our empirical evidence addressing this point. Figure 3.1 shows how much less likely someone in the lowest education third is to vote than someone in the middle third, how much less likely someone in the lowest education third is to vote than someone in the top third, and how much less likely someone in the middle third is to vote than someone in the top third.13 Each point on the graph represents our estimate of the marginal effect of being in the higher rather than lower education group in a specific year. The vertical line associated with each point represents the 95 percent confidence interval for that estimated effect. And the horizontal trend line connecting the points represents the trend over time of the effect.14
While the effects of education are positive, we can also see that the effect of education is substantially _higher_ in 1972 and 1976 than it was for subsequent years. In 1972 someone in the middle third of the education distribution was approximately 17 percentage points more likely to vote than someone in the bottom third, ceteris paribus. However, by 1992 that effect had dropped to approximately 12 percentage points before rebounding a bit in subsequent elections. The graph clearly shows that the difference in turnout from the bottom to middle group was unusually large in 1972 and 1976. Another way to interpret the graph is to infer that respondents paid a particularly high price for being in the bottom of the education distribution in 1972 and 1976. Thus, in looking to determine the impact of education on turnout, Wolfinger and Rosenstone (1980) seem to have looked at an unusual year.
**Figure 3.1.** (facing page) Marginal Effect of Education as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between an individual in the higher education group and an individual in the lower education group, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, an individual with more education is more likely to vote than an individual with less education. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
It is important to note that while the marginal effect of education has decreased, it has decreased by a comparatively small amount. The magnitude of the effect has been reasonably stable and _large_ : the estimated difference in the probability of voting between otherwise similar individuals in the middle third and the bottom third of the education distribution varies from 12 percentage points to 17 percentage points. Over the entire time period, the difference in turnout between someone in the bottom education group and someone in the top education group with otherwise the same demographic characteristics never drops below 25 percentage points. And the difference in turnout between someone in the bottom education group and middle education group, ceteris paribus, never drops below 12 percentage points.
We next look at the conditional relationship between income and turnout over the time period. In figure 3.2 we show the difference in predicted turnout between persons in different pairs of income distributions. The first four graphs in figure 3.2 show the difference in predicted turnout between persons in the bottom quintile and each of the other four income quintiles, conditional on all other characteristics in our model. Comparing respondents in the bottom income quintile to respondents in the second through fourth income quintiles, we see that, conditioning on other characteristics, members of the bottom income quintile seemed to fall _farther_ behind members of quintiles 2–4 from 1972 through 2000. For example, while in 1972 we predict a 4 percentage-point gap between persons in the first and second income quintile, by 1984 that gap had risen to approximately 6 percentage points. Then, in 2004 and 2008, the conditional relationships between income and turnout seem to weaken: in each of the graphs comparing the bottom quintile to the other quintiles, we see smaller effects in 2004 and 2008 than we did in earlier elections.
In contrast, when comparing respondents in the bottom quintile to those in the top quintile, the conditional relationship between income and turnout varies somewhat from election to election from 1972 to 2008, sometimes increasing and sometimes decreasing. In 2004 and 2008, however, the marginal effects actually decrease to levels _lower_ than those we estimated for 1972 and 1976. The relatively flat trend lines suggest that the effects of income did not systematically increase or decrease between 1972 and 2008.
**Figure 3.2.** Marginal Effect of Income as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between an individual in the higher income group and an individual in the lower income group, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, an individual with more income is more likely to vote than an individual with less income. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
In the next two graphs of figure 3.2 we compare respondents in quintiles 2 and 3 to respondents in the top quintile, and in the last graph we compare the respondents in quintile 4 to respondents in quintile 5. In each case we see the same effect: the conditional relationship between respondents in quintiles 2–4 and respondents in quintile 5 has _decreased over time_. That is, once we condition on other observable characteristics, the gap in turnout rates between respondents in the top quintile and the other quintiles is smaller today than it was in 1972. This gap is still large, however: it is over 10 percentage points when comparing respondents in the second quintile to the top quintile, and approximately 7 percentage points when comparing the middle quintile to the top quintile. But it has been decreasing.15
To summarize, we find no change in income bias, conditioning for other demographics, when comparing the bottom and top quintiles. Conditional on other characteristics, respondents in income quintiles 2–4 are voting _more_ relative to respondents in the very top income quintile than they were thirty years ago. Thus, while those in the top quintile have gotten a larger share of _income_ , they have not gotten a larger share of _votes_ conditional on their other demographic characteristics (education, ethnicity, and age). This finding of no change in share of the votes for the top income quintile when conditioning on other demographic characteristics is the same as our bivariate finding in chapter 2 comparing income and turnout directly.
Since there is particular interest in the question of whether education or income is more important to turnout, we report the predicted probability of voting for different combinations of education and income in both 1972 and 2008 (see table 3.1). Here again we are showing the predicted probability of voting for a hypothetical respondent, with other demographic characteristics set as they are for the graphs in this chapter. We discuss the results for 2008 here, but the discussion would be virtually identical for 1972 given the similar estimates.
We first consider the effect of changes in level of education. In 2008, the probability of voting for our hypothetical individual in the top income quintile increased from 0.64 to 0.87 (an increase in probability of 0.24) as her position in the education distribution improved from the bottom third to the top third. We next consider the effect of changes in level of income. The predicted probability of our 2008 hypothetical individual in the top education group but lowest income group is 0.74, and increases to 0.87 when shifted to the highest income group (an increase in probability of 0.13).
It thus appears that education still trumps income as a predictor of turnout. A hypothetical individual in the highest education group is anywhere from 23 to 30 percentage points more likely to vote than a similar individual in the lowest education group, whereas a hypothetical individual in the highest income group is anywhere from 13 to 20 percentage points more likely to vote than a similar individual in the lowest income group.16
**Table 3.1. The Relative Effects of Education and Income on Turnout for a Hypothetical Respondent, 1972 and 2008.**
_Note_ : Cell entries represent the probability that a hypothetical respondent with the specific combination of education and income characteristics will vote in either the 1972 presidential election (top panel) or the 2008 presidential election (bottom panel). The probabilities are computed by the authors using data from the Current Population Survey to estimate the demographic model described in section 3.2 for a married white woman, age thirty-one to forty-five, who lives outside the South.
**3.4 Race and Ethnicity**
Most analyses of participation that consider race and ethnicity do _not_ focus on the relative turnout rates of blacks, whites, and Hispanics conditioning on the demographic characteristics of these groups. Instead, they try to explain what drives participation in the different groups, or consider the relative participation of members of the groups conditioning on different sets of political attitudes such as political interest or partisanship. We are interested in answering questions of a different type, such as: Are Anglo and Hispanic citizens of equal ages, income, and education levels equally likely to vote?
To answer these questions, we focus exclusively on demographics. If we include attitudinal factors in our model of turnout, they can mask the underlying relationship between demographic characteristics and turnout. For instance, if highly educated people are also highly interested in politics, than including interest in politics in our models will mask the true underlying relationship between education and turnout.17 Since we are confident that the demographic characteristics precede the development of attitudes, we are not in danger of reporting spurious relationships between demographic characteristics and attitudes with our model specification.18
Empirical evidence on the effects of race and ethnicity on voter turnout _independent of attitudinal factors_ is sparse, as many typical data sources simply contain too few African Americans, Latinos, or Asians to allow for accurate comparisons across racial/ethnic groups. Ramakrishnan (2005) points to census data from 1984 and 1994 to show that Hispanics and blacks voted at much lower rates than did Anglos in those years.
The few analyses that examine race-related differences in turnout produce mixed results, but these studies typically include political, legal, and attitudinal characteristics as predictors of turnout. As a result, their substantive conclusions regarding race-related differences solely as a result of demographics are tentative. Rosenstone and Hansen's (1993) bivariate evidence indicates that blacks are less likely than whites to vote, comparing self-reported National Election studies (NES) data from 1972 through 1988 for blacks and whites. Their multivariable analyses of changes in black turnout include other demographic variables and various attitudinal mobilization measures, and also suggest that blacks are less likely to vote than whites conditional on those attitudinal characteristics.19 In contrast, Leighley and Nagler (1992b) find that blacks were more likely than whites to vote, based on data from the 1984 Current Population Survey (CPS), when controlling for other demographic characteristics such as education, income, and age, along with election law characteristics of the states in which citizens reside.20
For Hispanics, we have only a few studies of demographics and turnout, and these studies, too, reach different conclusions.21 Wolfinger and Rosenstone (1980) report that controlling for demographics (and election laws, but not attitudes), Chicanos in 1972 were slightly more likely than Anglos to vote (by about three percentage points), while Puerto Ricans in 1974 were less likely than Anglos to vote.22 Two more recent studies also lead to conflicting findings. Abrajano and Alvarez (2010, 79) describe how turnout rates of Hispanics have lagged behind those of blacks in every election since 1972 (though the gap narrows slightly in recent elections), and document lower levels of Hispanic turnout compared to black and Anglo turnout by education, by income, by age, and by residential mobility (separately); yet they do not consider a more complete demographic model of turnout.23 In contrast, Highton and Burris (2002), who examine the 1996 CPS data, conclude that Hispanics vote as often as whites once demographic characteristics and region are controlled for.24
Hence, studies of turnout of Hispanics compared to non-Hispanics do not provide a convincing answer as to whether Hispanic turnout, independent of attitudes or context, has changed over time. That means we do not know whether—or the extent to which—the gap between the voting rates of Hispanics and Anglos has shrunk over time. Based on the few conflicting studies that we discuss above, we remain agnostic as to what we will find regarding turnout of Hispanics in presidential elections over this time period when focusing on demographics alone.25
**Figure 3.3.** Marginal Effect of Race as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between a black individual and a white individual, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, a black individual is more likely to vote than a white individual. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
Figure 3.3 shows the difference in voting between whites and blacks, conditional on the other demographic characteristics (education, income, age, and gender) in our model. Figure 3.3 shows that not only do blacks not vote less than whites once we control for demographic factors, but that in every election since 1984 blacks have been voting at substantially higher rates than whites, conditioning on other demographic characteristics. The two largest black-white differences are seen in 1984 and 2008, years when Jesse Jackson and Barack Obama ran for president, respectively. But even if we were to remove these outliers, we would still see the same trend of a gap increasing from pre-1984 to post-1984, with the three largest values being in the three most recent non-Obama elections.
This, of course, differs from our findings in chapter 2, where we compared the voting rates of whites and blacks, _not_ conditioning on other demographic characteristics, and found that whites have voted at higher rates than blacks in every election except 2008. For questions relating to the _marginal_ difference in voting rates between whites and blacks, after controlling for basic characteristics such as education and income, the evidence is clear: blacks vote at higher rates than whites. It is the other characteristics of blacks (most likely lower levels of education and income) that result in a lower overall voting rate as a group than whites.
Our analyses of Hispanic turnout are presented in figure 3.4, in which we plot the differences in the probability of voting for Hispanic whites versus non-Hispanic whites, conditioning on other demographics. Unlike the analyses in the previous figure comparing whites and blacks, here we consider only non-Hispanic whites (i.e., Anglos) rather than including Hispanics in the group of whites. And because there were so few nonwhite Hispanics in our data, we are only examining the turnout of white Hispanics here. The figure suggests that, conditioning on other demographics, in every election since 1976 Hispanic whites were less likely to vote than non-Hispanic whites. In fact we see _no increase_ in turnout of white Hispanics relative to Anglos, conditioning on demographics, in the last thirty years. Thus it would be wrong to think that participation differences between Hispanics and Anglos are disappearing. And these results suggest that the turnout gap between Hispanics and Anglos would not disappear even if Hispanic and Anglo demographic characteristics such as education, income, and age become more similar.
**Figure 3.4.** Marginal Effect of Ethnicity as a Predictor of Turnout, 1976–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between a white Hispanic individual and a white non-Hispanic individual, holding all other variables in the multivariable model constant. Negative values indicate that, ceteris paribus, a Hispanic individual is less likely to vote than a white non-Hispanic individual. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
The implication of this is that the gap in Hispanic and Anglo turnout we observed in chapter 2 is _not_ spurious. Even conditioning on demographic characteristics such as income, education, and age, white Hispanics are still substantially less likely to vote than are Anglos. That this gap has not changed over time is contrary to what we might expect based on claims made by many political consultants and activists regarding the importance of the increasingly large Hispanic electorate.
**3.5 Age**
Although Wolfinger and Rosenstone (1980) emphasize the importance of education and income as predictors of turnout, they also confirmed and clarified the important role of age, especially for those individuals with lower levels of education and income. They suggest that aging is a proxy for life experience, and what one does not gain in formal education can be obtained through the experiences of daily life. Moreover, they find that the conventional wisdom of the day, that there is a drop-off in turnout in old age, is incorrect: lower levels of voter turnout over the age of sixty actually reflect differences in education, income, and marital status (see also Jennings and Markus [1988]).
Similarly, Rosenstone and Hansen (1993) find that age is the only demographic characteristic estimated to have a greater (conditional) effect on voter turnout than education and income and find no evidence to suggest that the oldest age group psychologically or physically disengages from politics. They conclude that the positive effect of age on turnout reflects the reduction of information costs associated with life experience (see also Strate et al. [1989]).
Of particular interest to Wolfinger and Rosenstone was a life-cycle model of age effects that was dominant at the time. The life-cycle model posits that participation increases as individuals mature and take on adult roles, and then decreases in the later years, when the physical and psychological costs of voting are likely greater. Highton and Wolfinger (2001), however, test adult role theory using a model of participation in which residential stability, marriage, home ownership, full-time employment, being a student, and leaving home are used as indicators of taking on adult roles. Among individuals age twenty four and under, they find little support for any of these role-takings increasing turnout.26
They conclude that theoretical arguments regarding the differences in the turnout of the young and the elderly must look elsewhere. They provide some evidence that aging as learning or experience is likely one possibility: in their sample, the probability of voting increased by about 5 percentage points as individuals aged from eighteen to twenty four, with other roles and demographics controlled for. A second possibility they raise, but provide no evidence for, is that age-related differences reflect generational differences—that is, factors unique to the political and social context in which each generation entered the political world, and which result in high turnout among older generations and lower turnout among younger generations.27
Another possibility is that differences in political mobilization help to account for the relationship between age and participation. As Rosenstone and Hansen (1993) note, political elites are more likely to mobilize those who have a record of voting, and those who are easily reached and socially connected. This suggests that political elites are less likely to target younger individuals. Indeed, Beck and Jennings (1979) demonstrate that in the 1970s, the relationship between age and participation in protests was reversed, with the younger more likely to participate than the elderly, and attribute this variation to the intense mobilizing efforts of antiwar interests.
Alternatively, the critical role of political elites in structuring participation, as noted by Rosenstone and Hansen, also suggests that over the past decade, with the increasing strength and visibility of organized interests such as the American Association of Retired Persons (AARP), we should likely see less of a decline in turnout of the oldest age groups, or that such a decline, if it exists, is delayed. We examine this possibility using the marginal effects of age over time (see fig. 3.5). We compare the turnout of each age group to that of forty-six to sixty-year-olds, whom we chose as an arbitrary comparison group.
The first graph in figure 3.5 shows the marginal impact of going from ages 18–24 to ages 46–60. Whereas, conditioning for other demographic characteristics in 1972 young people were almost 25 percentage points less likely than 46- to 60-year-olds to vote, that difference had dropped to approximately 12 percentage points in the 2004 and 2008 elections (though we note that it remained over 20 percentage points in each election prior to 2000).28 At the other end of the age spectrum, the fourth and fifth graphs in figure 3.5 show that, ceteris paribus, 61- to 75-year-olds have been voting more than 46- to 60-year-olds at a fairly steady rate. What is perhaps surprising is that they also show that it is the very old, 76- to 84-year-olds, whose turnout relative to 46- to 60-year-olds has been increasing the most.
**Figure 3.5.** Marginal Effect of Age as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between an individual in the older age group identified and an individual in the younger age group identified, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, an older individual is more likely to vote than a younger individual. Each vertical bar provides a 95 percent confidence interval. The trend line is an OLS regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
Generally, these age-related patterns in turnout affirm previous findings on the positive relationship between age and turnout. While increases in turnout rates, ceteris paribus, among the youngest age group and the oldest age group might reflect changes in the mobilization patterns of parties, candidates, or the AARP targeting these two groups, how enduring—rather than election-specific—these patterns are is not clear. Anecdotal discussions of the 2004 and 2008 elections have emphasized the importance of the Obama campaign's priority on youth mobilization, but more systematic analyses will be required to determine whether these mobilization efforts, as opposed to other differences in this youth cohort, account for the age patterns we observe.
Since mobilization by a group such as the AARP would target all respondents over 50, and we notice the turnout of the 76- to 84-year-old group increasing relative to the 61- to 75-year-old group, the higher levels of turnout of this older group, ceteris paribus, are likely _not_ caused by increased mobilization efforts. It might be an increase in the physical well-being of the oldest age group, or it could be the result of reforms such as no-excuse absentee voting making it easier for the very old to vote. But the contrast in behavior between the oldest group and the second oldest group is striking.
**3.6 Gender and Marital Status**
Wolfinger and Rosenstone (1980, 37–39) reported that women in 1972 were only slightly (by approximately 2 percentage points) less likely than men to report having voted, and that married individuals voted at substantially higher rates than single individuals.29 According to the Center for American Women in Politics at Rutgers University, prior to 1980, a smaller percentage of women reported voting in presidential elections than did men, but since 1980, a greater percentage of women, compared to men, have reported voting. Consistent with this data on self-reported turnout rates of men and women, we have found that an important difference between the 1972 and 1984 elections in the determinants of voter turnout was the marginal effect of gender: controlling for other demographic characteristics such as education, income, and age, women were significantly more likely than men to vote in 1984 (Leighley and Nagler 1992b). What we expect to find, then, is that, controlling for other demographics, women are more likely than men to vote in presidential elections.
Differences in turnout based on marital status are typically explained as a result of the increased stability and social integration associated with marriage, or as a reflection of the social networks and information contexts introduced by a new partner (Kingston & Finkel 1987, Stoker & Jennings 1995). Stoker and Jennings (1995) conclude that marital transitions are especially important influences on voter turnout compared to other types of participation: because the decision to vote is typically made within a fixed time period and is a relatively discrete act, the decision is more likely to be a joint one, influenced by a spouse.30 Based on this evidence, then, we expect to find a consistent (and positive) effect of being married, rather than single, on voter turnout.
In figure 3.6 we can see that the marginal difference in voting rates between women and men, conditioning on other demographic characteristics, increased sharply between 1972 and 1984—and has remained substantial ever since. The steadiness of the gap suggests that it does not depend on short-term electoral context. What is interesting here is not the trend over thirty years, but the fact that there has been such a substantively large effect for the last twenty years. When we control for demographic factors women are about 4 percentage points more likely to vote than men. Given the size of the two groups in the electorate, this gap translates to a large difference in the share of total votes. Specifically, it means that, given equivalent levels of education and income, women would have 52 percent of the votes and men would have only 48 percent of the votes in a presidential election.
**Figure 3.6.** Marginal Effect of Gender as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between a woman and a man, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, a woman is more likely to vote than a man. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
Figure 3.7 shows the difference in voting rates between single and married people, conditioning on other demographic characteristics. While we see no trend over time in the data, what is interesting is that married people are _substantially_ more likely to vote than single people, with the difference generally within the range of 8 to 10 percentage points. In 2008, a married person was over 7 percentage points more likely to vote than a single person with otherwise similar demographic characteristics. While we see year-to-year variation in the magnitude of this effect, it is nonetheless large and positive in every election.
**Figure 3.7.** Marginal Effect of Marital Status as a Predictor of Turnout, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between a married individual and an unmarried individual, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, a married individual is more likely to vote than a single individual. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
**3.7 Conclusion**
The analyses we report in this chapter assess whether what we have observed in the bivariate cases (chapter 2) also holds when we control for other demographics characteristics. The demographics of interest here—education, income, age, race, ethnicity, gender, and marital status—are all highly correlated. Thus, as a precursor to even thinking about any causal connection, we need to know if each variable is related to turnout _when_ we condition on other demographics, or if any of these variables are instead reflecting other important characteristics of the demographic subgroup. As an important example, education and income are the canonical pair of variables that are highly correlated with each other, _and_ with many outcome variables of interest to political scientists. We naturally want to know if the relationship between education and turnout is spurious, merely an artifact of education's correlation with income. And conversely, we want to know the same
about the relationship between income and turnout, as well as the relationships between ethnicity and turnout, age and turnout, and gender and turnout.
The overall pattern of our results is consistent with the relationships we observed in chapter 2, and so we conclude that these bivariate findings were _not_ spurious. Even conditioning on other characteristics, the rich vote more than the poor, the better educated vote more than the less educated, the old vote more than the young, Hispanics vote less than Anglos, women vote more than men, and married individuals vote more than singles.
One notable exception in which the marginal effects we report in this chapter differ from the bivariate results reported in chapter 2 is the turnout of blacks. Blacks vote less than whites, but the multivariable analysis shows that all of this difference can be accounted for by the other demographic characteristics of blacks compared to whites. Blacks as a group have overall lower levels of income and education than whites. But when we condition on our other demographic characteristics, we see that blacks are _more_ likely to vote than whites, and have been since 1984.
With a few exceptions, these relationships (again, conditioning on the other demographic characteristics included in our model) have generally been stable from 1972 through 2008. The effect of income and education remain most striking. The gap in turnout between an individual in the bottom third of the education distribution and an individual in the top third is approximately 25 percentage points once we condition on other demographic characteristics. That is a chasm in expected turnout rates for otherwise demographically similar individuals. For income, the relationship between the bottom fifth and top fifth of the income distribution once we condition on other demographic characteristics is approximately 20 percentage points. The stability of this relationship affirms one of Wolfinger and Rosenstone's (1980) central findings: education is a more influential predictor of turnout than is income.
In our bivariate comparisons in chapter 2 we saw a stable relationship between income and turnout. The rich vote more than the poor, and that relationship has not changed substantially. In the multivariable context, we see that, conditional on our other demographic measures, the same conclusion holds for the most part. Yes, turnout of the bottom quintile has dropped somewhat over time, but the increase in turnout for moving from quintiles 2, 3, or 4 into quintile 5 (the top quintile) has decreased somewhat. And moving from the bottom of the education distribution to the middle third is not worth as much as it used to be, nor is a move from the middle to the top. But the magnitude of these changes—which cut in different directions as to the import of education and income in predicting turnout—is quite small. We conclude that the most important findings regarding education and income from the multivariable analysis is that they remain critical and overwhelming in their roles as correlates of turnout.
We discuss the broader implications of these findings in chapter 7. But here we note that the stability of education and income as predictors of turnout over a period of tremendous changes in inequality of income is striking. We find that conditional income bias remains essentially the same over the entire period. To the extent that institutional stability is important to a functioning democracy, this is may be perceived as good news. The concern, of course, is that it may suggest something about the unresponsiveness of American electoral politics.
Another surprising finding of stability between 1972 and 2008 is that of the lower turnout rates of Hispanics relative to Anglos, even when controlling for demographic differences between the two groups. We saw in chapter 2 that Hispanic turnout was substantially lower than Anglo turnout (figs. 2.3 and 2.4). But we would expect much of that gap to be explained by the demographic differences between the two groups (levels of education and income, as well as age). Popular commentary suggests that Hispanics comprise a group that has potential political power because of its size and mobilization potential, especially in light of the salience of immigration as a political issue over the past decade. Yet the persistent gap in turnout between Hispanics and Anglos, even after conditioning on demographic characteristics, is not easily explained despite its political and normative importance.31
An important exception to patterns of stability that we have identified is that of gender. In chapter 2 we found that women vote more than men, and that this gap has been increasing steadily over time. In this chapter we have seen that the magnitude of the difference in turnout between men and women, when conditioning on other demographic characteristics, is striking. According to our estimates, since 1996 a woman is 5 percentage points more likely to vote than a man of comparable income, education, and age.32 That is an extremely large difference. To put that in perspective, a woman in the third income quintile is approximately as likely to vote as a man of similar age and education in the fourth income quintile.
**_3.7.1 A Note on Bivariate and Multivariable Results_**
In chapters 2 and we have represented the turnout of different demographic groups in three different ways: (1) the level of turnout of the group (figs. 2.1–2.7); (2) the representativeness ratio for the group (figs.
2.8–2.11 and appendix figs. A2.2.1–2.2.6; and in the chapter appendix table A2.2.1); and (3) the marginal difference in turnout between each group and other groups, _conditional_ on all other observed demographic characteristics (figs. 3.1–3.7 and chapter appendix fig. A3.2.1). Each representation of turnout tells a different story. The graphs of the level of turnout for a demographic group over time show whether turnout for the group has been increasing or decreasing over time, as well as how much it varies from election to election. And since each demographic group's level of turnout is graphed along with the level of turnout of other groups based on the same demographic characteristic (e.g., the bottom education group compared to the middle education group), these graphs show the difference in turnout over time across different subgroups of each demographic characteristic that we measure.
The representativeness graphs show whether a group is over- or underrepresented among the voters relative to its share of the population or to its share of the citizen population. The representativeness ratios are the statistics that indicate the over- or underrepresentation of each group.
Note that the representativeness ratio is different from a group's _share_ of the votes, as the latter depends not just on the turnout rate for the group but also on the group's size. This means that a group can maintain the same turnout rate while also gaining or losing in the share of votes if its group size changes. We generally do not focus on a group's share of the votes. But table 2.2 lists the vote shares for each group (the percentage of voters) for 1972 and 2008. Here we can see that whereas in 1972 the eighteen- to twenty-four-year old age group had 14.3 percent of the votes, the group's share of the votes had shrunk to 9.3 percent in 2008. This is not because the members of the group were voting less but because the group had simply decreased in size as a proportion of the electorate.
Finally, the graphs of conditional differences between groups presented in this chapter show how much, if any, of the bivariate relationships presented in chapter 2 remain after we condition on other observable demographics. So while in chapter 2 we saw difference in turnout rates between the bottom and top income quintile of almost 30 percentage points, in the present chapter we see that the difference in turnout rates for two _otherwise similar_ people from the bottom income quintile versus the top income quintile was less than 20 percentage points in most years.
Having extensively examined the demographics of turnout, we shift now to investigating several other factors associated with voter turnout. Chapter 4 focuses on changes in election laws governing registration and election administration. Changes in these laws since 1972 have been immense, yet few studies offer definitive, national-level evidence as to whether these legal changes have influenced turnout in presidential elections. Chapter 5 then considers one aspect of the political determinants of turnout: the policy choices offered by candidates. Demographics aside, if "there's not a dime's worth of difference" between presidential candidates, citizens might reasonably and rationally choose to stay home. These analyses offer some insight as to what candidates offer citizens as policy choices, and whether citizens recognize these choices as they decide whether to vote or not. These chapters are especially important in light of the findings of the centrality and stability of demographics as predictors of turnout, for if these relationships rarely change, the question is when, and if, they might. Perhaps changes in electoral laws and changes in policy choices offered by candidates could provide a mechanism for change.
**Appendix 3.1: Estimation Results for the Demographic Models of Voter Turnout**
**Table A3.1.1. Logit Estimates of Demographic Model without Hispanic Variables, 1972–1988.**
_Note_ : Entries are estimated logit coefficients followed by the associated t-statistic. See text for a description of variable coding. Estimated by the authors using data from the Current Population Survey.
**Table A3.1.2. Logit Estimates of Demographic Model without Hispanic Variables, 1992–2008.**
_Note_ : Entries are estimated logit coefficients followed by the associated t-statistic. See text for a description of variable coding. Estimated by the authors using data from the Current Population Survey.
**Table A3.1.3. Logit Estimates of Demographic Model with Hispanic Variables, 1976–1988.**
_Note_ : Entries are estimated logit coefficients followed by the associated t-statistic. See text for a description of variable coding. Estimated by the authors using data from the Current Population Survey.
**Table A3.1.4. Logit Estimates of Demographic Model with Hispanic Variables, 1992–2008.**
**Appendix 3.2: Additional First Differences for Income**
**Figure A3.2.1.** Marginal Effect of Income Using Different Income Comparisons, 1972–2008.
_Note_ : Each point in the graph represents the estimated difference in the probability of voting between an individual in the higher income group identified and an individual in the lower income group identified, holding all other variables in the multivariable model constant. Positive values indicate that, ceteris paribus, an individual in the higher income group is more likely to vote than an individual in the lower income group. Each vertical bar provides a 95 percent confidence interval. The trend line is an ordinary least squares regression line fitted to the points. Estimated by the authors using data from the Current Population Survey; see section 3.2 for model details.
1. See, for example, Gay (2001); Tate (1993); and Barreto (2005).
2. See, for example, Abramson, Aldrich, & Rohde (2003); Campbell et al. (1960); Lazarsfeld, Berelson, & Gaudet (1948); Rosenstone & Hansen (1993); Verba & Nie (1972); and Verba, Schlozman, & Brady (1995).
3. Examples of these arguments can be found in Engelen (2006); Ferejohn & Fiorina (1974, 1975); Fiorina (1976); Hinich (1981); Palfrey & Rosenthal (1985); Riker & Ordeshook (1968); and Schuessler (2000). See also Jackman (1993) and Miller (1986) on these points.
4. Coding details for each demographic variable are included in appendix 2.1.
5. In the multivariable models which include Hispanic ethnicity we use a more detailed coding. We include dummy variables for white non-Hispanic, white Hispanic, and black non-Hispanic in the model, with other non-Hispanic being the omitted category.
6. We note that given the small proportion of Hispanics in the population in 1972, differences between the two sets of model estimates are quite small.
7. Models were estimated using _R_ , and with standard errors clustered at the state level.
8. In 1976 the CPS does not identify the specific state a respondent is in for every state, as several states shared values for the state identifier variable. But we can identify respondents in every state except Arkansas, Deleware, Kentucky, Louisiana, Maryland, Oklahoma, Tennessee, and Virginia as either being in the South or North. Respondents in the states listed above are dropped from the analysis here for 1976. We include the indicator for the South since the region has long had politics distinct from the rest of the United States, and we want to condition on any such regional effect.
9. Complete model estimation results for both demographic models are included in appendix 3.1.
10. For studies of more recent elections, see Leighley & Nagler (1992a, 1992b, 2007); Nownes (1992); Teixeira (1992); and Timpone (1998).
11. See also Henderson & Chatfield (2011); Kam & Palmer (2008, 2011); Mayer (2011); and Tenn (2007).
12. This would be consistent with Rosenstone's (1982) analysis of "economic adversity," which suggested that poor people are less likely to vote.
13. The first differences in probabilities presented in the graphs are computed for a hypothetical respondent where all variables we are conditioning on are set to their mode or mean. So our hypothetical respondent is a white married woman between the ages of thirty-one and forty-five in the middle income quintile, living outside the South.
14. The horizontal line is the ordinary least squares regression of the estimated effects against a time trend.
15. We show the relationships between respondents in quintiles 2, 3, and 4 in appendix 3.2. Those relationships have been stable over time, and in the expected direction.
16. And note that this is a conservative conclusion, as the comparison is across only education thirds, but across five categories of income, which should bias our test in favor of a larger effect for income.
17. Alternatively, if we consider ethnicity as the treatment, then the attitudinal variables are all posttreatment, and we do not want to consider those in examining the relationship between demographic characteristics and turnout.
18. Education attainment and income are also, of course, posttreatment here, but they are part of our question of interest.
19. They also suggest in a footnote that their estimated gap in turnout between whites and blacks may underestimate the difference due to blacks possibly overreporting at a higher rate than whites; see Rosenstone and Hansen (1993, 60n 20).
20. Also inconsistent with Rosenstone and Hansen's finding that blacks vote less than whites conditioning on attitudinal characteristics is Verba, Schlozman, Brady, and Nie's (1993) conclusion, using a cross-sectional study of political participation conducted in 1985, that any differences in African American and Anglo self-reported turnout rates in presidential elections reflect differences in civic resources (i.e., education, income, language skills, and opportunities for civic involvement).
21. We should note that many analyses of Latino turnout are more complex than those used in the study of African American turnout. As in studies of Asian Americans, studies of Latino turnout often include four demographic or legal characteristics other than socioeconomic status: citizenship status, immigrant status (i.e., generation), country of origin and English-language proficiency, see, for example, Arvizu & Garcia (1996); Cho (1999); DeSipio, Masuoka, & Stout (2006); Leighley & Vedlitz (1999); Ramakrishnan (2005); Rocha et al. (2010); and Wong (2006). Since being a citizen is a necessary condition for voting, our models only include citizens, and thus our analyses compare Hispanics citizens with other citizens.
22. Rosenstone and Hansen say little about Hispanic participation relative to Anglo participation, most likely due to the small sample sizes of Hispanics in the NES. Cassel (2002), who uses validated NES data, finds no statistically significant differences between Anglos and Hispanics while controlling for demographic characteristics and party contact. However, she makes this claim based on a sample of only 255 Hispanic voters from 1984 and 1988.
23. See also Bullock & Hood (2006), who report that Hispanics in several southeastern states vote at lower rates than Anglos.
24. Highton & Burris (2002) examined political participation of Hispanics in 1996 and concluded that Hispanic versus non-Hispanic turnout differences disappeared when adding state dummies to a model similar to the one we use. We added state dummies to our model, and we found this not to be the case in 1996, _and_ that 1996 was an unusual year for Hispanic versus non-Hispanic turnout in that the conditional difference between the two groups reported in the CPS was noticeably less than in other years.
25. We recognize, of course, that some elections might witness distinct patterns of race-related differences in turnout. For example, Tate (1991, 1993) documents the mobilizing effects of Jesse Jackson's presidential bids in 1984 and 1988 on the African American community, while some recent discussions emphasize higher turnout of Latinos in California in response to a heightened anti-immigrant (i.e., Hispanic) political climate; see Barreto (2005).
26. There is limited and mixed evidence regarding the effects of parenthood on voter turnout. Jennings (1979) finds that being a parent increases voter turnout of women in school (but not national) elections, while Pacheco and Plutzer (2007) find that adolescents who become parents are less likely to vote.
27. Plutzer's (2002) analysis of turnout as a developmental process provides some evidence on generational differences in initial starting points. He notes that declining levels of partisanship might well result in longer-term declines in voter turnout, as youth are less likely to be exposed to partisan parents and political behaviors.
28. This should not be interpreted to mean that youth are better represented than they were previously. This is a ceteris paribus result, conditioning on other demographic characteristics. Also, this result is for youth turnout _relative_ to the turnout of a specific age group (46- to 60-year-olds). To see if people ages 18–24 are actually more represented among the voters in 2008 than they were in previous years, one should refer to the representativeness ratios presented in chapter 2.
29. Studies examining gender differences in voter turnout explained lower levels of female voting with reference to older women being socialized prior to the adoption of the constitutional amendment granting women the right to vote (Wolfinger & Rosenstone 1980, 43). More recent studies point to the mobilizing effects of the women's rights movement and the increasing prominence of women in national government as increasing the likelihood of women's political engagement (see, for example, Atkeson 2003; Conway, Ahren, & Steuernagel 2004; Rosenstone & Hansen 2003). Other studies note the importance of women being elected to office for stimulating the political interests of women, which tends to be lower, on average, than men's (Verba, Burns, & Schlozman 1997). Finally, another factor distinguishing men and women is women's lower probability of being mobilized (Rosenstone & Hansen 1993, 140–41n15), part of which is explained by working at home or working in lower-status jobs (Schlozman, Burns, & Verba 1999).
