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It was actually Kid Cudi’s announcement about his upcoming project that brought to light two collaboration with Andre 3000. Besides him, the 18-track project titled Passion, Pain & Demon Slayin’ which was expected to drop this summer and which was confirmed for release on September 30 features collaborations with Willow Smith, Pharrell Williams and Travi$ Scott. Passion, Pain & Demon Slayin’ is broken in four acts, with different titles: Tuned, Prophecy, Niveaux de l’Amour and It’s Bright and Heaven Is Warm.
Andre 3000 & Kaytranada in the studio 👀👀👀👀👀 pic.twitter.com/ZqraMhA7Gz
— scottiebeam (@ScottieBeam) September 26, 2016
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\begin{document}
\setcounter{page}{1}
\title[Joint Spectra]{Joint spectra of representations \\
of Lie algebras by compact operators}
\author {Enrico Boasso}
\begin{abstract}Given $X$ a complex Banach space, $L$ a complex nilpotent
finite dimensional Lie algebra, and
$\rho\colon L\to L(X)$, a representation of $L$ in $X$ such that
$\rho (l)\in K(X)$ for all $l\in L$, the Taylor, the S\l odkowski,
the Fredholm, the split and the Fredholm split joint spectra of the representation
$\rho$ are computed. \par
\vskip.2truecm
\noindent 2000 Mathematics Subject Classification: Primary 47A13, 47A10; Secondary 17B15, 17B55
\vskip.2truecm
\noindent Keywords: Taylor, S\l odkowski, Fredholm, split and Fredholm split joint spectra
\end{abstract}
\maketitle
\section{Introduction}
\indent Many of the well known joint spectra defined for commuting
tuples of operators were extended to solvable Lie algebras of operators, or more generally,
to representations of solvable Lie algebras in Banach spaces. For example, among such
joint spectra,
it is important to recall the Taylor, the S\l odkowski, the Fredholm, the split and
the Fredholm split joint spectra; see [14], [13], [8], [10], [7], [9],
[5], [2], [11], [12] and [4]. \par
\indent In the work [3] nilpotent Lie algebras of linear
transformations defined in finite dimensional Banach spaces were considered and the
S\l odkowski joint spectra, in particular the Taylor joint spectrum, of such an algebra were computed. In
[1] this characterization was extended to representations of nilpotent Lie
algebras by compact operators defined in Banach
spaces, though only for the Taylor joint spectrum.\par
\indent In this article representations of nilpotent Lie algebras by compact
operators defined
in a finite or infinite dimensional Banach space are considered and all the above mentioned joint spectra of such a representation are computed. These results extend the
characterizations of [3] and [1].\par
\indent The paper is organized as follows. In section 2 some
definitions and results needed for the present work are recalled, and in section 3
the main characterizations are proved.\par
\section{ Joint spectra of representations of Lie algebras}
\indent In this section the definitions and the main
properties of the Taylor, the S\l odkowski, the Fredholm, the split and
the Fredholm split joint spectra are reviewed; for a complete exposition see [9], [5],
[2], [11], [12] and [4].\par
\indent From now on $X$ denotes a complex Banach space, $L(X)$ the algebra of all
operators defined in $X$, $K(X)$ the ideal of all compact operators,
and $L(X,Y)$ the algebra of all bounded linear maps
from $X$ to $Y$, where $Y$ is another Banach space. If $X$
and $Y$ are two Banach spaces and $T\in L(X,Y)$, then the range and the null space of $T$
are denoted by $R(T)$ and $N(T)$ respectively.
In addition, if $L$ is a complex solvable finite dimensional
Lie algebra, then $\rho\colon L\to L(X)$ denotes a representation of $L$ in $X$.
Now well, if $X$, $L$ and $\rho$ are as above, it is possible to consider the
Koszul complex of the representation $\rho$, i.e.
$(X\otimes\wedge L, d(\rho))$, where $\wedge L$ denotes the
exterior algebra of $L$ and
$d_p (\rho)\colon X\otimes\wedge^p L\rightarrow X\otimes
\wedge^{p-1} L$ is the map defined by
\begin{align*}
d_p (\rho)&( x\otimes\langle l_1\wedge\dots\wedge l_p\rangle) = \sum_{k=1}^p (-1)^{k+1}\rho (l_k)(x)\otimes\langle l_1
\wedge\ldots\wedge\hat{l_k}\wedge\ldots\wedge l_p\rangle\\
+ & \sum_{1\le i< j\le p} (-1)^{i+j-1}x\otimes\langle [l_i, l_j]\wedge l_1\wedge\ldots\wedge\hat{l_i}\wedge\ldots\wedge
\hat{l_j}\wedge\ldots\wedge l_p\rangle ,
\end{align*}
where $\hat{ }$ means deletion. If $\dim L=n$, then
for $p$ such that $p\le 0$ or $p\ge {n+1}$, $d_p (\rho) =0$.\par
\indent In addition, if $f$ is a character of $L$, i.e. $f\in L^*$ and
$f(L^2) = 0$, where $L^2 = \{ [x,y]: x, y \in L\}$ and $[\cdot ,\cdot ]$ denotes
the Lie bracket of $L$, then it is possible to
consider the representation of $L$ in $X$ given by $\rho-f\equiv \rho-f\cdot I$,
where $I$ is the identity map of $X$. Now well, if $H_*(X\otimes
\wedge L , d(\rho-f))$ denotes the homology of the Koszul complex of
the representation $\rho-f$, then it is possible to introduce the sets
$$
\sigma_p(\rho)= \{ f\in L^*: f(L^2)=0, \hbox{ }H_p(X\otimes\wedge L,
d(\rho-f))\ne 0\},
$$
and
$$
\sigma_{p , e}(\rho)=\{f\in L^*: f(L^2)=0, \hbox{ }\dim \hbox{ }H_p(X\otimes
\wedge L,d(\rho-f))=\infty\}.
$$
\indent Next follow the definitions of the Taylor, the S\l odkowski, and the Fredholm joint
spectra; see [14], [13], [8], [10], [9], [5], [2], [11], [12] and [4].\par
\begin{df} Let $X$ be a complex Banach
space, $L$ a complex solvable finite dimensional Lie algebra,
and $\rho\colon L\to L(X)$ a representation of $L$
in $X$. Then,
the Taylor joint spectrum of $\rho$
is the set
$$\sigma(\rho)= \cup_{p=0}^n\sigma_p(\rho)=\{f\in L^*: f(L^2) =0,
\hbox{ }H_*(X\otimes\wedge L,d(\rho-f))\neq 0\}.
$$
\indent In addition, the $k$-th $\delta$-S\l odkowski joint spectrum of $\rho$ is the
set
$$
\sigma_{\delta ,k}(\rho)= \cup_{p=0}^k\sigma_p(\rho) ,
$$
and the $k$-th $\pi$-S\l odkowski joint spectrum of $\rho$ is the set
$$
\sigma_{\pi ,k}(\rho)= \cup_{p=n-k}^n\sigma_p(\rho)\cup \{f
\in L^*: f(L^2)=0, \hbox{ }R(d_{n-k}(\rho-f)) \hbox{ is not closed}\},
$$
for $k=0,\ldots , n=\dim L$. \par
\indent Observe that $\sigma_{\delta ,n}(\rho)= \sigma_{\pi ,n}
(\rho)= \sigma(\rho)$.\par
\indent On the other hand, the Fredholm or essential Taylor joint spectrum of $\rho$ is
the set
$$
\sigma_e(\rho)=\cup_{p=0}^n \sigma_{p , e}(\rho).
$$
\indent In addition, the $k$-th Fredholm or essential $\delta$-S\l odkowski joint spectra of $\rho$
is the set
$$
\sigma_{\delta , k ,e}(\rho)=\cup_{p=0}^k \sigma_{p ,e}(\rho),
$$
and the $k$-th Fredholm or essential $\pi$-S\l odkowski joint spectrum of $\rho$ is
the set
$$
\sigma_{\pi ,k ,e}(\rho)=\cup_{p=n-k}^n \sigma_{p ,e}(\rho)
\cup \{f\in L^*: f(L^2)=0,\hbox{ }
R(d_{n-k}(\rho))\hbox{ is not closed}\},
$$
for $k=0,\ldots , n$.\par
\indent Observe that $\sigma_e(\rho)=\sigma_{\delta ,n ,e}(\rho)=
\sigma_{\pi ,n ,e}(\rho)$.\par
\end{df}
\indent In order to state the definition of the split and Fredholm split joint spectra,
some preliminary facts are needed.\par
\indent A finite complex of Banach spaces and bounded linear operators
$(X,d)$ is a sequence
$$
0\to X_n\xrightarrow{d_n}X_{n-1}\to\ldots\to X_1\xrightarrow{d_1}X_0\to 0,
$$
where $n\in \Bbb N$, $X_p$ are Banach spaces, and the maps
$d_p \in L(X_p ,X_{p-1})$ are such that $d_{p-1}
\circ d_p=0$, for $p=1,\ldots , n$. \par
\indent Now well, given a fixed integer $p$, $0\le p\le n$,
the complex $(X,d)$ is said split (resp. Fredholm split) in degree $p$, if there are continous
linear operators
$$
X_{p+1}\xleftarrow{h_p}X_p\xleftarrow{h_{p-1}}X_{p-1},
$$
such that $d_{p+1}h_p+h_{p-1}d_p=I_p$ (resp. $d_{p+1}h_p+h_{p-1}d_p=I_p -k_p$, for
some $k_p\in K(X_p)$), where $I_p$ denotes the identity map of $X_p$,
$p=0, \ldots ,n$; see [7; 2].\par
\indent In addition, if $L$, $X$ and $\rho$ are as above, and if $p$
is such that $0\le p\le n$, then it is possible to introduce the sets
$$
sp_p (\rho )= \{ f\in L^*: f(L^2)=0,\hbox{ } (X\otimes\wedge L,d(\rho-f))
\hbox{ is not split in degree p} \}
$$
and
\begin{align*}
sp_{p,e} (\rho )= &\{ f\in L^*: f(L^2)=0, \hbox{ }(X\otimes\wedge L,d(\rho-f))
\hbox{ is not Fredholm split } \\
&\hbox{ in degree p} \}.
\end{align*}
\indent Next follow the definitions of the split and the Fredholm split joint spectra; see
[7], [12] and [4].\par
\begin{df}Let $X$ be a complex Banach
space, $L$ a complex solvable finite dimensional Lie algebra,
and $\rho\colon L\to L(X)$ a representation of $L$
in $X$. Then,
the split spectrum of $\rho$ is the set
$$
sp(\rho)=\cup_{p=0}^n sp_p(\rho).
$$
\indent In addition, the $k$-th $\delta$-split joint spectrum of $\rho$ is the set
$$
sp_{\delta ,k}(\rho)= \cup_{p=0}^k sp_p(\rho) ,
$$
and the $k$-th $\pi$-split joint spectrum of $\rho$ is the set
$$
sp_{\pi ,k}(\rho)= \cup_{p=n-k}^n sp_p(\rho),
$$
for $k=0,\ldots , n=\dim L$.\par
\indent Observe that $sp_{\delta ,n}(\rho)= sp_{\pi ,n}(\rho)=
sp(\rho)$.\par
\indent On the other hand, the Fredholm or essential split spectrum of $\rho$ is the set
$$
sp_e(\rho)=\cup_{p=0}^n sp_{p,e}(\rho).
$$
\indent In addition, the $k$-th Fredholm or essential $\delta$-split joint spectrum of $\rho$
is the set
$$
sp_{\delta ,k,e}(\rho)= \cup_{p=0}^k sp_{p,e}(\rho) ,
$$
and the $k$-th Fredholm or essential $\pi$-split joint spectrum of $\rho$ is the set
$$
sp_{\pi ,k,e}(\rho)= \cup_{p=n-k}^n sp_{p,e}(\rho),
$$
for $k=0,\ldots , n$.\par
\indent Observe that $sp_{\delta ,n,e}(\rho)=
sp_{\pi ,n,e}(\rho)= sp_e(\rho)$.\par
\end{df}
\indent It is clear that $\sigma_{\delta ,k}(\rho)\subseteq
sp_{\delta ,k}(\rho)$, $\sigma_{\pi ,k}(\rho)\subseteq
sp_{\pi ,k}(\rho)$, $\sigma(\rho)\subseteq sp(\rho)$, and
$\sigma_{\delta ,k, e}(\rho)\subseteq
sp_{\delta ,k, e}(\rho)$, $\sigma_{\pi ,k,e}(\rho)\subseteq
sp_{\pi ,k,e}(\rho)$, $\sigma_e(\rho)\subseteq sp_e(\rho)$.
Moreover, if $X$ is a Hilbert space, then the above inclusions are equalities.\par
\indent All the above considered joint spectra are defined
for representations of complex solvable finite dimensional Lie algebras
in complex Banach spaces and have the main spectral
properties. That is, they are compact nonempty subsets of
characters of the Lie algebra $L$ and they have
the projection property for ideals, i.e.
if $L$ is such a Lie algebra, $I$ is an ideal of $L$, $\sigma_*$
is one of above joint spectra, and $\pi\colon L^*\to I^*$ denotes the
restriction map from $L^*$ to $I^*$, then
$$
\pi (\sigma_*(L))=\sigma_*(I).
$$
For a complete exposition see the works [9], [5], [2], [11], [12] and [4].\par
\indent In the following section representations of nilpotent Lie algebras by compact operators
in finite or infinite dimensional Banach spaces will be considered and all the above mentioned joint spectra of such a representation
will be computed. However,
to this end, first it is necessary to
recall a result from [4].\par
\indent If $L$, $X$, and $\rho\colon L
\to L(X)$ are as above, then it is possible to consider the
representation
$$
L_{\rho}\colon L\to L(L(X)),\hskip1cm l\to L_{\rho(l)},
$$
where $L_{\rho(l)}$ denotes the left multiplication
operator associated to $\rho(l)$, $l\in L$.
In addition, since $L_{\rho(l)}( K(X))\subseteq
K(X)$, it is possible to consider the representation
$$
\tilde{L}_{\rho}\colon L\to L(C(X)),
$$
where $C(X)=L(X)/ K(X)$,
and $\tilde{L}_{\rho}(l)$ is the quotient operator defined in
$C(X)$ associated to $L_{\rho(l)}$, $l\in L$.\par
\indent Similarly, if $L^{op}$ is the Lie algebra which as vector space
coincides with $L$ but whose Lie bracket is the opposite one,
then it is possible to consider the representation
$$
R_{\rho}\colon L^{op}\to L( L(X)),\hskip1cm l\to
R_{\rho(l)},
$$
where $R_{\rho(l)}$ denotes the right multiplication
operator associated to $\rho(l)$, $l\in L^{op}$.
Furthermore, since $R_{\rho(l)}(K(X))\subseteq K(X)$, it is possible to consider the
representation
$$
\tilde{R}_{\rho}\colon L^{op}\to L( C(X)),
$$
where $\tilde{R}_{\rho}(l)$ is the quotient
operator defined in
$C(X)$ associated to $R_{\rho(l)}$, $l\in L^{op}$.\par
\indent Now well, if $L$ is a nilpotent Lie algebra, then according to [4; 8]
and [12; 0.5.8]:
\noindent i- $sp_{\delta ,k,e}(\rho)=\sigma_{\delta ,k}
(\tilde{L}_{\rho})$,\par
\noindent ii- $sp_{\pi ,k,e}(\rho)=\sigma_{\delta , k}
(\tilde{R}_{\rho})$,\par
\noindent iii- $sp_e(\rho)=\sigma
(\tilde{L}_{\rho})=\sigma(\tilde {R}_{\rho})$.\par
\section{The main results}
\indent In this section, given a
representation of a complex nilpotent finite dimensional
Lie algebra by compact operators in a finite or infinite
complex Banach space, the Taylor the S\l odkowski, the Fredholm,
the split, and the Fredholm split joint spectra of such a representation are characterized.
Since the example
which follows Theorem 5 in [3] shows that for a solvable non-nilpotent
Lie algebra of compact operators the characterization fails,
only nilpotent Lie algebras are considered.
In first place the infinite dimensional case is studied. \par
\indent In the following theorem the Taylor and the
S\l odkowski
joint spectra are described.\par
\begin{thm} Let $X$ be an infinite dimensional complex Banach space,
$L$ a complex nilpotent finite dimensional
Lie algebra, and $\rho\colon L\to L(X)$ a representation
of $L$ in $X$ such that $\rho (l)\in K(X)$ for each $l\in L$.
Then, the sets $\sigma(\rho)$,
$\sigma_{\delta ,k}(\rho)\cup \{ 0\}$,
and $\sigma_{\pi ,k}(\rho)\cup \{ 0\}$ coincide with the set
\begin{align*}
\{0\}\cup\{& f\in L^*: f(L^2)=0, \hbox{ such that
there is }x\in X,\hbox{ }x\neq 0, \hbox{ with the} \\
&\hbox{property: } \rho(l)(x)=f(l)x,\hbox{ } \forall\hbox{ } l\in L\},
\end{align*}
for $k=0, \ldots , n=\dim L$,
\end{thm}
\begin{proof}
\indent First of all recall that, according to [1; 3.8], [1; 3.9] and [9; 2.6],
\begin{align*}
\sigma(\rho) =\{ 0 \}\cup&\{f\in L^*: f( L^2)=0,
\hbox{ such that there is }x\in X,\hbox{ }x\neq 0,\\
& \hbox{ with the property: }
\rho(l)(x)=f(l)x,\hbox{ } \forall\hbox{ } l\in L \}.\end{align*}
\indent Now, since $L$ is a nilpotent Lie algebra, according to the
above equality and Definition 1,
$\sigma (\rho)=\{ 0 \}\cup \sigma_{\pi ,0}(\rho)$. However, since
$\sigma_{\pi ,0}(\rho)\cup \{ 0 \}\subseteq\sigma_{\pi ,k}(\rho)\cup \{ 0 \}
\subseteq \sigma (\rho)$, $0\le k\le n$,
$$
\sigma(\rho) =\sigma_{\pi ,k}\cup\{ 0\}=\sigma_{ \pi ,0}(\rho)\cup\{ 0\}.
$$
\indent On the other hand, if
the adjoint representation of $\rho$ is considered, i.e.
the representation $\rho^*\colon L^{op}
\to L(X{'})$, $\rho^*(l)=(\rho (l))^*$, where $X{'}$ denotes the dual space
of $X$, then since $L^{op}$ is a nilpotent Lie algebra, according to [12; 0.5.8] and [12; 2.11.4],
$$
\{0\}\cup\sigma_{\delta ,k}(\rho)=\{0\}\cup
\sigma_{\pi ,k}(\rho^*)=\{0\}\cup \sigma_{\pi ,0}(\rho^*)=
\sigma(\rho^*)=\sigma (\rho).
$$
\end{proof}
\indent In the following theorem the Fredholm joint spectra
are computed.\par
\begin{thm} Let $X$ be an infinite dimensional
complex Banach space, $L$ a complex nilpotent finite dimensional
Lie algebra, and $\rho\colon L\to L(X)$ a representation
of $L$ in $X$ such that $\rho(l)\in K(X)$ for each $l\in L$.
Then
$$
\sigma_e(\rho)=\sigma_{\delta ,k ,e}(\rho)=\sigma_{\pi ,k ,e}(\rho)=
\{0\},
$$
for $k=0, \ldots , n=\dim L$.\par
\end{thm}
\begin{proof}
\indent The proof is based on an induction argument on the dimension of the algebra.\par
\indent If $\dim L=1$, then consider $l\in L$ such that $<l>=L$. Now,
since $\rho (l)$ is a compact operator,
$\sigma_e(\rho(l))=\{0 \}$.
However, since $\dim L=1$, according to Definition 1, $\sigma_e (\rho)=\{ 0\}$.
Now, since $\sigma_{\delta ,k, e}(\rho)$ and $\sigma_{\pi ,k ,e}(\rho)$
are nonempty subsets of $\sigma_e(\rho)$, $0\le k\le 1=\dim L$,
$$
\sigma_e(\rho)=\sigma_{\delta ,k ,e}(\rho)=
\sigma_{\pi ,k ,e}(\rho)=\{0\},
$$
$0\le k\le 1=\dim L$.\par
\indent Now suppose that for every nilpotent Lie algebra of
dimension less than $n$ and for every representation of the algebra
by compact operators in an infinite dimensional Banach space $X$
$$
\sigma_e(\rho)=\sigma_{\delta ,k,e}(\rho)=
\sigma_{\pi ,k ,e}(\rho)=\{0\}.
$$
\indent Now well, if $L$ is a nilpotent Lie algebra of dimension $n$,
then according to [6; 5.1] there is a Jordan-H\" older sequences of ideals $(L_i)_{0\le i
\le n}$ such that \par
\noindent i- $L_0=0$, and $L_n=L$, \par
\noindent ii- $L_i\subseteq L_{i+1}$, $0\le i\le n-1$,\par
\noindent iii- $[L_i, L_j]\subseteq L_{i-1}$, for $i<j$.\par
\indent It is easy to prove that $L^2\subseteq L_{n-2}$.