30. We also found that married individuals were more likely than single individuals to report voting in the 1984 presidential election (Leighley & Nagler 1992b).
31. Our model omits factors that others have argued explain Hispanic political behavior, such as national origin, language, and generational differences. Including these could reduce our estimate for the ceteris paribus difference in Hispanic and Anglo turnout.
32. We use the phrase "5 percentage points more likely" to indicate an increase of 0.05 in the probability of an individual voting. While the graphs in this chapter represent the change in the probability of voting for a single hypothetical individual, we discuss percentage point increases in the text to maintain comparability with discussions of increases in group turnout.
**Four**
* * *
**The Legal Context of Turnout**
_Voter Registration and Voting Innovations_
W _ho Votes?_ (Wolfinger & Rosenstone 1980) provided new evidence of the effects of registration rules on the probability of voting. Examining numerous characteristics of state registration laws and their implementation, Wolfinger and Rosenstone's results underscored the critical negative effect of the registration closing date in the 1972 election: the greater the number of days prior to the election that voter registration closes, the lower an individual's probability of voting in presidential elections. This finding has been confirmed in subsequent analyses over the past several decades.1
However, since 1972 the states have engaged in a wide variety of policy innovations governing presidential elections, including, for example, election day registration, registering and voting by mail, increased access to absentee voting, and early voting. Many of these innovations were adopted with the explicit intention of increasing voter turnout, and often with the intention of increasing voter turnout of traditionally underrepresented groups (e.g., the poor or racial and ethnic minorities).
While some of these reforms—in particular, absentee voting and early voting—have been quite popular with voters, existing research provides no evidence that they have actually raised turnout or affected the disparities in voting rates between different groups of voters. Most research on the topic has concluded that voters using these methods of voting are simply switching the way in which they vote; were these new modes of voting not available, they would vote by other (i.e., traditional) means (Fitzgerald 2005; Gronke, Galanes-Rosenbaum & Miller 2007).
In this chapter we return to these fundamental questions, considering the electoral impact of this new, wider array of voter registration and election administration laws using a new data set we collected on state electoral rules between 1972 and 2008. We first describe the major features of changes in electoral laws during this period, and then use two analytical approaches to examine whether states that adopt easier registration and voting laws subsequently witness higher aggregate turnout. We first provide standard bivariate intervention analyses, comparing changes in turnout in states that adopted reforms with changes in turnout over the same period in states that did not adopt the same reforms. If a costs and benefits theory of the demographics of turnout is correct, then we should see that making voting easier should make those individuals with fewer demographic resources such as education (or, more broadly, traditionally underrepresented groups) more likely to vote, and therefore reduce group-based differences in turnout. Lowering the costs should allow more people with fewer resources to clear the cost hurdle. We examine this proposition with respect to election day registration.
Second, we also estimate cross-sectional time series models at the state level. This allows us to test whether these policy innovations are associated with increases in turnout once we condition on other factors related to turnout.
**4.1 Electoral Innovation in the United States**
The United States is unique among modern democracies in the burden it puts on citizens seeking to exercise their right to vote. Much of this burden over the past several decades has resulted from state voter registration rules that require citizens to formally apply for voting eligibility certification prior to election day. In contrast, citizens in most other democracies are automatically registered to vote through other identification or residential registration procedures (often borne by the national or local government), or indeed may even be _required_ to vote by national compulsory voting laws. Thus the costs of voting are higher in the United States, and this simple observation is often used to explain lower levels of voter turnout in the United States compared to other Western European democracies (Blais 2006; Franklin 2004; Jackman 1987; Powell 1986).2
Yet the last thirty years have seen substantial reductions in restrictions on registration and voting. States have eased the registration process in response to the 1993 National Voter Registration Act (NVRA), which mandated that they make available the opportunity to register to vote wherever drivers licenses were issued and wherever states provide public benefits. At the same time, many states have independently adopted election reforms intended to make voting easier. Early voting, where citizens may vote in person before election day, is now available in numerous states, as is the option of voting at locations other than traditional precinct polling places. And many states now allow citizens to vote without going to any polling place by returning an absentee ballot, independent of whether or not the voter is in the state and would have no serious impediment to voting in person on election day.
With the support of the Pew Charitable Trusts, we compiled data on voter registration and election administration statutes in the states for all presidential election years between 1972 and 2008; details about this data collection are provided in appendix 4.1 and can also be found in Cemenska, et al. (2009). These data show a diversity in the complex set of election laws adopted by the state since 1972, and they underscore how many of the states have adopted laws to make registering to vote and casting a ballot easier for citizens.
Table 4.1 lists the number of states in which various registration laws enacting these reforms had been adopted in each year since 1972: election day registration, availability of registration at departments responsible for motor vehicle and drivers licenses (DMV), and the number of days before an election that registration closes. Being able to register on election day, register at public agencies, and register closer to election day are considered legal features that reduce the costs of voting. They are also the registration reforms that have received the greatest amount of attention by lawmakers and the media.
**Table 4.1. State Adoption of Registration Reforms, 1972–2008.**
_Note_ : Entries are number of states in which the indicated registration provision was in effect in the given year. See appendix 4.1 for source and coding details.
As shown in table 4.1, between 1972 and 1976 three states (Maine, Minnesota, and Wisconsin) adopted election day registration (EDR), and this basically remained constant until three additional states (Idaho, New Hampshire, and Wyoming) adopted EDR between 1992 and 1996 rather than comply with the NVRA requirement that state agencies provide registration materials.3 Between 2004 and 2008 an additional three states adopted it, resulting in nine states offering election day registration in 2008.
A more dramatic increase is seen in the availability of DMV registration by 2008, largely due to the requirement being imposed by passage of the NVRA. Prior to the 1980s, only two states provided any sort of voter registration when citizens applied for drivers licenses. Between 1980 and 1992 this number rose to eighteen, and basically became universal (if one excludes the EDR states) with the passage of the NVRA.
In contrast, reforms associated with reducing the number of days prior to election day by which citizens must register to vote (i.e., the registration closing period) are far from universal, and changes in these laws are more accurately described as modest. In 1972, 28 states required citizens to be registered to vote 30 days or more before election day. While this number dropped to 13 by 2008, this does not mean that states have substantially reduced the registration closing period. Instead, what has happened is that many states with registration closing periods longer than 30 days have reduced that requirement to somewhere between 25 and 30 days, as suggested in the last column in table 4.1 (the number of states with registration closing period greater than or equal to 25 days). In 1972, 33 states required registration 25 or more days prior to the election. By 2008 this number had only dropped to 25 states. So many of the legal changes consisted in moving states to a registration deadline of within 30 days of the election but not to a deadline of closer than 25 days.
Perhaps the most dramatic change in state electoral laws over the past several decades has been in rules regarding when or where individuals (once registered) may vote. While absentee voting for citizens meeting certain requirements (such as being out of state or being physically unable to get to the polls on election day) has long been available in many states, since 1972 a large number of states have relaxed absentee voting requirements, effectively providing absentee voting on demand (no-fault absentee voting). Usually such ballots can be returned by mail or in person a certain number of days prior to election day.
Early voting is another electoral reform that has been adopted by many states in the post-1972 period. Like absentee voting, early voting reduces the costs of balloting for registered voters. Early voting periods in most states last up to two weeks, and close several days prior to election day. During this period citizens can go to the polls (sometimes traditional polling places, sometimes at fewer, yet more centralized, locations) to cast their votes in person. According to the U.S. Election Assistance Commission,4 nearly one third of ballots in the 2008 presidential election were cast prior to election day.5
Table 4.2 presents the number of states allowing two forms of alternative voting: no-fault absentee voting and in-person early voting. Here we see striking change. In 1972 only 2 states allowed no-fault absentee voting, in contrast to 27 states in 2008. While in 1972 only 5 states allowed early voting, that number increased to 31 by 2008. Of course, neither of these reforms do anything to reduce the costs associated with registration, but they do reduce the costs associated with having a specified day and finite time period in which a registered citizen must appear to cast a ballot.
**Table 4.2. State Adoption of Early and Absentee Voting Reforms, 1972–2008.** | **No-Fault Absentee Voting** | **In-Person Early Voting**
---|---|---
1972 | 2 | 5
1976 | 3 | 6
1980 | 6 | 7
1984 | 6 | 7
1988 | 6 | 9
1992 | 12 | 11
1996 | 16 | 14
2000 | 22 | 22
2004 | 24 | 27
2008 | 27 | 31
_Note_ : Entries are number of states in which the indicated voting provision was in effect in the given year. See appendix 4.1 for source and coding details.
**4.2 Previous Research on Electoral Rules and Turnout**
The increasing diversity and popularity of these alternative voting procedures does not, of course, demonstrate that their adoption has increased voter turnout. Without exception, reformers have argued that decreasing the costs of voting by adopting these reforms will increase the number of citizens casting ballots. And nearly as often, supporters seeking the adoption of such reforms have claimed that individuals least able to bear the costs of voting and thus those typically underrepresented among voters (i.e., the poor, the less-educated, and minorities) benefit more than those most able to bear such costs. And so adopting these electoral reforms should lead not just to a larger but also to a more representative electorate.
These arguments were clearly evident in the protracted debates and political maneuvering preceding passage of the National Voter Registration Act of 1993. The NVRA required states to provide the opportunity for citizens to register to vote at all state agencies dispensing public benefits, including state agencies providing drivers' licenses. While the latter provision is the best known of the bill (and generates its common name, "motor voter") the bill's provision requiring states to provide voter registration assistance in agencies providing aid to poor people generated much of the political conflict. Hence, the normative claims of a larger and perhaps more representative electorate were often met with concerns over the partisan implications of the bill's passage.
Once these substantial political hurdles were overcome, the states had some time before being required to implement various aspects of the legislation. At the same time the NVRA was being adopted, other electoral innovations that we have described—absentee voting and early voting—were being adopted by the states. However, most of the empirical studies evaluating whether these reforms have increased turnout or made the electorate more representative have focused on only one reform at a time, often looking at only brief time periods. The conclusion drawn from many of these existing studies is that these reforms have had at best a modest effect on turnout levels, and that they have either had no effect on the representativeness of the electorate or possibly made it slightly less representative.
Using aggregate-level data, a number of scholars have investigated the effects of motor voter on registration, turnout, and the composition of the electorate (Brown & Wedeking 2006; Franklin & Grier 1997; Hanmer 2007; Hill 2003; Knack 1995). While the particular estimates vary a bit, many of these studies suggest that this reform significantly increased registration in the states but did not increase turnout substantially. Knack (1995) is an exception to this set of null findings. Looking at data for 1976 through 1992, and estimating a multivariable model of turnout that considered how long motor voter implementation had been in effect, Knack concluded that motor voter _did_ increase turnout. None of these studies find any significant group-specific differences in turnout as a consequence of the law. Hence, motor voter was modestly effective in two of its initial goals, to increase registration and to reduce the costs of registration to underrepresented groups, but possibly ineffective in increasing turnout.
Most studies of the impact of EDR suggest that being able to register at the polls on election day increases turnout by about 3 to 5 percentage points, with the early adopting states experiencing a greater boost (Brians & Grofman 1999; Fenster 1994; Fitzgerald 2005; Hanmer 2007; Highton 1997; Knack 2001; Knack & White 2000). Fenster (1994) estimated the effect EDR had on aggregate turnout in the three early adopters—Maine, Minnesota, and Wisconsin—to be about 5 percentage points by comparing the change in turnout in those states to the change in turnout in other states preadoption and postadoption. Knack (2001) has estimated the change for the second wave of states—Idaho, New Hampshire, and Wyoming—to be about 3 percentage points using a similar technique.
Hanmer (2007) and Knack and White (2000) conclude that the adoption of EDR has had limited effects on the representativeness of the electorate. Hanmer does report that the effect of EDR is larger for persons with lower levels of education than those with higher levels. Knack and White (2000) report that the greatest effects of EDR are evidenced for younger citizens and recent movers, and only slight effects are shown across income groups. They find no effect on turnout differences, however, across education groups. Brians and Grofman (1999) are a bit more optimistic about group-related differences in turnout as a result of EDR, but their analysis is based on comparing only two elections. Hence, this evidence suggests that class effects of EDR on the composition of the electorate are likely minimal.
Research on absentee voting has also produced somewhat inconsistent findings on whether this electoral reform actually increases overall turnout. Demographically, absentee voters tend to be more educated and older than election day voters (Barreto et al. 2006; Karp & Banducci 2000, 2001). And because absentee voters tend to be more politically active, partisan, and psychologically engaged in politics than election day voters, these studies also suggest that absentee voting does not increase turnout overall but instead makes it more convenient for those who would have otherwise voted on election day to vote at another time. Consistent with this interpretation, Fitzgerald (2005), Giammo and Brox (2010), and Gronke et al. (2007) conclude that absentee voting does not increase turnout. The only studies to conclude that absentee voting increases turnout, Francia and Herrnson (2004) and Oliver (1996), do so based only on cross-sectional analyses.
Studies of early voting also suggest that the individuals who are most likely to take advantage of this electoral innovation are those who would otherwise vote on election day: those who are politically engaged and highly partisan (Neeley & Richardson 2001; Stein 1998; Stein & Garcia-Monet 1997). These studies, as well as Francia and Herrnson (2004) and Gronke et al. (2007), are all limited by research design in time, location, or both. Yet they are consistent with Fitzgerald's (2005) more comprehensive analysis of turnout between 1972 and 2002, which indicates that early voting has no significant effect on overall turnout. Only two studies conclude that early voting increases turnout, but these findings are somewhat tentative. Giammo and Brox (2010) find that early voting leads to greater turnout when first adopted, but that the effect disappears within two elections, while Lyons and Scheb (1999) claim that early voting increases turnout, using evidence based on early voting in one county in Tennessee in one year.
In sum, there is no causal evidence that the two electoral reforms adopted by approximately half the states and used by approximately a third of voters has actually affected net turnout. Below we provide a more definitive answer to the question of the effects of these electoral reforms. Our findings show that absentee voting _has_ in fact increased turnout, and suggest that early voting _can_ increase turnout when implemented aggressively, both of which are important advances in what we know about the consequences of electoral reforms in the United States
**4.3 Research Design and the Search for Effects**
We begin with a discussion of research design issues, for even the modest effects of election law reforms identified in some previous research might be questioned due to concerns regarding how previous studies have drawn causal inferences and how they have estimated these causal effects. Many studies either utilize a single cross-section (using either individual or aggregate data based on either national-, state-, or county-level data) or change over time in a single state (again, using either individual or aggregate data based on either national- or state- or county-level data). But when we ask whether election laws affect turnout, we consider this to be a causal question. That means that our research designs should maximize our ability to draw causal inferences about the relationship between the adoption of an electoral reform and the level of voter turnout.
A standard and common approach to establishing causality in contemporary social science research examines variations in individual behavior using cross-sectional, multivariable models, where the dependent variable of interest (voter turnout) is modeled as a function of a set of independent variables (demographics and state election laws). Many studies examining the turnout effects of in-person early and absentee voting use this approach. But the causal inferences we can draw from this approach are limited because we do not necessarily know why some states have adopted these measures (while others have not), or whether those states are as a group different from the states that have not adopted the election reforms. If we were to compare turnout in states with in-person early voting to turnout in states that do not have it, the comparison would only be valid for drawing an inference on the effect of early voting _if the adoption of in-person early voting were not correlated with other factors related to turnout_. This means that the causal inferences from any cross-sectional analysis are suspect. It may be that states where turnout is higher independent of the electoral rules also choose to adopt reforms designed to increase turnout.
An alternative approach to assessing the effects of an electoral reform on voter turnout is to analyze it as a policy innovation problem, comparing turnout in a state before the innovation to turnout in a state after the adoption of the electoral reform. But because turnout varies from election to election, comparing turnout in a single state in a single year to turnout in the same state in the first election after reform, we would not know if we were observing the impact of the electoral reform or instead observing a secular (independent) change in turnout that affected all states in that election year.
Solving this latter problem requires using a _difference-in-difference_ approach—that is, comparing the change in turnout for the state adopting the reform to the change in turnout of other states that did not adopt the reform. Provided that other things have remained constant, the difference between the differences would reflect the impact of the reform. We begin our analyses below using this difference-in-difference approach.
However, if other factors affecting turnout change across the states, and those changes are correlated with adoption of reforms, then the changes observed via the difference-in-difference approach could be the result of changes in those other factors and not the result of the electoral reform of interest. This possibility also leads us to believe that the most robust approach is to combine the cross-sectional and difference-in-difference analyses into a cross-sectional time series (CSTS) approach. Thus, our second analytical approach treats the adoption of absentee voting and the adoption of in-person early voting as interventions, and assesses whether such interventions raised or lowered turnout in the states. This approach allows us to condition on the values of other observable factors known to influence turnout, such as the closeness of elections, and demographic characteristics.
In this model, the unit of analysis is the state-year; the dependent variable is turnout percentage in the state-year; and independent variables include demographic and political characteristics argued to affect voter turnout, along with variables for each state-year indicating whether the state had in that year adopted either absentee or in-person early voting. The model thus incorporates both changes over time and differences across states. The strength of the CSTS approach is that we are able to condition on observable factors and avoid drawing inferences that require the assumption that no other factors related to turnout changed in ways correlated with the adoption of electoral reforms. However, the cost of more thoroughly isolating the causal impact of electoral reforms is that we lose statistical power.
In principle, we can use both of our approaches to determine the impact of reforms on different demographic groups. The difference-in-difference model can be used by examining differences in turnout before and after electoral reform adoptions by different education, income, or age groups, for example. Unfortunately, while disaggregating by demographic groups could be accomplished with the cross-sectional time series model, when we did this we were not able to recover precise enough estimates to distinguish effects between different demographic groups. Thus, in the demographic-specific analyses below, we only present the difference-in-difference estimates, and only for the effect of EDR.
We use the difference-in-difference estimates to compare the effects of election day registration across the five income groups to determine whether EDR has differential effects across these age, income, and education-level groups. This allows us to test whether the claims of supporters of this electoral reform—that it would increase the turnout of underrepresented groups and therefore make the electorate more representative—are substantiated. While the CSTS estimates would offer more robust estimates of effects, that the difference-in-difference estimates are similar to the CSTS estimates in the aggregate data gives us some confidence that the demographic estimates hold as well.
Our data are drawn from three sources: the Current Population Survey (CPS), official vote returns and census population estimates (as described in appendix 4.2), and a new data set on state registration and election administration laws (as described in appendix 4.1). We use the CPS (as described in appendix 2.1), and compute a measure of voter turnout that is the proportion of the citizen voting age population in each state that casts a vote for president. We also use CPS data for measures of turnout in each state for distinct demographic groups of the population. We supplement the CPS data with our state voter registration and election administration data (as described in appendix 4.1).
**4.4 The Effects of Electoral Reforms: Difference-in-Difference Estimates**
We first perform several difference-in-difference tests to examine the impact of election day registration (EDR) for the three states that first adopted it—that is, wave 1 states, and then analyze the later adopters (i.e., wave 2 and wave 3 states) separately to avoid contaminating the effects estimates due to differences in the timing of the adoptions over a thirty-five-year period. If EDR really raises turnout, then we should see a larger increase in turnout from pre–EDR adoption (i.e., 1972) to post–EDR adoption (i.e., 1976 through 2008) for the three states that adopted it prior to 1976 than for the states that did not adopt it. Since the CPS did not include a state variable in 1976, we compare 1972 turnout to turnout in the postadoption period of 1980–2008 with CPS data. But to utilize 1976, and extend our baseline comparison beyond one election, we also do the comparison using turnout based on official vote tabulations, comparing 1960–72 turnout to 1976–2008 turnout for the EDR states and the non-EDR states.6
Table 4.3 reports turnout for two groups of states, EDR wave 1 states (Maine, Minnesota, and Wisconsin) and non-EDR states, for two periods for each group. Prior to the adoption of EDR, the three EDR states had substantially higher turnout than the non-EDR states (69.9 percent compared to 64.1 percent). But what matters here is that adoption of EDR _increased_ this advantage even more. The key columns are the increases in turnout from 1972 to the postadoption period of 1980– 2008. If EDR is effective, we should see larger increases in the first column of one-time increases in turnout (EDR states) than for the second column of one-time increases in turnout (non-EDR states). That is exactly what we see. Since 1980 the EDR states have had turnout on average 3.0 percentage points higher than their 1972 turnout, whereas turnout in the non-EDR states has decreased by 1.5 percentage points since 1972.7
However, we note that we are only using one election year, 1972, as our pre-EDR baseline here. To avoid an error in inference that might be caused if 1972 was an unusual year for the three EDR states, we also average over the 1960–1972 presidential elections to create an alternative pre-EDR baseline for comparison. These numbers are reported in the second row of the table.8 Here we see that for the three EDR states, turnout averaged 69.1 percent for the four presidential elections from 1960–1972, whereas it averaged 68.5 percent for the presidential elections from 1976–2008. Thus these states saw a drop in turnout of 0.6 percentage points. However, the corresponding figures for the comparison set of non-EDR states are 63.8 percent and 57.2 percent, a drop in turnout of 6.7 percentage points. Thus, the net effect on turnout that can be attributed to EDR here is a 6.1 percentage-point _increase_.
Having concluded that EDR does in fact raise turnout, the question remains as to _who_ is voting more. To address this question, we use measures of turnout of different demographic groups for each state in each election and repeat the difference-in-difference analysis we initially did for overall turnout. Rows 3 through 7 in table 4.3 report the change in turnout for citizens in different income groups. EDR increased the turnout of the second and third income quintiles by 5.8 percentage points and 4.7 percentage points, respectively, while only increasing turnout for the bottom income quintile group by 0.6 percentage points. EDR increased the turnout of the top two income quintile groups by 2.3 percentage points and 2.2 percentage points, respectively. So it appears that EDR has almost twice as large an impact on the second and third income quintiles as on the top two income quintiles; the smallest effect is for those in the first income quintile. Thus, these results suggest that adoption of EDR leads to an increase in the representation of voters in the second and third quintiles relative to the fifth (i.e., wealthiest) quintile, while harming the representation of voters in the first quintile (i.e., the poorest).
**Table 4.3. The Effect of the Adoption of Election Day Registration on Turnout (Wave 1 EDR States).**
_Notes_ : Notes: Except for row 2, entries are computed by the authors using self-reported turnout from the Current Population Survey, or increase in reported turnout, for states that adopted EDR between 1972 and 1976, or for states that have never adopted EDR. The wave 1 EDR states are: Maine, Minnesota, and Wisconsin.
_a_ a Entries are the post-1972 change in turnout for the three EDR wave 1 states minus the post-1972 change in turnout for the non-EDR states.
_b_ Entries are the percentage of nonvoters converted to voters based on the net effect of EDR. See the text for a discussion of at-risk effects.
_c_ Entries in this row compare change in turnout from the period 1960–72 in the EDR states to changes in turnout over the same period in non-Southern, non-EDR states. Turnout figures used in this row are actual aggregate turnout figures from public sources (not reported turnout from the Current Population Survey); see appendix 4.2 for details.
However, it is important to realize that these are results about _groups_ of voters, not inferences about individuals. While the adoptation of EDR may result in a larger percentage increase in turnout among persons in the second income quintile than the fifth income quintile, that does _not_ mean that a given individual in the second quintile is more likely to vote because of EDR than is a given individual in the fifth quintile. One reason for the lower percentage-point increase for the fifth quintile is that there are simply not many nonvoting persons in the fifth quintile available to become voters—that is, to be receptive to the treatment of EDR.
Prior to the introduction of EDR we can think of the nonvoters as the people at risk of being converted to voters via the treatment of EDR. The eighth column of table 4.3 gives the impact of EDR on the at-risk population of citizens among any group. This is the set of people who could be converted to voters based on the availability of EDR. We can think of this as an individual-level average treatment effect. The at-risk effect is really an individual-level effect; this gives the increased likelihood that a single nonvoting person would become a voter based on adoption of EDR. These are the estimates that provide the most direct evidence regarding individual differences in the effects of electoral reforms across demographic subgroups.
Consider aggregate turnout first. In 1972, aggregate reported turnout was 69.9 percent in EDR states; thus, 31.1 percent of citizens were not voting. These are the citizens who were at risk to be converted to voters by EDR. Since EDR had a net impact of 4.5 percentage points on turnout, the effect on at-risk voters was that 4.5 out of 31.1 were converted to voters by EDR. Thus, the effect on at-risk voters was 15.0 percent (i.e., 4.53÷31.1). In other words, between one out of six and one out of seven nonvoters was converted to voting via EDR.
Now consider the at-risk effects for the different quintiles. If we look at the last column in table 4.3 we see that persons in quintile 5 were slightly _more_ likely to take advantage of EDR (15.1 percent) than were citizens in quintile 2 (14.6 percent). Thus, the increased net group turnout effect of EDR for quintiles 2 and 3 over quintiles 4 and 5 is _not_ because any individual person in income quintiles 2 or 3 is more likely to take advantage of EDR than any individual person in income quintile 4 or 5. It is simply because there are more persons eligible for the treatment. For the purpose of understanding the effect of EDR on income bias in turnout, the net group effect is what we care about. But if we want to understand individual behavior, it is the at-risk effect that we need to look at.
The next four rows in table 4.3 give the change in turnout for persons of different levels of education. For citizens with different levels of education, we see the largest net group effect for the middle two groups: aggregate turnout of citizens with a high school degree or some college goes up by 7.6 and 4.3 percentage points, respectively, while turnout of those with college degrees or higher barely changes. For the least educated, those with less than a high school degree, aggregate turnout only rose 3.0 percentage points. In the last column of the table, we see huge differences between the at-risk effect on individuals in the bottom group and the at-risk effect on individuals in the two middle groups. Whereas 7.4 percent of those eligible in the bottom group become voters through EDR, fully 24.7 percent and 22.5 percent of previously nonvoting members of the middle education group become voters. This challenges previous assertions that people with low levels of education do not vote because the process is too confusing for them. Here we see that a change designed to make registration easier is more likely to be taken advantage of by non-voters with a high school education than by nonvoters without a high school education.
The next four rows of table 4.3 give the corresponding values of people in different age groups. In contrast to the relatively small variation in the net group effect of EDR we see across socioeconomic classes, we see stark differences across age groups: younger citizens receive a much greater benefit from EDR than do older voters. Whereas the aggregate turnout of eighteen- to twenty-four-year-olds rose 11.9 percentage points based on the adoption of EDR, we cannot even observe a positive effect for those ages sixty-one through seventy-five, and the net group effects decrease as we look at older citizens.9 These comparisons hold for both the net group benefit of EDR being larger for younger than older age groups, _and_ for the at-risk effect being largest for younger individuals. Younger nonvoters are simply much more likely to become voters from EDR than are older nonvoters.
Table 4.4 reports similar analyses for the wave 2 EDR states (Idaho, New Hampshire, and Wyoming). These states adopted EDR between 1992 and 1996. Here the results are much less impressive. There has in fact been a decrease in turnout of 3 percentage points in the elections since adoption of EDR for the wave 2 states, and a similar decrease of 3.1 percentage points for the non-EDR states during this period. This suggests a net group EDR effect of just 0.1 percentage points. Of course, these are small states, but more important while they were adopting EDR, the non-EDR states were following the mandates of the NVRA and adopting other reforms intended to increase turnout. Thus, the correct inference here is not that EDR had no effect for these states but that it did not have a substantially larger effect than did the adoption of NVRA provisions in the other states.10
**Table 4.4. The Effect of the Adoption of Election Day Registration on Turnout (Wave 2 EDR States).**
_Notes_ : Entries are computed by the authors using self-reported turnout from the Current Population Survey, or increase in reported turnout, for states that adopted EDR between 1992 and 1996, or for states that have never adopted EDR. The wave 2 EDR states are: Idaho, New Hampshire, and Wyoming.
_a_ Entries are the post-1992 change in turnout for the three EDR wave 2 states minus the post-1992 change in turnout for the non-EDR states.
_b_ Entries are the percentage of nonvoters converted to voters based on net effect of EDR. See the text for a discussion of at-risk effects.
In table 4.5 we report the effect of EDR for the wave 3 states (Iowa, Montana, and North Carolina), which first used EDR in the 2008 presidential election. Obviously these results are based on only one election, and generalizing from them should be done with extreme caution. But these states did experience an increase of 1.5 percentage points relative to the non-EDR states.
We next turn to an analysis of early voting and absentee voting. As we documented in table 4.2, states adopted these two reforms over an extended period. The adoption of election reforms by so many states over such a long period makes it reasonably straightforward to determine the impact of the reforms. For each state that adopted no-fault absentee voting between the years 1976 and 2008, we looked at the change in turnout for the state from the election immediately preceding adoption of no-fault absentee voting to the election immediately following its adoption.11 To control for any nationwide changes in turnout between elections, we also compute the matching change in turnout between the same election years for the set of states that did not change their no-fault absentee voting laws between the relevant elections. Thus, in the election of 1980, we compare the change in turnout of the three states that adopted no-fault absentee voting between 1976 and 1980 to the change in turnout of the forty-seven states that did not alter their absentee voting laws between 1976 and 1980.
Table 4.6 presents these results. For 1980, the states that adopted no-fault absentee voting had an average increase in turnout of 0.2 percentage points, while the forty seven states that did not adopt no-fault absentee voting between 1976 and 1980 had an average _decrease_ in turnout of 0.4 percentage points. Thus, the difference between them, or the estimate of the impact of adoption of no-fault absentee voting, was 0.6 percentage points. We then compute a weighted average of these comparisons for each election from 1976 to 2008, and estimate the impact of adoption of no-fault absentee voting to be 1.4 percentage points.12
**Table 4.5. The Effect of the Adoption of Election Day Registration on Turnout (Wave 3 EDR States).**
_Notes_ : Entries are computed by the authors using self-reported turnout from the Current Population Survey, or increase in reported turnout, for states that adopted EDR between 2004 and 2008, or for states that have never adopted EDR. The wave 3 EDR states are: Iowa, Montana, and North Carolina.
_a_ Entries are the post-2004 change in turnout for the three EDR wave 3 states minus the post-2004 change in turnout for the non-EDR states.
_b_ Entries are the percentage of nonvoters converted to voters based on the net effect of EDR. See the text for a discussion of at-risk effects.
**Table 4.6. The Effect of the Adoption of No-Fault Absentee Voting, 1972–2008.**
_Notes_ : See appendix 4.2 for data sources.
_a_ Numbers in parentheses are the number of states adopting no-fault absentee voting in that year.
_b_ Entries are the average change in actual turnout from the previous presidential election to the current (row) election in those states adopting no-fault absentee voting between the previous presidential election and the current (row) year.
_c_ Entries are the average change in actual turnout from the previous presidential election to the current (row) election in those states that had no change in their absentee voting laws between the two elections.
_d_ Entries are the difference in change in turnout experienced by states adopting no-fault absentee voting between elections and those states that did not change absentee voting laws between elections.
_e_ See text for explanation of pre- and postadoption comparison involving all election results from 1972–2008.
We performed an identical analysis for the adoption of early voting by the states. The results are reported in table 4.7. Here the weighted average of the effects over the years is only a 0.4 percentage-point difference in turnout between states adopting no-excuse early voting and the comparison set of states. Thus these two sets of analyses suggest that the states adopting no-fault absentee voting and those adopting no-excuse early voting had small increases in turnout relative to those states that did not.
These comparisons are very direct analyses of the effect of no-fault absentee voting and early voting. We are using all the data from the election immediately preceding a state's adoption of early or absentee voting and the election immediately following the state's adoption of early or absentee voting. The values we present are then basic statements of fact: states adopting absentee voting and states adopting early voting experienced an increase in turnout in the elections immediately following compared to states that did not adopt these reforms.
**Table 4.7. The Effect of the Adoption of Early Voting, 1972–2008.**
_Notes_ : See appendix 4.2 for data sources.
_a_ Numbers in parentheses are the number of states adopting no-excuse early voting in that year.
_b_ Entries are the average change in actual turnout from the previous presidential election for those states adopting no-excuse early voting between the previous presidential election and the current (row) year.
_c_ Entries are the average change in actual turnout from the previous presidential election for those states that had no change in their early voting laws between the two elections.
_d_ Entries are the difference in change in turnout experienced by states adopting no-excuse early voting between elections, and those states that did not change early voting laws between elections.
_e_ See the text for an explanation of the pre- and postadoption comparison involving all election results from 1972 to 2008.
However, while the comparisons we report above are perfectly valid, we note that they are limited in that they only utilize one pair of elections per state that adopted a reform: the election immediately prior to reform, and the election immediately after the reform. Since either of those elections may have been unusual for the state, we might ask a more comprehensive question: if we compare all elections from 1972 to the year prior to adoption, to all elections from the year of adoption to 2008, was the increase in turnout higher for those states that adopted no-fault absentee voting (or early voting) than for those states that did not adopt these measures in the entire period?
Thus, for each state that adopted no-fault absentee voting, we calculated average turnout for the state for each election since 1972 _prior_ to adoption, and for each election _since_ adoption of no-fault absentee voting up to 2008. We look at the change in average turnout for the state between the two periods—preadoption versus postadoption—and compare it to the change in turnout _averaged over those same two periods_ for the set of states that never adopted no-fault absentee voting. We performed an identical analysis for states that adopted early voting. The results of these calculations are reported in the final rows of tables 4.6 and 4.7.
Here we get slightly different answers. Over the entire period from 1972 to 2008, the states that adopted no-fault absentee voting during this period had an average turnout increase of 1.6 percentage points, considering all preadoption elections versus all postadoption elections. However, over the same period the control group of states also had a turnout increase of 1.6 percentage points when we compare the same turnout in the same time periods. For early voting, the comparable figures are a 1.3 percentage-point increase for the states that adopted early voting, but a 3.4 percentage-point increase over the same time period for states that did not adopt early voting.
Thus, either absentee voting and early voting have led to initial increases in turnout that then subside, or they simply have not led to any increases in turnout in the states that have adopted them. Again, this is based on looking at all the votes cast and not cast, in presidential elections since 1972. Thus we can concur with conventional wisdom that suggests that the vast majority of votes cast by either of these methods simply represent voters switching their mode of casting a ballot, not additional votes.
An additional observation we would add to these results is that while we have shown that states adopting no-fault absentee voting and early voting did _not_ experience an increase in turnout relative to states that did not adopt these reforms, this does not answer the counterfactual question as to what would have happened had these states _not_ adopted no-fault absentee voting or early voting. While it is tempting to infer that no-fault absentee voting and no-excuse early voting do not raise turnout, it may be the case that the set of states adopting them were not random, and that other factors were at work to depress turnout in those states, thus masking the impact of the reforms.
The simple conjecture is that it is possible that the states adopting these reforms did so in order to combat other trends negatively affecting turnout. Or, in a less nefarious conjecture, it is possible that the states adopting these reforms may have been states that had less competitive elections postadoption, or changes in other turnout-related characteristics, leading to lower turnout. The data presented here simply do not control for other things that could be happening to affect turnout in both states that adopted and states that did not adopt these reforms. Thus, to draw a causal inference about the effect of these reforms on turnout, we next present a cross-sectional time-series analysis to control for observable characteristics of the states—such as demographic characteristics, other elections laws, and competitiveness of elections—that were changing over this period.