Then, it is possible to consider the ideals $I_1=L_{n-1}$ and $I_2=
L_{n-2}\oplus <x>$, where $x\in L$ is such that
$L_{n-1}\oplus<x>=L$.\par
\indent On the other hand, if the representations $\rho_1
=\rho\mid I_1\colon I_1\to L(X)$ and $\rho_2=\rho\mid I_2
\colon I_2\to L(X)$ are considered, then, according to the assumption under
consideration,
$$
\sigma_e(\rho_1)=\sigma_{\delta ,k ,e}(\rho_1)=
\sigma_{\pi ,k, e}(\rho_1)=\{0\},
$$
and
$$
\sigma_e(\rho_2)=\sigma_{\delta ,k ,e}(\rho_2)=
\sigma_{\pi ,k ,e}(\rho_2)=\{0\}.
$$
However, according to the projection property of the Fredholm joint spectra,
[4; 3.2] and [4; 3.5],
$$
\sigma_e(\rho)=\sigma_{\delta ,k ,e}(\rho)=
\sigma_{\pi ,k ,e}(\rho)=\{0\},
$$
for $k=0,\ldots , n=\dim L$.\par
\end{proof}
\indent In the following theorem the split and the
Fredholm split joint spectra are computed.\par
\begin{thm} Let $X$ be an infinite dimensional
complex Banach space, $L$ a complex nilpotent finite dimensional
Lie algebra, and $\rho\colon L\to L(X)$ a representation
of $L$ in $X$ such that $\rho(l)\in K(X)$ for each $l\in L$.
Then, $\sigma (\rho)=sp(\rho)$, $\sigma_{\delta ,k}(\rho)=sp_{\delta ,k}(\rho)$
and $\sigma_{\pi ,k}(\rho)=sp_{\pi ,k}(\rho)$,where $k=0,\ldots , n=\dim L$.\par
\indent In particular, the sets $\sigma (\rho)$, $\sigma_{\delta ,k}(\rho)\cup \{0\}$,
$\sigma_{\pi ,k}(\rho)\cup \{0\}$, $sp(\rho)$, $sp_{\delta ,k}(\rho)\cup \{0\}$
and $sp_{\pi ,k}(\rho)\cup \{0\}$ coincide with the set
\begin{align*}
\{0\}\cup\{& f\in L^*: f(L^2)=0, \hbox{ such that there
is }x\in X,\hbox{ }x\neq 0, \\
&\hbox{with the property }\rho(l)(x)=f(l)x,\hbox{ } \forall\hbox{ } l\in L\}.\end{align*}
\indent In addition,
$$
sp_{e}(\rho)=sp_{\delta ,k ,e}(\rho)=sp_{\pi ,k,e}(\rho)=\{0\},
$$
where $k=0,\ldots , n=\dim L$.\par
\indent In particular, all the Fredholm and Fredholm split joint spectra
coincide with the set $\{0\}$.
\end{thm}
\begin{proof}
\indent First of all, since $\rho (L)\subseteq K(X)$, $\tilde{L}_{\rho}=0$
and $\tilde{R}_{\rho}=0$. Thus, since $L$ is a nilpotent Lie algebra,
according to [4; 8] and [12; 0.5.8], \par
\noindent i- $sp_{\delta ,k,e}(\rho)=\sigma_{\delta ,k}(\tilde{L}_{\rho})=\{0\}$,\par
\noindent ii- $sp_{\pi ,k,e}(\rho)=\sigma_{\delta, k}(\tilde{R}_{\rho})=\{0\}$,\par
\noindent iii- $sp_e(\rho)=\sigma(\tilde{L}_{\rho})=\sigma(\tilde{R}_{\rho})=\{0\}$.\par
\indent Now well, in order to prove the first assertion of the Theorem, observe that,
according to [7; 2.7], \par
\noindent i- $sp(\rho)=\sigma (\rho)\cup {\mathcal C}$,
$sp_e(\rho)=\sigma_e (\rho)\cup \tilde{{\mathcal C}}$,
\noindent ii- $sp_{\delta ,k}(\rho)=\sigma_{\delta ,k} (\rho)
\cup {\mathcal A}_k$,
$sp_{\delta ,k ,e}(\rho)=\sigma_{\delta ,k,e} (\rho)
\cup\tilde{ {\mathcal A}}_k$,\par
\noindent iii- $sp_{\pi ,k}(\rho)=\sigma_{\pi ,k} (\rho)\cup {\mathcal B}_k$,
$sp_{\pi ,k,e}(\rho)=\sigma_{\pi ,k,e} (\rho)\cup\tilde{ {\mathcal B}}_k$,
where $k=0,\ldots , n$ and\par
\noindent iv- ${\mathcal A}_k=\{f\in L^*: f(L^2)=0,\hbox{ } f\notin
\sigma_{\delta ,k}(\rho), \hbox{ and there is } p,\hbox{ } p=1,\ldots
, k+1,\hbox{ such that } N(d_p(\rho-f))
\hbox{ is not complemented in } X\otimes \wedge^p L\}$,
\par
\noindent v- $\tilde{{\mathcal A}}_k=\{f\in L^*: f(L^2)=0, \hbox{ }f\notin
\sigma_{\delta ,k ,e}(\rho), \hbox{ and there is } p,\hbox{ }
p=1,\ldots ,k+1, \hbox{ such that }N(d_p(\rho-f))
\hbox{ is not complemented in }X\otimes \wedge^p L\}$,
\par
\noindent vi- ${\mathcal B}_k=\{f\in L^*: f(L^2)=0, \hbox{ }f\notin\sigma_{\pi ,k}
(\rho), \hbox { and there is } p,\hbox{ } p=n,\ldots , n-k,
\hbox{ such that }R(d_p(\rho-f))\hbox{ is not complemented in } X\otimes \wedge^{p-1}L \}$, \par
\noindent vii- $\tilde{{\mathcal B}}_k=\{f\in L^*: f(L^2)=0, \hbox{ }f\notin
\sigma_{\pi ,k ,e}(\rho), \hbox { and there is } p, \hbox{ } p=n,\ldots ,n-k,
\hbox{ such that } R(d_p(\rho-f))\hbox{ is not complemented in }X\otimes \wedge^{p-1} L
\}$,\par
\noindent viii- ${\mathcal C}={\mathcal A}_n={\mathcal B}_n$,
$\tilde{{\mathcal C}}=\tilde{{\mathcal A}}_n=\tilde{{\mathcal B}}_n$.\par
\indent In addition,
${\mathcal A}_k\subseteq\tilde{{\mathcal A}}_k$, and
${\mathcal B}_k\subseteq\tilde{{\mathcal B}}_k$, for $k=0,\ldots , n$.\par
\indent However, since $\sigma_{\delta ,k ,e}(\rho)\cap
\tilde{\mathcal A}_k=\emptyset$ and $\sigma_{\pi ,k ,e}(\rho)\cap
\tilde{\mathcal B}_k=\emptyset$, according to Theorem 2 and to the second
part of the Theorem, which have already been proved,
$\tilde{\mathcal A}_k=\emptyset$ and $\tilde{\mathcal B}_k=\emptyset$.
In particular, ${\mathcal A}_k=\emptyset$ and ${\mathcal B}_k=\emptyset$,
which proves the first assertion of the Theorem.\par
\end{proof}
\indent Next representations of nilpotent Lie algebras in finite
dimensional Banach spaces are considered. Observe that in this case the
spectra to be studied are the Taylor and the S\l odkowski
joint spectra.\par
\begin{thm} Let $X$ be a complex finite dimensional Banach space,
$L$ a complex nilpotent finite dimensional Lie algebra, and $\rho\colon L\to L(X)$
a representation of $L$ in $X$. Then
\begin{align*}
\sigma(\rho)=\sigma_{\delta ,k}(\rho)=\sigma_{\pi, k}(\rho)= &\{f\in L^*: f(L^2)=0,
\hbox{ such that there is } x\in X,\\
& x\neq 0, \hbox { with the property: }\rho (l)x=f(l)x,\hbox{ }\forall\hbox{ } l\in L\}.
\end{align*}
\end{thm}
\begin{proof}
\indent Since $L$ is a nilpotent Lie algebra, according to [3; 4] and [9; 2.6],
\begin{align*}
\sigma(\rho)= &\{f\in L^*: f(L^2)=0,\hbox{ such that there is } x\in X,\hbox{ } x\neq 0,
\hbox { with the}\\
&\hbox{property: }\rho (l)x=f(l)x,\forall\hbox{ } l\in L\}.\end{align*}
\indent Therefore, since $L$ is a nilpotent Lie algebra, $\sigma(\rho)=
\sigma_{\pi , 0}(\rho)$. Now well, in order to finish the proof, it is possible to use an argument
similar to the one developed in Theorem 1.\par
\end{proof}
\bibliographystyle{amsplain}
| 7,873
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"As Of" Date: Monday, January 7, 2008.
Deadline for Completion of SRS Updates by Ex Officio Trustees or their assistants via the Web Interface each year: March 15th.. This booklet outlines the process of registering students for this program.
If you have questions about Federal Quota Registration or the Student Registration System (SRS) software, please contact Cindy Amback, Department of Field Services, camback@aph.org or 800-223-1839, ext. 257.
In order for students to be eligible for registration in the Federal Quota Program, they MUST meet the requirements as outlined in An Act to Promote the Education of the Blind. Students must:
The educational programs providing services to these students can include public, private, and parochial schools.
There is NO chronological age limit for eligibility. The federal law limits registration to persons working at less than college level, but places no restriction on the ages of eligible students.
The Board of Trustees of the American Printing House for the Blind,."
Eligible infants, preschool children, and homebound students can be registered if they are enrolled in a formally organized, regularly scheduled educational or training program and have a written education plan.
In order for blind students enrolled in home school programs to prove eligible for registration in the Federal Quota Program, the home school program should meet guidelines and/or procedures in effect within each individual state.
For the purposes of the Federal Quota Registration, school age is to be determined by the state law of each respective state.
The following documents are not to be submitted to APH, but must be on file with the agency or school where students are attending in the event that an audit requires evidence of the student's educational program or visual acuity:
Students cannot be registered by two accounts. If a student is attending two programs, the agencies involved must work together to determine which account will register the student. The agency providing the major portion of educational services is usually the account with which the student is registered.
Students who are enrolled in nonpublic schools must, by law, be registered in a separate Private Nonprofit (PNP) account. This requires a separate account number and registration forms.
If students in nonpublic schools are to be registered by you and your agency does not have a separate PNP account, please contact APH in order to establish one.
For a student who moves from a public school to a private school as of January 7, 2008, it is necessary to delete the student record from the public listing and add it to the private school registration--or vice versa.
The "as of" date for registration in the Federal Quota Program is traditionally the first Monday in January. Students must be enrolled on this date. The first Monday in 2008 is January 7th.
The method for reporting the Federal Quota Registration to APH is data entry via the SRS Web interface into the SRS SQL database.
IMPORTANT: All agencies are requested to use the SRS Web interface to submit their updated Census information. Please contact the APH Census Support Specialist, Cindy Amback, if you are unable to use the Web interface for any reason.
Please note: Adobe Reader 5.0 or above is required for reports developed through the new SRS Web interface. Adobe Reader is available as a free download.
The data to be reported and the reporting codes to be used are outlined later in this instruction booklet.
To update student information:
In order to edit a student's data or delete a students data, you must be logged in and execute a SEARCH. To edit or delete a record, select the DETAILS column of the desired record to bring up the edit screen. At this point, a record may be edited or deleted.
To Save Your Updates/Edits:
To save the results of any edits, the SAVE button must be clicked at the end each edit. At that time, various edit checks against the data will be performed and you will be notified of any fields in which an error occurred. THESE MUST BE CORRECTED before the data can be saved.
To Add a Student Not Enrolled the Previous Year:
Choose the "Add New Student" button on the Student Search Screen. Enter the student information and click the SAVE button..
For a Student No Longer Enrolled
To delete a student who was enrolled the previous year: Select the "Details" column for the record to be deleted and click the DELETE button to remove the data. When a student's information is deleted through the Web interface, it is only marked for deletion. Once marked for deletion, the information will not appear in standard reports and will not be completely deleted from the database until APH closes the Census for the year.
Students Moving from One Account to Another Administered by the Same Trustee
Trustees who administer more than one account may transfer a student's data from one account they administer to another by utilizing the account drop down list on the student's Details Screen. This eliminates the need to delete and re-enter the same data twice
Note: When entering visual acuities/reporting codes through the SRS Web interface into the SRS SQL database, a drop down list of available codes will be provided for your use. Any additional qualifying measurements/codes not listed in the drop down list may be added by Cindy Amback at APH. Submit your written requests to Cindy Amback at camback@aph.org. You may also contact Cindy at (800) 223-1839, extension 257, with any other questions regarding the Census.
The preliminary student registration report is a tool used to gather your data before the actual Census begins in January and, can be generated by accessing the new SRS Web interface/Report Menu. The SRS program is available to Trustees and their assistants to make changes to their data the day after the first Monday in January.
In addition to the master Preliminary Student Registration Report form, a blank preliminary student registration report form will be e-mailed to Ex Officio Trustees and their assistants prior to the first Monday in January each year. When gathering data on the Preliminary Student Registration Report form, the following steps are to be used:
To update student information:
Type or print clearly all corrections or changes on the white line underneath the previous year's data on the Preliminary Student Registration Form for each respective student.
To add a student not enrolled the previous year:
Type or print clearly the entire information for a new student on the white lines of the blank preliminary student registration form provided to you via email. List new students alphabetically to assist with later checking of data entered on the SRS website.
To delete a student who was enrolled the previous year:
Type or print clearly the word "DELETE" on the white line under the student's name.
For each eligible student, the following data must be reported: programs, but not enrolled in grade 12 or below, must "have a written instruction plan and be enrolled in and attend, on a regular basis, an instructional program of at least 20 hours of instruction per week. Social and leisure programs do not qualify as instructional programs."
Does not include any eligible participants over school age.
The primary reading medium is to be reported for each student using the following reporting codes. Only these codes will be accepted.
Note: Infants and preschoolers identified as visual, braille, or auditory readers should be reported using the appropriate media codes.
In addition to listing a primary reading medium, a secondary reading medium is also a required category/field. Only one medium can be chosen and entered in the secondary reading medium field to provide more accurate statistics.
Note: Please remember that you may not duplicate a reading medium in the primary and secondary reading media fields.
An optional other (or third) reading medium field has been added for your convenience. This field is not required and defaults to NA, or not applicable. You may choose not to include an other (or third) reading medium.
Note: Please remember that you may not duplicate a reading medium in the second and third reading media fields with the exception of NA.
When all of your data has been collected and recorded on the registration forms and all of the changes, additions, and deletions have been entered through the SRS Web interface, go to the Report Menu and run the Duplicate Student Report.
If the report opens empty, or with no students listed, then you do not have any duplicates within your own account. If names appear on this report, you have duplicate students within your account. You will need to adjust your data prior to submitting it to APH.
While in the Report Menu, open the Student Listing by Name Report, which provides a total count of your students and verify that the number corresponds with your records. Keep in mind that as you change your data, the data listed on the reports in the Report Menu changes as well.
Last, but not least, while still in the Report Menu, open the Deleted Students Report and verify that the students listed as marked for deletion are truly the students you wish to delete. These students will be deleted later in the census process.
Suggestion: After all of the updates for the current year have been made to your data in Phase One of the Census, print a copy of the Student Listings by Name Report (found on the Report Menu of the SRS Web interface) which lists all of your students, or save a copy to your hard drive for reference in Phase Two. If you choose to save the report to your computer, be sure to label it "2008 Census Data-Phase I (and your account number)" so it will not be confused with the reports you save at the end of the year for historical purposes.
Notify Cindy by e-mail when you have completed each phase of the Census. This will allow her to track completion of each phase not only by account, but as a whole for the United States. Be sure to include your account number in all e-mails to Cindy in order to expedite service.
**Remember that at the end of Phase One you cannot add any additional students. At the end of Phase Two, you may not delete any students.
Questions regarding the Federal Quota Census should be directed to:
CINDY AMBACK, SUPPORT SPECIALIST
DEPARTMENT OF FIELD SERVICES
AMERICAN PRINTING HOUSE FOR THE BLIND
P.O. BOX 6085
LOUISVILLE, KENTUCKY 40206-0085
DEADLINE--Phase One: March 15, 2008
Registrations will not be accepted after March 15!
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When you think about hope and encouragement, what do you picture in your mind? Can you visualize hope and encouragement? What does it look like to you?
The great Alice Walker wrote the book “The Color Purple” for seemingly many reasons. Of course, the most glaring purpose was to bring awareness to the plight of black women during the turn of the century. The title of the book doesn’t seem to make sense, though, until close to the end. But that’s only if you catch it.
“I think it pisses God off if you walk by the color purple in a field somewhere and don’t notice it. People think pleasing God is all God cares about. But any fool living in the world can see it always trying to please us back.” ~ Shug Avery
In a book full of turmoil and strife, Alice Walker named the book for what she believed was hope and encouragement. Think about it, how often do you see purple naturally in the world? Purple is so rare that people would pay to find specific flowers to mix and make fabrics for the royals. Of course, now we have many ways to fabricate it, but seeing purple in its natural element is a gift from God.
I think Shug is right. We do tend to always work to please God without stopping to recognize that He’s left us little love notes of hope and encouragement throughout the earth.
Enjoy the blooms
Here’s another example. God gave cherry blossoms to the Asian countries, who in turn shared them with the United States. We gather and celebrate these beautiful blooms because we understand that they are temporary and we want to enjoy them while we can. But we also know that they will return the next year. These trees give us hope and encourage us to celebrate and enjoy God’s gift because it is rare. Just like the color purple.
In a few days, we will celebrate the death and resurrection of our Lord and Savior Jesus Christ. For most Christians, it is not difficult to understand how those three days over 2000 years ago give us hope and encouragement. For others, Easter has become another commercialized holiday focused on meaningless symbols. Thankfully God has created us all to be a beacon of light for Him to give others hope and encourage each other to remain strong. I pray that this month’s guide strengthens you and your light for Christ so that others can be encouraged through you.
As always, I’m also including a cell phone wallpaper/calendar for April, and a new Gospel Spotify playlist so we can praise Him together through song. If you have any music suggestions, leave a comment and let me know what you’d like me to include for next month.
He is Risen!
Click to save or just listen to the playlist
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TITLE: How to prove the cofactor formula for determinants, using a different definition of the determinant?
QUESTION [8 upvotes]: So, I am very interested on this theorem (Laplace expansion), but I am still a high school student. I have four books about matrices, but only one of them have proo and that doesn't start with my definition of determinant. Also I don't understand the proof of Laplace expansion in wikipedia, because there is too much symbol and terms I didn't learn and that proof dosn't start with my definition. So may someone give me the outline of the proof start with my definition?
Here's my definition of the $n \times n$ determinant:
The value of the determinant of a matrix of order $n$ is defined as the sum of $n!$ terms of the form $(-1)^k a_{1 i_1} a_{2 i_2} \cdots a_{n i_n}$. Each term contains one and only one element from each row and one and only element from each column; i.e., the second subscripts $i_1, i_2 , \ldots, i_n$ are equal to $1,2, \ldots, n$ taken in some order. The exponent $k$ represents the number of interchanges of two elements necessary for the second subscripts to be placed in the order $1,2, \ldots, n$. For example, consider the term containing $a_{13} a_{21} a_{34} a_{42}$ in the evaluation of the determinant of a matrix of order four. The value of $k$ is $3$ since three interchanges of two elements are necessary for the second subscripts to be placed in the order $(1,2,3,4)$.
REPLY [0 votes]: Arturo Magidin, thanks for your answer. I now understand the steps, namely:
1) All determinants of a matrix of size n will be calculated by a sum of n! terms (easy to prove by induction)
2) these n! terms will each be a product of n numbers (evident)
3) By the definition of how a determinant is defined, these n numbers will be taken from different rows and columns, for a total of n! possible terms, that is: from looking at the matrix, we can know exactly which terms will appear in the end sum of the determinant. All we need is to find out the sign of each term.