**4.5 Cross-Sectional Time Series Analysis of Aggregate Turnout**
Our second analytical approach is to estimate a multivariable cross-sectional time series model of turnout where we condition the effect of the institutional changes on observable characteristics of the states known to affect turnout. By including year-specific fixed effects and state-specific fixed effects we are only measuring the impact of variables that change over time in the model while allowing for unobserved factors (incorporated in the year fixed effects) leading to secular changes in turnout for each year. Assuming our model is well specified, we will have measures of the causal impact of the institutional variables of interest on turnout.13
We estimate aggregate turnout in each election as a function of aggregate turnout in the previous presidential election, the presence or absence of the five registration or election administration characteristics of interest (EDR, absentee voting, early voting, the number of days before the election that registration closes, and whether voter registration is available in motor vehicle offices), demographic characteristics of the state, and measures of electoral context. To condition on the demographic characteristics of the state we include the state per capita income, the age distribution of the citizens as measured by the proportion of citizens in each of six age categories, and the education distribution of the state as measured by the proportion of citizens in each of four education categories. To condition on the electoral context, we include a measure for the closeness of the presidential race in the state, and dummy variables for the presence of a gubernatorial or senate race in the state, as well as measures of the closeness of those races.14
To allow for the effect of EDR to be contingent on how soon before election day registration closes, we include an interaction term between the two variables. We expect EDR to have less of an effect in states where registration has closed well before the election than in states where registration is available until very close to the election.15 We also allow
for the effect of early voting to depend on the length of the early voting period by including the length of the period.
Thus, we estimate a model of the following form, where _s_ and _t_ index state and time, respectively:
We estimated the model above using data for turnout in the fifty states from 1972 to 2008. We measured aggregate state turnout as the proportion of the voting-age citizen population casting votes for the highest office. Since turnout is bounded between 0 and 1, we used the log-odds ratio of turnout as the dependent variable, and we compute panel corrected standard errors. In table 4.8 we report the parameter estimates for the model; we do not report the estimated coefficients for the year or state fixed effects.
Because of the inclusion of several interactive terms, and the nature of the model with log-odds of turnout as the dependent variable, we do not discuss these coefficients but instead focus our discussion on the first differences below. However, note that the coefficients generally have the expected sign. The specification we use implies that the impact of early voting is contingent on the length of the early voting period, and that the impact of EDR depends upon the number of days to closing of registration.
**Table 4.8. Cross-Sectional Time Series Model of Turnout by State, 1972–2008.** | **Coefficient** | **T-statistic**
---|---|---
Log-odds Turnout (t – 1) | 0.553** | (6.74)
Registration Closing Period | –0.0018* | (–1.81)
Early Voting | –0.075** | (–2.54)
Early Voting Period | 0.0028** | (2.15)
No-fault Absentee Voting | 0.056** | (3.38)
Election Day Registration (EDR) | –0.025 | (–0.38)
EDR × Registration Closing Period | 0.0049 | (1.45)
DMV Registration | 0.001 | (0.04)
State Per-Capita Income | –4.08 _e_ -06 | (–0.59)
Proportion of Citizens Age 25–30 | 0.258 | (0.38)
Proportion of Citizens Age 31–45 | 0.843 | (1.64)
Proportion of Citizens Age 46–60 | 0.429 | (0.76)
Proportion of Citizenss Age 61–75 | 0.758 | (1.37)
Proportion of Citizens Age 76–89 | 0.518 | (0.70)
Proportion of Citizens with High School Degree | 0.366 | (1.40)
Proportion of Citizens with Some College | –0.049 | (–0.14)
Proportion of Citizens with College Plus | 0.837** | (2.42)
Closeness of Presidential Election | 0.0035 | (0.67)
Closeness of Gubernatorial Election | 0.0055 | (0.79)
Closeness of Senate Election | –0.0000 | (–0.05)
Gubernatorial Election in State | 0.050 | (1.40)
Senate Election in State | 0.013* | (1.92)
Constant | –0.548 | (–0.98)
|
|
Observations | 450
|
_R_ 2 | 0.92
|
| |
_Notes_ : Table entries are cross-sectional time series coefficient estimates of voter turnout by state in each year, where the dependent variable is the log-odds of turnout. We use panel-corrected standard errors and report the associated t-statistics. State and year fixed effects are included but not reported here. See appendix 4.2: for data sources.
According to the estimates in table 4.8, an early voting period of twenty-seven days is required for early voting to increase turnout. Also note that while we found no differences in aggregate turnout based on absentee voting using the difference-in-difference estimates reported in table 4.6, in the multivariable model we see that when controlling for the state-level characteristics listed above, no-fault absentee voting has a statistically significant, positive impact on turnout.
To gauge the magnitude of the effects of the institutional reforms, we estimated the long-run impact on turnout in a state with a 50 percent turnout rate if the state adopted a reform, and all other variables were held constant. Thus, we incorporate both the immediate impact of the change in each institutional reform on turnout as well as the long-run impact picked up through the lagged log-odds turnout term in the model. These first differences are presented in table 4.9. These are the most precise estimates we have of the real causal effect of these reforms, they make use of data from all fifty states over ten presidential elections. And these are not based on samples, but on actual recorded votes.
According to our estimates, adoption of no-fault absentee voting leads to a 3.2 percentage-point increase in turnout. We note that this estimate is quite different from the conventional wisdom, which claims that absentee voting has had no appreciable impact on turnout. However, this finding, reflecting a more rigorous analytical approach than previous studies, suggests that no-fault absentee voting is one of—if not _the_ —single most important of the changes made to election laws since the Civil Rights Act. We also note that this finding comports more with the reality of millions of votes cast via absentee voting than a finding that _all_ of those votes would have been cast by other means if absentee voting were not available.
As reported in table 4.9, the adoption of early voting is estimated to lead to a 3.1 percentage-point increase in turnout if the early voting period were as long as forty-five days. Since this is a longer voting period than states allow, we have less confidence that early voting is actually increasing turnout. As we indicated above, our estimates suggest that a voting period of as long as twenty-seven days is required to see any positive effect of early voting.
Consistent with previous estimates on the effect of EDR, we estimate that EDR leads to an increase in turnout of 2.8 percentage points in a state with a fifteen-day registration closing period.16 The estimated effect would be higher in states with longer registration closing periods, and lower in states with shorter registration closing periods. These estimates bolster our earlier conclusion that the adoption of EDR does indeed increase turnout.
**Table 4.9. The Marginal Effect of Legal Reforms as Predictors of Turnout, 1972–2008.**
**Change in Law** | **Effect on Turnout** | **95% CI**
---|---|---
Absentee Voting | 3.2 | [1.7, 5.1]
Early Voting with a 45-Day Period: | 3.1 | [–1.5, 8.1]
Registration Closing Period | 1.0 | [0.1, 2.1]
(10-Day Decrease, no EDR): | |
EDR (Registration Closing Period = 15 days): | 2.8 | [–0.8, 6.5]
EDR (Registration Closing Period = 29 days): | 6.6 | [1.1, 12.3]
_Note_ : Entries are the long-run expected percentage-point increase in turnout for the adoption of the indicated reform, with 95 percent confidence intervals in brackets. Estimates are based on the model reported in table 4.8.
Finally we estimate that a ten-day decrease in the length of the registration closing period itself would lead to a 1.0 percentage-point increase in turnout (for states without EDR). Again, this is broadly consistent with previous work going all the way back to Wolfinger and Rosenstone (1980) that has suggested that the length of the registration period is crucial to turnout. Registration is a prerequisite for voting, and having to register further in advance of election day predictably decreases turnout.
We also estimated our cross-sectional time series model on data disaggregated by demographic groups. Here the unit of analysis is turnout _of persons with particular demographic characteristics_ by state and year. We did this for persons in each of five quintile groups, each of four education categories, and each of six age groups. We were not able to distinguish between the impact of the reforms on different groups with any reasonable level of precision (though it does appear that EDR is more important for younger voters than for older voters).17
**4.6 Conclusion**
Our analyses in this chapter have focused primarily on three different electoral reforms—EDR, absentee voting, and early voting—and advances what we know about them in important ways. First, we provided a comprehensive set of tests regarding the effects of the adoption of EDR over the past forty years and concluded that EDR has an important effect on turnout in the states that adopt it.
Second, we estimated these effects by considering the impact of EDR on the entire state population, but also on the at-risk population—the set of nonvoters. This is a major modification to prior approaches to estimating the impact of electoral reforms. In order to understand the impact of electoral reforms it is important to develop an understanding of the behavioral mechanism through which they operate. Authors of previous work have inferred that larger net effects of EDR on the turnout of people with low levels of education mean that such people have a harder time dealing with registration barriers than people with higher levels of education. However, our difference-in-difference analysis as presented in table 4.3 suggests that the net effect of EDR on the probability of poorly educated individuals voting is not larger than for highly educated individuals, _and_ that the individual-level at-risk effect of EDR on poorly educated individuals is substantially less than it is for individuals in the two middle education categories. The evidence in table 4.3 suggests that inability to deal with registration barriers is _not_ what accounts for the lower turnout of poorly educated individuals, given that a higher proportion of nonvoting higher-educated people become voters when the advance registration restriction is lifted than do nonvoting lower-educated people. But we caution that further analysis using a multivariable approach is required to confirm this.
Third, confirming differences in the impacts of EDR in the wave 1, wave 2, and wave 3 states, our results suggest that variation in implementation of EDR, as well as variation in the strategic use of EDR by the parties and candidates, can determine how effective EDR is at increasing turnout.
Our empirical evidence also strongly refutes most earlier research on the effect of absentee voting. Identifying a positive, substantial, and statistically significant effect of no-fault absentee voting adds substantially to what we know about electoral reforms and their effects.
Our finding on the effect of early voting is more tenuous. Early voting could increase turnout, given a long enough voting period. However, the length of the period required suggests that previous scholars who inferred that early voters are individuals who simply shifted the day on which they voted—rather than new voters who cast early ballots instead of staying home and not voting—may have been correct. The sensitivity of the result to the length of the early voting period suggests that we may need to more accurately model other aspects of implementation, such as the number and location of polling places available, to have a better understanding of the impact of early voting on turnout.
Our evidence also demonstrates the importance of state laws on the closing of registration prior to the election. Previous research on the effects of registration laws yielded fairly consistent conclusions: the longer the period, the lower the turnout, but this cross-sectional research could not make strong claims as to the causal relationship between closing days and turnout. Our analyses provide original and persuasive evidence showing that the relationship between the length of the registration period and turnout is a true causal effect. By isolating the effects of the length of the registration closing period on turnout in a well-specified model with data over time and including a variety of more contemporary reforms, we are far more confident that we have identified a true causal effect of these laws on turnout, thus validating inferences drawn from earlier, more limited, analyses.
Our findings suggest where those interested in increasing voter turnout might find some hope. Increasing the number of states with no-fault absentee voting, decreasing the registration closing period to fifteen days, and increasing the number of states with EDR could all increase turnout in presidential elections. Whether voters, candidates, or parties think that the efforts devoted to accomplishing these policies changes are worth it or not is, of course, a different question.
Taken together, our results confirm the cost part of a cost and benefit theory of voting: if the cost of voting is lowered, more people will vote. This does not contradict the claim that many people will not vote even if costs are zero. Nor does it suggest that the entire difference between the level of turnout in the United States and the level of turnout in other countries can be explained by differences in the cost of registering and voting. But we would be foolish to deny that costs influence the turnout decision.
We would also be foolish to ignore the _benefit_ part of the cost and benefit theory of voting, which we consider more fully in chapter 5.
**Appendix 4.1: Voter Registration and Election Law Data Set**
Our data set on voter registration and election administration in the states, 1972–2008, was made possible with a generous grant from the Pew Charitable Trusts to Daniel Tokaji, Nathan Cemenska, Jan Leighley, and Jonathan Nagler. The data set of laws is available from the Pew Trusts as both a spreadsheet and a Stata document. The details that follow are drawn from our 2009 report (Cemenska et al. 2009).
Our documentation of state laws included whether election day registration was available in a state; how many days prior to the election registration closed; and whether voter registration was available in motor vehicle agencies. In addition, we focused on two types of nonprecinct voting available: absentee voting and in-person early voting. These terms are sometimes used interchangeably by policy makers as well as academics, and state legislative statutes governing their availability sometimes overlap as well. Consequently, it was necessary to first clearly define each term so that states could be accurately categorized on various dimensions of these policies. We define _absentee voting_ as the option of requesting, completing, and returning a ballot prior to election day, and being able to do so without being present in person at an election office or precinct.
We define _in-person early voting_ as a one-stop transaction in which the voter requests a ballot, completes it, and returns it. If any portion of this three-part transaction does not occur simultaneously, or if it occurs at another location, it is not in-person early voting by our definition. Therefore, states were only classified as allowing early voting if the relevant statutory language explicitly permited the voter to complete the ballot in the presence of election officials. This definition may exclude some states in which early voting takes place where those operations are not explicitly described in state statutes. We use the terms _in-person early voting_ and _early voting_ interchangeably.
One case that often causes confusion in the use of the terms _absentee voting_ and _early voting_ is that of Oregon, which conducts all statewide elections primarily by mail (both ballot delivery and ballot returns). Because voting by mail can be done before election day and does not take place in traditional precincts, the state might be considered as having a form of early or absentee voting. In our data set, however, we code Oregon (like the other states) based on the above definitions of absentee and in-person early voting. Under these definitions, voting by mail is neither early nor absentee voting. Instead, in accordance with our definitions, we code Oregon's absentee and early voting rules based on state laws specific to those individuals who do not vote via the regular vote-by-mail system rules, for example, those individuals who will be away from their residence during the voting period.
The primary research method used to produce this data set was review of relevant state statutes and administrative codes (hereinafter referred to simply as laws) identified using standard search procedures in LexisNexis and Westlaw. The goal was to identify the contours of the laws according to their plain meaning, even if other sources suggested that actual practice may sometimes deviate from that meaning. The review did not take into account any case law that might have interpreted these laws in a way that deviated from their plain meaning. We emphasize that the data we have collected are based entirely on state statutes and administrative law. We have no data on how state or local officials implement these state laws.
After identifying the relevant laws, researchers coded each state on fifty three variables associated with state absentee and early voting laws. For a description of each variable and its values, see Cemenska et al. (2009).
**Appendix 4.2: Sources of State-Level Turnout and Demographic Data**
For analyses that use state-level turnout data by a specific demographic group, we have aggregated reported turnout rates from U.S the Census Bureau's Current Population Survey. For overall state turnout rates we have relied on two sources. For the period 1972 through 1980 we use figures on turnout reported by the Congressional Research Service (Crocker 1996). We combine this with data for 1980 through 2008 with figures for turnout made available by Michael McDonald (2011). Splicing the two series together gives a value for turnout for highest office in each state, with the best available estimate of the citizen voting age population for 1980 through 2008 as the denominator, and an estimate of the voting-age population of the state as the denominator for 1972 and 1976. In analyzing the series where overlap was available, we do not think this is a significant source of error. In the one case in which we use data for the 1960–68 period, we also use aggregate turnout data reported by the CRS.
State-level demographic data was estimated from the Census Bureau Current Population Survey for each year (with 1976 interpolated).
1. See Leighley and Nagler 1992a, Nagler (1991; 1994), and Rosenstone and Hansen (1993).
2. The centrality of registration to the act of voting is underscored by Erikson's (1981) study equating registration with voting, but more recent analyses have challenged this concept; see Lloyd (2001) as well as Brown & Wedeking (2006); Fitzgerald (2005).
3. The NVRA relieved states of the obligation to provide registration materials at state agencies offering public services if they adopted election day registration.
4. U.S. Election Assistance Commission. "Ballots Cast Before Election Day Expected to Increase as Early Voting Trend Continues," September 30, 2012; available at <http://www.eac.gov/ballots_cast_before_election_day_expected_to_increase_as_early_voting_trend_continues_/>.
5. While absentee votes are also cast early (i.e., before election day), we use the term _early voting_ to refer to voting that occurs in person prior to election day.
6. Since the non-EDR states were affected by a major institutional change during the 1960–72 period, the Voting Rights Act of 1965, we only used non-Southern states for this analysis.
7. We note that our unit of analysis here is _the state_ ; thus we average over states, not over population. This difference between the increase for the EDR states and the non-EDR states is the net effect of adoption of EDR on turnout, and is reported in column 7 (Net EDR Effect) of table 4.3.
8. Here we only use non-Southern states to avoid conflating increases in turnout in Southern states due to the passage of the 1965 Voting Rights Act with increases in turnout caused by the electoral reforms we are studying.
9. Note, however, 1972 was the first election following adoption of the twenty-sixth Amendment which gave eighteen-year-olds the right to vote, so the comparisons at the very bottom of the age distribution should be interpreted with caution.
10. We also note that these three states appear to have had relatively high registration rates prior to adoption of EDR. Thus, the impact of adoption of EDR could be mitigated, though we would not expect it to be zero.
11. Since our initial year of data is 1972, we do not know if states with no-fault absentee voting in 1972 also had it in 1968, so we cannot use turnout in 1972 as a measure of the impact of adoption of no-fault absentee voting. Also note that we are only considering elections in presidential years. So for a state that adopted no-fault absentee voting in 1979, the election preceding adoption would be 1976, not 1978.
12. We weight each year by the number of states adopting no-fault absentee voting for that year.
13. We attempted to use this methodology to estimate the effects of the reforms on different demographic groups, and we explain the result of that separately below. Our estimation strategy for the group-specific effects differs from previous research in that we estimate the turnout rates of different subgroups separately rather than relying on model specifications that include interaction terms between the demographic characteristic of interest (e.g., registration income) and the electoral reform of interest (e.g, registration closing).
14. We measure closeness of an election as the reciprocal of the absolute value of the difference between the two-party vote shares. For closeness of senate and gubernatorial elections when there was no election, we assign a closeness value corresponding to an extremely uncompetitive race. We experimented with different values of closeness for non-races, and the results are not sensitive to the value chosen.
15. A common misperception is that states with EDR have no closing of registration (i.e., a closing period of zero). That is not the case. States with EDR typically close normal registration anywhere from seven to fifteen days _prior_ to election day, and then allow unregistered persons to register on election day itself.
16. These estimates are based on the adoption of EDR by all nine current EDR states.
17. The CSTS model we are using is saturated with year and state-level fixed effects, making it difficult to get statistically significant estimates of the parameters of interest for the electoral reforms we are studying (e.g., early voting, etc.). We also attempted to estimate an individual level model that included an imputed value for the likelihood that the respondent voted in the previous election to take advantage of the intervention of the state law change and attempted to compare behavior before and after the change. This also did not yield sufficiently precise estimates to distinguish between demographic groups, though we think the method could prove fruitful in future work.
**Five**
* * *
**Policy Choices and Turnout**
A fundamental shift in the study of political participation since Wolfinger and Rosenstone's classic study (1980) has been to acknowledge the role of political elites in stimulating (or at times, depressing) voter turnout. Rosenstone and Hansen (2003) offered a theoretical interpretation of the role of elites in affecting turnout similar to that of the demographics model of turnout: in some circumstances political elites reduce or subsidize the costs of participating, and this is likely to increase turnout. Their evidence that the activities of party elites (such as contacting voters) and election characteristics (such as competitiveness) account for more than half of the decline of voter turnout since 1960 sustains their more fundamental claim: "Explanations of political involvement that have focused exclusively on the personal attributes of individual citizens—their demographic characteristics and political beliefs—have missed at least half the story" (2003, 213).1
Existing research has shown that contextual factors such as electoral competitiveness (Cox & Munger 1989) and unionization (Leighley & Nagler 2007) affect turnout. In this chapter we address a relatively overlooked aspect of this other half of the story: candidate position-taking in presidential elections. As Key (1966) famously observed, "Voters are not fools." But they are constrained by the electoral choices that they are offered. And it would be more foolish to sit out an election where the choices differ than to sit out an election in which the choices are not choices, but echoes.
Zipp (1985) was perhaps the first to empirically document the importance of candidates' relative policy positions as determinants of voter turnout, but little research has been done more recently to assess whether the choices that candidates offer voters in presidential elections matter. Consistent with a cost and benefit framework of voter turnout, we argue that individuals will be more likely to vote when offered more distinctive positions between the candidates. To test this claim, we add to our basic individual-level demographic model the relative policy positions of the leading presidential candidates in each election year from 1972 through 2008. These analyses demonstrate whether the choices elites offer to individuals matter as they decide to vote or not.
We also consider how these choices have changed over time, and whether the nature of the choices is equally important to the turnout decisions of people in different positions of the income distribution. That is, do the policy positions offered by elites have a larger influence on the turnout of poor people than rich people? Our evidence on this question allows us to determine whether the policy choices offered by candidates affect who votes as well as influence the representativeness of voters.
**5.1 Policy Choices and the Costs and Benefits of Voting**
Discussions of _why_ elite activities and characteristics such as party competitiveness, union mobilization, and voter contact increase turnout typically emphasize how such factors reduce the costs associated with voting. More generally, most theoretical interpretations likewise focus almost exclusively on the costs rather than the benefits of voting. Certain demographic characteristics, such as higher levels of education and income, for example, are often interpreted as reducing information costs, while research on electoral reforms focusing on voter registration requirements and election administration frames these legal restrictions as increasing the costs of voting.
In a review of some of this work, Aldrich (1993) suggests that one of the fundamental assumptions motivating many rational choice models of voter turnout—that voter turnout is an example of a collective goods problem—approaches being incorrect. Or, at least, he portrays voter turnout as so low-cost and so low-benefit that it likely takes little in the way of reduced costs _or_ enhanced benefits to get voters to the polls.
Aldrich suggests the reason turnout appears higher in close elections is that strategic politicians invest more heavily in close races, and that citizens respond to these mobilization efforts.2 Most aggregate-level studies, and many individual-level studies, report significant effects of competitiveness on voter turnout (see, for example, Cox & Munger [1989], Endersby, Galatas, & Rackaway [2002], Leighley & Nagler [1992b], and Rosenstone and Hansen 2003]). But as Aldrich suggests, the causal mechanism remains uncertain: do competitive races yield high turnout because voters calculate that they are more likely to be decisive in such contests, or because elites invest more heavily in mobilization activities?
Discussions of benefits as integral to the decision to vote are few, in part due to the basic logic of the calculus of voting, where one's benefit must be discounted by the probability that one casts the decisive ballot; because this probability is so small, the benefits approach zero (Riker & Ordeshook 1968). Alternatively, several different types of psychological benefits associated with voting, such as feelings of civic duty or partisan loyalty, as well as benefits associated with the social rewards from voting, would not depend on the perceived probability of being pivotal. Yet interpretations of these factors as benefits are often discounted for they seem to be posthoc adjustments to the formal model.
Several scholars, however, have considered a different type of benefit in empirical analyses of voter turnout: the choices that are available to voters, as represented by the candidates. Zipp (1985) argues that individuals' decisions to _not_ vote likely reflect the choices that they are offered as opposed to any particular individual characteristics typically used to explain not showing up at the polls.3 Zipp's empirical analysis of voter turnout in presidential elections from 1968 to 1980 confirms this argument, as do numerous more recent studies of turnout in legislative elections (see, for example, Adams, Dow, & Merrill [2006] , Adams & Merrill [2003]; Ashenfelter & Kelley [1975]; and Plane & Gershtenson [2004].
Zipp conceptualizes individuals' distance from candidates in each presidential election as resulting in alienation (i.e., the distance between the individual's preferred policy position and the closest candidate's policy position) and indifference (i.e., the difference in distance between the respondent's preferred policy position and the policy position of each of the candidates). Estimating cross-sectional models of turnout consisting of demographic characteristics along with measures of issue-specific alienation and indifference, Zipp finds that both alienation and indifference significantly influence individuals' decisions to vote. Though which particular issue-based measures of indifference and alienation are significant varies each election year, he concludes that indifference has a slightly larger effect on turnout than alienation.
Using similar measures of candidate issue positions in the 1988–92 Senate elections, Plane and Gershtenson (2004) find that individuals are less likely to vote when they feel indifferent to or alienated from candidates' ideological positions. They report that alienation has a greater potential effect on citizens' turnout decisions than does indifference. Analyzing presidential elections from 1980 to 1988, Adams, Dow, and Merrill (2006) suggest that the effect of alienation on turnout is slightly larger than the effect of indifference on turnout.
Following Zipp (1985) we argue that an individual will be more likely to vote when candidates take policy positions providing the voter with more distinct choices, and when candidates offer policy choices that more closely match the individual's preferences. The hypothesized relationships we examine are the same as what these previous studies have examined. First, when one candidates' policy positions are more appealing to an individual than the other candidates' policy positions, the resulting perceived policy difference increases the probability of voting. Second, when candidates' policy positions are distant from those of the individual, then the resulting perceived policy alienation of the individual decreases the probability of voting.
Our evidence in this chapter begins with descriptive data on individuals' perceptions of candidates' policy positions, and how these change over time and vary by income group. We then report multivariable tests of the relationships between these perceived policy choices and voter turnout. Our interest in evaluating possible differences across income groups is motivated by our broader interest in understanding the political consequences of economic inequality. An increase in economic inequality is likely accompanied by an increasing divergence in the economic needs and priorities of poorer and wealthier individuals, and this divergence might well be reflected in increasingly distinctive policy preferences across income groups (Schlozman et al. 1999). A critically important empirical question over a period of increasing economic inequality, then, is whether citizens both poor and rich are offered equally satisfying policy options by presidential candidates as they decide whether to vote or not.
**5.2 Policy Choices: Conceptualization and Measurement**
In our analyses below, we use data from the American National Election Study to test the effects of policy choices on voter turnout in each presidential election year from 1972 to 2008. Our measures of policy choices are similar to those used by Zipp (1985), who derived measures of indifference and alienation based on the individual's self-reported policy positions on 7-point scales on a variety of issues (e.g., urban unrest, the Vietnam War, government guarantee of jobs, minority rights, the role of women, and ideology) compared to the individual's perception (i.e., placement) of each candidate's location on the same set of 7-point scales. Our analysis includes measures based on ideology and the government guaranteeing jobs issue.
To avoid confusion with psychological approaches to alienation and mass political behavior, we have adopted different terminology for the policy-based concepts of indifference and alienation introduced by Zipp.4 Instead of using the term _indifference_ we use the term _perceived policy difference_ (PPD), and instead of using _alienation_ , we use _perceived policy alienation_ (PPA).
Our measures of perceived policy difference and perceived policy alienation are based on the two 7-point scales that are available in every presidential election year between 1972 and 2008: ideology and government guaranteeing jobs. For each topic, respondents are asked to identify their preferred position on the 7-point scale. They are also asked to identify the positions of the presidential candidates. The two endpoints of the ideology scale are "extremely liberal" and "extremely conservative"; the two endpoints for the government guaranteeing jobs scale are "Some people feel that the government in Washington should see to it that every person has a job and a good standard of living" and "Others think that the government should just let each person get ahead on his/her own." (For complete question wording, see appendix 6.1.)
Our measure of perceived policy difference on each question is the absolute value of the _difference_ in the distance between a respondent's own placement and her placement of each of the candidates. A perceived ideology/jobs policy difference score of 0 thus means that a respondent is equidistant from both candidates, and a positive perceived policy difference score of high magnitude suggests that the respondent is substantially closer to one candidate than the other.5 To illustrate: if the respondent places herself at 2, and places the candidates at 1 and 5, then the perceived policy difference score is 2 (||2 − 1| − |2 − 5||).
Our measure of perceived policy alienation on each question is the minimum of the absolute value of the distance from the respondent to each candidate. A score of 0 indicates that one of the candidates (i.e., quadratic vs. linear vs. log). These results and the details on the different measurement decisions are presented in appendix 5.1. is placed at the same point as the respondent's preferred position on the ideology/jobs scale, while a positive perceived alienation scores means that the closest candidate's position is farther way from the respondent's preferred position. Using the same scenario as described above, if the respondent places herself at 2 on the 7-point scale, and places the candidates at 1 and 5, the perceived policy alienation score is 1 ( **min** (|2 − 1|,|2 − 5|)).
**5.3 Perceived Policy Choices, 1972–2008**
We begin with a preliminary description of individuals' perceptions of presidential candidates from 1972 to 2008 by graphing (in figs. 5.1 and 5.2) the average placement of Republican and Democratic candidates by each income group, first for ideology and then for government jobs. We provide the graphs broken down by income group because we ultimately want to investigate how the impact of candidate choice varies by income. As the lines represent equal portions of respondents, the aggregate placement would just be the average of the five lines. This descriptive detail provides some useful information about how policy choices vary by election as well as whether the poor and the rich see the same or different policy options over time.
**Figure 5.1.** Perception of Democratic Candidate Ideology.
_Note_ : Higher values indicate more conservative perceptions. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
What we see is _substantial_ divergence over time between the placement of the candidates by the different groups of respondents. Whereas from 1972 through 1980 the groups basically agreed on the placement of the Democratic and Republican candidates, they diverged quite sharply and systematically thereafter. After 1980, respondents in the lower-income quintile consistently saw the Democratic candidate as less liberal than did respondents in the upper-income quintiles. At the same time, respondents in the lower-income quintile consistently saw the Republican candidate as less conservative than did respondents in the upper-income quintiles.6
Respondents in the lower-income groups were also less likely to be able to offer a placement of the candidates. The nonresponse rates for the bottom quintile for placing the Democratic candidate and the Republican candidate on the ideology scale ranged between 12 and 17 percent in the three most recent elections, while the nonplacement rate ranged between only 2 and 6 percent in the three most recent elections for respondents in the top quintile. For placement of candidates on the guaranteed jobs scale, the nonresponse rate for those in the bottom quintile is again substantially higher than those in the top quintile, ranging from 8 to 24 percent over the last three elections for the bottom quintile but only 5 to 11 percent for those in the top quintile. Those respondents in the bottom income quintile who do not know where the candidates stand on issues will not be motivated to vote by a perception that the candidates offer distinct choices. Failure to place the candidates is equivalent to seeing both candidates as having identical positions for the purposes of motivating turnout.
**Figure 5.2.** Perception of Republican Candidate Ideology.
_Note_ : Higher values indicate more conservative perceptions. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
It is well documented that the parties have become polarized in Congress over this time period.7 But what has not been documented is whether these elite-level changes are recognized equally among various demographic groups. Our evidence shows that those in the lower-income groups are much less aware of this than those in the higher-income groups. Regardless of the reasons for this gap in perceptions, we want to examine how individuals' perceptions of the policy choices offered them in presidential elections influence whether they decide to vote or not.
**5.4 Multivariable Analysis: Perceived Policy Alienation and Perceived Policy Difference**
To test for the effects of perceived policy difference and perceived policy alienation on voter turnout, we estimate an individual-level multivariable logit model of turnout for each presidential election year from 1972 to 2008, using data from the American National Election Study. Our primary interest focuses on the estimated effect of our four measures: two of perceived policy difference (one based on ideology, one based on government guaranteeing jobs) and two of perceived policy alienation (one based on ideology, one based on government guaranteeing jobs).
We refer to the perceived policy difference measure based on ideology as PPD _ **Ideology**_ and the perceived policy difference measure based on
government guaranteeing jobs as PPD _ **Jobs**_. Similar terms are used for the measures of perceived policy alienation: PPA _ **Ideology**_ and PPA _ **Jobs**_.
We condition on the demographic characteristics of respondents by including a series of dummies for the level of education of the respondent; the position in the income distribution of the respondent; the age of the respondent, and the respondent's gender, marital status, and race.8 We also include a dummy variable for respondents living in the South.
We estimate the model using logit for each of the ten election years, including in each year the four measures of perceived difference and perceived alienation described above, as well as the control variables listed above. We provide graphs of the estimates for the effect of the two PPD and two PPA variables in figures 5.3 through 5.6. We expect to see that increases in perceived policy difference will lead to higher turnout (positive effects), and that increases in Perceived Policy alienation will lead to lower turnout (negative effects). Figure 5.3 graphs the effect of one-standard-deviation change in perceived policy difference, measured on ideology (PPD _ **Ideology**_ ), on turnout for each election from 1972 through 2008.
**Figure 5.3.** Marginal Effect of Perceived Policy Difference (Ideology) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard deviation change in perceived policy difference (ideology) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
The solid dot in the figure represents the estimated magnitude of the marginal effect of a one-standard-deviation change in PPD _ **Ideology**_ , while the vertical line represents the 95 percent confidence interval for the estimated effect in each year.9 The effect is estimated conditional on all other demographic characteristics included in the model.10 The figure shows that respondents who have a perceived policy difference one standard deviation higher on ideology, ceteris paribus, are approximately 5 percentage points more likely to vote in each election from 1972 to 2008.11 The estimated effect is quite stable and in nearly every election reaches traditional levels of statistical significance.
That the estimated effect does not reach traditional levels of statistical significance in two of the ten elections does not suggest that the hypothesized effect is not confirmed at the 95 percent confidence level. It is true that we are not 95 percent confident that perceived policy difference affected turnout _in the 2008 election_ based on our estimate for that year. However, given the distribution of _p_ -values for our ten estimated coefficients, we can reject the null hypothesis that perceived policy difference does not affect turnout at well over the 99 percent confidence level. In other words, the probability that we would observe the distribution of estimates we do (with the ratio of positive to negative estimates, and associated standard errors) if the hypothesized relationship did not exist is less than 1 percent.
Figure 5.4 graphs the effect of a one-standard-deviation change in perceived policy difference, measured this time on the respondent's view of the role of government in guaranteeing jobs ( _P P D **J obs**_ ) on turnout. The effect of perceived policy difference based on the jobs question does not seem as large as the effects of perceived policy difference based on ideology until the most recent elections. In 2004 and 2008 the estimated effect of perceived policy difference on government jobs on turnout was _larger_ than the estimated effect of perceived policy difference on ideology on turnout.
**Figure 5.4.** Marginal Effect of Perceived Policy Difference (Government Jobs) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy difference (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
Having shown that increases in perceived policy differences between the candidates lead to increases in turnout, we next look at the effect of perceived policy alienation. Figure 5.5 graphs the effect of one-standard-deviation change on the level of perceived policy alienation, measured on ideology (PPA _ **Ideology**_ ), on turnout for each election from 1972 through 2008. Here we expect that higher levels of alienation would lead to _lower_ levels of turnout. But we see almost no relationship between levels of alienation on ideology and turnout in each election. Each estimated effect is very close to zero, or statistically indistinguishable from zero.
Figure 5.6 graphs the effect of one-standard-deviation change in perceived policy alienation on the respondent's view of the role of government in guaranteeing jobs. Here we see, in contrast to the results based on the perceived policy alienation measure based on ideology, that for most elections prior to 1996, increases in alienation did indeed lead to lower levels of turnout. However, in each election from 1996 through 2008 it appears that there was no effect of _PPA **Jobs**_ on turnout; each dot for those years is extremely close to zero.