I believe that a crucial element is missing: how can we be sure that the sign calculus will be the same, no matter the order that we choose to calculate it? In order to prove that we can begin with any column/row and reach the same determinant, this seems necessary. For example, in a n=4 matrix
x x b x
a x x x
x c x x
x x x d
let's ignore the x's, which can be any number, and say we are calculating the sign of the abcd term that will appear in the determinant sum. How can we be sure that we will end up with the same sign if we choose any order of abcd? that is, take b and calculate its cofactor's sign, then to calculate b's cofactor's sign, take a, and then c and d, namely b-a-c-d, or do the same for c-b-a-d.
Because as the matrices for the new calculations become smaller, the sign of the cofactors of a, b, c, d can also change. Is there a proof in this regard, that shows that multiplying the resulting sign here is the same for any sequence that we choose to use? This is evident only for the main diagonal. This seems to me to transcend the scope of Linear Algebra and an answer might come from a different field of Mathematics.
| 200,298
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Tuesday, May 137:30 pm1968 REVISTED Liberation, Imagination & the Black Panther Party George Katsiaficas, Ashanti Alston Omowali & James PonceContinuing our year-long events series on the 40th anniversary of 1968, we will look at the impact of the Black Panther Party, which inspired thousands to join their movement to transform “the system.” Liberation, Imagination, and the Black Panther Party will offer a fresh and realistic recounting of the tumultuous history of what arguably became the most significant revolutionary organization in the US during the late 20th century.Ashanti Alston Omowali is an anarchist activist, speaker, and writer, and former member of the Black Panther Party, the Black Liberation Army, and spent more than a decade in prison after government forces captured him and the official court system convicted him of armed robbery. Alston is the former northeast coordinator for Critical Resistance, a current co-chair of the National Jericho Movement (to free U.S. political prisoners), a member of pro-Zapatista people-of-color U.S.-based Estación Libre, and is on the board of the Institute for Anarchist Studies.George Katsiaficas has been active in social movements since 1969 when he participated in the anti-Vietnam movement. A target of the FBI’s COINTELPRO program (Counterintelligence), he was honored to be classified “Priority 1 ADEX” meaning in the event of a national emergency, people like him were to be immediately arrested. After living in Berlin for 1 1/2 years and learning first-hand about the autonomous movement there, he wrote about that movement (The Subversion of Politics: European Autonomous Social Movements and the Decolonization of Everyday Life). He is the author of several other books including the classic: The Imagination of the New Left: A Global Analysis of 1968.
Tags: No Comments.
| 33,732
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jam sync
In audio (sound) production, jam sync is a mode of device synchronization using SMPTE time code in which a slave device can furnish its own timing during the time that a master device is temporarily unstable. Jam sync is usually an optional mode that the user can select.
In a basic SMPTE synchronization scenario, time code is fed from one device (the master) to a second device (the slave). The slave device follows the time locations as transmitted by the master device. If the master device goes to one hour and thirteen minutes (expressed as 01:13:00:00; Hours:Minutes:Seconds:Frames), the slave device follows it there.
The problem arises when the master device's SMPTE time code has an error in it, caused by a small bad portion of tape, for example. In most situations, the slave device would not know what to do for a split second since it has missing instructions from its host. It would stutter or stop completely.
By using jam syncing, the slave device can be instructed to ignore small dropouts of time code from the master, thus ensuring smooth synchronization. To do this, it generates its own time code whenever the master's time code is missing.
Start the conversation
| 89,422
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\begin{document}
\maketitle
\thispagestyle{empty}
\begin{abstract}
Random increasing $k$-trees represent an interesting, useful class
of strongly dependent graphs for which analytic-combinatorial tools
can be successfully applied. We study in this paper a notion called
connectivity-profile and derive asymptotic estimates for it; some
interesting consequences will also be given.
\end{abstract}
\section{Introduction}
A $k$-tree is a graph reducible to a $k$-clique by successive
removals of a vertex of degree $k$ whose neighbors form a
$k$-clique. This class of $k$-trees has been widely studied in
combinatorics (for enumeration and characteristic
properties~\cite{beineke_number_1969,rose_simple_1974}), in graph
algorithms (many NP-complete problems on graphs can be solved in
polynomial time on $k$-trees~\cite{arnborg_complexity_1987}), and in
many other fields where $k$-trees were naturally encountered
(see~\cite{arnborg_complexity_1987}). By construction, vertices in
such structures are remarkably close, reflecting a highly strong
dependent graph structure, and they exhibit with no surprise the
scale-free property~\cite{gao_degree_2009}, yet somewhat
unexpectedly many properties of random $k$-trees can be dealt with
by standard combinatorial, asymptotic and probabilistic tools, thus
providing an important model of synergistic balance between
mathematical tractability and the predictive power for
practical-world complex networks.
While the term ``$k$-trees" is not very informative and may indeed
be misleading to some extent, they stand out by their underlying
tree structure, related to their recursive definition, which
facilitates the analysis of the properties and the exploration of
the structure. Indeed, for $k=1$, $k$-trees are just trees, and for
$k\ge 2$ a bijection~\cite{darrasse_limiting_2009} can be explicitly
defined between $k$-trees and a non trivial simple family of trees.
The process of generating a $k$-tree begins with a $k$-clique, which
is itself a $k$-tree; then the $k$-tree grows by linking a new
vertex to every vertex of an existing $k$-clique, and to these
vertices only. The same process continues; see Figure~\ref{fg-2-tr}
for an illustration. Such a simple process is reminiscent of several
other models proposed in the literature such as
$k$-DAGs~\cite{devroye_long_2009}, random
circuits~\cite{arya_expected_1999}, preferential
attachment~\cite{barabsi_emergence_1999,bollobs_degree_2001,
hwang_profiles_2007}, and many other models (see, for
example,~\cite{boccaletti_complex_2006,
durrett_random_2006,newman_structure_2003}). While the construction
rule in each of these models is very similar, namely, linking a new
vertex to $k$ existing ones, the mechanism of choosing the existing
$k$ vertices differs from one case to another, resulting in very
different topology and dynamics.
\begin{figure}[!h]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
\ding{172} & \ding{173} & \ding{174} & \ding{175} & \ding{176} \\
\includegraphics{figure-3-1} &
\includegraphics{figure-3-2} &
\includegraphics{figure-3-3} &
\includegraphics{figure-3-4} &
\includegraphics{figure-3-5} \\
\includegraphics{figure-4-1} &
\includegraphics{figure-4-2} &
\includegraphics{figure-4-3} &
\includegraphics{figure-4-4} &
\includegraphics{figure-4-5} \\ \hline
\end{tabular}
\end{center}
\caption{\emph{The first few steps of generating a $3$-tree and a
$4$-tree. Obviously, these graphs show the high connectivity of
$k$-trees.}} \label{fg-2-tr}
\end{figure}
Restricting to the procedure of choosing a $k$-clique each time a
new vertex is added, there are several variants of $k$-trees
proposed in the literature depending on the modeling needs. So
$k$-trees can be either labeled~\cite{beineke_number_1969},
unlabeled~\cite{labelle_labelled_2004},
increasing~\cite{zhang_high-dimensional_2006},
planar~\cite{zhang_high-dimensional_2006},
non-planar~\cite{beineke_number_1969}, or
plane~\cite{palmer_number_1973}, etc.
For example, the family of random Apollonian networks, corresponding
to planar 3-trees, has recently been employed as a model for complex
networks~\cite{andrade_apollonian_2005,zhang_high-dimensional_2006}.
In these frameworks, since the exact topology of the real networks
is difficult or even impossible to describe, one is often led to the
study of models that present similarities to some observed
properties such as the degree of a node and the distance between two
nodes of the real structures.
For the purpose of this paper, we distinguish between two models of
random labeled non-plane $k$-trees; by non-plane we mean that we
consider these graphs as given by a set of edges (and not by its
graphical representation):
\begin{itemize}
\item[--] \emph{random simply-generated $k$-trees}, which correspond
to a uniform probability distribution on this class of $k$-trees,
and
\item[--] \emph{random increasing $k$-trees}, where we consider the
iterative generation process: at each time step, all existing
$k$-cliques are equally likely to be selected and the new vertex is
added with a label which is greater than the existing ones.
\end{itemize}
The two models are in good analogy to the simply-generated family of
trees of Meir and Moon~\cite{meir_altitude_1978} marked specially by
the functional equation $f(z) = z\Phi(f(z))$ for the underlying
enumerating generating function, and the increasing family of trees
of Bergeron et al.~\cite{bergeron_varieties_1992}, characterized by
the differential equation $f'(z) = \Phi(f(z))$. Very different
stochastic behaviors have been observed for these families of trees.
While similar in structure to these trees, the analytic problems on
random $k$-trees we are dealing with here are however more involved
because instead of a scalar equation (either functional, algebraic,
or differential), we now have a system of equations.
\begin{table}
\begin{center}
\begin{tabular}{|r||c|c|} \hline
\backslashbox{Properties}{Model} & Simply-generated structures&
Increasing structures \\ \hline Combinatorial description &
$\mathcal{T}_s = \mbox{Set}(\mathcal{Z}\times\mathcal{T}_s^k)$ &
$\mathcal{T} =
\mbox{Set}(\mathcal{Z}^\square \times\mathcal{T}^k)$\\
\hline Generating function & $T_s(z) = \exp(z T_s^k(z))$
& $ T'(z) = T^k(z)$ \\
\hline Expansion near singularity& $T_s(z) = \tau -
h\sqrt{1-z/\rho}+\ldots$ & $ T(z) = (1-kz)^{-1/k}$ \\ \hline Mean
distance of nodes & $O(\sqrt{n})$ & $O(\log n)$ \\ \hline Degree
distribution & Power law with exp.\ tails & Power
law~\cite{gao_degree_2009}
\\\hline Root-degree distribution & Power law with exp.\ tails &
Stable law~(Theorem~\ref{thm-ld}) \\\hline Expected Profile &
Rayleigh limit law & Gaussian limit law (\ref{Ecp-LLT})
\\ \hline
\end{tabular}
\end{center}
\caption{\emph{The contrast of some properties between
random simply-generated $k$-trees and
random increasing $k$-trees. Here $\mathcal{Z}$ denotes a node
and $\mathcal{Z}^\square$ means a marked node.}} \label{tb1}
\end{table}
It is known that random trees in the family of increasing trees are
often less skewed, less slanted in shape, a typical description
being the logarithmic order for the distance of two randomly chosen
nodes; this is in sharp contrast to the square-root order for random
trees belonging to the simply-generated family; see for
example~\cite{bergeron_varieties_1992,drmota_random_2009,
fuchs_profiles_2006,marckert_families_2008,meir_altitude_1978}. Such
a contrast has inspired and stimulated much recent research. Indeed,
the majority of random trees in the literature of discrete
probability, analysis of algorithms, and random combinatorial
structures are either $\log n$-trees or $\sqrt{n}$-trees, $n$ being
the tree size. While the class of $\sqrt{n}$-trees have been
extensively investigated by probabilists and combinatorialists,
$\log n$-trees are comparatively less addressed, partly because most
of them were encountered not in probability or in combinatorics, but
in the analysis of algorithms.