**Figure 5.5.** Marginal Effect of Perceived Policy Alienation (Ideology) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy alienation (ideology) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Figure 5.6.** Marginal Effect of Perceived Policy Alienation (Government Jobs) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy alienation (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
We note that the estimated effects above actually understate the potential impact of perceived policy difference as we are showing the effect of changing only one of our two measures of PPD at a time. However, respondents' views of government jobs and ideology, as well as candidate placement on the issues, are likely to be highly correlated. And the estimates above are holding one issue fixed as we estimate the effect of change on the other issue alone. So a more realistic exercise is to see what would happen if a respondent perceived larger levels of policy difference on _both_ issues simultaneously. Thus, in figure 5.7 we show what would happen if a respondent's level of perceived policy difference on _both_ ideology and government job guarantees increased by one standard deviation each. As we can see, the estimated cumulative effect on a single respondent is between 5 and 10 percentage points, substantially larger than we saw for each issue alone.
We also looked at the effect of the respondent's level of perceived policy alienation changing on _both_ ideology and government job guarantees, again moving the respondent on both issues simultaneously by one standard deviation. Here we see in figure 5.8 that the impact of alienation appears to be decreasing over time, mirroring our finding for the individual estimates above.
**Figure 5.7.** Marginal Effect of Perceived Policy Difference (Ideology and Government Jobs) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy difference (ideology) and a one-standard-deviation change in perceived policy difference (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Figure 5.8.** Marginal Effect of Perceived Policy Alienation (Ideology and Government Jobs) on Turnout, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy alienation (ideology) and a one-standard-deviation change in perceived policy alienation (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is a middle-income, married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
The size of the effects of increased perceived policy differences based on both ideology and government guaranteeing jobs is striking. To put this in a broader context, a 5 to 10 percentage-point impact is larger than our estimates of the impact of any legal change that we observed in chapter 4. We emphasize that is likely a conservative estimate of the effect of perceived policy difference. We are only looking at the perception of the candidates on two issues. If we included more measures of candidate positions, we might well see an even larger effect. This suggests that choices offered by candidates matter for elections, and that strategic decisions by candidates to try to alter voter perceptions of them can have significant effects on turnout (if such attempts at altering perceptions are successful).
Thus, our analyses confirm the theory that respondents' perceptions of the policy differences between candidates do influence turnout. Respondents who perceive a greater difference between the candidates, and presumably have a stronger preference for one over the other, are more likely to vote. And the difference in the findings between perceived policy difference on ideology, and perceived policy difference on the government role in guaranteeing jobs, is suggestive. The recent increase in the effect of perceived policy difference on the government's role in guaranteeing jobs suggests that economic concerns could be becoming more important, and that respondents could be putting more emphasis on candidates' positions on economic issues.
**5.5 Perceived Policy Difference and Perceived Policy Alienation across Income Groups**
Our initial evidence, then, suggests that levels of perceived policy difference and perceived policy alienation do affect voter turnout. Yet we also observed differences in the perceptions of the poor and wealthy as to where they place Republican and Democratic candidates on ideology and the government guaranteeing jobs. Our interest in the political consequences of increasing economic inequality thus leads us to consider whether levels of perceived policy difference and perceived policy alienation vary across income groups over time, as well as whether perceived policy difference and perceived policy alienation influence individuals' turnout decisions differently depending on income levels.
Above we have shown that perception of candidate positions on issues varies across income groups. In figure 5.9 we graph the level of perceived policy difference for respondents in each of the five income quintiles on ideology (PPD _ **Ideology**_ ) from 1972 through 2008. For the
period 1972 through 2000 the levels of perceived policy difference reported by each quintile group basically move together. The gap between the groups is never very large, nor is the level of perceived policy difference generally monotonically related to income. However, in 2004 and 2008 we see a divergence between the groups. In both of those years, the perceived policy difference between the two candidates for the top quintile is substantially higher than that of the bottom quintile. Note what this means: those in the top income quintile see a _larger_ difference between the candidates on ideology than do those in the bottom quintile.
Figure 5.10 graphs the level of perceived policy difference for respondents in each of the five income quintiles on the government's role in guaranteeing jobs for 1972 through 2008. We see a similar pattern here in 2008: respondents in the highest income quintile see a much larger difference between the candidates on the government jobs issue than do respondents in the lowest income group. Since respondents in the top income quintile observe larger policy differences between candidates than those in the low income quintile observe, we would expect an increase in income bias in 2008. Since we did not see such an increase in 2008, this suggests that other election-specific factors worked to mitigate income bias.
**Figure 5.9.** Perceived Policy Difference (Ideology) by Income, 1972–2008.
_Note_ : Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Figure 5.10.** Perceived Policy Difference (Government Jobs) by Income, 1972–2008.
_Note_ : Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
We also examined the levels of perceived policy alienation of respondents for each income group. Figure 5.11 graphs the mean level of perceived policy alienation on ideology for each quintile, and figure 5.12 graphs the mean level of perceived policy alienation on guaranteed jobs for each quintile. Here we see that respondents in the highest income quintile are almost always the least alienated on ideology, and that those in the bottom income quintile generally are more alienated on guaranteed jobs than are respondents in the other income quintiles.
But differences in the impact of perceived policy difference and perceived policy alienation across income groups on turnout can come from two causes: different _levels_ of perceived policy difference and perceived policy alienation, or varying magnitudes of _the effects_ of perceived policy difference and perceived policy alienation on turnout. We now turn to examine the possibility that aside from differences in _levels_ of perceived policy difference and perceived policy alienation between the poor and wealthy, the _effects_ of these two predictors of turnout may be greater for the poor than for the wealthy. We anticipate that it is the poor for whom distinctive policy positions on redistribution will have the greatest appeal as a benefit of voting. We test this possibility by estimating the multivariable model used above, disaggregated by income group. This allows the effects of perceived policy alienation and perceived policy difference to vary by the income of the respondent.12
**Figure 5.11.** Perceived Policy Alienation (Ideology) by Income, 1972–2008.
_Note_ : Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Figure 5.12.** Perceived Policy Alienation (Government Jobs) by Income, 1972–2008.
_Note_ : Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
Figure 5.13 provides the estimates of the effect of a one-standard-deviation change in the level of perceived policy difference on ideology on turnout for respondents in each of the five income quintiles, for each election from 1972 through 2008. We are not able to estimate the effect of perceived policy difference on ideology for each income group very precisely for each election. But we do observe that when averaging the estimates for any one group over time, the effects do not differ substantially across the income groups.
Figure 5.14 provides similar estimates for the effect of a one-standard-deviation change in the level of perceived policy difference on government's role in guaranteeing jobs on turnout. Once again, we cannot estimate effects precisely enough to make meaningful comparisons of the effects across the quintile groups for each election, but the graphs do suggest that the role of the perceived policy differences between the candidates on government jobs as a predictor of turnout is likely larger for both the first, and especially the fifth, quintile in the last three elections.
We also present estimates of the effect of perceived policy alienation on ideology and perceived policy alienation on guaranteed jobs by income quintile in figures 5.15 and 5.16. Again, our caveats about the precision of the estimates apply.
However, while we could not determine that equal levels of perceived policy difference and perceived policy alienation would have different effects on rich and poor individuals, we note that we _did_ see above in figures 5.9 and 5.10 that poorer respondents have perceived less of a difference between candidates in recent elections. To examine the effects of these differences across the income groups, we simulated what turnout would be for the bottom quintile in 2008 if members of the bottom quintile perceived the same differences that the top quintile perceived. We adjusted the perceived policy difference on ideology and guaranteed jobs for each member of the bottom quintile to achieve mean perceived policy differences on ideology and guaranteed jobs equal to the mean perceived policy differences on both issues for the top quintile. We then computed the probability of voting for each respondent in the bottom quintile under the hypothetical scenario. According to our estimates, turnout of the bottom quintile would have risen by 3.5 percentage points under this scenario. This is a substantial effect, similar to the effects we estimated for several electoral reforms in the chapter 4. But it would still leave turnout of the bottom quintile approximately 20 percentage points below the turnout of the top income quintile.
**5.6 Conclusion**
Our interest in this chapter was to make more prominent the role of policy choices as a determinant of turnout in presidential elections. The blur of modern campaign politics often seems to overlook, or even forget, that elections are choices not just between candidates or parties, but also between issue positions.
Our evidence affirms that the policy choices offered by candidates matter for voter turnout. We have shown that the choices offered individuals affect their likelihood of voting. Citizens who perceive a substantial difference between the candidates are more likely to vote than are respondents who are indifferent between the candidates.
**Figure 5.13.** Marginal Effect of Perceived Policy Difference (Ideology) by Income, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy difference (ideology) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is of given income, a married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
Our analyses of the role of policy choices also included comparisons of perceived policy difference, perceived policy alienation and perceptions of candidate policy positions across income groups. We found that respondents in the bottom income quintile had substantially lower levels of perceived policy difference than did respondents in the higher-income quintiles in the 2004 and 2008 elections. This suggests that theoretical expectations that people at the bottom of the income distribution will be motivated to vote by a desire for economic redistribution are not likely to be met, as those persons are precisely the ones least likely to see the candidates as offering meaningful choices on issues.
**Figure 5.14.** Marginal Effect of Perceived Policy Difference (Government Jobs) by Income, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy difference (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is of given income, a married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
Previous theories and research on the political implications of economic inequality have ignored the key role of whether parties or candidates offer relevant choices to those hurt by increasing inequality. Our answer to this point for the United States over the past decade is that the poor are less likely to perceive these differences than are the wealthy—and the poor cannot respond to policy choices they do not see. As a result, increasing economic inequality is unlikely to be met by increased turnout on the part of the poor unless one or both of the major parties offers a distinctive and compelling policy choice.
**Figure 5.15.** Marginal Effect of Perceived Policy Alienation (Ideology) by Income, 1972–2008.
_Note_ : The vertical axis represents the effect of a one-standard-deviation change in perceived policy alienation (ideology) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is of given income, a married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Figure 5.16.** Marginal Effect of Perceived Policy Alienation (Government Jobs) by Income, 1972–2008.
_Note_ : The vertical axis represents the effect of a one standard deviation change in perceived policy alienation (government jobs) on the probability of voting. The vertical line through each point gives a 95 percent confidence interval about the estimate. The probabilities are calculated using the demographic model described in section 5.4 for a hypothetical respondent who is of given income, a married, white woman, a high school graduate, age thirty-one to forty-five, who lives outside the South. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008.
**Appendix 5.1: Comparing Alternative Measures of Alienation and Indifference**
When trying to measure alienation and indifference one is faced with (at least) one measurement choice and (at least) one modeling choice. We define a respondent as being indifferent between two candidates if his utility for each candidate is equal. If we restrict ourselves to spatial (proximity) models of utility, this means that we need to place the respondent and the candidates on an issue scale and come up with a functional form for utility.
In the NES surveys, respondents are asked to place themselves on the issue scale. If we take this as a starting point, the question becomes where to place the candidates. We have two choices: the respondent's placement of each candidate, or some external measure of placement such as the mean placement by all respondents of the candidate or expert placement of the candidates. Using self-placement can be problematic because (1) many respondents simply fail to place one or more candidates; and (2) respondents may place their most preferred candidate closer to themselves (or vice versa, place themselves closer to their most preferred candidate) than they would otherwise do in order to minimize cognitive dissonance. They might simply be rationalizing their voting decision.
While the second problem could jeopardize the validity of estimates of the impact of issue positions on vote choice, we argue that it should not affect estimates of the impact of issue positions on turnout. The decision to turn out to vote is, after all, based on the voter's perception of indifference between the candidates. However, we find it unlikely that the voter feels any need to justify the decision to turn out by rigging the placement of candidates to suggest a large preference for one candidate over the other. Thus, even if the respondent is minimizing cognitive dissonance in vote choice, this is what we would want to take into account in measuring his level of indifference between candidates.
However, we are still left with the problem when using the respondent's placement of the candidates that many respondents simply fail to place the candidates. This means that we could (1) impute the respondents' placement of the candidate; (2) assume that respondents who cannot place either candidate are indifferent between them; (3) use the mean placement of the candidate given by all respondents who can place the candidate; or (4) omit the respondents from the analysis who cannot place the candidates. Imputation is obviously problematic here because the failure to place the candidate is almost certainly not done at random; presumably people who cannot place the candidate fail to do so because they do not know where the candidate is in the issue space. Imputing an opinion to the respondent that the respondent has explicitly denied having is probably not a good idea.
If respondents cannot place either candidate, they might be indifferent between them. However, the failure to place the candidates might be because they find the task of placement on the 7 point scale cognitively challenging. Or, perhaps they really are indifferent?
Using the mean placement of the candidate generates measurement error in two ways. First, of respondents who can place the candidate, we are replacing their own placement with a mean of placement by others, and not all respondents interpret the scale the same way. Thus, if a respondent places himself at 2, and the candidate at 3, we _know_ the respondent believes himself to be one unit from the candidate. However, the mean placement of the candidate could represent a mean given by respondents who interpret the scale differently from this respondent.
Second, for respondents who could not place the candidate we have the same problem we would with imputation—we are giving the respondent an opinion that the respondent has explicitly denied having. Omitting the respondents who fail to place the candidates from the analysis means that our analysis only generalizes to respondents able to place the candidates, and thus limits its usefulness. In the analysis presented here we have chosen the third option—omitting these respondents.
But even after measuring respondent placement and candidate placement, we still need to relate those positions to utility and indifference, and to alienation. When specifying the actual utility function for the voter, there are two common Euclidean choices we can make. First, we can specify utility as a quadratic loss function of the distance between the voter and each candidate. This is the most commonly used function. And it has a very real substantive implication. It suggests that voters put a greater value on the difference in distances between candidates going from 3 to 4 than from 1 to 2. Or, we can use a linear loss function of the distance between the voter and each candidate.
Consider how utility is measured: Define 13
• **X** _i_ = Respondent Position
• **D** _i_ = Democratic Candidate Position
• **R** _i_ = Republican Candidate Position
• **U** _i D_ = -( **X** **i** − **D** _i_ )2
• **U** _i R_ = -( **X** **i** − **R** _i_ )2
• **U** _i D_ − **U** _i R_ = − ( **X** **i** − **D** _i_ )2 \+ ( **X** **i** − **R** _i_ )2
• **Pr** ( **Vote** ) = **F** ( **abs** ( **U** _i D_ − **U** _i R_ ))
• **U** _i D_ − **U** _i R_ = − ( **X** **i** − **D** _i_ )2 \+ ( **X** **i** − **R** _i_ )2
• **Indifference** = − **abs** (−( **X** **i** − **D** _i_ )2 \+ ( **X** **i** − **R** _i_ )2)
The equations below give an example of the implication of a quadratic loss function for indifference.
In the case above, with quadratic utility, for case 1: Indifference = − **abs** ((3 − 4)2 − (3 − 5)2) = − **abs** (1 − 4) = −3. For case 2, Indifference = − **abs** ((3 − 5)2 − (3 − 6)2) = − **abs** (4 − 9) = −5. Thus, with quadratic utility the respondent is _less_ likely to vote in case 1 than in case 2, as they have a higher value of indifference (the closer the value of indifference is to zero, the less likely someone is to vote). This does _not_ seem to be intuitively appealing: in case 2 the Republican candidate is only 50 percent farther from the respondent than the Democratic candidate is, whereas in case 1 the Republican candidate is twice as far from the respondent as the Democratic candidate is. Thus, it is appealing to think that the respondent is more likely to vote in case 1 than in case 2, not less likely. If we are to make these proportional comparisons, it suggests taking the log of the squared distances.
Now consider alienation.
In case A1, the respondent is at position 3, and the Democratic candidate is initially at position 4. Using quadratic distance, the alienation for the respondent is (3− 4)2 = 1. If the Democratic candidate moved to 5, the level of alienation would then be (3 − 5)2 = 4, for an increase of 3 units of alienation. If we consider case A2, the Democratic candidate is again going to move 1 unit. But here, the level of alienation goes from (1 − 6)2 = 25 to (1 − 7)2 = 36, for an increase of 11 units of alienation. However, we might logically think that whereas in case A1 the Democratic candiate moved to being _twice_ as far from the respondent as he started, and in case A2 the distance between the Democratic candidate and the respondent only went up by 20 percent, that alienation should have increased more in the first case. We could capture this notion of proportional increases in distance being equivalent by simply taking the log of the squared distance.
We opt for brute force empiricism: we estimate four sets of models of turnout. In one model we use absolute value of distances and respondent placement of candidates; in the second model we use squared value for distances, and respondent placement of candidates; in the third model we use squared values for distances, and the mean placement for candidates; and in the fourth model we use the squared values for distance for indifference but take the log of the squared value for alienation and use respondent placement of candidates.
One of our measures is thus what was used by Zipp (1985). Using American National Election Study data from 1968 to 1980, Zipp derives measures of alienation and indifference based on the individual's self-reported position on 7-point scales on a variety of issues (e.g., urban unrest, the Vietnam, War, government guarantee of jobs, minority rights, the role of women, and ideology) compared to the individual's placement of where each candidate was located on the same set of 7-point scales. The alienation measure is the absolute value of the minimum distance between the individuals' issue position and either of the candidate's issue positions. Higher values thus represent greater alienation.
The indifference measure Zipp used is the absolute value of the difference of the distance between the respondent's self-placement and the perceived location of the Democratic candidate, and the distance between the respondent's self-placement and the perceived location of the Republican candidate, with this value being reversed in sign so that higher values represent greater indifference. That is, if individuals are equally close to both candidates, regardless of the direction of the preferred policy differences, then they should be more indifferent to which candidate is selected; but if individuals are quite close to one candidate and very far away from the other, then they should have a strong preference as to who is elected, and thus have low levels of indifference.
We computed the same measures of alienation and indifference used by Zipp for each election year from 1972 to 2008. As described above, we also computed three additional measures of alienation and indifference, reflecting differences in how the candidate's policy position was computed (e.g., the individual's perception vs. the mean placement of a candidate based on the entire sample's reported perception) and differences in the voter's utility function (i.e., quadratic vs. linear).
**Table A5.1.1. Alternative Model Specifications and Measures: Goodness-of-Fit Estimates.**
_Note:_ The dependent variable is self-reported voter turnout. Each model includes demographic characteristics of respondents and perceived policy difference and perceived policy alienation for two issues: ideology and government jobs. Estimated by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008. Goodness-of-fit measures are reported for five models.
_a_ Absolute Value-Respondent Placement of Candidates: Uses respondent placement of candidates, with the absolute value of the difference in perceived policy placements on the seven-point scale as the functional form for measuring perceived policy difference and perceived policy alienation.
_b_ Squared Distance—Respondent Placement of Candidates: Uses respondent placement of candidates, with the squared value of the difference in perceived policy placements on the seven-point scale as the functional form for measuring perceived policy difference and perceived policy alienation.
_c_ Squared Distance—Mean Placement of Candidates: Uses the mean sample placement of candidates, with the squared value of the difference in perceived policy placements on the seven-point scale as the functional form for measuring perceived policy difference and perceived alienation.
_d_ Squared/Log(Squared) Distance—Respondent Placement of Candidates: Uses respondent placement of candidates, with the squared value of the difference in perceived policy placements on the seven-point scale for perceived policy difference, and the log of squared value for perceived policy alienation as the functional forms.
_e_ Log(Squared)—Respondent Placement of Candidates: Uses respondent placement of candidates, with the log of squared values of difference in perceived policy placements on the seven-point scale as the functional form for perceived policy difference and perceived policy alienation.
Table A5.1.1 presents the log-likelihood values and measures of fit (ePCP) for a model of turnout estimated on each presidential election from 1972 to 2008, but using the four different sets of measures we describe above. The model includes basic demographic variables for each respondent—education, income, and age—as well as a dummy variable for living in the South. Education is included as a series of dummy variables (high school graduates, some college, and college and beyond—with no high school degree being the omitted category). Income is also measured by a series of dummies for which income quintile the respondent is in, with the bottom quintile being the omitted category. Age is also measured as a series of dummies, with the oldest group being the omitted category. Gender, marital status, and race are included. Finally, there is a dummy variable for living in the South.
Looking across the rows, one can see almost no difference in fit in any year between the models. Clearly whether one chooses to use the respondent placement of the candidate (the Squared-Resp column) versus the mean placement of the candidate (the Squared-Mean column) makes no difference for the fit of the model. The log likelihood values and ePCP values barely change.14 Similarly, comparing the models using squared distance versus linear (absolute value) distance, there is again virtually no difference in model fit. We do note that in only two years do the models based on mean placement of the candidate, rather than respondent placement of the candidate, fare better. Thus, consistent with our theoretical view, the model using respondent placement appears to be preferred. However, this model has a severe practical shortcoming as many voters cannot place the candidate on the issues. Examining how to treat respondents who cannot place one or both of the candidates remains an ongoing research question.
1. A renewed wave of research focusing on field experiments has demonstrated anew the role of elite and nonelite contacting activity in stimulating turnout. See Green and Gerber (2008) for an introduction to this work.
2. See Jackman (1993) for a commentary on Aldrich's essay.
3. Implicitly Zipp assumes, as do we, that voters are behaving as if they are pivotal, or as if they are not completely discounting the likelihood of being pivotal.
4. See, for example, Dennis & Owen (2001), Gibson (1991); Schildkraut 2005; Theiss-Morse (1993); and Weatherford (1991; 1992).
5. We also computed additional measures of perceived policy alienation and perceived policy difference, reflecting differences in how the candidate's ideology/jobs policy position was computed (e.g., the individual's perception vs. the mean placement of a candidate based on the entire sample's reported perception) and differences in the voter's utility function
6. We note that this is also consistent with respondents in the lower-income quintile as seeing _both_ candidates as being more centrist, which could be observed if these respondents are simply guessing the midpoint more often than are respondents in higher-income quintiles.
7. See Poole & Rosenthal (1997) and Han & Brady (2007) for evidence on polarization over time in Congress.
8. Education was coded as less than high school, high school graduate, some college, and college and beyond. Income was coded based on the income quintile the respondent was in. Age was coded as: 18–24, 25–30, 31–45, 46–60, 61–75, and 76–89.
9. If any one vertical line (i.e., the estimated effect in any one election) does not cross zero, then the interpretation is that we are 95 percent sure that the the hypothesized effect is larger than zero.
10. The graph shows the first difference estimated for a single hypothetical respondent who is a high school graduate in the middle income quintile, age thirty-one to forty-five, a married white woman not living in the South.
11. We use the phrase "5 percentage points more likely" to indicate an increase of 0.05 in the probability of an individual voting. While the graphs in this chapter represent the change in the probability of voting for a single hypothetical individual, we discuss percentage-point increases in the text to maintain comparability with discussions of increases in group turnout.
12. We are thus allowing all the parameters in the model to vary over both income quintile, and year. We are generating fifty (10 years × 5 income groups) distinct sets of estimates here.
13. Note that in the definition here we use _indifference_ , rather than perceived policy _difference_ , to be consistent with standard notation on loss functions in spatial utility models.
14. See Herron (1999) for a discussion of ePCP.
**Six**
* * *
**On the Representativeness of Voters**
In this chapter we consider what we believe is a critical aspect of the potential _consequences_ of turnout, and that is whether voters are representative of nonvoters with respect to their preferred policy positions. Most discussions of the consequences of turnout focus on whether changes in the partisanship of the voters lead to changes in who wins the election. We believe it is also important to consider the governance consequences of turnout. Who wins an election is obviously important in a representative democracy. But once elected, officials have some flexibility to define their policy agendas and their policy priorities in ways that go beyond partisanship. We have argued that elected officials respond to their electoral constituencies by pursuing the issues or policy preferences of those who cast ballots for them. In this respect, presidents respond not only to fellow partisans, but also to the more specific policy preferences of their supporters. This argument shifts the focus from how representative voters are of nonvoters with respect to demographic characteristics to how representative voters are of nonvoters with respect to policy preferences.1
The empirical evidence presented in the last several chapters suggests that the relative turnout rates of the wealthy and poor have been fairly constant over the past several decades, with perhaps a slight decrease in the relative turnout of the poor in the 1990s that has recovered since 2000. As we pointed out at the beginning of this book, theory suggests that poor individuals (specifically those below the median income level) will be inclined to favor policies that redistribute income, whereas rich voters (those above the median income level) will be opposed to policies that redistribute income. Given that nonvoters are disproportionately poor relative to voters, and have been since 1972, we expect to find sustained differences in the policy preferences of voters and nonvoters in presidential elections since 1972 on redistributive issues.
We briefly review the handful of studies that have addressed the question of the representativeness of voters, and then replicate some of Wolfinger and Rosenstone's (1980) evidence for 1972 with 2008 data. We then test our expectations regarding the distinctiveness of voters' preferences using data from the 1972–2008 American National Election Studies (NES), as well as the 2004 National Annenberg Election Study (NAES), comparing the policy preferences of voters and nonvoters on redistributive issues, as well as a variety of other policy issues.
**6.1 The Conventional Wisdom**
The centrality of elections to representative democracy—along with concerns regarding low turnout in American elections—would suggest that scholars might well pay special attention to whether voters' policy positions are representative of nonvoters'. Yet aside from Wolfinger and Rosenstone (1980), we have identified few studies that consider this key question, and their conclusions are fairly consistent with each other: there are surprisingly few and, in any case, only modest, differences in the policy preferences of voters and nonvoters.2
Conventional wisdom seems to have interpreted those findings as indicating that there are no differences between voters and nonvoters. This strict interpretation certainly emerges from Wolfinger and Rosenstone's (1980) description of their data from 1972. After reporting a "slight" overrepresentation of Republicans among the voters, Wolfinger and Rosenstone examine citizens' preferences on seven issues (government guaranteeing jobs, medical insurance, bussing, abortion, legalizing marijuana, the role of women, and ideology) and observe, "All other political differences between voters and the general population are considerably smaller than this [partisan] gap of 3.7 percentage points. Moreover, these other differences, as slight as they are, do not have a consistent bias toward any particular political orientation... . In short, on these issues voters are virtually a carbon copy of the citizen population" (1980, 109).
Bennett and Resnick's (1990) analysis of General Social Survey (1985), Gallup poll (1987), and American National Election Studies (1968–1988) data mirrors these conclusions for the most part, though they offer some evidence that conflicts with Wolfinger and Rosenstone's (1980) observations of "small and statistically insignificant" differences between voters and the citizen population. Bennett and Resnick's analysis considers a broader range of the attitudinal characteristics of voters, such as patriotism and other measures of system support, attitudes toward political and social groups, and levels of political information. On these items, they too report that there are few differences and that nonvoters thus do not represent a threat to democracy.
However, on some of the same issue positions that Wolfinger and Rosenstone examined, as well as some additional policy preference measures, they note that findings are mixed. Few differences are observed on partisanship, ideology, and foreign policy positions. But on some domestic policies, "nonvoters and voters do not see eye to eye. Nonvoters are slightly more in favor of an increased government role in the domestic arena. They are more likely to oppose curtailing government spending for health and education services, and they are more likely to support government guarantees that everyone has a job and a good standard of living" (Bennett & Resnick 1990, 789–94).
In addition, Bennett and Resnick's analyses of voters' and nonvoters' opinions on spending for a set of eight domestic programs indicates that nonvoters are significantly more likely than voters to favor spending. Thus, the conventional wisdom that who votes does not matter in the representation of citizens' policy views to elected officials is clearly situated in a substantial amount of data and in the analyses of Wolfinger and Rosenstone's (1980) work, with the refinements provided by Bennett and Resnick (1990) somewhat obscured.
We find these somewhat inconsistent conclusions—coupled with the common claim that voters are representative of nonvoters—to be troubling. The substantive conclusion that it does not matter who votes seems especially inconsistent with our basic beliefs about how representative politics work: it is not just that these differences in policy preferences _should_ matter in a normative sense but also that common political sense suggests that they _must_ matter to some degree for policy outcomes.
Moreover, these conclusions (based on the policy preferences of voters vs. nonvoters) that who votes does not matter contrast with several studies that argue that who votes _does_ matter in terms of policy benefits. Hill and Leighley (1992), for example, find that states in which the poor vote as frequently as the wealthy provide significantly higher welfare benefits. Similarly, Martin (2003) finds that members of
Congress allocate federal grant awards to areas where turnout is highest. And, Bartels (2008) finds that elected officials pay more attention to the preferences of the wealthy than the poor, suggesting there is not anonymity among the electorate: not everyone's preferences count the same (see also Gilens [2012]; and Soroka & Wlezien [2010]). It would not be a great leap to suggest that elected officials pay less attention to the preferences of nonvoters than the preferences of voters.
But the key point is that the elected officials are aware of the preferences of their supporters. As we suggested in chapter 1, the poor and the wealthy might well support the same candidate and elect her; but when in office, she will pursue the policies preferred by voters (who are disproportionately wealthy) rather than those preferred by nonvoters (who are disproportionately poor). That means that for poor nonvoters; to achieve substantive representation it is not sufficient for rich voters to prefer the same _candidate_ as poor nonvoters. they must also share the same _issue positions_. This possibility demands that we clearly understand whether voters hold the same policy positions as nonvoters if we are to understand the representational consequences of turnout.
We underscore the importance of this argument by noting that significant differences between voters and nonvoters have important electoral consequences even if voters and nonvoters have identical distributions of preferences across _candidates_. Imagine a world with two dimensions, and that voters who prefer the Republican candidate to the Democratic candidate do so based on economic issues, and voters who prefer the Democratic candidate to the Republican candidate do so based on social issues; but non-voters who prefer the Republican candidate to the Democratic candidate do so based on social issues, and nonvoters who prefer the Democratic candidate to the Republican candidate do so based on economic issues. Assuming that equal proportions of voters and nonvoters prefer Republican candidates to Democratic candidates, it would make no difference to the _electoral outcome_ whether the nonvoters stay home or whether they choose to become voters. But if we assume that elected officials know the preferences of those who vote for them and respond to those preferences, then it would make a tremendous difference to _governing outcomes_ if the nonvoters choose to vote.
Below we address the question of whether who votes matters, first assessing the representativeness of the policy preferences of voters in the 1972 and the 2008 elections to provide an initial assessment of the extent to which Wolfinger and Rosenstone's classic findings (1980) remain true. We then provide a more detailed assessment regarding trends in the representativeness of voters by examining the policy differences between voters and nonvoters in each presidential election year since 1972. These analyses provide some insight as to whether such representation varies by issue type and whether any variations we observe reflect election-specific factors or instead reflect more enduring compositional characteristics of voters relative to nonvoters. The latter is especially important from a normative perspective given the notable changes in inequality since the 1980s, while the former is valuable as well in terms of identifying contextual sources—such as candidate positions or the varying salience of different issues over time—of the representativeness of the policy preferences of voters.
We also consider additional data on the representativeness of the policy preferences of voters, relying on the 2004 National Annenberg Election Survey (NAES). This analysis complements our findings based on the time series available in the American National Election Studies (NES) in that it focuses on an additional set of more contemporary policy issues than what the NES time series allows. Our analyses of NES policy positions are drawn from the standard set of 7-point issue scales, along with questions on party identification, political ideology, and presidential candidate thermometer scores available in the NES. Our analyses of the NAES data focus on the set of policy issues asked in the postelection wave of the general election panel survey. In categorizing voters and nonvoters, we rely on the postelection self-report for the NES and on NAES respondents' self-reports on whether they voted in the 2000 election.3 Specific question wording for both the NES and the NAES policy questions is provided in appendix 6.1.
**6.2 Political Differences between Voters and Nonvoters: 1972 and 2008**
We begin our comparison of the preferences of voters and nonvoters by considering attitudes on party identification and ideology, and then consider citizens' attitudes on specific issues. Table 6.1 reports the distributions of partisanship for 1972 and 2008 for nonvoters and voters using both the traditional 7-point party identification scale and a collapsed, 3-point scale.4 Wolfinger and Rosenstone's basic observations (1980) for 1972 remain, with the most notable points being the underrepresentation of independents and the overrepresentation of Republicans among voters relative to nonvoters. More specifically, while independents comprised 22 percent of nonvoters in 1972, they comprised only about 9 percent of voters; Republicans comprised about 26 percent of nonvoters and almost 40 percent of voters; and Democrats comprised about 52 percent of both nonvoters and voters.
**Table 6.1. Political Attitudes of Nonvoters and Voters, 1972 and 2008 (NES).**
_Notes_ : Entries in the first two columns for each year are column percentages. Entries in the "Difference" column are the difference between the group's share of voters and nonvoters. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972 and 2008; see appendix 6.1 for question-wording details.
_a_ Positive numbers indicate overrepresentation among voters, while negative numbers indicate underrepresentation among voters.
_b_ The moderate category includes only respondents who place themselves at the midpoint of the ideology scale.
In 2008, these same patterns can be observed. Independents represented almost 19 percent of nonvoters but only 5.6 percent of voters; Republicans represented nearly 27 percent of nonvoters but over 42 percent of voters; and Democrats comprised between 50 and 55 percent of both voters and nonvoters. Thus, the underrepresentation of independents and overrepresentation of Republicans is slightly greater in 2008 than in 1972.
Turning to a comparison of the distributions of ideology in 1972 and 2008 we see that moderates are underrepresented among voters in both years, though the underrepresentation is greater in 2008 than in 1972. Liberals and conservatives are overrepresented in both elections, with the overrepresentation of conservatives increasing somewhat more than overrepresentation of liberals in 2008.
To begin our analysis of voters and nonvoters on specific attitudes, in table 6.2 we reexamine the preferences of voters and nonvoters on the four issues that Wolfinger and Rosenstone (1980) presented from 1972 for which we have data in 2008. On the two economic issues (the government guaranteeing jobs and providing health insurance), nonvoters have more liberal views than voters in both elections, and the gap between them has increased for both issues. In 2008, there is a
10.2 percentage-point difference between nonvoters and voters believing that it is the government's responsibility to guarantee jobs, and a 12.5 percentage-point difference between voters and nonvoters believing that people should "get by on their own." In 1972 these gaps were only 7.8 percentage points and 4.6 percentage points, respectively.
While overall opinion changed on abortion from 1972 to 2008, representativeness on this issue did not change very much. The proportion of individuals taking extreme positions on abortion has increased, and in these extreme positions voters are least representative of nonvoters. For example, from 1972 to 2008, underrepresentation of extreme pro-life positions increased: the gap between nonvoters and voters taking this position was 3.6 percentage points in 1972, compared to a 9.5 percentage-point difference in 2008.
Finally, the issue on which there was the greatest improvement in the representativeness of voters is that of women's roles. Between 1972 and 2008, all segments of the electorate seem to have converged on the response in favor of women's equality. Although there were substantial differences between voters and nonvoters on whether "women's place is in the home" and whether "women are equal," in 1972, in 2008 these differences had all but disappeared. The largest representational bias on this issue remained a liberal one, with almost 87 percent of voters believing that women are equal, compared to only 80 percent of nonvoters.
Thus, in comparing the differences between voters and non-voters, we see both expected and interesting changes between 1972 and 2008. Opinion on the role of women in society has become more widely supportive of equality (at least in the voicing of public policy views), and in contrast to liberal fears that social issues now serve to mobilize conservatives, it is _liberal_ views on abortion that are overrepresented among the voters.5 For our purposes, however, the more interesting differences between 1972 and 2008 relate to the role of government in providing jobs or health insurance because these issues relate most directly to the possibly distinctive preferences of voters and nonvoters on redistributive issues. We observe here a greater underrepresentation of nonvoters' more liberal positions on these issues in 2008 as compared to 1972. Whether this difference is merely a function of the two particular time points we selected for observation or instead reflects a more fundamental difference between voters and nonvoters is addressed in the next section.