Table~\ref{tb1} presents a comparison of the two models: the classes
${\mathcal{T}}_s$ and $\mathcal{T}$, corresponding respectively to
simply-generated $k$-trees and increasing $k$-trees. The results
concerning simple $k$-trees are given
in~\cite{darrasse_limiting_2009, darrasse_unifying_????}, and those
concerning increasing $k$-trees are derived in this paper (except
for the power law distribution~\cite{gao_degree_2009}). We start
with the specification, described in terms of operators of the
symbolic method~\cite{flajolet_analytic_2009}. A structure of
${\mathcal{T}}_s$ is a set of $k$ structures of the same type, whose
roots are attached to a new node: $\mathcal{T}_s =
\mbox{Set}(\mathcal{Z}\times\mathcal{T}_s^k)$, while a structure of
${\mathcal{T}}$ is an increasing structure, in the sense that the
new nodes get labels that are smaller than those of the underlying
structure (this constraint is reflected by the box-operator)
$\mathcal{T} = \mbox{Set} (\mathcal{Z}^\square\times\mathcal{T}^k)$.
The analytic difference immediately appears in the enumerative
generating functions that translate the specifications: the
simply-generated structures are defined by $T_s(z)= \exp(z
T_s^k(z))$ and corresponding increasing structures satisfy the
differential equation $ T'(z)= T^k(z)$. These equations lead to a
singular expansion of the square-root type in the simply-generated
model, and a singularity in $(1 - kz)^{-1/k}$ in the increasing
model. Similar analytic differences arise in the bivariate
generating functions of shape parameters.
The expected distance between two randomly chosen vertices or the
average path length is one of the most important shape measures in
modeling complex networks as it indicates roughly how efficient the
information can be transmitted through the network. Following the
same $\sqrt{n}$-vs-$\log n$ pattern, it is of order $\sqrt n$ in the
simply-generated model, but $\log n$ in the increasing model.
Another equally important parameter is the degree distribution of a
random vertex: its limiting distribution is a power law with
exponential tails in the simply-generated model of the form
$d^{-3/2} \rho_k^d$, in contrast to a power-law in the increasing
model of the form $d^{-1-k/(k-1)}$, $d$ denoting the
degree~\cite{gao_degree_2009}. As regards the degree of the root,
its asymptotic distribution remains the same as that of any vertex
in the simply-generated model, but in the increasing model, the
root-degree distribution is different, with an asymptotic stable law
(which is Rayleigh in the case $k=2$); see Theorem~\ref{thm-ld}.
Our main concern in this paper is the connectivity-profile.
Recall that the profile of an usual tree is the sequence
of numbers, each enumerating the total number of nodes with the same
distance to the root. For example, the tree
\includegraphics{tree}
has the profile $\{1,2,2,1,3\}$. Profiles represent one of the
richest shape measures and they convey much information regarding
particularly the silhouette. On random trees, they have been
extensively studied recently; see~\cite{chauvin_martingales_2005,
drmota_profile_1997,drmota_functional_2008,fuchs_profiles_2006,
hwang_profiles_2007,marckert_families_2008,park_profiles_2009}.
Since $k$-trees have many cycles for $k\ge2$, we call the profile of
the transformed tree (see next section) \emph{the
connectivity-profile} as it measures to some extent the connectivity
of the graph. Indeed this connectivity-profile corresponds to the
profile of the ``shortest-path tree'' of a $k$-tree, as defined by
Proskurowski~\cite{proskurowski_k-trees:_1980}, which is nothing
more than the result of a Breadth First Search (BFS) on the graph.
Moreover, in the domain of complex networks, this kind of BFS trees
is an important object; for example, it describes the results of the
\texttt{traceroute} measuring
tool~\cite{stevens_chapter_1994,viger_detection_2008} in the study
of the topology of the Internet.
We will derive precise asymptotic approximations to the expected
connectivity-profile of random increasing $k$-trees, the major tools
used being based on the resolution of a system of differential
equations of Cauchy-Euler type (see~\cite{chern_asymptotic_2002}).
In particular, the expected number of nodes at distance $d$ from the
root follows asymptotically a Gaussian distribution, in contrast to
the Rayleigh limit distribution in the case of simply-generated
$k$-trees. Also the limit distribution of the number of nodes with
distance $d$ to the root will be derived when $d$ is bounded. Note
that when $d=1$, the number of nodes at distance $1$ to the root is
nothing but the degree of the root.
This paper is organized as follows. We first present the definition
and combinatorial specification of random increasing $k$-trees in
Section~\ref{sec:def}, together with the enumerative generating
functions, on which our analytic tools will be based. We then
present two asymptotic approximations to the expected
connectivity-profile in Section~\ref{sec:profile}, one for $d=o(\log
n)$ and the other for $d\to\infty$ and $d=O(\log n)$. Interesting
consequences of our results will also be given. The limit
distribution of the connectivity-profile in the range when $d=O(1)$
is then given in Section~\ref{sec:ld}.
\section{Random increasing $k$-trees and generating functions}
\label{sec:def}
Since $k$-trees are graphs full of cycles and cliques, the key step
in our analytic-combinatorial approach is to introduce a bijection
between $k$-trees and a suitably defined class of trees (\emph{bona
fide} trees!) for which generating functions can be derived. This
approach was successfully applied to simply-generated family of
$k$-trees in~\cite{darrasse_limiting_2009}, which leads to a system
of algebraic equations. The bijection argument used there can be
adapted \emph{mutatis mutandis} here for increasing $k$-trees, which
then yields a system of differential equations through the bijection
with a class of increasing trees~\cite{bergeron_varieties_1992}.
\begin{figure}
\begin{center}
\includegraphics{figure-bij}
\end{center}
\caption{\emph{A $2$-tree (left) and its corresponding increasing
tree representation (right).}}\label{fig:bij}
\end{figure}
\paragraph{Increasing $k$-trees and the bijection.}
Recall that a $k$-clique is a set of $k$ mutually adjacent vertices.
\begin{definition}
An increasing $k$-tree is defined recursively as follows. A
$k$-clique in which each vertex gets a distinct label from
$\{1,\dots,k\}$ is an increasing $k$-tree of $k$ vertices. An
increasing $k$-tree with $n > k$ vertices is constructed from an
increasing $k$-tree with $n-1$ vertices by adding a vertex labeled
$n$ and by connecting it by an edge to each of the $k$ vertices in
an existing $k$-clique.
\end{definition}
By \emph{random increasing $k$ trees}, we assume that all existing
$k$-cliques are equally likely each time a new vertex is being
added. One sees immediately that the number $T_n$ of increasing
$k$-trees of $n+k$ nodes is given by $T_n = \prod_{0\le i<n}(ik+1)$.
Note that if we allow any permutation on all labels, we obtain the
class of simply-generated $k$-trees where monotonicity of labels
along paths fails in general.
Combinatorially, simply-generated $k$-trees are in
bijection~\cite{darrasse_limiting_2009} with the family of trees
specified by $\mathcal{K}_s = \mathcal{Z}^k \times \mathcal{T}_s$,
where $\mathcal{T}_s = \mbox{Set}(\mathcal{Z} \times
\mathcal{T}_s^k)$. Given a rooted $k$-tree $G$ of $n$ vertices, we
can transform $G$ into a tree $T$, with the root node labeled
$\{1,\dots,k\}$, by the following procedure. First, associate a
white node to each $k$-clique of $G$ and a black node to each
$(k+1)$-clique of $G$. Then add a link between each black node and
all white nodes associated to the $k$-cliques it contains. Each
black node is labeled with the only vertex not appearing in one of
the black nodes above it or in the root. The last step in order to
complete the bijection is to order the $k$ vertices of the root and
propagate this order to the $k$ sons of each black node. This
constructs a tree from a $k$-tree (see Figure~\ref{fig:bij});
conversely, we can obtain the $k$-tree through a simple traversal of
the tree.
Such a bijection translates directly to increasing $k$-trees by
restricting the class of corresponding trees to those respecting a
monotonicity constraint on the labels, namely, on any path from the
root to a leaf the labels are in increasing order. This yields the
combinatorial specification of the class of increasing trees
$\mathcal{T} = \mbox{Set}(\mathcal{Z}^\square \times
\mathcal{T}^k)$. An increasing $k$-tree is just a tree in
$\mathcal{T}$ together with the sequence $\{1,\dots,k\}$
corresponding to the labels of the root-clique\footnote{We call
\textit{root-clique} the clique composed by the $k$ vertices
$(1,\ldots,k)$. The increasing nature of the $k$-trees guarantees
that these vertices always form a clique. We call
\textit{root-vertex} the vertex with label $1$.}. A tree in
$\mathcal{K}$ is thus completely determined by its $\mathcal{T}$
component, giving $\mathcal{K}_{n+k} \equiv \mathcal{T}_n$. For
example figure~\ref{fig:bij} shows a $2$-tree with $19$ vertices and
its tree representation with $17$ black nodes. In the rest of this
paper we will thus focus on class $\mathcal{T}$.
\paragraph{Generating functions.}
Following the bijection, we see that the complicated dependence
structure of $k$-trees is now completely described by the class of
increasing trees specified by $\mathcal{T} =
\mbox{Set}(\mathcal{Z}^\square \times \mathcal{T}^k)$. For example,
let $T(z) := \sum_{n\ge0} T_n z^n/n!$ denote the exponential
generating function of the number $T_n$ of increasing $k$-trees of
$n+k$ vertices. Then the specification translates into the equation
\[
T(z) = \exp\left(\int_0^z T^k(x) \dd x\right),
\]
or, equivalently, $T'(z) = T^{k+1}(z)$ with $T(0)=1$, which is
solved to be
\[
T(z) = (1-kz)^{-1/k},
\]
we then check that $T_n = \prod_{0\le i<n}(ik+1)$.