**6.3 Who Votes Matters: Policy Differences between Voters and Nonvoters**
In this part of the analysis we seek to document more broadly the contours of voters' policy representativeness over time. We want to overcome the potential hazards of comparing the 1972 and 2008 elections as endpoints and instead comment on changes in voter representativeness across the entire time period. This also allows us to assess whether such representativeness shifts slowly—as one might expect were policy views largely structured by the longer-term, enduring demographic predictors of turnout—or whether it reflects more short-term, election-specific factors. To the extent to which we observe the latter we would likely draw some inferences regarding the importance of election specific factors such as elite mobilization and candidate positioning.
According to Meltzer and Richard (1981), periods of increasing inequality should be associated with increased demand for government redistribution. Because we believe that increasing economic inequality was likely accompanied by an increasing divergence in the economic needs and priorities of poorer and wealthier individuals, and that this divergence would be reflected in increasingly distinctive policy preferences across social class (Schlozman, Burns & Verba 1999), we expected to find voters to be less representative—especially on redistributive issues—in 2008 than they were in 1972.
Our evidence on this point is drawn from the biennial American National Election Studies (NES) surveys between 1972 and 2008, which include a series of policy preference questions asked of voters and nonvoters.6 We consider citizens' preferences and attitudes as reflected in their political views, their preferences on redistributive issues, and their preferences on values-based issues. Figure 6.1 documents the policy differences between voters and nonvoters in presidential election years between 1972 and 2008, focusing on three redistributive policy questions: support for government spending on health; support for providing services; and support for government guaranteeing jobs (see appendix 6.1 for precise question wording).
**Table 6.2. Issue Preferences of Nonvoters and Voters, 1972 and 2008 (NES).**
_Notes_ : Entries in the first two columns for each year are column percentages. Entries in the "Difference" column are the difference between the group's share of voters and nonvoters. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972 and 2008; see appendix 6.1 for question-wording details.
_a_ Positive numbers indicate overrepresentation among voters, while negative numbers indicate underrepresentation among the voters.
_b_ The NES asked different questions on abortion in 1972 and 2008, both with four response categories. In both years, the question included "Always" and "Never" as response categories. In 1972, the additional two categories were: "Only if the mother's life and health of the mother was in danger" and "If, due to personal reasons, the woman would have difficulty caring for the child." In 2008, the additional two response categories were "Only in the case of rape, incest or when the woman's life is in danger" and "For reasons other than rape, incest or danger to the woman's life, but only after the need for the abortion has been clearly established." See appendix 6.1 for additional question-wording details.
We describe these questions as redistributive because they indicate the degree to which respondents support governmental services or policies that redistribute resources to the poor. For each question, respondents are asked to place themselves on a 7-point scale, with the high point indicating the most conservative policy position (opposing redistribution) and the low point indicating the most liberal policy position (supporting redistribution). In figure 6.1 we plot the difference between the mean score of voters and the mean score of nonvoters on each issue. Positive values thus indicate that voters are more conservative than nonvoters, while negative values indicate that voters are more liberal than nonvoters.
**Figure 6.1.** Differences between Voters' and Nonvoters' Attitudes on Redistributive Policies, 1972–2008.
_Note_ : Plotted values are the weighted mean difference between voters' and nonvoters' attitudes on each issue in the specified year. Values greater than 0 indicate that voters (as a group) are more conservative than nonvoters (as a group) on the specific policy question; values less than 0 indicate that voters are more liberal than nonvoters. All mean differences are significant at p < .05, except 1972 for guaranteed jobs, 1984 for government health insurance, and 2000 for government service. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008; see appendix 6.1 for question-wording details.
As shown in figure 6.1, we find consistent differences between voters and nonvoters on each of these issues. In each year since 1972, voters are more conservative than are nonvoters in their beliefs regarding how much the government should do to provide jobs, health insurance, and services. More specifically, except for the difference between voters and nonvoters on government health insurance and job guarantees in 1972 and on government health insurance in 1984, the difference between voters and nonvoters is statistically significant in each election. Substantively, the mean differences on all three issues are typically greater than .4 on a 7-point scale. This suggests that voters are about one-half a scale position more conservative than are nonvoters. As we expected, then, there are notable, consistent and substantial differences between voters and nonvoters on redistributive issues—and the conventional wisdom should be updated accordingly.
Next we consider the representativeness of voters on two different sets of issues that we refer to as values-based issues and political attitudes. We present these results in separate graphs. We expect the responses to the first set of questions, including party identification, party ideology, and candidate preference, to be most sensitive to the particular electoral context (i.e., the nature of the issues, campaign strategy, etc.). We therefore expect these attitudes—and candidate preference, in particular—to be most likely to change election by election.
In contrast, the second set of issues are largely motivated by some sense of "values": the role of women/women's equality, aid to blacks, and defense spending. While we are not arguing that this is a coherent set of opinions that share common demographic or attitudinal sources, we do believe that each of these likely reflects more personal, fundamental symbolic beliefs than the other issues we consider. As such, we expect them to likely exhibit little sensitivity to election-specific contexts.
We turn to the representativeness of voters and nonvoters on political attitudes first. The party identification and political ideology measures are based on the standard NES 7-point party identification and political ideology questions.7 The vote preference measure is based on respondents' thermometer rankings of the two major presidential candidates in each election year. We first compute the difference between voters' evaluations of the Republican and Democratic candidates and then compute the same value for nonvoters. We then take the difference between these two scores and then, for graphing purposes, rescale it to be comparable to values on a 7-point scale.
Figure 6.2 presents the mean differences between voters and nonvoters on party identification, political ideology, and candidate vote preference for 1972 through 2008.8 The values that are plotted for each attitude or preference (i.e., the vertical axis values) are the mean differences between voters and nonvoters for each attitude or preference. Based on our general knowledge of the interrelationships among partisanship, ideology, and vote choice, we expected these three measures to move largely in sync with each other, and that is mostly what we see. And because vote preference is necessarily tied to candidate characteristics, we see this difference between voters and nonvoters varying the most from election to election, as we would expect.
**Figure 6.2.** Differences between Voters and Nonvoters on Ideology, Partisanship, and Vote Preference.
_Note_ : Plotted values are the weighted mean difference between voters and nonvoters on each opinion item in the specified year. Values greater than 0 indicate that voters (as a group) are more conservative than nonvoters (as a group) on the specific attitude; values less than 0 indicate that voters are more liberal than nonvoters. Mean differences on partisanship (except 1992 and 1996) and vote preference (except 1988, 1992, 2000, and 2004) are significant at p < .05. Mean differences on ideology are significant at p < .05 only in 1988, 1996, and 2004. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008; see appendix 6.1 for question-wording details.
We note that in many of these elections we observe statistically significant differences between voters and nonvoters on partisanship and vote preference. These differences are statistically significant in six of ten elections for vote preference and eight of ten elections for partisanship. Statistically significant differences between voters and nonvoters on ideology are less common, observed in only two elections, 1988 and 1996, though in seven of the ten elections voters in our sample are ideologically more conservative than nonvoters.9 Thus we find that voters are more conservative than nonvoters on partisanship, candidate preference, and ideology, although the evidence is more robust for partisanship and candidate preference.
Figure 6.3 presents the mean differences between voters and nonvoters on values-based issues for 1972 through 2008. Our expectations of null findings here are generally supported. The magnitude of the difference between voters and non-voters on aid to blacks and defense spending is generally less than 0.1. Larger differences between voters and nonvoters on women's equality are observed, and are statistically significant in four years (1972, 1980, 1992, and 2000), but even these differences disappear in the two most recent elections. Generally, then, we find little or no systematic differences between voters and nonvoters on these values-based issues.
**6.4 A More Detailed Look at Preferences: 2004**
To compare the preferences of voters and nonvoters on a broader set of issues we utilize the 2004 NAES. Because the Annenberg policy questions are more timely queries regarding citizens' issue positions than the long-standing questions of political attitudes included on the NES, they provide another perspective on the representativeness of voters in 2004. They also offer the advantage of a substantially larger sample size than the NES.
**Figure 6.3.** Differences between Voters' and Nonvoters' Attitudes on Values-Based Issues.
_Note_ : Plotted values are the weighted mean difference between voters' and nonvoters' attitudes on each issue in the specified year. Values greater than 0 indicate that voters (as a group) are more conservative than nonvoters (as a group) on the specific policy question; values less than 0 indicate that voters are more liberal than nonvoters. Mean differences on women's equality are significant at p < .05 in 1972, 1980, 1992, and 2000. Mean differences on aid to blacks are significant at p < .05 in 1976. Computed by the authors using data from the American National Election Studies Time Series Cumulative File, 1972–2008; see appendix 6.1 for question-wording details.
Although the particular questions differ from those used in the NES and reported in earlier tables, we note that these issues, too, can be viewed more broadly as income-based (both tables 6.3 and 6.4), as politically-based (table 6.5), and as values-based (table 6.6). In addition, the NAES also included a series of questions regarding security issues (table 6.7), and legal issues (table 6.8), which we discuss as well.
The NAES offers five questions that we consider to be redistributive in nature: whether the respondent favors making union organizing easier, government health insurance for workers, government health insurance for children, more federal assistance to schools, or increasing the minimum wage. The differences between voters and nonvoters on these issues are reported in table 6.3. We also examine voter and nonvoter attitudes on seven economic policy questions, and these differences are reported in table 6.4. These issues include making permanent the tax cuts enacted during the administration of President George W. Bush, school vouchers, trade agreements, investing Social Security in the stock market, eliminating overseas tax breaks, eliminating the estate tax, and reimporting drugs.10 Note that the format of the Annenberg questions generally is to ask whether respondents favor or oppose a policy. When respondents were given the option to say whether they strongly favor or strongly oppose, we collapsed the categories into favor or oppose. Thus, in the tables that follow, we present the proportion of voters who say they favor the policy, compared to the proportion of nonvoters who favor the policy.
**Table 6.3. Preferences of Nonvoters and Voters on Redistributive Issues (NAES 2004).**
_Notes_ : Cell entries in columns 1 and 2 are the percentage of respondents favoring the policy identified in each row, with the number of cases in parentheses. Unless otherwise noted, the percentage favoring consists of all respondents who said they _somewhat favor_ , _favor_ , or _strongly favor_ the policy. The complement of each reported percentage is made of all respondents who said they _somewhat oppose_ , _oppose_ , _neither favor nor oppose_ , _strongly oppose_ , _don't know_ , or refused to answer. Cell entries in the "Difference" column are the difference between the percentage of nonvoters and voters favoring each issue. Computed by the authors from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
_a_ Higher values indicate that the nonvoters are more liberal than the voters on the issue. Each difference is statistically significant at the 95 percent confidence level.
_b_ This "favorable" category includes those who responded that they wanted more federal assistance to schools. The complement of this reported percentage is made of all respondents who said they wanted the same amount, less, or no federal assistance to schools and those who said _don't know_ or refused to answer.
**Table 6.4. Preferences of Nonvoters and Voters on Economic Policies (NAES 2004).**
_Notes_ : Cell entries in columns 1 and 2 are the percentage of respondents favoring the policy identified in each row, with the number of cases in parentheses. Unless otherwise noted, the percentage favoring consists of all respondents who said they _somewhat favor_ , _favor_ , or _strongly favor_ the policy. The complement of each reported percentage consists of all respondents who said they _somewhat oppose_ , _oppose_ , _neither favor nor oppose_ , _strongly oppose_ , _don't know_ , or refused to answer. Cell entries in the "Difference" column are the difference between the percentage of nonvoters and voters favoring each issue. Computed by the authors using data from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
_a_ Higher values indicate that the nonvoters are more favorable than voters on the issue. Each difference (except on Bush tax cuts) is statistically significant at the 95 percent confidence level.
The largest differences between voters and nonvoters across the entire set of issues we describe are almost all on redistributive issues (as reported in table 6.3). On every redistributive issue except one, the difference in support between voters and nonvoters is over 11 percentage points. The largest difference observed is 14.7 percentage points on support for government health insurance for workers. And on every issue it is the nonvoters who have the more liberal position overall. Only on support for increasing the minimum wage is the difference less than 11 percentage points, and that is mostly because even voters support this.
On most of the economic policy questions (reported in table 6.4) we also see differences between voters and nonvoters, but here it is more difficult to characterize these differences as either more liberal or more conservative. What we can report is that voters are significantly less favorable than nonvoters on school vouchers, creating more trade agreements, and having social security in the stock market. Voters are significantly more favorable than nonvoters on eliminating overseas tax breaks, eliminating the estate tax, and reimporting drugs.
The distribution of responses to the redistributive questions we saw on the previous table compared to the smaller difference between groups we see on the more complicated economic questions on this table do an excellent job of illustrating our point: the voters may be representative of the electorate on some issues, but they are _not_ representative of the electorate on issues that go to the core of the role of government in modern democracies.
Respondents' self-reports on party identification and ideology in the 2004 NAES, as shown in table 6.5, broadly follow the contours identified in the 2004 NES, with Republicans being overrepresented among voters compared to their proportion of nonvoters. As shown in table 6.5, Republicans make up 34.5 percent of voters compared to 18.4 percent of nonvoters, and conservatives comprise 38.2 percent of voters and 32.2 percent of nonvoters. Both surveys suggest that conservatives are also overrepresented among voters, but the magnitude of these differences is smaller than those for partisanship.11
**Table 6.5. Political Attitudes of Nonvoters and Voters (NAES 2004).**
_Notes_ : Entries are computed by authors using data from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
_a_ Respondents who identified as _strong_ , _moderate_ , or _weak_ Democrats/Republicans are coded as partisans. Respondents who said _don't know_ or refused to respond are classified as other.
_b_ Respondents who said they were _conservative_ or _very conservative_ are classified as conservative; respondents who said they were _liberal_ or _very liberal_ are classified as liberal. Respondents who said they were _moderate_ , _don't know_ , or refused to answer are classified as moderate.
Table 6.6 presents nine issues included in the NAES postelection study that measure what we have termed values-based issues, though this categorization is perhaps too broad for this large set of diverse issues. Five of these issues focus on abortion or stem cell research funding and legality, two focus on same-sex marriage issues, and two focus on gun control. The largest difference between voters and nonvoters on these issues is only 6 percentage points and is on the question of gun control, where over 62 percent of nonvoters favor increased gun control, compared to only 56 percent of voters. Every other observed difference is less than 5 percentage points, and the direction of bias is not consistent: sometimes more liberal positions are favored, sometimes more conservative positions are favored by the voters compared to the nonvoters. Thus, on these values issues, we do not see the set of voters as being unrepresentative of the electorate.
The NAES includes data on seven "security" issues ranging from those focusing on the war in Iraq, to military spending, the 9/11 Commission recommendations and the Patriot Act. We can see in table 6.7 that, with one exception, the differences between voters and nonvoters on these issues are, as in the case of the values-based issues from the NES, quite modest. The most dramatic exception is on support for keeping troops in Iraq or bringing them home, in which case there is more than a 15 percentage-point difference between nonvoters and voters, with voters preferring that the troops stay in Iraq. The difference in support for keeping troops in Iraq is striking; it might be because voters are less likely than nonvoters to have relatives serving there. But it is a fascinating example of the voters _not_ being representative of the electorate on a vitally important public policy issue.
**Table 6.6. Preferences of Nonvoters and Voters on Values-Based Issues (NAES 2004).**
_Notes_ : Entries are computed by the authors using data from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
Entries in columns 1 and 2 are the percentage of respondents favoring the policy identified in each row, with the number of cases in parentheses. Unless otherwise noted, the percentage favoring consists of all respondents who said they _somewhat favor_ , _favor_ , or _strongly favor_ the policy. The complement of each reported percentage consists of all respondents who said they _somewhat oppose_ , _oppose_ , _neither favor nor oppose_ , _strongly oppose_ , _don't know_ , or refused to answer. All differences are statistically significant at greater than 95 percent confidence level except for "Making Abortion More Difficult."
_a_ This percentage represents those who responded that they wanted more gun control. The complement of this reported percentage consists of all respondents who said they wanted the same amount, less, or no gun control and those who said don't know or refused to answer.
**Table 6.7. Preferences of Nonvoters and Voters on Security Issues (NAES 2004).**
_Notes_ : Entries are computed by the authors using data from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
Entries in columns 1 and 2 are the percentage of respondents favoring the policy identified in each row, with the number of cases in parentheses. Unless otherwise noted, the percentage favoring consists of all respondents who said they _somewhat favor_ , _favor_ , or _strongly favor_ the policy. The complement of each reported percentage is made of all respondents who said they _somewhat oppose_ , _oppose_ , _neither favor nor oppose_ , _strongly oppose_ , _don't know_ , or refused to answer. All differences are statistically significant at greater than 95 percent confidence level except for "911 Recommendations."
_a_ This percentage represents those who responded that they wanted more spending on the military. The complement of this reported percentage consists of all respondents who said they wanted the same amount, less, or no spending on the military and those who said _don't know_ or refused to answer.
_b_ This percentage represents those who responded that the Patriot Act was good for the country. The complement of this reported percentage consists of all respondents who said they thought the Patriot Act was _neither good nor bad_ , _bad_ , _don't know_ , or refused to answer.
**Table 6.8. Preferences of Nonvoters and Voters on Legal Policies (NAES 2004).**
_Notes_ : Computed by the authors using data from the National Annenberg Election Study, 2004; see appendix 6.1 for question-wording details.
Unless otherwise noted, the percentage favoring consists of all respondents who said they _somewhat favor_ , _favor_ , or _strongly favor_ the policy. The complement of each reported percentage consists of all respondents who said they _somewhat oppose_ , _oppose_ , _neither favor nor oppose_ , _strongly oppose_ , _don't know_ , or refused to answer. All differences are statistically significant at greater than 95 percent confidence level.
Finally, in table 6.8 we report significant and large differences in the legal policy attitudes of voters compared to nonvoters. There is a substantial difference between the two groups on limiting malpractice awards, with voters more strongly supportive of such limitations than nonvoters. An even larger difference appears on the question of limiting lawsuit awards, with nearly 70 percent of voters supporting such limitations, compared to only 56 percent of nonvoters. These figures are consistent with the more conservative positions taken by voters on the redistributive issues we examined earlier.
All told, our findings on the large and consistent differences in the redistributive policy positions of voters versus nonvoters that we observed in the case of a limited number of broad policy issues included in the NES are sustained when examining more timely and detailed policy positions drawn from the NAES data. The magnitude of the differences observed on values-based and policy-oriented questions pales in comparison to what we find for economic issues. And so we repeat: voters are _not_ representative of nonvoters on redistributive issues. This has been true since 1972, in every election and for every redistributive issue we examine.12
**6.5 Conclusion**
Our initial interest in evaluating the representativeness of voters in contemporary American politics was stimulated largely by normative concerns associated with representation in modern democratic societies. The conventional wisdom established by Wolfinger and Rosenstone (1980) is that it does not matter who votes because voters and nonvoters share similar policy preferences. However, they noted that this was based on a comparison of preferences across a wide range of issues, and that in fact voters and nonvoters did differ on particular issues. In our analysis of preferences of voters and nonvoters across ten elections, we, too, find that on some issues voters are more liberal than nonvoters; that on some issues voters are more conservative than nonvoters; and that on some issues, voters' and nonvoters' preferences are about the same. Yet the pattern Wolfinger and Rosentsonte described for 1972 obscures the distinct difference between the preferences of voters and non-voters on issues related to redistributive policies. We consistently find on issues of economic redistribution—those issues most related to income and economic inequality—voters have more conservative policy preferences than nonvoters.
Revising the conventional wisdom according to our evidence is thus important in both normative and practical empirical respects. Both elected officials and citizens alike tend to think of elections as mandates of sorts, though it is extremely difficult to know what policies they are mandates for. Elections merely record which candidate is preferred by a majority of the voters to the other candidate. Elections do _not_ record the reasons for the choices of those voters. The outcomes of elections do not tell us what specific preferences motivate voters to choose one candidate over the other. What our work has shown is that the electoral victory of any one candidate cannot be presumed to be reflective of the broader electorate in terms of preferences on redistributive policies.
We have offered in this chapter a more extended analysis of the extent to which voters represent nonvoters. We take issue with the claim that voters are indeed representative of nonvoters. Our evidence deviates from that offered by Wolfinger and Rosenstone (1980) in one very important respect: in every election year from 1972 through 2008, voters and nonvoters differ substantively on most issues relating to the role of government in redistributive policies. In addition to these differences being evident in every election since 1972, we also note that the nature of the electoral bias is clear as well: voters are substantially more conservative than nonvoters on redistributive issues.
**Appendix 6.1: Survey Question Wording**
**_A6.1.1 The American National Election Studies_**
Representative introductions to the 7-point scale responses are included below:
• Government Health Insurance: There is much concern about the rapid rise in medical and hospital costs. Some feel there should be a government insurance plan which would cover all medical and hospital expenses. Suppose these people are at one end of a scale, at point 1. Others feel that medical expenses should be paid by individuals, and through private insurance like Blue Cross or some other company paid plans. Suppose these people are at the other end, at point 7. And of course, some people have opinions somewhere in between at points 2, 3, 4, 5 or 6. Where would you place yourself on this scale, or haven't you thought much about this?
• Government Guaranteeing Jobs: Some people feel that the government in Washington should see to it that every person has a job and a good standard of living. Suppose these people are at one end of a scale, at point 1. Others think the government should just let each person get ahead on his/her own. Suppose these people are at the other end, at point 7. And, of course, some other people have opinions somewhere in between, at points 2, 3, 4, 5 or 6. Where would you place yourself on this scale, or haven't you thought much about this?
• Government Services: Some people think the government should provide fewer services, even in areas such as health and education, in order to reduce spending. Suppose these people are at one end of a scale, at point 1. Other people feel that it is important for the government to provide many more services even if it means an increase in spending. Suppose these people are at the other end, at point 7. And of course, some other people have opinions somewhere in between, at points 2, 3, 4, 5, or 6. Where would you place yourself on this scale, or haven't you thought much about this?
• Women's Roles: Recently there has been a lot of talk about women's rights. Some people feel that women should have an equal role with men in running business, industry, and government. Suppose these people are at one end of a scale, at point 1. Others feel that a woman's place is in the home. Suppose these people are at the other end, at point 7. And of course, some people have opinions somewhere in between, at points 2, 3, 4, 5, or 6. Where would you place yourself on this scale, or haven't you thought much about this?
• Aid to blacks: Some people feel that the government in Washington should make every effort to improve the social and economic position of blacks. Suppose these people are at one end of a scale, at point 1. Others feel that the government should not make any special effort to help blacks because they should help themselves. Suppose these people are at the other end, at point 7. And, of course, some other people have opinions somewhere in between, at points 2, 3, 4, 5 or 6). Where would you place yourself on this scale, or haven't you thought much about it?
• Defense Spending: Some people believe that we should spend much less money for defense. Others feel that defense spending should be greatly increased. And, of course, some other people have opinions somewhere in between at points 2, 3, 4, 5, or 6. Where would you place yourself on this scale, or haven't you thought much about this?
Representative question wording used to measure political views between 1972 and 2008:
• Party Identification: Generally speaking, do you usually think of yourself as a Republican, a Democrat, an independent, or what? (IF REPUBLICAN OR DEMOCRAT) Would you call yourself a strong (REP/DEM) or a not very strong (REP/DEM)? (IF INDEPENDENT, OTHER [1966 AND LATER: OR NO PREFERENCE]:) Do you think of yourself as closer to the Republican or Democratic Party?
• Ideology: We hear a lot of talk these days about liberals and conservatives. When it comes to politics, do you usually think of yourself as extremely liberal, liberal, slightly liberal, moderate or middle of the road, slightly conservative, extremely conservative, or haven't you thought much about this?
• Candidate/Vote Preference (1978 and later): I'd like to get your feelings toward some of our political leaders and other people who are in the news these days. I'll read the name of a person and I'd like you to rate that person using something we call the feeling thermometer. Ratings between 50 and 100 mean that you feel favorably and warm toward the person; ratings between 0 and 50 degrees mean that you don't feel favorably toward the person and that you don't care too much for that person. You would rate the person at the 50 degree mark if you don't feel particularly warm or cold toward the person. If we come to a person whose name you don't recognize, you don't need to rate that person. Just tell me and we'll move on to the next one.
• Abortion (1972 and 1976): There has been some discussion about abortion during recent years. Which one of the opinions on this page best agrees with your view?
1. Abortion should never be permitted.
2. Abortion should be permitted only if the life and health of the woman is in danger.
3. Abortion should be permitted if, due to personal reasons, the woman would have difficulty in caring for the child.
4. Abortion should never be forbidden, since one should not require a woman to have a child she doesn't want.
• Abortion (1980 and later): There has been some discussion about abortion during recent years. Which one of the opinions on this page best agrees with your view?
1. By law, abortion should never be permitted.
2. The law should permit abortion only in case of rape, incest, or when the woman's life is in danger.
3. The law should permit abortion for reasons other than rape, incest, or danger to the woman's life, but only after the need for the abortion has been clearly established.
4. By law, a woman should always be able to obtain an abortion as a matter of personal choice.
**_A6.1.2 The National Annenberg Election Study_**
Question wording for the policy preference questions is provided below. Possible response categories are included in tables 6.3 and 6.4.
• Making Union Organizing Easier: Do you favor or oppose making it easier for labor unions to organize?
• Government Health Insurance for Children: Do you favor or oppose the federal government helping to pay for health insurance for all children?
• Social Security in the Stock Market: Do you favor or oppose allowing workers to invest some of their Social Security contributions in the stock market?
• School Vouchers: Do you favor or oppose the federal government giving tax credits or vouchers to parents to help send their children to private schools?
• Favoring Federal Assistance for Schools: Providing financial assistance to public elementary and secondary schools—should the federal government spend more on it, the same as now, less, or no money at all?
• Favoring Military Spending: Military defense—should the federal government spend more on it, the same as now, less, or no money at all?
• Reinstateing the Draft: Do you think the United States should put the military draft back into operation?
• Spending to Rebuild Iraq: Rebuilding Iraq—should the federal government spend more on it, the same as now, less, or no money at all?
• Troops in Iraq: Do you think the United States should keep troops in Iraq until a stable government is established there, or do you think the United States should bring its troops home as soon as possible?
• Implementing 9/11 Commission Recommendations: As you may know, the 9/11 Commission has recently released its final report on what the government knew about potential terrorist attacks before 9/11, and made recommendations on what the government should do to prevent future attacks. Based on what you know about the report, do you think the government should adopt all of the commission's recommendations, most of them, just some of them, or none of them?
• Gun Control: Restricting the kinds of guns people can buy—should the federal government do more about it, do the same as now, do less about it, or do nothing at all?
• Assault Weapons Ban: The current federal law banning assault weapons is about to expire. Do you think the U.S. Congress should pass this law again, or not?
• Banning All Abortions: The federal government banning all abortions—do you favor or oppose the federal government doing this?
• Making Abortion More Difficult: Laws making it more difficult for a woman to get an abortion—do you favor or oppose this?
• Party Identification: Generally speaking, do you think of yourself as a Democratic, a Republican, an independent, or something else?
• Ideology: Generally speaking, would you describe your political views as very conservative, conservative, moderate, liberal, or very liberal?
1. See Erikson and Tedin (2011, fig. 7:1) for a simple demonstration of the differences between the preferences of voters and nonvoters on economic issues.
2. We emphasize that we are considering policy preferences here, not candidate preferences. See Bennett & Resnick (1990); Ellis, Ura, & Ashley-Robinson (2006); Shaffer (1982); and Studlar & Welch (1986).
3. We also conducted these analyses using the validated vote for those years when it is available for the NES data, 1976–88, and discuss these results below.
4. Note that the distribution for 1972 is not precisely the same as that reported by Wolfinger & Rosenstone (1980, table 6.2) because we compare the distribution of partisanship among voters with its distribution among nonvoters (rather than the entire population).
5. We are of course not claiming that the issue of abortion is causing this overrepresentation of liberal views. The result could simply be caused by wealthy citizens having more liberal views on abortion _and_ voting more frequently than poor voters.
6. The NES is restricted to citizens.
7. See appendix 6.1 for details on question wording.
8. Mean differences are computed using the NES supplied weights.
9. Ideology could be interpreted differently by different respondents, and differently across elections. Some respondents might be emphasizing a social dimension in their evaluation of ideology, others might be emphasizing an economic dimension in their evaluation of ideology.
10. See appendix 6.1 for a fuller description of the NAES questions and the notes in tables 6.3 and 6.4 for additional details on our coding of responses.
11. Given the difference in question design, we do not make too much of the small differences here.
12. We also examined voter/nonvoter preferences based on the NES validated vote measure for the limited number of years for which it is available. Our general findings hold when comparing validated voters and validated nonvoters: there are few differences on values-based issues for voters versus nonvoters; there are differences in partisanship and vote preference, but not ideology, for voters versus nonvoters; and for most observations, there are significant differences between voters and nonvoters on redistributive issues, though the magnitude of these differences is smaller than when we use the self-reported vote.
**Seven**
* * *
**Conclusion**
As we finished writing this book in January 2013 many aspects of the 2012 presidential election had yet to be carefully examined, including who voted, whether who voted mattered for who won, and, critically, how the winner would govern. Claims that who voted could drive the outcome were fairly common in October 2012, but concern with the validity of those claims had essentially disappeared by election day. Instead, the political banter shifted to what would now transpire, given that President Barack Obama had won reelection.
It is this disjuncture between observations of who votes, and what policies will be produced postelection, that we have sought to address in this volume. In the 2012 presidential election, the media devoted large amounts of space to reporting on Bayesian averages of horse-race polls in order to predict the winner of the election. This focus may have crowded out coverage of the implications of the election for policy outcomes. Little attention was being paid as to why these voters were behaving as they were. If one believes that it is important for government to be responsive to the preferences of all citizens, then our emphasis on the differences in preferences of voters and nonvoters on _policies_ rather than on _candidates_ should be incorporated in careful analyses of public opinion. And these analyses should include the opinions of both voters and nonvoters.
We began writing this book thinking that who votes matters in terms of the issues that officials address, the policies they enact, and how a democratic government responds to the preferences of its citizens. We also believed that each of these consequences has substantial normative significance.
Our analysis of turnout in U.S. presidential elections since 1972 has proceeded against a backdrop of seismic changes in economic inequality, and with a particular interest in the income bias of voters. The importance of the substantial and sustained income bias that we have documented is underscored by our findings on the consistent differences in the policy preferences of voters and nonvoters. Voters are significantly more conservative than nonvoters on redistributive issues, and they have been in every election since 1972. If we had to point to our most important empirical finding of the many that we report, this is it. Voters may be more liberal than nonvoters on social issues, but on redistributive issues they are not. These redistributive issues define a fundamental relationship between citizens and the state in a modern industrialized democracy and are central to ongoing conflicts about the scope of government. It is on these issues that voters offer a biased view of the preferences of the electorate.
And it is not just that voters have different preferences than nonvoters on redistributive issues. In addition, the parties are failing to convince lower-income voters that they are offering distinctive choices on these issues. Whether perception or reality, this perceived lack of choices undermines the extent to which elections function as a linkage mechanism between citizen preferences and government policies.
**7.1 The Politics of Candidate Choices and Policy Choices**
Key to our effort to advance the study of turnout to a more explicitly political perspective is to consider the role of the policy positions that candidates offer in motivating voters. The predominant emphasis in the study of voter turnout for the past decade or more has been about voter mobilization and political competitiveness. When voters choose a president, they may indeed be influenced by various aspects of the candidacy, party, and campaign. But what we have emphasized is that presidential elections are choices about the policies that will govern the nation.
These policies can become an important benefit for citizens who must decide whether voting is worth the effort. We have shown that when individuals believe candidates offer distinctive choices they are more likely to vote. But not all eligible citizens perceive candidates as offering distinctive positions. In chapter 5 we observed that individuals in the highest income quintile are consistently most likely to perceive ideological differences between the major party presidential candidates, while individuals in the lowest quintile are least likely to see these differences. Moreover, the magnitude of these differences in perception increased dramatically
in 2004 and 2008, with high-income individuals becoming increasingly likely to see differences between the two candidates.
Our initial reaction to this finding was to suspect that these changes may have had something to do with differences between wealthy and poor individuals in how they perceive the candidates' issue positions. This thought was at least partly true: since 1980, and especially in 2004 and 2008, individuals in the highest income quintile increasingly placed the Democratic candidate on the liberal end of the 7-point scale and also increasingly placed the Republican candidate on the conservative end of the scale. In other words, these individuals reported seeing the major candidates become more extreme, consistent with descriptions of polarization in contemporary American politics.1
At the same time, individuals in the poorest quintile increasingly placed both candidates closer to the middle of the 7-point scale. This moderate scale placement is quite stable from 1980 onward for the Democratic candidate, and only slightly less stable for the Republican candidate. In short, the poor do not seem to see as much of a difference in the policy positions of the two presidential candidates.
These observations are relevant to our argument about the political consequences of economic inequality for voter turnout. While some theories of political inequality might predict a decrease in income bias in turnout since 1972 because increasing economic inequality mobilizes the poor to vote, our empirical evidence of stability in income bias contradicts this expectation. The argument we presented in chapter 1 was that _if_ candidates do not offer relevant choices during a period of increasing economic inequality then the poor will not be mobilized, and therefore income bias in turnout will not decrease. We also argued that _if_ candidates do offer relevant choices during a period of increasing economic inequality, then the poor could be mobilized, and income bias in turnout would decrease. Our evidence suggests that, indeed, poor people are substantially less likely to perceive policy differences between the parties on redistributive issues, notably failing to see that one party offers a distinctive _left_ choice relative to the other. Given that we observe no change in income bias in turnout, this result is consistent with our first argument.
Why don't poor people perceive a larger ideological difference between the parties? We speculate on a number of relevant points that might be fruitful to consider. Numerous studies of public opinion in the United States suggest the importance of education, especially, but also income, to the level of information citizens acquire and how this information influences the development and influence of policy and partisan attitudes on other political orientations and behaviors such as vote choice (Delli Carpini & Keeter 1996). More specifically, Bartels (2008) finds that individuals with higher levels of income have higher levels of political knowledge. While the sources of these differences might be diverse—variations in political interest and media exposure are obvious suspects—the implications for our findings are clear. Since poor citizens have less information about politics generally, their perceptions of presidential candidate policy positions are likely based on less information and therefore might be less accurate or more uncertain than the perceptions of those with higher levels of information.2
We also argue that these information differences are not simply a function of individual taste and choice but also reflect the political environment. We know that individuals' engagement in politics is influenced by elite mobilization and political issues (Holbrook & McClurg 2005; Rosenstone & Hansen 1993), as well as institutional characteristics of elections and parties, as illustrated in comparative studies of voter turnout (Blais 2006; Franklin 2004). One could also argue that variations in information across income groups may directly result from the political campaign strategies adopted by both major parties, both of which focus most mobilization efforts on individuals who have voted previously.
Highton and Wolfinger (2001) report, for example, that in 1992 and 1996, nonvoters were far less likely to be contacted by phone or in person by a political party. Other research also suggests that both parties are more likely to contact more highly educated, wealthier individuals than they are to contact less-educated and poorer individuals (Goldstein & Ridout 2002; Panagopoulos & Wielhouwer 2008; Parry et al. 2008).3 While much of this party contact may be focused on fund raising, the contacts are also likely to include appeals that contain policy information. The problem here is that most of these efforts are targeted toward registered voters or habitual voters, and thus low-income individuals are least likely to be reached in this way. This seems to us to be a key feature of contemporary election campaigns that may result in low income individuals having less information than wealthy individuals.