If we mark the number of neighbors of the root-node in $\mathcal{T}$
by $u$, we obtain
\[
T(z,u) = \exp\left(u \int_0^z T(x) T^{k-1}(x,u) \dd x\right),
\]
where the coefficients $n![u^\ell z^n] T(z,u)$ denote the number of
increasing $k$-trees of size $n+k$ with root degree equal to
$k+\ell-1$. Taking derivative with respect to $z$ on both sides and
then solving the equation, we get the closed-form expression
\begin{align} \label{F-sol}
T(z,u) = \left(1-u(1-(1-kz)^{1-1/k})\right)^{-1/(k-1)}.
\end{align}
Since $k$-trees can be transformed into ordinary increasing trees,
the profiles of the transformed trees can be naturally defined,
although they do not correspond to simple parameters on $k$-trees.
While the study of profiles may then seem artificial, the results do
provide more insight on the structure of random $k$-trees. Roughly,
we expect that all vertices on $k$-trees are close, one at most of
logarithmic order away from the other. The fine results we derive
provide in particular an upper bound for that.
Let $X_{n;d,j}$ denote the number of nodes at distance $d$ from $j$
vertices of the root-clique in a random $k$-tree of $n+k$ vertices.
Let $T_{d,j}(z,u)= \sum_{n\ge0} T_n \mathbb{E}(u^{X_{n;d,j}})
z^n/n!$ denote the corresponding bivariate generating function.
\begin{theorem} The generating functions $T_{d,j}$'s satisfy the
differential equations
\begin{equation} \label{bgf-Tdj}
\frac{\partial}{\partial z} T_{d,j}(z,u) =
u^{\delta_{d,1}}T_{d,j-1}^j(z,u) T_{d,j}^{k-j+1}(z,u),
\end{equation}
with the initial conditions $T_{d,j}(0,u)=1$ for $1\le j\le k$,
where $\delta_{a,b}$ denotes the Kronecker function,
$T_{0,k}(z,u)=T(z)$ and $T_{d,0}(z,u) = T_{d-1,k}(z,u)$.
\end{theorem}
\begin{proof}\ The theorem follows from
\begin{equation*}
T_{d,j}(z,u) = \exp\left(u^{\delta_{d,1}}
\int_0^z T_{d,j-1}^j(x,u) T_{d,j}^{k-j}(x,u)\dd x\right),
\end{equation*}
with $T_{d,j}(z,1) = T(z)$.
\end{proof}
For operational convenience, we normalize all $z$ by $z/k$ and write
$\tilde{T}(z) := T(z/k) = (1-z)^{-1/k}$. Similarly, we define
$\tilde{T}_{d,j}(z,u) := T_{d,j}(z/k,u)$ and have, by
(\ref{bgf-Tdj}),
\begin{align} \label{Tdj}
\frac{\partial}{\partial z} \tilde{T}_{d,j}(z,u)
=\frac{u^{\delta_{d,1}}}{k}\tilde{T}_{d,j-1}^j(z,u)
\tilde{T}_{d,j}^{k-j+1}(z,u),
\end{align}
with $\tilde{T}_{d,j}(1,z) = \tilde{T}(z)$,
$\tilde{T}_{0,k}(z,u)=\tilde{T}(z)$ and $\tilde{T}_{d,0}(z,u) =
\tilde{T}_{d-1,k}(z,u)$.
\section{Expected connectivity-profile}
\label{sec:profile}
We consider the expected
connectivity-profile $\mathbb{E}(X_{n;d,j})$ in this section.
Observe first that
\[
\mathbb{E}(X_{n;d,j}) = \frac{k^n[z^n]\tilde{M}_{d,j}(z)}
{T_n} ,
\]
where $\tilde{M}_{d,j}(z) := \partial \tilde{T}_{d,j}(z,u)/(\partial
u)|_{u=1}$. It follows from (\ref{Tdj}) that
\begin{align}\label{Mdj}
\tilde{M}_{d,j}'(z) = \frac1{k(1-z)}
\left((k-j+1)\tilde{M}_{d,j}(z)
+j\tilde{M}_{d,j-1}(z) + \delta_{d,1}\tilde{T}(z)\right).
\end{align}
This is a standard differential equation of Cauchy-Euler type whose
solution is given by (see~\cite{chern_asymptotic_2002})
\[
\tilde{M}_{d,j}(z) = \frac{(1-z)^{-(k-j+1)/k}}k
\int_0^z (1-x)^{-(j-1)/k}\left( j\tilde{M}_{d,j-1}(x)
+\delta_{d,1}\tilde{T}(x) \right) \dd x,
\]
since $\tilde{M}_{d,j}(0)=0$. Then, starting from
$\tilde{M}_{0,k}=0$, we get
\begin{align*}
\tilde{M}_{1,1}(z)= \frac1{k-1}\left(\frac1{1-z}
- \frac1{(1-z)^{1/k}}\right)
= \frac{\tilde{T}^k(z) - \tilde{T}(z)}{k-1}.
\end{align*}
Then by induction, we get
\[
\tilde{M}_{d,j}(z)
\sim \frac{j}{(k-1)(d-1)!}\cdot\frac{1}{1-z}
\log^{d-1}\frac1{1-z} \qquad(1\le j\le k;d\ge1; z\sim1).
\]
So we expect, by singularity analysis, that
\[
\mathbb{E}(X_{n;d,j}) \sim \Gamma(1/k)\frac{j}{k-1}
\cdot \frac{(\log n)^{d-1}}{(d-1)!}\,n^{1-1/k},
\]
for large $n$ and fixed $d$, $k$ and $1\le j\le k$. We can indeed
prove that the same asymptotic estimate holds in a larger range.
\begin{theorem} \label{thm-E}
The expected connectivity-profile $\mathbb{E}(X_{n;d,j})$
satisfies for $1\le d= o(\log n)$
\begin{align}\label{Ecp-1}
\mathbb{E}(X_{n;d,j}) \sim \Gamma(1/k)\frac{j}{k-1}
\cdot \frac{(\log n)^{d-1}}{(d-1)!}\,n^{1-1/k},
\end{align}
uniformly in $d$, and for $d\to\infty$, $d=O(\log n)$,
\begin{align}\label{Ecp2}
\mathbb{E}(X_{n;d,j}) \sim
\frac{\Gamma(1/k)h_{j,1}(\rho)
\rho^{-d} n^{\lambda_1(\rho)-1/k}}
{\Gamma(\lambda_1(\rho))\sqrt{2\pi(\rho\lambda_1'(\rho)
+\rho^2\lambda_1''(\rho))\log n}}
\end{align}
where $\rho=\rho_{n,d}>0$ solves the equation $\rho\lambda_1'(\rho)=
d/\log n$, $\lambda_1(w)$ being the largest zero (in real part) of
the equation $\prod_{1\le \ell\le k}(\theta-\ell/k)- k! w/k^k=0$ and
satisfies $\lambda_1(1) =(k+1)/k$.
\end{theorem}
An explicit expression for the $h_{j,1}$'s is given as follows. Let
$\lambda_1(w),\ldots,\lambda_k(w)$ denote the zeros of the equation
$\prod_{1\le \ell\le k}(\theta-\ell/k)- k! w/k^k=0$. Then for $1\le
j\le k$
\begin{align}\label{hj1}
h_{j,1}(w) = \frac{j!w(w-1)}{(k\lambda_1(w)-1)\left(
\sum_{1\le s\le k}\frac1{k\lambda_1(w)-s}\right)
\prod_{k-j+1\le s\le k+1}(k\lambda_1(w)-s)}.
\end{align}
The theorem cannot be proved by the above inductive argument and our
method of proof consists of the following steps. First, the
bivariate generating functions $\mathscr{M}_j(z,w) := \sum_{d\ge1}
\tilde{M}_{d,j}(z) w^d$ satisfy the linear system
\[
\left((1-z)\frac{\mbox{d}}{\dd{z}}
- \frac{k-j+1}{k}\right)\mathscr{M}_j
= \frac{j}{k} \mathscr{M}_{j-1}
+ \frac{w\tilde{T}}{k}\qquad(1\le j\le k).
\]
Second, this system is solved and has the solutions
\[
\mathscr{M}_j(z,w) = \sum_{1\le j\le k}
h_{j,m}(w)(1-z)^{-\lambda_m(w)}
- \frac{w-(w-1)\delta_{k,j}}{k}\,\tilde{T}(z),
\]
where the $h_{j,m}$ have the same expression as $h_{j,1}$ but with
all $\lambda_1(w)$ in (\ref{hj1}) replaced by $\lambda_m(w)$. While
the form of the solution is well anticipated, the hard part is the
calculations of the coefficient-functions $h_{j,m}$. Third, by
singularity analysis and a delicate study of the zeros, we then
conclude, by saddle-point method, the estimates given in the
theorem.
\begin{corollary} The expected degree of the root
$\mathbb{E}(X_{n,1,j})$ satisfies
\[
\mathbb{E}(X_{n,1,j}) \sim \Gamma(1/k)\frac{j}{k-1}
\,n^{1-1/k}\qquad(1\le j\le k).
\]
\end{corollary}
This estimate also follows easily from (\ref{F-sol}).
Let $H_k := \sum_{1\le\ell\le k} 1/\ell$ denote the harmonic
numbers and $H_k^{(2)} := \sum_{1\le\ell\le k} 1/\ell^2$.
\begin{corollary} The expected number of nodes at distance
$d= \left\lfloor \frac{1}{kH_k}\log n + x\sigma\sqrt{\log
n}\right\rfloor$ from the root, where $\sigma =
\sqrt{H_k^{(2)}/(kH_k^3)}$, satisfies, uniformly for $x=o((\log
n)^{1/6})$,
\begin{align}\label{Ecp-LLT}
\mathbb{E}(X_{n;d,j})\sim
\frac{n e^{-x^2/2}}{\sqrt{2\pi\sigma^2\log n}}.
\end{align}
\end{corollary}
This Gaussian approximation justifies the last item corresponding to
increasing trees in Table~\ref{tb1}.