**7.2 Turnout and Institutions**
Our analyses span a period of major changes in electoral laws in the United States, many of which were passed with the intention of increasing turnout of underrepresented demographic groups. Our evidence indicates that both election day registration and absentee voting have positive effects on turnout. It appears that early voting's potentially positive effects are dependent on the length of the early voting period. When registration is closed farther in advance of election day it lowers turnout—a causal relationship that we have established with more confidence than earlier cross-sectional studies could do. Taken together, this evidence shows that electoral reforms over the past several decades have modestly increased voter turnout in presidential elections. Our findings on whether these reforms have succeeded in increasing the turnout of groups such as the poor or less educated are not as crisp. Our results suggest that election day registration may have its largest effect on those in the second and third income quintiles, and that members of the very lowest income group may be those least likely to take advantage of it.
For decades the United States has been identified as a unique case in the study of voter turnout worldwide, primarily based on the low level of U.S. turnout compared to that of other advanced industrial democracies (Jackman 1987; Lijphart 1997; Powell 1986). Most of these discussions attribute lower turnout in the United States to various aspects of election rules. Many other countries have either compulsory registration, or systems that put the primary burden of registration on the state, rather than one that places the primary burden for registration on the individual citizen, as is the case in the United States.
Claims of the burdens of the American electoral system have often been used as rationales for policy makers to adopt registration and balloting procedures that make it easier for individuals to overcome the cost of voting. As we discussed in chapter 4, some of the claims of reformers have been met, with electoral reforms such as absentee voting and election day registration leading to modest increases in voter turnout. But the effects of these reforms seem unlikely to overcome the turnout disparity between the United States and other Western democracies. What is less often discussed as an explanation for low turnout in the United States are the choices offered (or the lack thereof) by the candidates.
We caution, however, that these findings on the effects of electoral reforms and candidate choices are contingent on the politics of the period that we studied. To the extent that future campaigns and candidates seek to mobilize supporters more through these alternative voting methods, these reforms might offer greater potential for increasing turnout and reducing disparities in voter turnout across demographic groups. And to the extent that candidates offer distinctive policy choices that envision different levels of redistribution or distinctive roles of government, the poor and rich alike might view casting a ballot as worth the effort.
We believe that it is important to note that other institutional changes have occurred over the period of our analysis—changes that are likely relevant to understanding the broad contours of voter turnout in the United States. For example, increasing proportions of individuals now live in one-party-dominant congressional districts, and presidential campaigns have increasingly honed their campaign strategies to targeting only states where their efforts could be pivotal to yielding an important set of electoral votes—battleground states. To the extent that citizens in many districts or states are not exposed to competitive parties seeking their votes, they will be less likely to be contacted to vote and less likely to be exposed to information about their choices.
Organized labor is another important institutional feature of American politics today that has undergone tremendous change (Hirsch & Macpherson 2003). Even highly competitive election campaigns offering truly distinctive policy options to citizens must somehow overcome the conflicting demands on citizens' time and attention to convey useful and relevant electoral information. For most of the twentieth century, labor unions provided an important linkage between the Democratic Party and its constituents, helping to translate and make real why Democratic candidates were offering relevant policies. That the proportion of unionized workers has declined so dramatically over the past forty years, and the political power of labor waned, suggests that an important means of translating politics to poorer and less educated individuals has weakened greatly in electoral politics today.
**7.3 On Turnout and Political Inequality**
The distinctiveness of the United States in comparative studies of turnout is largely based on its substantially lower level of turnout compared to other contemporary democracies. But it has also been distinctive with regard to the especially strong relationship between income and turnout (Beramendi & Anderson 2008). Not all countries have the strong level of income bias in turnout that the United States does. One potential explanation is the nature of the party system and the policy choices offered in other countries.
Our findings on the substantial income bias of voters in the United States are not new. Schlozman, Verba, and Brady's (2012) study of political engagement in the United States underscores the fact that income bias in political participation in the United States today is as evident as it was several decades ago—despite the legal reforms, despite the expanded role of government in citizens' lives, and despite the broader reach of modern political campaigns.
As we have shown that voters and nonvoters have different preferences on redistributive issues, our findings punctuate the importance of this income bias as it relates to turnout. Turnout has always been unequal in the United States: the wealthy vote more than the poor. But this does not mean that a high level of income bias is an unchangable characteristic of U.S. elections. If candidates took positions perceived by poor voters to offer more distinct choices, then income bias could decrease.
The potential power of the ballot is that each eligible person has one to cast—despite different levels of education and income or different levels of other social resources that are distributed unevenly in our society. The bulk of recent scholarship that has focused on elected officials being differentially responsive to different groups of constituents has focused on comparing responsiveness to the wealthy to responsiveness to the poor. And this literature largely confirms that elected officials are more responsive to the wealthy than to the poor (Bartels 2008; Gilens 2012). But literature looking specifically at responsiveness to voters versus responsiveness to nonvoters also suggests that there are indeed policy consequences of who votes, that elected officials are more responsive to voters than to nonvoters (Griffin 2005; Martin 2003).
Given this evidence that elected officials do respond more to voters than nonvoters, it is important to repudiate the conventional wisdom that who votes does not matter. Voters are less supportive of government redistribution than are nonvoters. To the extent that elected officials respond to voters, we expect that public policies regarding redistribution and inequality will be less generous than they would in the case of universal turnout. As Wolfinger and Rosenstone explained in their concluding comments, "Citizens who are underrepresented at the polls... are less able to command the attention of elected officials and affect their decisions on public policy" (1980, 111).
Who votes now? And does it matter? Our answer to the first question was a lengthy one. Our answer to the second is more direct: Yes!
1. See Han & Brady (2007) or Poole & Rosenthal (1997) for a description of polarization in the legislature.
2. We note that this does not mean that the perceptions of the wealthy are without uncertainty or error.
3. See Abrajano (2010) for an analysis of this specific to Hispanics.
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**Index**
abortion, , , 161n5,
Abrajano, Marisa,
absentee voting, , , , –, , –; effect of on voter turnout, –, , 107nn11–12, , –, ,
Adams, James, ,
African Americans, , ; bivariate evidence that African Americans are less likely to vote than whites, , 68n19; changes in voter turnout for, –; gap in voter turnout between whites and African Americans, , 68nn19–20, –, 70n25; higher levels of education and income among, ; increase in the population of, ; increase in voter turnout of,
age, as a factor in voter turnout, , , , –, , 76n28, 105n9; conditional effect of on voter turnout, , , , ; and the life-cycle model of age effects, –; net group effect of EDR across age groups in Wave , Wave , and Wave 3 states, , , 107n10. _See also_ demographics, of voter turnout
Aldrich, John H., , –
alienation, , –; measurement of, –, ; multivariate analysis of perceived policy alienation and policy difference, –, –; perceived policy alienation, , 125n5; perceived policy alienation and difference across income groups, , –; policy-based concepts of indifference and alienation,
Alvarez, Michael,
American Association of Retired Persons (AARP), ,
American National Election Studies (NES), , , , , , , , , , , ; NES-validated vote measures, , 20n12; question wording of survey items, –
_American Voter, The_ (Campbell, Converse, Miller, and Stokes),
Anglos (non-Hispanic whites), , 13n16, ; decline in the population of, ; differences in voter turnout compared to that of African Americans, , 68nn19–20, –, 70n25; voter turnout among, , , ,
Ashenfelter, Orley,
at-risk effects. _See_ election day registration (EDR): at-risk effects of on voter population
Bartels, Larry,
Beck, Paul Allen,
Bennett, Stephen Earl,
Berelson, Bernard,
Biden, Joe,
Brady, Henry E.,
Brians, Craig Leonard,
Brox, Brian J.,
Burris, Arthur L., , 69n24
Bush, George W.,
Campbell, Angus,
causality, establishment of in current social science research, , ,
Cemenska, Nathan,
Center for American Women in Politics,
citizens: citizens in the voting-age population (CVAP), , , ; percentage of the voting-age population who are not citizens,
citizenship, as a requirement for voting, ; and state electoral requirements,
Civil Rights Act (1964),
class bias, ; socioeconomic class bias, , ,
collective goods problem, voter turnout as an example of,
competitiveness: electoral competitiveness, , , ; party competitiveness,
Converse, Philip,
costs, of voting, , , –, , –; higher costs of in the United States, ; increase in, ; psychological costs, ; reduction in, ,
democracy, ; centrality of elections to representative democracy,
demographics, of income, –, 17nn6–7; increase in the median income of African Americans (1972 compared to 2008), ; increase in the median income of Hispanics (1972 compared to 2008), , 18n8; increase in the median income of whites (1972 compared to 2008), , 18n8; median income of all races (1972 compared to 2008), ; median income of citizens, 43n35, –; median income of men over (1972 compared to 2010), ; median income of women over (1972 compared to 2008), . _See also_ socioeconomic status: measurement of
demographics, of voter turnout, , , –, –, –, ; change in the demographics of the United States, ; demographic predictors and turnout, ; logit model of, –; model specification for the demographic model of voter turnout, –; multivariate results concerning, –; proportion of noncitizens in the voting-age population, ; turnout rates by age groups, –, , –, ; turnout rates by education and income, –, –; turnout rates by ethnicity, –, , , –; turnout rates by gender and marital status, –, , –, 77nn29–30, –; turnout rates by race, , , , –
Dow, Jay,
Downs, Anthony,
early voting, , , , 94n5, –; effect of on voter turnout, , –, –, ; in-person early voting, –, , ; no-excuse early voting, ,
education, , –, , , 68n17; conditional effect of on voter turnout, , , , , , –; as a factor in voter turnout, , , –, –, , –; and government policy, ; as a measure of class/status, , ; relative effects of education, –, 66n16. _See also_ demographics, of voter turnout
election day registration (EDR), , , , –; at-risk effects of on voter population, , –, ; impact of on voter turnout, , –, , –; limited effects of on the representativeness of the electorate, ; states that have adopted EDR, ; tests examining the impact of in Wave , Wave , and Wave 3 states, –, 101nn7–8, –,
election law reform, design of current research concerning, –, 100n6; cross-sectional time series (CSTS) approach to the research, , –; difference-in-difference approach to the research, –; sources of data used in the research, ,
election reforms, in the United States, , 10n13, –, –; cross-sectional time series analysis of election law reform on voter turnout, –, 112n13, 113nn14–15, –17n17; difference-in-difference estimates of the effects of electoral reform, –, –, , –; election administration laws, , ; sources of data concerning, ; voter identification laws, , , 11n14; and voter turnout, , –. _See also_ absentee voting; early voting; election day registration (EDR); election law reform, design of current research concerning; electoral innovation, in the United States; voter registration
electoral innovation, in the United States, , –; and forms of alternative voting, –; previous research concerning, –
Fenster, Mark J.,
Fitzgerald, Mary,
Francia, Peter L.,
Freeman, Richard,
Gallup poll,
Gaudet, Hazel,
gender, as a factor in voter turnout, , , , –, , ; conditional effects of on voter turnout, , , , . _See also_ demographics, of voter turnout
General Social Survey,
Gershtenson, Joseph,
Giammo, Joseph D.,
Grofman, Bernard,
Gronke, Paul,
gun control, ,
Hanmer, Michael J.,
Hansen, John Mark, , , , ; on the role of elites in affecting voter turnout,
Herrnson, Paul S.,
Highton, Benjamin, , 69n24,
Hill, Kim Quaile,
Hispanics, , , 26n23, , ; analysis of Hispanic voter turnout, –; decrease of voter turnout compared to that of African Americans, ; Hispanic ethnicity as a determinant of voter turnout, , 56nn5–6; increase in the population of, ; political participation of, 69n24, , 81n31; voter turnout among, –, 29n27, ; voter turnout of compared to that of non-Hispanics, –, 69n22, . _See also_ Latino voters
ideology, , , , 167n9, 176n12; distribution of, ; ideology/jobs policy position, –, 125n5, , , –, ; political ideology, , –
income, –, , 68n18; conditional effect of on voter turnout, , –, , , –, ; as a factor in voter turnout, , –, , –; marginal effect of on voter turnout, –; as a measure of class, ; measurement of five income groups (quintiles), , 24n18, 24n20; perceived policy alienation and difference across income groups, , –; reasons that income is associated with voter turnout, ; relation of government policy to income, . _See also_ demographics, of voter turnout; income bias; socioeconomic status
income bias, 1n1, –, , , , –, , , –, ; analysis of changes in, , –; changes in share of income, ; conditional income bias, , ; decrease in, , , ; different tests of changes in, –; income bias measured over a series of elections, –; increase in, , , 10n13, , , ; stability of, , ; substantial income bias in the United States, –
indifference, –, –; measurement of, –,
inequality, economic, –, 6n10, ; association of increased periods of inequality with increased demand for government redistribution, ; differential income changes over time, ; increase in, , , 10n12; low income of nonvoters compared to that of voters, ; and the median voter model of candidate choice, –; overrepresentation of the wealthy in voting, ; and voting inequality, –
inequality, political, , , –
Jackson, Jesse, Sr., , 70n25
Jennings, M. Kent, , , 73n26,
Junn, Jane,
Kelley, Stanley,
Key, V. O., ,
Knack, Steven,
Latino voters, 69n21
law/legal issues, voter preferences concerning, –
Lazarsfeld, Paul,
Leighley, Jan E., , , ,
Lyons, William,
marital status, ; conditional effects of on voter turnout, , , ; as a factor in voter turnout, , , , , , , , , –; same-sex marriage, . _See also_ demographics, of voter turnout
Markus, Gregory B.,
Martin, Paul S., –
McDonald, Michael P., , ,
Meltzer, Allan H., , , 10n12,
Merrill, Samuel, ,
Miller, Warren E.,
Nagler, Jonathan, , ,
National Annenberg Election Study (NAES [2004]), , , , ; question wording of the survey, –, , –
national security, voter preferences concerning, , ,
National Voter Registration Act (NVRA 1993]), , , [93n3,
Nie, Norman H.,
nonvoters, , ; as an at-risk population, ; low income of nonvoters compared to that of voters, ; policy differences between nonvoters and voters, , –, –; policy preferences of nonvoters compared to those of voters, –, –, –; political differences between nonvoters and voters, –, 158n4; similar political preferences of voters and nonvoters, , –
Obama, Barack, , , ,
occupation, ; categorization of as either white or blue collar jobs, ; as a measure of class and socioeconomic status,
Oliver, J. Eric,
Pacheco, Julianna, 73n26
partisanship, , , , , , 176n12; declining levels of, 73n27; distribution of, , 158n4
party identification, ; and representativeness of voters for Democrats and Republicans, –, , ; of voters versus nonvoters,
_People's Choice, The_ (Lazersfeld, Berelson, and Gaudet),
perceived policy alienation, , 125n5; multivariate analysis of perceived policy alienation, –, –; perceived policy across income groups, , –
perceived policy difference, , , , 125n5; multivariate analysis of perceived policy difference, –, –; perceived policy difference across income groups, , –
Pew Charitable Trusts, 12n15, ,
Plane, Dennis,
Plutzer, Eric, 73nn, –
policy choices, of candidates, –, , , ; basis of candidate choice, ; conceptualization and measurement of, –; and the costs and benefits of voting, –; impact of on voters, –, ; measuring perceived policy alienation and difference, –, ; multivariate analysis of perceived policy alienation and policy difference, –, –; perceived policy alienation, , 125n5; perceived policy alienation across income groups, , –; perceived policy choices (1972–2008), –; perceived policy difference, ; perceived policy difference across income groups, , –; politics of, –
policy preferences, of voters, , 9n11; compared to those of nonvoters, –, –, –; conventional wisdom concerning, –, ; influence of economic standing on redistributive policies of candidates, –, ; on national security issues, , ; voters as not representative of nonvoters on redistribution issues, , , –; voters as representative of nonvoters on policy issues, , , –
Popkin, Samuel L.,
race/ethnicity: conditional effects of on voter turnout, , , , 69n24; and the demographics of voter turnout, , –, , , –; effects of on voter turnout independent of attitudinal factors, , 68n17, –; as a factor in voter turnout, , , , ; measurement of, –. _See also_ demographics, of voter turnout
redistribution, , 10n12, , , , ; policy positions concerning, ; salience of, –; voters as not representative of nonvoters on redistributive issues, , , –, 176n12,
representation, , , , , , , , , –, ; descriptive representation, ; differences in representation between the rich and the poor, –; of the eligible or the available, –
representativeness, , , , 40n32, , , –, –; alternative measure of, –; limited effects of EDR on, ; representativeness on the issue of abortion, , , 161n5; representativeness on the issues of health insurance and job guarantees, ; representativeness on the role of women in society, –; representativeness on value-based issues, –; representativeness of voters for Democrats and Republicans, –, , ; voters' policy positions as representative of nonvoters' positions, –
Resnick, David,
Richard, Scott F., , , 10n12,
Rigby, Elizabeth, 10n13
Rosenstone, Steven J., , , , , , , , 69n22, , , ; bivariate evidence of that African Americans are less likely to vote than whites, , 68n19; central findings of, ; on the central role of education in voter turnout, , ; on Hispanic voter participation relative to Anglo participation, 69n22; on the life-cycle model of age effects, –; reasons given by that income might be associated with voter turnout, ; on the role of elites in affecting voter turnout, ; theoretical argument involving education and voting, –, –; on the underrepresentation of citizens at the polls, ; on the underrepresentation of independents and the overrepresentation of Republicans, –
Scheb, John M.,
Schlozman, Kay Lehman,
social security, investment of in the stock market, , ,
socioeconomic status, , , , 69n21; income as a measure of socioeconomic status, ; measurement of, –; socioeconomic class bias, , ,
Springer, Melanie J., 10n13
Stehlik-Barry, Kenneth,
stem cell research,
Stoker, Laura,
Stokes, Donald E.,
Tenn, Steven,
Tokaji, Daniel,
Traugott, Santa, , 20n13
unions/unionization, , , ,
U.S. Census Bureau Current Population Survey (CPS), –, –, , 20n14, , ; educational categories of, , 25n22; income categories of, ; lack of "Hispanic Origin" question until 1976, ; racial categories of, ; sample and variable details of, –; state respondents to, , 57n8
U.S. Congress, polarization in, , 128n7
U.S. presidential campaign (2012), –, –,
U.S. presidential election (1972), , , , , 105n9, . _See also_ voter turnout
Verba, Sidney,
voter eligibility, , ; legal issues involving,
voter participation, critical role of political elites in, , –, 121n1
voter populations, ; eligible population, ; resident population, . _See also_ citizens: citizens in the voting-age population (CVAP); voting-age population (VAP)
voter registration, , , ; availability of registration, –; centrality of to voting, 91n2; easing of the registration process, ; and the election law data set, –; and the registration closing period, , ; registration through the Department of Motor Vehicles (DMV), . _See also_ election day registration (EDR)
voters: eligible voters, –; ineligible voters (including felons), , 21nn15–16, 43n34; policy differences between voters and nonvoters, , –, –; policy preferences of voters compared to nonvoters, –, –, –; political differences between voters and nonvoters, –, 158n4; similar political preferences of voters and nonvoters, , –; voters as not representative of nonvoters on redistribution issues, , , –, 176n12
voter turnout, –, –, , –; argument that voter turnout does not matter in elections, ; bivariate turnout rates, , –; and candidate policy choices, –; cross-sectional time series analysis of aggregate turnout, –, 112n13, 113nn14–15; effect of parenthood on, 73n26; estimates of (1972–2008), –; and institutions, –; among low-income voters, , , 6n10, , ; measurement of, –; and motor voter implementation, ; as pivotal to presidential election outcomes, ; previous research on electoral rules and voter turnover, –; and the problem of self-reporting, –; reasons that income is associated with voter turnout, ; relative importance of education and income to voter turnout, –; role of elites in affecting voter turnout, ; sources of data concerning, –; stability in turnout patterns, ; voting-age turnout, ; among wealthy voters, , 1n1, , 6n10, , , –. _See also_ African Americans: gap in turnout between whites and African Americans; demographics, of voter turnout; voter turnout, marginal effects on; voting-eligible turnout; women/women voters: voter turnout of
voter turnout, marginal effects on, , , ; marginal effects of age, –, ; marginal effects of education, , ; marginal effects of ethnicity, –; marginal effects of gender, ; marginal effects of income, –
vote share, , ,
voting: benefits of, , , , –, –, –, –; as a low-cost, low-benefit activity, ; as a rational rather than an expressive act, –. _See also_ absentee voting; costs, of voting; early voting; voting-eligible turnout
voting-age population (VAP), , 21n15,
voting-eligible turnout: official, , , ; self-reported, , 20n13, , , , , 68n20, ; validated, –, 20n13, 176n12
White, James,
Wolfinger, Raymond, , , , , , , , , , , ; central findings of, ; on Hispanic voter participation relative to Anglo participation, 69n22; on the life-cycle model of age effects, –; reasons given by that income might be associated with voter turnout, ; theoretical argument involving education and voting, –, –; on the underrepresentation of citizens at the polls, ; on the underrepresentation of independents and the overrepresentation of Republicans, –
women/women voters, , –; representativeness on the role of women in society, –; tendency of to vote Democratic at higher rates than men, –; voter turnout of, , 73n26
Zipp, John F., , , , 123n3, , ; and the policy-based concepts of indifference and alienation,
| {
"redpajama_set_name": "RedPajamaBook"
} | 2,758 |
Пем Шрайвер була чинною чемпіонкою, але того року не брала участі.
Перша сіяна Мартіна Навратілова виграла титул, перемігши у фіналі Лорі Макніл з рахунком 6–7, 6–3, 7–6.
Сіяні гравчині
Сіяну чемпіонку виділено жирним, тоді як для інших сіяних прописом вказано коло, в якому вони вибули. Перші четверо сіяних гравчинь виходять без боротьби в друге коло.
Мартіна Навратілова (переможниця)
Кріс Еверт (чвертьфінал)
Габріела Сабатіні (чвертьфінал)
Наташа Звєрєва (2-ге коло)
Зіна Гаррісон (півфінал)
Клаудія Коде-Кільш (чвертьфінал)
Лорі Макніл (фінал)
Мері Джо Фернандес (півфінал)
Сітка
Фінал
Секція 1
Секція 2
Посилання
Toray Pan Pacific Open 1989 Draw
1989
Toray Pan Pacific Open, одиночний розряд | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,270 |
NARS Everglades Full Vinyl Lip Lacquer (LELimited Edition. $26.00/0.17 oz.) dupes are shown below with comparison swatches (when available). Refine by price to find cheaper dupes or by brand and availability for a product more accessible for you.
How similar is Dirty Dark Martini to Everglades?
How similar is 503 Black to Everglades?
How similar is 502P Beetle to Everglades?
How similar is Vanity to Everglades? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,618 |
Q: Sort command in bash script does not work when called from Jenkins I am trying to execute a shell script on a windows node using Jenkins.
The bash script uses sort -u flag in one of the steps to filter out unique elements from an existing array
list_unique=($(echo "${list[@]}" | tr ' ' '\n' | sort -u | tr '\n' ' '))
Note - shebang used in the script is #!/bin/bash
On calling the script from command prompt as - bash test.sh $arg1
I got the following error -
-uThe system cannot find the file specified.
I understand the issue was that with the above call, sort.exe was being used from command prompt and not the Unix sort command. To get around this I changed the path variable in Windows System variables and moved \cygwin\bin ahead of \Windows\System32
This fixed the issue and the above call gave me the expected results.
However, When the same script is called on this node using Jenkins, I get the same error again
-uThe system cannot find the file specified.
Jenkins stage calling the script
stage("Run Test") {
options {
timeout(time: 5, unit: 'MINUTES')
}
steps {
script {
if(fileExists("${Test_dir}")){
dir("${Test_dir}"){
if(fileExists("test.sh")){
def command = 'bash test.sh ${env.arg1}'
env.output = sh(returnStdout: true , script : "${command}").trim()
if (env.output == "Invalid"){
def err_msg = "Error Found."
sh "echo -n '" + err_msg + " ' > ${ERR_MSG_FILE}"
error(err_msg)
}
sh "echo Running tests for ${env.output}"
}
}
}
}
}
}
Kindly Help
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,111 |
Q: Set horizontal width on custom listings environment I want to create a custom listings environment and want to set its width as something like 90% of textwidth, centred.
Per an earlier question for algorithms, the first thing I tried was to embed the listing in a minipage environment. However, custom listing environments must be done through the \lstnewenvironment command. I found a way to still make standard environment work, but it requires to use the escape command every time before ending the environment.
The second thing I tried was this latter command, but as far as I can see, it doesn't offer a feature to set the width of the listing or centre it.
\documentclass{article}
\usepackage{listings}
\usepackage{lipsum}
\lstnewenvironment{queryl} %% APPROACH 1
{\lstset{frame=shadowbox,escapechar=`}}
{}
\newenvironment{query} %% APPROACH 2
{\begin{minipage}{4cm}\centering\begin{queryl}}
{\end{queryl}\end{minipage}}
\begin{document}
%% APPROACH 1
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis.
\begin{queryl}
begin
{ do nothing }
end ;
\end{queryl}
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis.
%% APPROACH 2
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis.
\begin{query}
begin
{ do nothing }
end ;`%<- needs escape-to-LaTeX character to work
\end{query}
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis.
\end{document}
One can note that for the first environment, the frame extends beyond the width of the text. Also, puzzlingly the minipage environment doesn't seem to want to centre and swallows space (though I guess I could perhaps at least get that to work with tinkering).
In any case, does anyone have any ideas on how to set a width for a custom listings environment and centre it without requiring use of the escape character each time to end the environment?
A: You can specify the width with \lstset{linewidth=<length>}.
To incorporate this into your custom environment:
\lstnewenvironment{queryl}
{\lstset{frame=shadowbox,escapechar=`,linewidth=6cm}}
{}
However, I would actually suggest you define the queryl environment to be able to accept an optional first parameter in case you need to adjust any parameters locally:
\lstnewenvironment{queryl}[1][]
{\lstset{frame=shadowbox,escapechar=`,linewidth=8cm, #1}}
{}
Now when you use this as
\begin{queryl}
...
\end{queryl}
you get the first image below, but with
\begin{queryl}[linewidth=10cm]
...
\end{queryl}
you would obtain the second version:
Code:
\documentclass{article}
\usepackage{listings}
\lstnewenvironment{queryl}[1][]
{\lstset{frame=shadowbox,escapechar=`,linewidth=8cm, #1}}
{}
\begin{document}
Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Ut purus elit,
vestibulum ut, placerat ac, adipiscing vitae, felis.
\begin{queryl}
begin
{ listing with default line width }
end ;
\end{queryl}
\begin{queryl}[linewidth=10cm]
begin
{ listing with width adjust locally }
end ;
\end{queryl}
\end{document}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,566 |
/**
* DocuSign REST API
* The DocuSign REST API provides you with a powerful, convenient, and simple Web services API for interacting with DocuSign.
*
* OpenAPI spec version: v2.1
* Contact: devcenter@docusign.com
*
* NOTE: This class is auto generated. Do not edit the class manually and submit a new issue instead.
*
*/
(function(root, factory) {
if (typeof define === 'function' && define.amd) {
// AMD. Register as an anonymous module.
define(['ApiClient'], factory);
} else if (typeof module === 'object' && module.exports) {
// CommonJS-like environments that support module.exports, like Node.
module.exports = factory(require('../ApiClient'));
} else {
// Browser globals (root is window)
if (!root.Docusign) {
root.Docusign = {};
}
root.Docusign.AdminMessage = factory(root.Docusign.ApiClient);
}
}(this, function(ApiClient) {
'use strict';
/**
* The AdminMessage model module.
* @module model/AdminMessage
*/
/**
* Constructs a new <code>AdminMessage</code>.
* @alias module:model/AdminMessage
* @class
*/
var exports = function() {
var _this = this;
};
/**
* Constructs a <code>AdminMessage</code> from a plain JavaScript object, optionally creating a new instance.
* Copies all relevant properties from <code>data</code> to <code>obj</code> if supplied or a new instance if not.
* @param {Object} data The plain JavaScript object bearing properties of interest.
* @param {module:model/AdminMessage} obj Optional instance to populate.
* @return {module:model/AdminMessage} The populated <code>AdminMessage</code> instance.
*/
exports.constructFromObject = function(data, obj) {
if (data) {
obj = obj || new exports();
if (data.hasOwnProperty('baseMessage')) {
obj['baseMessage'] = ApiClient.convertToType(data['baseMessage'], 'String');
}
if (data.hasOwnProperty('moreInformation')) {
obj['moreInformation'] = ApiClient.convertToType(data['moreInformation'], 'String');
}
}
return obj;
}
/**
*
* @member {String} baseMessage
*/
exports.prototype['baseMessage'] = undefined;
/**
*
* @member {String} moreInformation
*/
exports.prototype['moreInformation'] = undefined;
return exports;
}));
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,326 |
{"url":"https:\/\/tutorial.math.lamar.edu\/Solutions\/CalcI\/Tangents_Rates\/Prob2.aspx","text":"Paul's Online Notes\nHome \/ Calculus I \/ Limits \/ Tangent Lines and Rates of Change\nShow Mobile Notice Show All Notes\u00a0Hide All Notes\nMobile Notice\nYou appear to be on a device with a \"narrow\" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.\n\n### Section 2-1 : Tangent Lines And Rates Of Change\n\n2. For the function $$g\\left( x \\right) = \\sqrt {4x + 8}$$ and the point $$P$$ given by $$x = 2$$ answer each of the following questions.\n\n1. For the points $$Q$$ given by the following values of $$x$$ compute (accurate to at least 8 decimal places) the slope, $${m_{PQ}}$$, of the secant line through points $$P$$ and $$Q$$.\n1. 2.5\n2. 2.1\n3. 2.01\n4. 2.001\n5. 2.0001\n1. 1.5\n2. 1.9\n3. 1.99\n4. 1.999\n5. 1.9999\n2. Use the information from (a) to estimate the slope of the tangent line to $$g\\left( x \\right)$$ at $$x = 2$$ and write down the equation of the tangent line.\n\nShow All Solutions\u00a0Hide All Solutions\n\na For the points $$Q$$ given by the following values of $$x$$ compute (accurate to at least 8 decimal places) the slope, $${m_{PQ}}$$, of the secant line through points $$P$$ and $$Q$$. Show Solution\n1. 2.5\n2. 2.1\n3. 2.01\n4. 2.001\n5. 2.0001\n1. 1.5\n2. 1.9\n3. 1.99\n4. 1.999\n5. 1.9999\n\nThe first thing that we need to do is set up the formula for the slope of the secant lines. As discussed in this section this is given by,\n\n${m_{PQ}} = \\frac{{g\\left( x \\right) - g\\left( 2 \\right)}}{{x - 2}} = \\frac{{\\sqrt {4x + 8} - 4}}{{x - 2}}$\n\nNow, all we need to do is construct a table of the value of $${m_{PQ}}$$ for the given values of $$x$$. All of the values in the table below are accurate to 8 decimal places.\n\n$$x$$ $${m_{PQ}}$$ $$x$$ $${m_{PQ}}$$\n2.5 0.48528137 1.5 0.51668523\n2.1 0.49691346 1.9 0.50316468\n2.01 0.49968789 1.99 0.50031289\n2.001 0.49996875 1.999 0.50003125\n2.0001 0.49999688 1.9999 0.50000313\n\nb Use the information from (a) to estimate the slope of the tangent line to $$g\\left( x \\right)$$ at $$x = 2$$ and write down the equation of the tangent line. Show Solution\n\nFrom the table of values above we can see that the slope of the secant lines appears to be moving towards a value of 0.5 from both sides of $$x = 2$$ and so we can estimate that the slope of the tangent line is : $$\\require{bbox} \\bbox[2pt,border:1px solid black]{{m = 0.5 = \\frac{1}{2}}}$$.\n\nThe equation of the tangent line is then,\n\n$y = g\\left( 2 \\right) + m\\left( {x - 2} \\right) = 4 + \\frac{1}{2}\\left( {x - 2} \\right)\\hspace{0.5in} \\Rightarrow \\hspace{0.5in}\\,\\require{bbox} \\bbox[2pt,border:1px solid black]{{y = \\frac{1}{2}x + 3}}$\n\nHere is a graph of the function and the tangent line.","date":"2021-10-20 04:06:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7086367607116699, \"perplexity\": 283.6457841429395}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585302.56\/warc\/CC-MAIN-20211020024111-20211020054111-00084.warc.gz\"}"} | null | null |
\section{Introduction}
We consider the problem
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta u=\left\vert u\right\vert ^{p-2}u & \text{in }\Omega,\\
\hspace{0.6cm}u=0 & \text{on }\partial\Omega,
\end{array}
\right. \tag{$\wp_p$}\label{prob
\end{equation}
where $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N\geq3,$ and
$p\geq2^{\ast},$ where $2^{\ast}:=\frac{2N}{N-2}$ is the critical Sobolev exponent.
It is well known that the existence of a solution depends on the domain.
Pohozhaev's identity \cite{po} implies that (\ref{prob}) does not have a
nontrivial solution if $\Omega$ is strictly starshaped. On the other hand,
Kazdan and Warner \cite{kw} showed that infinitely many radial solutions exist
if $\Omega$ is an annulus.
For $p=2^{\ast}$ a remarkable result obtained by Bahri and Coron \cite{bc}
establishes the existence of at least one positive solution to problem
$(\wp_{2^{\ast}})$ in every domain $\Omega$ having nontrivial reduced homology
with $\mathbb{Z}/2$-coefficients. Multiplicity results are also available,
either for domains which are small perturbations of a given one, as in
\cite{gmp}, or for domains which have enough, but possibly finite, symmetries,
as in \cite{cf}. A more detailed discussion may be found in these papers.
Unlike the critical case, in the supercritical case the existence of a
nontrivial cohomology class in $\Omega$ does not guarantee the existence of a
nontrivial solution to problem (\ref{prob}). In fact, for each $1\leq k\leq
N-3,$ Passaseo \cite{pa1,pa2} exhibited domains having the homotopy type of a
$k$-dimensional sphere $\mathbb{S}^{k}$ in which problem (\ref{prob}) does not
have a nontrivial solution for any $p\geq2_{N,k}^{\ast}:=\frac{2(N-k)
{N-k-2}.$ We call $2_{N,k}^{\ast}$ the $(k+1)$-st critical exponent. It is the
critical exponent for the Sobolev embedding $H^{1}(\mathbb{R}^{N-k
)\hookrightarrow L^{q}(\mathbb{R}^{N-k}).$ Nonexistence of bounded positive
solutions for $p>2_{N,k}^{\ast}$ in a thin enough tubular neighborhood of a
$k$-dimensional submanifold of $\mathbb{R}^{N}$\ was recently shown in
\cite{pps}.
The first nontrivial existence result for $p>2^{\ast}$ was obtained by del
Pino, Felmer and Musso \cite{dfm} in the slightly supercritical case, i.e. for
$p>2^{\ast}$ but close enough to $2^{\ast}.$ This case was also considered in
\cite{mpa,pr} where multiplicity was established. In \cite{dw} existence was
established in a domain with a small enough hole for a.e. $p>2^{\ast}$,
whereas in \cite{bcgp, pps} solutions of a particular type were constructed in
a tubular neighborhood of fixed radius of an expanding manifold for every $p.$
The problem for $p$ slightly below the second critical exponent was considered
in \cite{dmp} where solutions for $p=2_{N,1}^{\ast}-\varepsilon$ concentrating
at a boundary geodesic as $\varepsilon\rightarrow0$ have been constructed in
certain domains. Quite recently, positive and sign changing solutions for
$p=2_{N,k}^{\ast}-\varepsilon$ which concentrate at $k$-dimensional
submanifolds of the boundary as $\varepsilon\rightarrow0$ were exhibited in
\cite{acp}, while in \cite{kp} positive and sign changing solutions for $p$
large which concentrate at $(N-2)$-dimensional submanifolds of the boundary as
$p\rightarrow+\infty$ have been constructed.
In a recent work Wei and Yan \cite{wy} exhibited domains $\Omega$ in which
problem (\ref{prob})\ has infinitely many positive solutions for
$p=2_{N,k}^{\ast}$. They considered domains $\Omega$ of the form
\begin{equation}
\Omega:=\{(y,z)\in\mathbb{R}^{k+1}\times\mathbb{R}^{N-k-1}:\left( \left\vert
y\right\vert ,z\right) \in\Theta\}, \label{rotOmega
\end{equation}
where $\Theta$ is a bounded smooth domain in $\mathbb{R}^{N-k}$ with
$\overline{\Theta}\subset\left( 0,\infty\right) \times\mathbb{R}^{N-k-1}$
which satisfies certain geometric assumptions.
For domains of this type we give a geometric condition which guarantees nonexistence.
\begin{definition}
\label{**}We shall say that $\Theta$ is doubly starshaped with respect to
$\mathbb{R}\times\left\{ 0\right\} $ if there exist two numbers
$0<t_{0}<t_{1}$ such that $t\in(t_{0},t_{1})$ for every $(t,z)\in$ $\Theta$
and $\Theta$ is strictly starshaped with respect to $\xi_{0}:=(t_{0},0)$ and
to $\xi_{1}:=(t_{1},0)$, i.e.
\[
\left\langle x-\xi_{i},\nu_{\Theta}(x)\right\rangle >0\qquad\forall
x\in\partial\Theta\smallsetminus\left\{ \xi_{i}\right\} ,
\]
for each $i=0,1,$ where $\nu_{\Theta}(x)$ is the outward pointing unit normal
to $\partial\Theta$ at $x.$
\end{definition}
For $\Omega$ as in (\ref{rotOmega}) and $K\in\mathcal{C}^{1}(\overline{\Omega
})$ we consider the proble
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta u=K(y,z)\left\vert u\right\vert ^{p-2}u & \text{in }\Omega,\\
\hspace{0.6cm}u=0 & \text{on }\partial\Omega.
\end{array}
\right. \label{probK
\end{equation}
We assume $K$ to be strictly positive on $\overline{\Omega}$ and radially
symmetric in $y,$ i.e. $K(y,z)=K(\left\vert y\right\vert ,z).$ We prove the
following result.
\begin{theorem}
\label{thmmain1}If $\Theta$ is doubly starshaped with respect to
$\mathbb{R}\times\left\{ 0\right\} \ $and if $\left\langle y,\partial
_{y}K(y,z)\right\rangle \leq0$ and $\left\langle z,\partial_{z
K(y,z)\right\rangle \leq0$ for all $(y,z)\in\Omega,$ then problem
\emph{(\ref{probK})} does not have a nontrivial solution for $p\geq
2_{N,k}^{\ast}$ and has infinitely many solutions for $p\in(2,2_{N,k}^{\ast
}),$ where $0\leq k\leq N-3.$
\end{theorem}
The domains in Passaseo's examples \cite{pa1,pa2} are defined as in
(\ref{rotOmega}) with $\Theta$ being a ball centered at some point $(\tau,0),$
which is obviously doubly starshaped with respect to $\mathbb{R}\times\left\{
0\right\} .$ We stress that it is not enough for $\Theta$ to be strictly
starshaped to guarantee nonexistence: the domains considered by Wei and Yan
\cite{wy} are obtained from a domain $\Theta$ which is not doubly starshaped
with respect to $\mathbb{R}\times\left\{ 0\right\} $, but which may be
chosen to be strictly starshaped.
The domains in Passaseo's examples \cite{pa1,pa2}, as well as those in Theorem
\ref{thmmain1},\ have the homotopy type of $\mathbb{S}^{k}.$ One may ask
whether there are examples of domains having a richer topology for which a
similar nonexistence result holds true. We prove the following result.
\begin{theorem}
\label{thmmain4}Given $k=k_{1}+\cdots+k_{m}$ with $k_{i}\in\mathbb{N}$ and
$k\leq N-3,$ and $\varepsilon>0$ there exists a bounded smooth domain $\Omega$
in $\mathbb{R}^{N},$ which has the homotopy type of $\mathbb{S}^{k_{1}
\times\cdots\times\mathbb{S}^{k_{m}},$ in which problem \emph{(\ref{prob})}
does not have a nontrivial solution for $p\geq2_{N,k}^{\ast}+\varepsilon$ and
has infinitely many solutions for $p\in(2,2_{N,k}^{\ast}).$
\end{theorem}
In particular, if we take all $k_{i}=1,$ the domain $\Omega$ is homotopy
equivalent to the product of $k$ circles. So not only the homology of $\Omega$
is nontrivial but there are $k$ different cohomology classes in $H^{1
(\Omega;\mathbb{Z})$ whose cup-product is the generator of $H^{k
(\Omega;\mathbb{Z})$. Hence, the cup-length of $\Omega$ equals $k+1.$
We also obtain an existence result for a different type of domains, arising
from the Hopf fibrations. We are specifically interested in the cases where
$N=4,8,16.$ In these cases $\mathbb{R}^{N}\mathbb{=K}\times\mathbb{K}$, where
$\mathbb{K}$ is either the complex numbers $\mathbb{C}$, the quaternions
$\mathbb{H}$ or the Cayley numbers $\mathbb{O}.$ The set of units
$\mathbb{S}_{\mathbb{K}}:=\{\zeta\in\mathbb{K}:\left\vert \zeta\right\vert
=1\},$ which is a group if $\mathbb{K=C}$ or $\mathbb{H}$ and a quasigroup
with unit if $\mathbb{K=O}$, acts on $\mathbb{R}^{N}$ by multiplication on
each coordinate, i.e. $\zeta(z_{1},z_{2}):=(\zeta z_{1},\zeta z_{2}).$ The
orbit space of $\mathbb{R}^{N}$ with respect to this action turns out to be
$\mathbb{R}^{\dim\mathbb{K}+1}$ and the projection onto the orbit space is the
Hopf map $\pi:\mathbb{R}^{N}=\mathbb{K}\times\mathbb{K}\rightarrow
\mathbb{K}\times\mathbb{R}=\mathbb{R}^{\dim\mathbb{K}+1}$ given b
\[
\pi(z_{1},z_{2}):=(2\overline{z_{1}}z_{2},\,\left\vert z_{1}\right\vert
^{2}-\left\vert z_{2}\right\vert ^{2})\text{.
\]
We consider domains of the form $\Omega=\pi^{-1}(U)$ where $U$ is a bounded
smooth domain in $\mathbb{R}^{\dim\mathbb{K}+1}.$ We assume that $U$ is
invariant under the action of some closed subgroup $G$ of the group
$O(\dim\mathbb{K}+1)$ of linear isometries of $\mathbb{R}^{\dim\mathbb{K}+1}.$
We denote by $Gx:=\{gx:g\in G\}$ the $G$-orbit of a point $x\in\mathbb{R
^{\dim\mathbb{K}+1}$ and by $\#Gx$ its cardinality. Recall that $U $ is called
$G$-invariant if $Gx\subset U$ for all $x\in U,$ and a function
$u:U\rightarrow\mathbb{R}$ is called $G$-invariant if $u$ is constant on every
$Gx.$
Fix a closed subgroup $\Gamma$ of $O(\dim\mathbb{K}+1)$ and a nonempty
$\Gamma$-invariant bounded smooth domain $D$ in $\mathbb{R}^{\dim\mathbb{K
+1}$ such that $\#\Gamma x=\infty\ $for all $x\in D.$ We prove the following result.
\begin{theorem}
\label{thmmain2}There exists an increasing sequence $(\ell_{m})$ of positive
real numbers, depending only on $\Gamma$ and $D$, with the following property:
If $U$ contains $D$ and if it is invariant under the action of a closed
subgroup $G$ of $\Gamma$ for whic
\[
\min_{x\in U}\left( \#Gx\right) \left\vert x\right\vert ^{\frac
{\dim\mathbb{K}-1}{2}}>\ell_{m
\]
holds, then, for $p=2_{N,\dim\mathbb{K}-1}^{\ast},$ problem \emph{(\ref{prob
)} has at least $m$ pairs of\ solutions $\pm u_{1},\ldots,\pm u_{m}$ in
$\Omega:=\pi^{-1}(U),$ which are constant on $\pi^{-1}(Gx)$ for each $x\in U.$
In particular, they are $\mathbb{S}_{\mathbb{K}}$-invariant. Moreover, $u_{1}$
is positive and $u_{2},\ldots,u_{m}$ change sign.
\end{theorem}
For example, we may fix a bounded smooth domain $D_{0}$ in $\mathbb{R}^{2}$
with $\overline{D_{0}}\subset(0,\infty)\times\mathbb{R}$ and set
\[
D:=\{(z,t)\in\mathbb{K}\times\mathbb{R}:(\left\vert z\right\vert ,t)\in
D_{0}\}.
\]
Then $D$ is invariant under the action of the group $\Gamma:=\mathbb{S
_{\mathbb{C}}$ of unit complex numbers on $\mathbb{K}\times\mathbb{R}$ given
by $e^{i\theta}(z,t):=(e^{i\theta}z,t)$. If $G_{n}:=\{e^{2\pi ik/n
:k=0,...,n-1\}$ is the cyclic subgroup of $\Gamma$ of order $n,$ then
$\#G_{n}x=n$ for every $x\in(\mathbb{K}\smallsetminus\{0\}\mathbb{)
\times\mathbb{R}.$ Therefore, for every $G_{n}$-invariant bounded smooth
domain $U$ in $\mathbb{K}\times\mathbb{R}$ with
\[
D\subset U\subset(\mathbb{K}\smallsetminus\{0\}\mathbb{)}\times\mathbb{R
\text{\quad and\quad}n\left\vert x\right\vert ^{\frac{\dim\mathbb{K}-1}{2
}>\ell_{m},
\]
Theorem \ref{thmmain2} yields at least $m$ pairs of solutions to problem
(\ref{prob}) in $\Omega:=\pi^{-1}(U)$ for $p=2_{N,\dim\mathbb{K}-1}^{\ast}.$
In contrast to \cite{wy}, where multiplicity is established\ using
Lyapunov-Schmidt reduction, the proof of the Theorem \ref{thmmain2} uses
variational methods. It is based on the following result.
\begin{proposition}
\label{propHopf}Let $N=2,4,8,16$, $\ U$ be a bounded smooth domain in
$\mathbb{R}^{\dim\mathbb{K}+1}$ which does not contain the origin,
$a\in\mathbb{R}$, and $f:\mathbb{R}\rightarrow\mathbb{R}$. If $v$ solve
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta v+\frac{a}{2\left\vert x\right\vert }v=\frac{1}{2\left\vert
x\right\vert }f(v) & \text{in }U,\\
\qquad\qquad\quad v=0 & \text{on }U,
\end{array}
\right. \label{prob2
\end{equation}
then $u:=v\circ\pi$ is a solution o
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta u+au=f(u) & \text{in }\Omega:=\pi^{-1}(U),\\
\qquad\qquad u=0 & \text{on }\partial\Omega,
\end{array}
\right. \label{prob3
\end{equation}
where $\pi:\mathbb{R}^{N}\rightarrow\mathbb{R}^{\dim\mathbb{K}+1}$ is the Hopf
map. Conversely, if $u$ is an $\mathbb{S}_{\mathbb{K}}$-invariant solution of
\emph{(\ref{prob3})} and $u=v\circ\pi,$ then $v$ solves \emph{(\ref{prob2})}.
\end{proposition}
For $N=4$ this result was proved by Ruf and Srikanth in \cite{sr}\ by direct
computation. Here we derive it from the theory of harmonic morphisms (see
section \ref{sec:hm}).
Theorem \ref{thmmain4} does not apply to the case $p\in\lbrack2_{N,k}^{\ast
},2_{N,k}^{\ast}+\varepsilon)$. So the question remains open whether there
are\ examples of domains having the homotopy type of a product of spheres for
which nonexistence holds true for all $p\geq2_{N,k}^{\ast}$. We give a partial
answer as follows.
\begin{theorem}
\label{thmmain3}Let $N=4,8,16.$ Then there exist bounded smooth domains
$\Omega_{n}$ in $\mathbb{R}^{N}=\mathbb{K}\times\mathbb{K},$ which have the
homotopy type of $\mathbb{S}^{\frac{N-2}{2}}\times\mathbb{S}^{n}$ if $1\leq
n\leq\frac{N-4}{2}$ and of $\mathbb{S}^{\frac{N-2}{2}}$ if $n=0$, such that
problem \emph{(\ref{prob})} does not have a nontrivial $\mathbb{S
_{\mathbb{K}}$-invariant solution for $p\geq2_{N,k}^{\ast}$ and has infinitely
many $\mathbb{S}_{\mathbb{K}}$-invariant solutions for $p<2_{N,k}^{\ast}$
where $k:=\frac{N-2}{2}+n.$
\end{theorem}
The question remains open as to whether for such domains other solutions
exist, which are not $\mathbb{S}_{\mathbb{K}}$-invariant, particularly for
$p\geq2_{N,k}^{\ast}$.
This paper is organized as follows: in Section \ref{sec:hm} we present the
basic notions and results of the theory of harmonic morphisms and prove
Proposition \ref{propHopf}. Section \ref{sec:existence} is devoted to proving
Theorem \ref{thmmain2}. Theorems \ref{thmmain1}, \ref{thmmain4} and
\ref{thmmain3} are proved in Section \ref{sec:nonexistence}.
\section{Harmonic morphisms}
\label{sec:hm}We recall some basic notions and give examples of harmonic
morphisms. A detailed discusion is given e.g. in \cite{bw, er, w}.
Let $(M,\mathfrak{g})$ and $(N,\mathfrak{h})$ be Riemannian manifolds of
dimensions $m$ and $n$ respectively. A smooth map $\pi:M\rightarrow N$ is
called \emph{horizontally weakly conformal} if for each $x\in M$ at which
\textrm{d}$\pi_{x}\neq0$ the differential \textrm{d}$\pi_{x}:T_{x}M\rightarrow
T_{\pi(x)}N$ is surjective and horizontally conformal, i.e. there exists a
number $\lambda(x)\neq0$ such that
\[
\mathfrak{h}(\mathrm{d}\pi_{x}X,\mathrm{d}\pi_{x}Y)=\lambda^{2}(x)\mathfrak{g
(X,Y)\text{\qquad for all }X,Y\in T_{x}^{H}M\text{,
\]
where $T_{x}^{H}M$ denotes the orthogonal complement of $\ker\left(
\mathrm{d}\pi_{x}\right) .$ Defining $\lambda(x)=0$ if \textrm{d}$\pi_{x}=0$
we obtain a function $\lambda:M\rightarrow\lbrack0,\infty)$ called the
\emph{dilation of }$\pi.$ It is given by $\lambda^{2}(x)=\frac{\left\vert
\mathrm{d}\pi_{x}\right\vert ^{2}}{n},$ where $\left\vert \mathrm{d}\pi
_{x}\right\vert $ is the Hilbert-Schmidt norm of \textrm{d}$\pi_{x}.$ Hence,
it is a smooth function.
If $\pi$ has no critical points (i.e. \textrm{d}$\pi_{x}\neq0$ for all $x\in
M$) then it is called a \emph{conformal submersion}. If $\lambda\equiv1$ then
$\pi:M\rightarrow N$ is a \emph{Riemannian submersion}. Note that, if the
dilation is constant and non-zero, then $\pi$ is a Riemannian submersion up to
scale, i.e. it is a Riemannian submersion after a suitable homothetic change
of metric on the domain or codomain.
The \emph{tension field} $\tau(\pi)$ of a smooth map $\pi:M\rightarrow N$ is
defined a
\[
\tau(\pi):=\text{Trace}_{\mathfrak{g}}\nabla\mathrm{d}\pi.
\]
Thus, $\tau(\pi)$ is a vector field along $\pi,$ i.e. a section of the
pullback bundle $\pi^{-1}TN.$ In charts
\[
\tau(\pi)=\mathfrak{g}^{ij}(\nabla_{\partial_{i}}\mathrm{d}\pi)(\partial
_{j}),
\]
that is
\begin{align*}
\tau^{\gamma}(\pi) & =\mathfrak{g}^{ij}(\nabla\mathrm{d}\pi^{\gamma
)_{ij}+\mathfrak{g}^{ij}\Gamma_{\alpha\beta}^{N\,\gamma}\pi_{i}^{\alpha
\pi_{j}^{\beta}\\
& =\mathfrak{g}^{ij}\left( \frac{\partial^{2}\pi^{\gamma}}{\partial
x^{i}\partial x^{j}}-\Gamma_{ij}^{M\,k}\frac{\partial\pi^{\gamma}}{\partial
x^{k}}+\Gamma_{\alpha\beta}^{N\,\gamma}\pi_{i}^{\alpha}\pi_{j}^{\beta}\right)
\\
& =-\Delta_{M}\pi^{\gamma}+\mathfrak{g}^{ij}\Gamma_{\alpha\beta}^{N\,\gamma
}\pi_{i}^{\alpha}\pi_{j}^{\beta},\text{\qquad}1\leq\gamma\leq n,
\end{align*}
where $\Delta_{M}$ is the Laplace-Bertrami operator on $M$ (with the customary
sign convention of Riemannian geometry) and $\Gamma_{ij}^{M\,k}$ and
$\Gamma_{\alpha\beta}^{N\,\gamma}$ are the Christoffel symbols of\ $M $ and
$N$ respectively. The map $\pi:M\rightarrow N$ is called \emph{harmonic} if
$\tau(\pi)\equiv0.$ If, in addition, $\pi$ is horizontally weakly conformal,
then $\pi$ is called a \emph{harmonic morphism.} The main property of harmonic
morphisms is the following one.
\begin{proposition}
\label{prophm}A smooth map $\pi:M\rightarrow N$ is a harmonic morphism with
dilation $\lambda$ if
\[
\Delta_{M}(v\circ\pi)=\lambda^{2}\left[ (\Delta_{N}v)\circ\pi\right]
\]
for each smooth function $v:V\rightarrow\mathbb{R}$ defined on an open subset
$V$ of $N$ with $\pi^{-1}(V)\neq\emptyset.$
\end{proposition}
\begin{proof}
See \cite[Proposition 4.2.3]{bw}.
\end{proof}
\begin{corollary}
\label{correduction}Let $\pi:M\rightarrow N$ be a harmonic morphism with
dilation $\lambda$, $\ a:V\rightarrow\mathbb{R}$ be a function defined on an
open subset $V$ of $N$ with $\pi^{-1}(V)\neq\emptyset,$ and $f:\mathbb{R
\rightarrow\mathbb{R}$. Assume there exists $\mu:V\rightarrow(0,\infty) $ such
that $\mu\circ\pi=\lambda^{2}$ on $\pi^{-1}(V).$ If $v:V\rightarrow\mathbb{R}$
solve
\begin{equation}
\Delta_{N}v+\frac{a(y)}{\mu(y)}v=\frac{1}{\mu(y)}f(v), \label{eqN
\end{equation}
then $u:=v\circ\pi:\pi^{-1}(V)\rightarrow\mathbb{R}$ solves
\begin{equation}
\Delta_{M}u+\left( a\circ\pi\right) u=f(u). \label{eqM
\end{equation}
Conversely, if $\pi:\pi^{-1}(V)\rightarrow V$ is surjective and
$v:V\rightarrow\mathbb{R}$ is such that $u:=v\circ\pi:\pi^{-1}(V)\rightarrow
\mathbb{R}$ solves \emph{(\ref{eqM})} then $v$ solves \emph{(\ref{eqN})}.
\end{corollary}
\begin{proof}
This follows easily from Proposition \ref{prophm}.
\end{proof}
Next we give some examples of harmonic morphisms.
\begin{proposition}
Let $\pi:M\rightarrow N$ be a Riemannian submersion. Then $\pi$ is a harmonic
map iff each fiber $\pi^{-1}(y)$ is a minimal submanifold of $M$ (i.e. the
mean curvature of $\pi^{-1}(y)$ in $M$ is zero).
\end{proposition}
\begin{proof}
See \cite[(1.12)]{er}.
\end{proof}
Consequently, harmonic morphisms with constant non-zero dilation are simply
Riemannian submersions with minimal fibres, up to scale. Some interesting
examples are the Hopf fibrations.
\begin{example}
\label{exHfib}The Hopf fibrations $\mathbb{S}^{n}\rightarrow\mathbb{R}P^{n}$,
$\mathbb{S}^{2n+1}\rightarrow\mathbb{C}P^{n}$, $\mathbb{S}^{4n+3
\rightarrow\mathbb{H}P^{n}$ and $\mathbb{S}^{15}\rightarrow\mathbb{S}^{8}$ are
Riemannian submersions (up to scale) with totally geodesic, and so minimal,
fibres, see \cite[Examples 2.4.14-17]{bw}.
\end{example}
\begin{example}
The Hopf fibration $\mathbb{S}^{2n+1}\rightarrow\mathbb{C}P^{n}$ factors
through the double covering $\mathbb{S}^{2n+1}\rightarrow\mathbb{R}P^{2n+1}$
to give a Riemannian submersion $\mathbb{R}P^{2n+1}\rightarrow\mathbb{C}P^{n}$
with totally geodesic fibres. Similarly, one obtains a Riemannian submersion
$\mathbb{C}P^{2n+1}\rightarrow\mathbb{H}P^{n}$ with totally geodesic fibres.
\end{example}
The main example for our purposes is the following one.
\begin{example}
\label{exHmap}The Hopf maps $\pi:\mathbb{R}^{N}=\mathbb{K}\times
\mathbb{K}\rightarrow\mathbb{K}\times\mathbb{R}=\mathbb{R}^{\dim\mathbb{K}+1}$
given b
\[
\pi(z_{1},z_{2}):=(2\overline{z_{1}}z_{2},\left\vert z_{1}\right\vert
^{2}-\left\vert z_{2}\right\vert ^{2}),
\]
with $\mathbb{K=R}$, $\mathbb{C}$, $\mathbb{H}$, or $\mathbb{O}$ respectively,
are harmonic morphisms \cite[Corollary 5.3.3]{bw} with dilation $\lambda
(x,y)=\sqrt{2(\left\vert x\right\vert ^{2}+\left\vert y\right\vert ^{2})}.$
Their restrictions to the unit sphere are the Hopf fibrations of \emph{Example
\ref{exHfib}} with $n=1.$ A simple computation shows that $\left\vert
\pi(x,y)\right\vert =\left\vert x\right\vert ^{2}+\left\vert y\right\vert
^{2}.$ Hence, $\lambda^{2}(x,y)=2\left\vert \pi(x,y)\right\vert .$
\end{example}
\bigskip
\noindent\textbf{Proof of Proposition \ref{propHopf}.}\emph{\qquad}Apply
Corollary \ref{correduction} to Example \ref{exHmap}. \qed\noindent
\section{Existence}
\label{sec:existence}Proposition \ref{propHopf} suggests considering the
proble
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta v=K(x)\left\vert v\right\vert ^{2^{\ast}-2}v & \text{in }U,\\
\hspace{0.6cm}v=0 & \text{on }U,
\end{array}
\right. \tag{$\wp_U^\ast$}\label{prob4
\end{equation}
where $U$ is a bounded smooth domain in $\mathbb{R}^{M}$, $K\in\mathcal{C
^{0}(\mathbb{R}^{M})$ is strictly positive on $\overline{U}$ and $2^{\ast
}:=\frac{2M}{M-2}$ is Sobolev's critical exponent. We assume that $U$ and $K$
are $G$-invariant for some closed subgroup $G$ of $O(M)$. Then, the principle
of symmetric criticality \cite{p}\ asserts that the $G$-invariant solutions of
problem (\ref{prob4}) are the critical points of the restriction of the
functiona
\[
J(v):=\frac{1}{2}\int_{U}\left\vert \nabla v\right\vert ^{2}-\frac{1}{2^{\ast
}}\int_{U}K(x)\left\vert v\right\vert ^{2^{\ast}
\]
to the space of $G$-invariant functions
\[
H_{0}^{1}(U)^{G}:=\{v\in H_{0}^{1}(U):v(gx)=v(x)\text{ \ for all }g\in
G,\text{ }x\in U\}.
\]
We shall say that $J$ satisfies the Palais-Smale condition $(PS)_{c}^{G}$ in
$H_{0}^{1}(U)$ if every sequence $(v_{n})$ such that
\[
v_{n}\in H_{0}^{1}(U)^{G},\qquad J(v_{n})\rightarrow c,\qquad\nabla
J(v_{n})\rightarrow0,
\]
contains a convergent subsequence. Let $S$ be the best Sobolev constant for
the embedding $D^{1,2}(\mathbb{R}^{M})\hookrightarrow L^{2^{\ast}
(\mathbb{R}^{M}).$ The following result was proved in \cite[Corollary 2]{c}.
\begin{proposition}
\label{propPS}$J$ satisfies condition $(PS)_{c}^{G}$ in $H_{0}^{1}(U)$ for
ever
\[
c<\left( \min_{x\in\overline{U}}\frac{\#Gx}{K(x)^{\frac{M-2}{2}}}\right)
\frac{1}{M}S^{M/2}.
\]
In particular, if $\#Gx=\infty$ for all $x\in\overline{U},$ then $J$ satisfies
condition $(PS)_{c}^{G}$ in $H_{0}^{1}(U)$ for every $c\in\mathbb{R}$.
\end{proposition}
Fix a closed subgroup $\Gamma$ of $O(M)$ and a nonempty $\Gamma$-invariant
bounded smooth domain $D$ in $\mathbb{R}^{M}$ such that $\#\Gamma x=\infty
\ $for all $x\in D.$ Then, the following holds.
\begin{theorem}
\label{thmcfK}Assume that $K$ is $\Gamma$-invariant. Then, there exists an
increasing sequence $(\ell_{m})$ of positive real numbers, depending only on
$\Gamma,$ $D$ and $K$, with the following property: if $U$ contains $D$ and if
it is invariant under the action of a closed subgroup $G$ of $\Gamma$ for
whic
\[
\min_{x\in\overline{U}}\frac{\#Gx}{K(x)^{\frac{M-2}{2}}}>\ell_{m
\]
holds, then problem \emph{(\ref{prob4})} has at least $m$ pairs of\ $G
-invariant solutions $\pm v_{1},\ldots,\pm v_{m}$ such that $v_{1}$ is
positive, $v_{2},\ldots,v_{m}$ change sign, and
\[
\int_{U}\left\vert \nabla v_{k}\right\vert ^{2}\leq\ell_{k}S^{M/2}\text{\qquad
for every }k=1,\ldots,m.
\]
\end{theorem}
\begin{proof}
For $K=1$ this was proved in \cite[Theorem 1]{cf}. The proof for general $K$
goes through with minor modifications. We sketch it here for the reader's
convenience. Let $\mathcal{P}_{1}(D)$ be the set of all nonempty $\Gamma
$-invariant bounded smooth domains contained in $D,$ and defin
\[
\mathcal{P}_{k}(D):=\{(D_{1},\mathcal{\ldots},D_{k}):D_{i}\in\mathcal{P
_{1}(D)\text{, \ }D_{i}\cap D_{j}=\emptyset\text{ if }i\neq j\}.
\]
Note that $\mathcal{P}_{k}(D)\neq\emptyset$ for every $k\in\mathbb{N}.$ Since
$\#\Gamma x=\infty$ for all $x\in D_{i},$ Proposition \ref{propPS} allows to
apply the mountain pass theorem \cite{ar} to obtain a nontrivial least energy
$\Gamma$-invariant solution $\omega_{D_{i}}\ $to problem $(\wp_{D_{i}}^{\ast
}).$ Extending $\omega_{D_{i}}$ by zero outside $D_{i}$ we have that
$\omega_{D_{i}}\in H_{0}^{1}(\Omega)^{G}$ an
\begin{equation}
J(\omega_{D_{i}})=\max_{t\geq0}J(t\omega_{D_{i}}). \label{mountainpass
\end{equation}
We defin
\[
c_{k}:=\inf\left\{
{\textstyle\sum\limits_{i=1}^{k}}
J(\omega_{D_{i}}):(D_{1},\mathcal{\ldots},D_{k})\in\mathcal{P}_{k}(D)\right\}
\text{\qquad and\qquad}\ell_{k}:=\left( \frac{1}{M}S^{M/2}\right) ^{-1
c_{k}.
\]
Note that $c_{1}=J(\omega_{D})>0$ and that $J(\omega_{D_{i}})\geq c_{1}.$
Therefore
\[
c_{k-1}+c_{1}\le
{\textstyle\sum\limits_{i=1}^{k}}
J(\omega_{D_{i}})
\]
for every $(D_{1},\mathcal{\ldots},D_{k})\in\mathcal{P}_{k}(D)$, $k\geq2.$ It
follows that
\[
c_{k-1}+c_{1}\leq c_{k}\text{\qquad and\qquad}\ell_{k-1}+\ell_{1}\leq\ell
_{k}.
\]
Let $m\in\mathbb{N}$ and let $\Omega$ be a bounded smooth domain containing
$D,$ which is invariant under the action of a closed subgroup $G$ of $\Gamma$
for whic
\begin{equation}
\min_{x\in\overline{U}}\frac{\#Gx}{K(x)^{\frac{M-2}{2}}}>\ell_{m}
\label{hypothesis
\end{equation}
holds. Given $\varepsilon\in(0,c_{1})$ with $c_{m}+\varepsilon<\left(
\min_{x\in\overline{U}}\frac{\#Gx}{K(x)^{\frac{M-2}{2}}}\right) \frac{1
{M}S^{M/2},$ we choose $(D_{1},\mathcal{\ldots},D_{m})\in\mathcal{P}_{m}(D)$
such tha
\[
c_{m}\le
{\textstyle\sum\limits_{i=1}^{m}}
J(\omega_{D_{i}})<c_{m}+\varepsilon.
\]
For each $k=1,\ldots,m,$ let $W_{k}$ be the subspace of $H_{0}^{1}(\Omega
)^{G}$ generated by $\{\omega_{D_{1}},\ldots,\omega_{D_{k}}\}$ and
$d_{k}:=\sup_{W_{k}}J.$ Then, $\dim W_{k}=k$ and identity (\ref{mountainpass})
implies that
\[
d_{k}=\sup_{W_{k}}J\le
{\textstyle\sum\limits_{i=1}^{k}}
J(\omega_{D_{i}})<\left( \min_{x\in\overline{U}}\frac{\#Gx}{K(x)^{\frac
{M-2}{2}}}\right) \frac{1}{M}S^{M/2}.
\]
Then, by Proposition \ref{propPS}, $J$ satisfies $(PS)_{c}^{G}$ in $H_{0
^{1}(\Omega)$ for all $c\leq d_{k},$ so the mountain pass theorem \cite{ar}
yields a positive critical point $v_{1}\in H_{0}^{1}(\Omega)^{G}$ of $J$ such
that $J(v_{1})\leq d_{1},$ and Theorem 3.7 in \cite{cp}, conveniently adapted
to the functional we are considering here, yields $m-1$ pairs of sign changing
critical points $\pm v_{2},\ldots,\pm v_{m}\in H_{0}^{1}(\Omega)^{G}$ such
tha
\[
J(v_{k})\leq d_{k}\text{\qquad for every }k=2,\ldots,m.
\]
The proof that $v_{k}$ may be chosen so that $J(u_{k})\leq c_{k}$ for every
\ $k=1,\ldots,m,$ follows just as in \cite{cf}.
\end{proof}
\bigskip
\noindent\textbf{Proof of Theorem \ref{thmmain2}.}\emph{\qquad}This follows
from Theorem \ref{thmcfK} and Proposition \ref{propHopf}. \qed\noindent
\section{Nonexistence}
\label{sec:nonexistence}Fix $k_{1},\ldots,k_{m}\in\mathbb{N}\cup\{0\}$ with
$k:=k_{1}+\cdots+k_{m}\leq N-3$ and a bounded smooth domain $\Theta$ in
$\mathbb{R}^{N-k}$ with $\overline{\Theta}\subset\left( 0,\infty\right)
^{m}\times\mathbb{R}^{N-k-m}$. Set
\begin{equation}
\Omega:=\{(y^{1},\ldots,y^{m},z)\in\mathbb{R}^{k_{1}+1}\times\cdots
\times\mathbb{R}^{k_{m}+1}\times\mathbb{R}^{N-k-m}:\left( \left\vert
y^{1}\right\vert ,\ldots,\left\vert y^{m}\right\vert ,z\right) \in\Theta\}.
\label{omega
\end{equation}
Let $G:=O(k_{1}+1)\times\cdots\times O(k_{m}+1).$ We think of $G$ as a
subgroup of $O(N)$ acting on $\mathbb{R}^{k_{1}+1}\times\cdots\times
\mathbb{R}^{k_{m}+1}\times\mathbb{R}^{N-k-m}$ in the obvious way, i.e.
\begin{equation}
(g_{1},\ldots,g_{m})(y^{1},\ldots,y^{m},z):=(g_{1}y^{1},\ldots,g_{m}y^{m},z)
\label{Gaction
\end{equation}
for $g_{i}\in O(k_{i}+1),$ $y^{i}\in\mathbb{R}^{k_{i}+1},$ $z\in
\mathbb{R}^{N-k-m}.$ Then $\Omega$ is $G$-invariant. For $K\in\mathcal{C
^{0}(\overline{\Omega})$ we consider the proble
\begin{equation}
\left\{
\begin{array}
[c]{ll
-\Delta u=K(x)\left\vert u\right\vert ^{p-2}u & \text{in }\Omega,\\
\hspace{0.6cm}u=0 & \text{on }\partial\Omega.
\end{array}
\right. \label{probK2
\end{equation}
\begin{proposition}
\label{propexK}If $K$ is positive and $G$-invariant in $\overline{\Omega}$ and
$0\leq k\leq N-3,$ then problem \emph{(\ref{probK2})} has infinitely many
$G$-invariant solutions for $p\in(2,2_{N,k}^{\ast}).$
\end{proposition}
\begin{proof}
A $G$-invariant function $u(y^{1},\ldots,y^{m},z)=v(\left\vert y^{1
\right\vert ,\ldots,\left\vert y^{m}\right\vert ,z)$ solves problem
(\ref{probK2}) if and only if $v$ solves
\[
-\Delta v-\sum_{i=1}^{m}\frac{k_{i}}{x_{i}}\frac{\partial v}{\partial x_{i
}=K(x)|v|^{p-2}v\quad\text{in}\ \Theta,\qquad v=0\quad\text{on}\ \partial
\Theta.
\]
This problem can be rewritten a
\begin{equation}
-\text{div}(a(x)\nabla v)=Q(x)|v|^{p-2}v\quad\text{in}\ \Theta,\qquad
v=0\quad\text{on}\ \partial\Theta, \label{eqdiv
\end{equation}
where $a(x_{1},\ldots,x_{N-k}):=x_{1}^{k_{1}}\cdots x_{m}^{k_{m}}$ and
$Q(x):=a(x)K(x).$ Note that both $a$ and $Q$ are continuous and strictly
positive in $\overline{\Theta}.$ Hence, the norm
\[
\left\Vert v\right\Vert _{a}:=\left( \int_{\Theta}a(x)\left\vert \nabla
v\right\vert ^{2}\right) ^{1/2}\text{\qquad and\qquad}\left\vert v\right\vert
_{Q,p}:=\left( \int_{\Theta}Q(x)\left\vert v\right\vert ^{p}\right) ^{1/p
\]
are equivalent to those of $H_{0}^{1}(\Theta)$ and $L^{p}(\Theta)$
respectively. Since $H_{0}^{1}(\Theta)$ is compactly embedded in $L^{p
(\Theta)$ for $p<2_{N-k}^{\ast},$ the functiona
\[
J(v):=\frac{1}{2}\left\Vert v\right\Vert _{a}^{2}-\frac{1}{p}\left\vert
v\right\vert _{Q,p}^{p},\text{\qquad}v\in H_{0}^{1}(\Theta),
\]
satisfies the Palais-Smale condition. It clearly satisfies all other
hypotheses of the symmetric mountain pass theorem \cite{ar}. Hence, it has an
unbounded sequence of critical values. The critical values of $J$ are the
solutions of (\ref{eqdiv}).
\end{proof}
Next, fix $\tau_{1},\ldots,\tau_{m}\in(0,\infty),$ and let $\varphi_{i}$ be
the solution to the proble
\[
\left\{
\begin{array}
[c]{ll
\varphi_{i}^{\prime}(t)t+(k_{i}+1)\varphi_{i}(t)=1, & t\in(0,\infty),\\
\varphi_{i}(\tau_{i})=0. &
\end{array}
\right.
\]
Explicitly, $\varphi_{i}(t)=\frac{1}{k_{i}+1}\left[ 1-(\frac{\tau_{i}
{t})^{k_{i}+1}\right] .$ Note that $\varphi_{i}$ is strictly increasing in
$(0,\infty).$ For $y^{i}\neq0$ we defin
\begin{equation}
\chi(y^{1},\ldots,y^{m},z):=(\varphi_{1}(\left\vert y^{1}\right\vert
)y^{1},\ldots,\varphi_{m}(\left\vert y^{m}\right\vert )y^{m},z). \label{vf
\end{equation}
\begin{lemma}
\label{lemvf}$\chi$ has the following properties:
\begin{enumerate}
\item[(a)] \emph{div}$\chi=N-k,$
\item[(b)] $\left\langle \mathrm{d}\chi(y^{1},\ldots,y^{m},z)\left[
\xi\right] ,\xi\right\rangle \leq\max\left\{ 1-k_{1}\varphi_{1}(\left\vert
y^{1}\right\vert ),\ldots,1-k_{m}\varphi_{m}(\left\vert y^{m}\right\vert
),1\right\} \left\vert \xi\right\vert ^{2}$ \ for every $y^{i}\in
\mathbb{R}^{k_{i}+1}\smallsetminus\{0\},$ $z\in\mathbb{R}^{N-k-m},$ $\xi
\in\mathbb{R}^{N}.$
\end{enumerate}
\end{lemma}
\begin{proof}
(a) Write $y^{i}=(y_{1}^{i},\ldots,y_{k_{i}+1}^{i}).$ Then,
\[
\text{div}\chi(y^{1},\ldots,y^{m},z)
{\textstyle\sum\limits_{i=1}^{m}}
\left[
{\textstyle\sum\limits_{j=1}^{k_{i}+1}}
\varphi_{i}^{\prime}(\left\vert y^{i}\right\vert )\frac{(y_{j}^{i})^{2
}{\left\vert y^{i}\right\vert }+(k_{i}+1)\varphi_{i}(\left\vert y^{i
\right\vert )\right] +N-k-m=N-k.
\]
(b) $\chi$ is $G$-equivariant for the $G$-action defined in (\ref{Gaction}),
that is,
\[
\chi(gy,z)=g\chi(y,z)
\]
for every $g\in G,$ $y=(y^{1},\ldots,y^{m}),$ $y^{i}\in\mathbb{R}^{k_{i
+1}\smallsetminus\{0\},$ $z\in\mathbb{R}^{N-k-m}.$ Therefore, $g\circ
\mathrm{d}\chi(y,z)=\mathrm{d}\chi(gy,z)\circ g$ and, hence
\[
\left\langle \mathrm{d}\chi\left( y,z\right) \left[ \xi\right]
,\xi\right\rangle =\left\langle g\left( \mathrm{d}\chi\left( y,z\right)
\left[ \xi\right] \right) ,g\xi\right\rangle =\left\langle \mathrm{d
\chi\left( gy,z\right) \left[ g\xi\right] ,g\xi\right\rangle
\]
for all $\xi\in\mathbb{R}^{N}.$ Thus, it suffices to show that the inequality
(b) holds for $y^{i}=(y_{1}^{i},0,\ldots,0)$ with $y_{1}^{i}>0.$ Set $\chi
_{i}(y^{i}):=\varphi_{i}(\left\vert y^{i}\right\vert )y^{i}.$ A
straightforward computation shows that, for such $y^{i},$ $\mathrm{d}\chi
_{i}(y^{i})$ is a diagonal matrix whose diagonal entries are $a_{11
=1-k_{i}\varphi_{i}(y_{1}^{i})$ and $a_{jj}=\varphi_{i}(y_{1}^{i})$ for
$j=2,\ldots,k_{i}+1.$ Since $\varphi_{i}(t)<\frac{1}{k_{i}+1}$ for all
$t\in(0,\infty),$ (b) follows.\bigskip
\end{proof}
\noindent\textbf{Proof of Theorem \ref{thmmain1}.}\emph{\qquad}The variational
identity (4) in Pucci and Serrin's paper \cite{ps} implies that, if
$u\in\mathcal{C}^{2}(\Omega)\cap\mathcal{C}^{1}(\overline{\Omega})$ is a
solution of (\ref{probK}) and $\chi\in\mathcal{C}^{1}(\overline{\Omega
},\mathbb{R}^{N}),$ the
\begin{align}
\frac{1}{2}\int_{\partial\Omega}\left\vert \nabla u\right\vert ^{2
\left\langle \chi,\nu_{\Omega}\right\rangle d\sigma & =\int_{\Omega}\left(
\text{div}\chi\right) \left[ \frac{1}{p}K\left\vert u\right\vert ^{p
-\frac{1}{2}\left\vert \nabla u\right\vert ^{2}\right] dx\nonumber\\
& +\frac{1}{p}\int_{\Omega}\left\vert u\right\vert ^{p}\left\langle
\chi,\nabla K\right\rangle dx+\int_{\Omega}\left\langle \mathrm{d}\chi\left[
\nabla u\right] ,\nabla u\right\rangle dx \label{idPS
\end{align}
where $\nu_{\Omega}$ is the outward pointing unit normal to $\partial\Omega$.
Take $\chi$ to be the vector field defined in (\ref{vf}) for $m=1,$ $0\leq
k\leq N-3$ and $\tau_{1}=t_{0}$ as in Definition \ref{**}, that is
\[
\chi(y,z):=(\varphi(\left\vert y\right\vert )y,z),\text{\qquad}(y,z)\in\left(
\mathbb{R}^{k+1}\smallsetminus\{0\}\right) \times\mathbb{R}^{N-k-1
\]
with $\varphi(t)=\frac{1}{k+1}\left[ 1-(\frac{t_{0}}{t})^{k+1}\right] .$
Then, by Lemma \ref{lemvf},
\begin{equation}
\text{div}\chi=N-k. \label{fact2
\end{equation}
Note that, since $\varphi(t)\geq0$ for $t\in(t_{0},\infty)$ and $\left\vert
y\right\vert >t_{0}$ if $(y,z)\in\Omega,$ we have that
\begin{equation}
\left\langle \chi(y,z),\nabla K(y,z)\right\rangle =\varphi(\left\vert
y\right\vert )\left\langle y,\partial_{y}K(y,z)\right\rangle +\left\langle
z,\partial_{z}K(y,z)\right\rangle \leq0\text{\quad}\forall(y,z)\in\Omega.
\label{fact1
\end{equation}
Moreover, since $1-k\varphi(t)<1$ for $t\in(t_{0},\infty)$, Lemma
\ref{lemvf}\ yields
\begin{equation}
\left\langle \mathrm{d}\chi\left( x\right) \left[ \xi\right]
,\xi\right\rangle \leq\left\vert \xi\right\vert ^{2}\qquad\forall x\in
\Omega,\text{ }\xi\in\mathbb{R}^{N}. \label{fact3
\end{equation}
We claim that \
\begin{equation}
\left\langle \chi(x),\nu_{\Omega}(x)\right\rangle >0\qquad\forall x\in
\partial\Omega\smallsetminus\left\{ g\xi_{0},g\xi_{1}:g\in O(k+1)\right\} .
\label{fact4
\end{equation}
Since $\Omega$ is $O(k+1)$-invariant, $\nu_{\Omega}$ is $O(k+1)$-equivariant.
Thus, it suffices to show tha
\begin{equation}
\left\langle (\varphi(t)t,z),\nu_{\Theta}(t,z)\right\rangle >0\qquad\text{for
all }(t,z)\in\partial\Theta\smallsetminus\left\{ \xi_{0},\xi_{1}\right\} ,
\label{fact4a
\end{equation}
where $\nu_{\Theta}(t,z)$ is the outward pointing unit normal to
$\partial\Theta$ at $(t,z)$ which we write as $\nu_{\Theta}(t,z)=\left(
\nu_{1}(t,z),\nu_{2}(t,z)\right) \in\mathbb{R}\times\mathbb{R}^{N-k-1}.$ Let
$(t,z)\in\partial\Theta.$ Since $\Theta$ is doubly starshaped we have tha
\[
(t-t_{i})\nu_{1}(t,z)+\left\langle z,\nu_{2}(t,z)\right\rangle >0\text{\qquad
if }(t,z)\neq(t_{i},0),\text{ for }i=0,1,
\]
with $t_{0},t_{1}$ as in Definition \ref{**}. Therefore
\[
\left\langle (\varphi(t)t,z),\nu_{\Theta}(t,z)\right\rangle =\varphi
(t)t\nu_{1}(t,z)+\left\langle z,\nu_{2}(t,z)\right\rangle >(\varphi
(t)t-t+t_{i})\nu_{1}(t,z).
\]
Set $\psi(t):=\varphi(t)t-t.$ Note that $\psi^{\prime}(t)=-k\varphi(t)<0$ if
$t>t_{0}.$ So, since $t\in(t_{0},t_{1})$ for every $(t,z)\in$ $\Theta,$ we
have tha
\[
\varphi(t_{1})t_{1}-t_{1}=\psi(t_{1})\leq\psi(t)\leq\psi(t_{0})=-t_{0
\text{\qquad}\forall(t,z)\in\partial\Theta.
\]
If $\nu_{1}(t,z)\leq0$, the
\[
\left\langle (\varphi(t)t,z),\nu_{\Theta}(t,z)\right\rangle >(\psi
(t)+t_{0})\nu_{1}(t,z)\geq0
\]
and if $\nu_{1}(t,z)\geq0$, then
\[
\left\langle (\varphi(t)t,z),\nu_{\Theta}(t,z)\right\rangle >(\psi
(t)+t_{1})\nu_{1}(t,z)\geq\varphi(t_{1})t_{1}\nu_{1}(t,z)\geq0.
\]
This proves (\ref{fact4a}).
\noindent Combining properties (\ref{fact2}), (\ref{fact1}), (\ref{fact3}) and
(\ref{fact4}) with identity (\ref{idPS}) give
\begin{align*}
0 & <\int_{\Omega}\left( \text{div}\chi\right) \left[ \frac{1
{p}K\left\vert u\right\vert ^{p}-\frac{1}{2}\left\vert \nabla u\right\vert
^{2}\right] dx+\int_{\Omega}\left\vert \nabla u\right\vert ^{2}dx\\
& =(N-k)\left( \frac{1}{p}-\frac{1}{2}+\frac{1}{N-k}\right) \int_{\Omega
}\left\vert \nabla u\right\vert ^{2}dx
\end{align*}
which implies that $p<2_{N,k}^{\ast}$ if $u\neq0.$
\noindent Proposition \ref{propexK} yields infinitely many solutions for
$p<2_{N,k}^{\ast}$. \qed\noindent
\bigskip
\noindent\textbf{Proof of Theorem \ref{thmmain4}.}\qquad Choose $\alpha
\in(1,\frac{N-k}{2})$ with $2_{N,k}^{\ast}+\varepsilon\geq\frac{2(N-k)
{N-k-2\alpha}.$ Fix $\tau_{1},\ldots,\tau_{m}\in(0,\infty)$ and, for the given
$k_{1},\ldots,k_{m},$ define $\chi$ as in (\ref{vf})$.$ Let $0<\varrho
<\tau_{i}$ be defined b
\[
\max\left\{ 1-k_{1}\varphi_{1}(\tau_{1}-\varrho),\ldots,1-k_{m}\varphi
_{m}(\tau_{m}-\varrho)\right\} =\alpha,
\]
$\Theta:=B_{\varrho}^{N-k}(\tau)$ be the ball of radius $\varrho$ centered at
$\tau=(\tau_{1},\ldots,\tau_{m},0)$ in $\mathbb{R}^{m}\times\mathbb{R
^{N-k-m}$ and $\Omega$ be defined as in (\ref{omega}). Then $\Omega$ has the
homotopy type of $\mathbb{S}^{k_{1}}\times\cdots\times\mathbb{S}^{k_{m}}.$
Moreover, Lemma \ref{lemvf}\ asserts that
\begin{equation}
\text{div}\chi=N-k\text{\qquad and\qquad}\left\langle \mathrm{d}\chi\left(
x\right) \left[ \xi\right] ,\xi\right\rangle \leq\alpha\left\vert
\xi\right\vert ^{2}\quad\forall x\in\Omega,\text{ }\xi\in\mathbb{R}^{N}.
\label{fact5
\end{equation}
Since $\varphi_{i}(t)<0$ if $t<\tau_{i}$ and $\varphi_{i}(t)>0$ if $t>\tau
_{i}$ we have that, for all but a finite number of points $(x,z)\in
\partial\Theta,$
\[
\left\langle (\varphi_{1}(x_{1})x_{1},\ldots,\varphi_{m}(x_{m})x_{m
,z),\,\nu_{\Theta}(t,z)\right\rangle
{\textstyle\sum\limits_{i=1}^{m}}
\varphi_{i}(x_{i})x_{i}(x_{i}-\tau_{i})+\left\vert z\right\vert ^{2}>0.
\]
Hence,
\begin{equation}
\left\langle \chi,\nu_{\Omega}\right\rangle >0\qquad\text{a.e. on
\partial\Omega. \label{fact6
\end{equation}
Combining properties (\ref{fact5}) and (\ref{fact6}) with identity
(\ref{idPS}) for $K=1$ we obtai
\begin{align*}
0 & <\int_{\Omega}\left( \text{div}\chi\right) \left[ \frac{1
{p}\left\vert u\right\vert ^{p}-\frac{1}{2}\left\vert \nabla u\right\vert
^{2}\right] dx+\alpha\int_{\Omega}\left\vert \nabla u\right\vert ^{2}dx\\
& =(N-k)\left( \frac{1}{p}-\frac{1}{2}+\frac{\alpha}{N-k}\right)
\int_{\Omega}\left\vert \nabla u\right\vert ^{2}dx
\end{align*}
which implies that $p<\frac{2(N-k)}{N-k-2\alpha}\leq2_{N,k}^{\ast
+\varepsilon$ if $u\neq0.$ Consequently, problem (\ref{prob}) does not have a
nontrivial solution in $\Omega$ for $p\geq2_{N,k}^{\ast}+\varepsilon$, whereas
Proposition \ref{propexK} yields infinitely many solutions for $p<2_{N,k
^{\ast}$. \qed\noindent
\bigskip
\noindent\textbf{Proof of Theorem \ref{thmmain3}.}\emph{\qquad}For $0\leq
n\leq\dim\mathbb{K}-2,$ let $\Theta_{n}$ be a bounded smooth domain in
$\mathbb{R}^{\dim\mathbb{K}-n+1}$ with $\overline{\Theta_{n}}\subset\left(
0,\infty\right) \times\mathbb{R}^{\dim\mathbb{K}-n},$ which is doubly
starshaped with respect to $\mathbb{R}\times\{0\}$. Define $U_{0}:=\Theta_{0}$
and
\[
U_{n}:=\{(y,z)\in\mathbb{R}^{n+1}\times\mathbb{R}^{\dim\mathbb{K}-n}:\left(
\left\vert y\right\vert ,z\right) \in\Theta_{n}\}\subset\mathbb{R
^{\dim\mathbb{K}+1
\]
if $n\geq1.$ Theorem \ref{thmmain1} asserts that problem
\[
-\Delta v=\frac{1}{2\left\vert x\right\vert }\left\vert v\right\vert
^{p-2}v\quad\text{in }U_{n},\qquad v=0\quad\text{on }\partial U_{n},
\]
has infinitely many solutions for $p<2_{\dim\mathbb{K}+1,n}^{\ast}$ and no
nontrivial solutions for $p\geq2_{\dim\mathbb{K}+1,n}^{\ast}.$ Hence,
Proposition \ref{propHopf} implies that proble
\[
-\Delta u=\left\vert u\right\vert ^{p-2}u\quad\text{in }\Omega_{n}:=\pi
^{-1}(U_{n}),\qquad u=0\quad\text{on }\partial\Omega_{n},
\]
has infinitely many $\mathbb{S}_{\mathbb{K}}$-invariant solutions if
$p<2_{N,k}^{\ast}$ and does not have a nontrivial $\mathbb{S}_{\mathbb{K}
$-invariant solution if $p\geq2_{N,k}^{\ast},$ where $k:=\dim\mathbb{K}-1+n.$
\noindent Finally, since the restriction of the Hopf map $\pi:\mathbb{R
^{N}\smallsetminus\{0\}\rightarrow\mathbb{R}^{\dim\mathbb{K}+1}\smallsetminus
\{0\}$ is a fibration and $U_{n}$ is contractible in $\mathbb{R
^{\dim\mathbb{K}+1}\smallsetminus\{0\},$ the domain $\Omega_{n}$ is fiber
homotopy equivalent to $\mathbb{S}_{\mathbb{K}}\times U_{n}$ \cite[Chap.2,
Sec.8, Theorem 14]{s}. Hence, it has the homotopy type of $\mathbb{S
_{\mathbb{K}}\times\mathbb{S}^{n}$ if $n\geq1$ and of $\mathbb{S}_{\mathbb{K
}$ if $n=0.$ \qed\noindent
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,750 |
I make drawings and paintings that scrutinize our conception of nature. Sometimes this involves 'unnaturalizing' the natural world and articulating the distance that we put between ourselves and nature. I'm drawn to the fact that nature is something that we are a part of, yet simultaneously we are capable of seeing it as something outside of ourselves. The word has many definitions for many people, and has both potent literal and figurative meanings. Sifting through all these definitions is at the core of my art practice.
For the drawings, I use a brush and ink, including a lot of homemade walnut ink. I like the activity of big, labor-intensive drawing.
I grew up around Chapel Hill, North Carolina. My family comes from the rural south and I feel like that region is present in a lot of what I do, what with all the hot overgrowth. Moreover, my family is full of amateur naturalists, birdwatchers, gardeners, farmers, etc. I always liked to draw. Out of high school, I studied agriculture and worked on a pig farm. That did not stick, but I learned an enormous amount and gained a deep appreciation for the complexity of land management. I have also worked as a cook, cashier, dishwasher, flower deliverer, artist assistant, census-taker (an incredible experience), and dog walker, among other odd and unglamorous jobs. These jobs often took me places I did not expect to be, I think that is pretty valuable.
I actually toil away, alone in a room. I try and get to the woods as often as possible. Other than that, an expansive definition of my studio would include frequent long walks and reading.
I am not sure I was anticipating the ins and outs of entrepreneurship that is part of becoming an artist.
At this point I work full-time, so weekends are usually my best times to grab a solid day of drawing. Nights, mornings if I can get up for it. Other than that, I like to have areas of a drawing that I can work on for brief moments (textures, etc) that I can hop into without thinking too much about it.
I think my work has changed considerably in the last five years. Going back to school was really helpful for me. Five years ago I wasn't entirely sure what direction I was headed in, and I decided to focus on drawing and painting. This was in some ways at the expense of other interests, but I'm happy with that decision. Looking closely at the natural world and an interest in detail have been there since all this started.
I am lucky in this respect, there are a lot of them. My family and friends have been continual sources of support and encouragement. Within that lot, there are a number of artists who have been important role models. My wife Katarina has an extremely good eye and throwing ideas around with her is extremely valuable. In college, I learned a great deal about drawing and painting by working for Michael Brown, a muralist and sign-painter in Chapel Hill. I am often thinking about conversations with my professors Joan Linder, Adele Henderson, and Reinhard Reitzenstein. And I've got a cadre of artists, musicians, and writers that I treasure. It includes John Cage, Vija Celmins, Robert Smithson, John J. Audubon, Frederic Church, Toba Khedoori, Breugel the elder, Durer, Amy Cutler, Charles Burchfield, David Hockney, Roger Tory Peterson, Kent Monkman, Catherine Murphy, Lynette Yiadom-Boakye, Raymond Pettibon, Rackstraw Downes, Tacita Dean, Phil Ross, The Mustard Seed Garden Manual of Painting (not a person per se), Walter T. Foster, Henry Thoreau, John McPhee, William Cronin, Borges, Nick Cave, and Joni Mitchell.
I'd be a small forward in the NBA. And since we're in fantasyland, I'd be Kevin Durant. I believe Dave Hickey was correct when he said basketball is "civilized complexity incarnate".
Ripley Whiteside was born in North Carolina. He lived in a number of corners of that state prior to graduating with a BFA from UNC-CH in 2008. In 2012, he completed an MFA at SUNY-Buffalo. He lives and works in Montreal.
This entry was posted in Uncategorized and tagged brush and ink, Drawing, landscape, Montreal, nature. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,601 |
{"url":"https:\/\/mathemerize.com\/differentiation-of-infinite-series\/","text":"# Differentiation of Infinite Series Class 12\n\nHere you will learn what is differentiation of infinite series class 12 with examples.\n\nLet\u2019s begin \u2013\n\n## Differentiation of Infinite Series\n\nSometimes the value of y is given as an infinite series and we are asked to find $$dy\\over dx$$. In such cases we use the fact that if a term is deleted from a infinite series, it remains unaffected. The method of finding $$dy\\over dx$$ is explained in the following examples.\n\nExample 1 : If y = $$x^{x^{x^{\u2026\\infty}}}$$, find $$dy\\over dx$$.\n\nSolution : Since by deleting a single term from an infinite series, it remains same. Therefore, the given function may be written as\n\ny = $$x^y$$\n\nTaking log on both sides,\n\n$$\\implies$$ log y = y logx\n\nDifferentiating both sides with respect to x,\n\n$$1\\over y$$$$dy\\over dx$$ = $$dy\\over dx$$ log x + y $$d\\over dx$$ (log x)\n\n$$1\\over y$$$$dy\\over dx$$ = $$dy\\over dx$$ log x + $$y\\over x$$\n\n$$dy\\over dx$${$${{1\\over y} \u2013 log x}$$} = $$y\\over x$$\n\n$$\\implies$$ $$dy\\over dx$$$$(1 \u2013 y log x)\\over y$$ = $$y\\over x$$\n\n$$\\implies$$ $$dy\\over dx$$ = $$y^2\\over {x(1 \u2013 ylog x)}$$\n\nExample 2 : If y = $$\\sqrt{sinx + \\sqrt{sinx + \\sqrt{sinx + \u2026\u2026. to \\infty}}}$$, find $$dy\\over dx$$.\n\nSolution : Since by deleting a single term from an infinite series, it remains same. Therefore, the given function may be written as\n\ny = $$\\sqrt{sin x + y}$$\n\nSquaring on both sides,\n\n$$\\implies$$\u00a0 $$y^2$$\u00a0 = sin x + y\n\nDifferentiating both sides with respect to x,\n\n2y $$dy\\over dx$$ =cosx +\u00a0 $$dy\\over dx$$\n\n$$\\implies$$ $$dy\\over dx$$$$(2y \u2013 1)$$ = cos x\n\n$$\\implies$$ $$dy\\over dx$$ = $$cos x\\over {2y \u2013 1}$$","date":"2023-03-23 11:10:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8984091877937317, \"perplexity\": 479.45523111710196}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296945144.17\/warc\/CC-MAIN-20230323100829-20230323130829-00632.warc.gz\"}"} | null | null |
It was a 30 day challenge but even before that I was using the same method and the brand so even if I did not take the challenge I had sufficient amount of experience to share.
This is easy and safe. There is no skin color change, patchiness or the thicker hair thing. They were just the myths which I have busted in my earlier posts as well.
So, being someone who would frequently use this method. I will suggest you to try this out. But remember to exfoliate the skin on regular basis is since this could lead to ingrown hair which happens with waxing or the hair removal creams that we use. So, that is important. I shave 2 times in a week and exfoliate almost every day.
Coming to Gillette Venus Women's razor. The razor color is pretty blue. The design and handle style is made with perfection to use it while the hands are wet so that the handle does not slip. The 3 blades on the shaver head makes sure that even the tiniest hair is taken off with ease. The head actually rotates therefore can be easily used over the contours like, elbows, knees etc.
The reason why I don't really like hair removal creams lies in the smell and the waiting time. Not just that when I am trying to spread the cream on the skin, it will not be even, doesn't matter whether I do not with a spatula or the fingers. Due to this some of the hair strands will still be there or will not be removed at all. The waiting time is not for me. We generally get rid of the unwanted hair when we are going for office, work, date, party, and function etc so, those are the times when you simply don't want to wait.
Now, let me share some shaving tips and proper method to shave since there can be some of us who are still new to this method or have been doing certain things in incorrect way. These tips will enhance the shaving experience.
2. Do stretch and shave - The underarm skin and behind your knees is supple and flexible to allow movement at these jointed areas, which can make it harder to shave as the skin moves with the razor. To keep these areas as taut as possible for a cleaner shave, stretch your arm up and reach your hand behind your shoulder, and pull your leg straight.
After the shaving you also apply some aloe vera gel or some moisturizer. Make sure when you go out in the sun you apply sunscreen for no skin damage through UV rays.
You can buy the Gillette Venus razor here and the blades here. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,840 |
const uuid = require('uuid/v4')
const _ = require('lodash')
module.exports = (list: Array<{_id: string, item: string}> = [], event: { type: string, payload: any } ): Array<{_id: string, item: string}> => {
switch(event.type){
case "manual-webhook":
switch(event.payload.action){
case "add":
return list.concat([{ _id: uuid(), item: event.payload.item }])
case "remove":
return _.reject(list, ['_id', event.payload.id])
default:
return list
}
case "popcorn-time":
const { title: epTitle, season, episode } = event.payload
return list.concat([{ _id: uuid(), item: `${epTitle} S${season}E${episode}` }])
case "github":
const { title: issueTitle, html_url: issueURL } = event.payload.issue || event.payload.pull_request
const { name: repoName } = event.payload.repository
const item = `${repoName} - PR: <a href="${issueURL}">${issueTitle}</a>`
if(_.find(list, ['item', item]))
return list
return list.concat([{ _id: uuid(), item}])
case "shipping":
return list.concat([{ _id: uuid(), item: 'Package status: ' + event.payload.status}])
default:
return list
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,983 |
package client
import (
"crypto/sha256"
"encoding/json"
"testing"
"time"
"github.com/Sirupsen/logrus"
tuf "github.com/endophage/gotuf"
"github.com/stretchr/testify/assert"
"github.com/endophage/gotuf/data"
"github.com/endophage/gotuf/keys"
"github.com/endophage/gotuf/signed"
"github.com/endophage/gotuf/store"
)
func TestRotation(t *testing.T) {
kdb := keys.NewDB()
signer := signed.NewEd25519()
repo := tuf.NewTufRepo(kdb, signer)
remote := store.NewMemoryStore(nil, nil)
cache := store.NewMemoryStore(nil, nil)
// Generate initial root key and role and add to key DB
rootKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating root key")
rootRole, err := data.NewRole("root", 1, []string{rootKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating root role")
kdb.AddKey(rootKey)
err = kdb.AddRole(rootRole)
assert.NoError(t, err, "Error adding root role to db")
// Generate new key and role. These will appear in the root.json
// but will not be added to the keyDB.
replacementKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating replacement root key")
replacementRole, err := data.NewRole("root", 1, []string{replacementKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating replacement root role")
// Generate a new root with the replacement key and role
testRoot, err := data.NewRoot(
map[string]data.PublicKey{replacementKey.ID(): replacementKey},
map[string]*data.RootRole{"root": &replacementRole.RootRole},
false,
)
assert.NoError(t, err, "Failed to create new root")
// Sign testRoot with both old and new keys
signedRoot, err := testRoot.ToSigned()
err = signed.Sign(signer, signedRoot, rootKey, replacementKey)
assert.NoError(t, err, "Failed to sign root")
var origKeySig bool
var replKeySig bool
for _, sig := range signedRoot.Signatures {
if sig.KeyID == rootKey.ID() {
origKeySig = true
} else if sig.KeyID == replacementKey.ID() {
replKeySig = true
}
}
assert.True(t, origKeySig, "Original root key signature not present")
assert.True(t, replKeySig, "Replacement root key signature not present")
client := NewClient(repo, remote, kdb, cache)
err = client.verifyRoot("root", signedRoot, 0)
assert.NoError(t, err, "Failed to verify key rotated root")
}
func TestRotationNewSigMissing(t *testing.T) {
logrus.SetLevel(logrus.DebugLevel)
kdb := keys.NewDB()
signer := signed.NewEd25519()
repo := tuf.NewTufRepo(kdb, signer)
remote := store.NewMemoryStore(nil, nil)
cache := store.NewMemoryStore(nil, nil)
// Generate initial root key and role and add to key DB
rootKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating root key")
rootRole, err := data.NewRole("root", 1, []string{rootKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating root role")
kdb.AddKey(rootKey)
err = kdb.AddRole(rootRole)
assert.NoError(t, err, "Error adding root role to db")
// Generate new key and role. These will appear in the root.json
// but will not be added to the keyDB.
replacementKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating replacement root key")
replacementRole, err := data.NewRole("root", 1, []string{replacementKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating replacement root role")
assert.NotEqual(t, rootKey.ID(), replacementKey.ID(), "Key IDs are the same")
// Generate a new root with the replacement key and role
testRoot, err := data.NewRoot(
map[string]data.PublicKey{replacementKey.ID(): replacementKey},
map[string]*data.RootRole{"root": &replacementRole.RootRole},
false,
)
assert.NoError(t, err, "Failed to create new root")
_, ok := testRoot.Signed.Keys[rootKey.ID()]
assert.False(t, ok, "Old root key appeared in test root")
// Sign testRoot with both old and new keys
signedRoot, err := testRoot.ToSigned()
err = signed.Sign(signer, signedRoot, rootKey)
assert.NoError(t, err, "Failed to sign root")
var origKeySig bool
var replKeySig bool
for _, sig := range signedRoot.Signatures {
if sig.KeyID == rootKey.ID() {
origKeySig = true
} else if sig.KeyID == replacementKey.ID() {
replKeySig = true
}
}
assert.True(t, origKeySig, "Original root key signature not present")
assert.False(t, replKeySig, "Replacement root key signature was present and shouldn't be")
client := NewClient(repo, remote, kdb, cache)
err = client.verifyRoot("root", signedRoot, 0)
assert.Error(t, err, "Should have errored on verify as replacement signature was missing.")
}
func TestRotationOldSigMissing(t *testing.T) {
logrus.SetLevel(logrus.DebugLevel)
kdb := keys.NewDB()
signer := signed.NewEd25519()
repo := tuf.NewTufRepo(kdb, signer)
remote := store.NewMemoryStore(nil, nil)
cache := store.NewMemoryStore(nil, nil)
// Generate initial root key and role and add to key DB
rootKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating root key")
rootRole, err := data.NewRole("root", 1, []string{rootKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating root role")
kdb.AddKey(rootKey)
err = kdb.AddRole(rootRole)
assert.NoError(t, err, "Error adding root role to db")
// Generate new key and role. These will appear in the root.json
// but will not be added to the keyDB.
replacementKey, err := signer.Create("root", data.ED25519Key)
assert.NoError(t, err, "Error creating replacement root key")
replacementRole, err := data.NewRole("root", 1, []string{replacementKey.ID()}, nil, nil)
assert.NoError(t, err, "Error creating replacement root role")
assert.NotEqual(t, rootKey.ID(), replacementKey.ID(), "Key IDs are the same")
// Generate a new root with the replacement key and role
testRoot, err := data.NewRoot(
map[string]data.PublicKey{replacementKey.ID(): replacementKey},
map[string]*data.RootRole{"root": &replacementRole.RootRole},
false,
)
assert.NoError(t, err, "Failed to create new root")
_, ok := testRoot.Signed.Keys[rootKey.ID()]
assert.False(t, ok, "Old root key appeared in test root")
// Sign testRoot with both old and new keys
signedRoot, err := testRoot.ToSigned()
err = signed.Sign(signer, signedRoot, replacementKey)
assert.NoError(t, err, "Failed to sign root")
var origKeySig bool
var replKeySig bool
for _, sig := range signedRoot.Signatures {
if sig.KeyID == rootKey.ID() {
origKeySig = true
} else if sig.KeyID == replacementKey.ID() {
replKeySig = true
}
}
assert.False(t, origKeySig, "Original root key signature was present and shouldn't be")
assert.True(t, replKeySig, "Replacement root key signature was not present")
client := NewClient(repo, remote, kdb, cache)
err = client.verifyRoot("root", signedRoot, 0)
assert.Error(t, err, "Should have errored on verify as replacement signature was missing.")
}
func TestCheckRootExpired(t *testing.T) {
repo := tuf.NewTufRepo(nil, nil)
storage := store.NewMemoryStore(nil, nil)
client := NewClient(repo, storage, nil, storage)
root := &data.SignedRoot{}
root.Signed.Expires = time.Now().AddDate(-1, 0, 0)
signedRoot, err := root.ToSigned()
assert.NoError(t, err)
rootJSON, err := json.Marshal(signedRoot)
assert.NoError(t, err)
rootHash := sha256.Sum256(rootJSON)
testSnap := &data.SignedSnapshot{
Signed: data.Snapshot{
Meta: map[string]data.FileMeta{
"root": {
Length: int64(len(rootJSON)),
Hashes: map[string][]byte{
"sha256": rootHash[:],
},
},
},
},
}
repo.SetRoot(root)
repo.SetSnapshot(testSnap)
storage.SetMeta("root", rootJSON)
err = client.checkRoot()
assert.Error(t, err)
assert.IsType(t, tuf.ErrLocalRootExpired{}, err)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,773 |
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