Note that $\lambda_1(1)=(k+1)/k$ and $\alpha = d/\log n \sim
1/(kH_k)$. In this case, $\rho=1$ and
\[
\rho\lambda_1'(\rho) = \frac1{\sum_{1\le \ell\le k}
\frac{1}{\lambda_1(\rho)-\frac \ell k}},
\]
which implies that $\lambda_1(\rho)-1/k -\alpha\log \rho \sim 1$.
\begin{corollary} \label{cor-height}
Let $\mathscr{H}_{n;d,j} := \max_d X_{n;d,j}$ denote the
height of a random increasing $k$-tree of $n+k$ vertices. Then
\[
\mathbb{E}(\mathscr{H}_n) \le \alpha_+\log n -
\frac{\alpha_+}{2(\lambda_1(\alpha_+)-\frac1k)}\log\log n
+ O(1),
\]
where $\alpha_+>0$ is the solution of the system of equations
\[
\left\{\begin{array}{l}
\displaystyle\frac{1}{\alpha_+} = \sum_{1\le \ell\le k}
\frac1{v-\frac \ell k},\\ \displaystyle
v-\frac1k-\alpha_+\sum_{1\le \ell \le k}
\log\left(\frac{k}{\ell} v-1\right) = 0.
\end{array}\right.
\]
\end{corollary}
Table~\ref{tab1} gives the numerical values of $\alpha_+$ for small
values of $k$.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|}\hline
$k$ & $2$ & $3$ & $4$ & $5$ & $6$ \\ \hline
$\alpha_+$ & $1.085480$ & $0.656285$ & $0.465190$ &
$0.358501$ & $0.290847$\\ \hline\hline
$k$ & $7$ & $8$ & $9$ & $10$ & $20$ \\ \hline
$\alpha_+$ & $0.244288$ & $0.210365$ & $0.184587$ & $0.164356$ &
$0.077875$\\ \hline
\end{tabular}
\end{center}
\caption{\emph{Approximate numerical values of $\alpha_+$.}}
\label{tab1}
\end{table}
For large $k$, one can show that $\alpha_+\sim 1/(k\log 2)$ and
$\lambda_1(\alpha_+) \sim 2$.
Corollary~\ref{cor-height} justifies that the mean distance of
random $k$-trees are of logarithmic order in size, as stated in
Table~\ref{tb1}.
\begin{corollary} The width $\mathscr{W}_{n;d,j} := \max_d X_{n;d,j}$
is bounded below by
\[
\mathbb{E}(\mathscr{W}_n) =
\mathbb{E}(\max_d X_{n,d}) \ge \max_d \mathbb{E}(X_{n,d})
\asymp \frac{n}{\sqrt{\log n}}.
\]
\end{corollary}
We may conclude briefly from all these results that \emph{in the
transformed increasing trees of random increasing $k$-trees, almost
all nodes are located in the levels with $d= \frac{1}{kH_k}\log n +
O(\sqrt{\log n})$, each with $n/\sqrt{\log n}$ nodes.}
\section{Limiting distributions}
\label{sec:ld}
With the availability of the bivariate generating functions
(\ref{bgf-Tdj}), we can proceed further and derive the limit
distribution of $X_{n;d,j}$ in the range where $d=O(1)$. The case
when $d\to\infty$ is much more involved; we content ourselves in
this extended abstract with the statement of the result for bounded
$d$.
\begin{theorem} \label{thm-ld}
The random variables $X_{n;d,j}$, when normalized by their mean
orders, converge in distribution to
\begin{align}\label{Xndj-cid}
\frac{X_{n;d,j}}{n^{1-1/k}(\log n)^{d-1}/(d-1)!}
\stackrel{d}{\to}\Xi_{d,j},
\end{align}
where
\begin{align*}
\mathbb{E}(e^{\Xi_{d,j} u}) &=\Gamma(\tfrac1k)
\sum_{m\ge0} \frac{c_{d,j,m}}{m!\Gamma(m(1-1/k)+1/k)}\,u^m\\
&= \frac{\Gamma(\frac1k)}{2\pi i}\int_{-\infty}^{(0+)} e^\tau
\tau^{-1/k}C_{d,j}\left(\tau^{-1+1/k} u\right) \dd \tau,
\end{align*}
and $C_{d,j}(u) :=1+ \sum_{m\ge1} c_{d,j,m} u^m/m!$ satisfies the
system of differential equations
\begin{align} \label{Cdju}
(k-1)uC_{d,j}'(u) + C_{d,j}(u) = C_{d,j}(u)^{k+1-j}
C_{d,j-1}(u)^j\qquad(1\le j\le k),
\end{align}
with $C_{d,0}=C_{d-1,k}$. Here the symbol $\int_{-\infty}^{(0+)}$
denotes any Hankel contour starting from $-\infty$ on the real axis,
encircling the origin once counter-clockwise, and returning to
$-\infty$.
\end{theorem}
We indeed prove the convergence of all moments, which is stronger
than weak convergence; also the limit law is uniquely determined by
its moment sequence.
So far only in special cases do we have explicit solution for
$C_{1,j}$: $C_{1,1}(u) = (1+u)^{-1/(k-1)}$ and
\[
C_{1,2}(u) = \left\{\begin{array}{ll}
\frac{e^{1/(1+u)}}{1+u},&\text{if } k=2;\\
\frac1{1+u^{1/2}\arctan(u^{1/2})},&
\text{if } k=3.
\end{array}\right.
\]
Note that the result (\ref{Xndj-cid}) when $d=0$ can also be derived
directly by the explicit expression (\ref{F-sol}). In particular,
when $k=2$, the limit law is Rayleigh.
\bibliographystyle{plain}
\bibliography{increasing_ktrees}
\end{document}
| 93,420
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Steven Staples to Finance Committee: “Canada’s future is in Bombardier’s passenger jets, not Lockheed Martin’s fighter jets”
The Rideau Institute presented pre-budget consultation recommendations to the House of Commons Finance Committe on October 24, 2012. The full minutes and transcripts of the committee meeting are available online.
Steven Staples (President, Rideau Institute):
Good afternoon, honourable members of the committee and guests. Thank you for inviting me back to present our pre-budget consultation recommendations. My name is Steve Staples. I am the president of the Rideau Institute.
The Rideau Institute is a non-profit, non-partisan research, advocacy, and consulting group founded in 2006 with an expertise in Canadian defence and foreign policy, and we do not receive funding from the government or from firms. We are funded from donations from individuals and from consulting services we provide to non-profits and trade unions. I, myself, have been working in the field of defence spending and disarmament for about 20 years.
Our pre-budget submission has three main points for the government: first, to further reduce defence expenditures with a goal of returning to pre-9/11/2001 levels over time; second, to improve parliamentary and therefore public oversight of the military procurement processes; and third, to invest in Canadian industry by providing targeted support to areas of the economy where Canada is a world leader.
Defence spending today in Canada, even with the modest reductions made recently, remains at historically high levels. The dramatic buildup of defence budgets in the last decade, sometimes exceeding 10% a year, has left defence spending roughly 40% higher than before the 9/11 terrorist attacks. I’ve provided a chart from 2011 that documents in adjusted dollars where the budget is in real spending. We are sixth highest in NATO right now. In adjusted dollars, our defence spending is higher than at any point since the Cold War with the Soviet Union. I’ve also included that chart in the packages. However, the security situation has changed. We have a budget deficit and economic challenges at home. Our Afghanistan combat mission is over, and Osama bin Laden is dead.
Despite this, the government continues to commit to expensive procurement programs such as the F-35 and the shipbuilding program. It’s evident from the analysis provided by the Parliamentary Budget Officer and the Auditor General that the costs of these programs have not been properly assessed by the Department of National Defence, and are not likely affordable, even with the increased spending commitments within the Canada First defence strategy. The National Defence budget must be brought into line.
As well, improved parliamentary oversight over major crown projects would increase transparency in defence procurement and accountability of the government and military contractors. A dedicated parliamentary committee for major crown projects would provide Canadians with additional confidence that public dollars are being used wisely, fairly, and efficiently.
For instance, on the F-35 stealth fighters, the public has been greatly confused by statements made by the government on costs that are frankly unpredictable, contracts that don’t exist, and job opportunities that are little more than a hope and a prayer.
Establishing a parliamentary committee or a subcommittee responsible for major crown projects would help avoid the mistakes and complications that have arisen in many of the department’s projects and help cut through some of the spin from the defence contractors and ministers.
Finally, defence procurement strategies have not reflected the government’s goal of economic recovery and growth in key areas of the economy. The lack of a competitive process in sole-sourcing contracts has not guaranteed the industrial regional benefits that are essential for Canadian employment in these projects. For instance, there are no requirements for investment and job creation in the F-35 program, and I ask, what about the equally large shipbuilding program now? Where are the guarantees for jobs and investment there? Investments should be made where they stand the greatest chance of long-term benefit.
Media reports have suggested that the government is interested in a defence industrial strategy. This just doesn’t make sense. With defence budgets declining globally, respected financial analysts such as PwC point to the commercial market as the best bet. Canada’s future is in Bombardier’s passenger jets, not Lockheed Martin’s fighter jets.
Thank you, and I look forward to any questions you may have.
| 89,708
|
TITLE: How can entropy increase in a reversible process?
QUESTION [1 upvotes]: According to Wikipedia, a reversible process is "a process whose direction can be "reversed" by inducing infinitesimal changes to some property of the system via its surroundings, with no increase in entropy."
However, for isothermal processes, any reversible heat added to the system at constant temperature increases entropy. So, a reversible process of heating is causing a change in entropy. This seems to contradict the definition above.
Where have I gone wrong?
REPLY [0 votes]: When one says that entropy does not increase for a reversible process, the entropy they are talking about is the total entropy of an isolated system. If we consider the universe as our isolated system total entropy is equal to the entropy of system + the entropy of the surroundings.
So, for a reversible process where heat is added into the system, the entropy of the system does increase however the entropy of the surroundings in any reversible process = -(entropy of the system)and ultimately, the total entropy is zero.
| 183,145
|
This guide is intended as a reference in the management, use, and understanding of PST Flight Deck. This document was written for a technical audience whom are anticipating a base deployment of a PST Flight Deck environment. It is intended to provide an overview and high-level understanding of the product. Large, complex, or constrained migrations should engage a PST Flight Deck Architect to take the circumstances of your project into consideration.
Introduction
PST Flight Deck is a scalable enterprise capable solution designed to address the issue of decades of unmanaged PST file creation, utilization, and proliferation throughout an organization. The objective of this solution is to identify and migrate the content of Personal Storage Table (PST) files into a target environment in order to permit the content to be subject to regulatory, retention, and organizational requirements. Upon ingestion, it can also be used to eliminate PST files identified throughout the organization.
PST Files
PST files were initially introduced to help Administrators manage data on Exchange 4.0 mail servers. Their availability was expanded to support the use by Outlook to archive or store mail items locally. Expensive and limited mailbox store data on early versions of Exchange made PST files popular with organizations. PST files permitted the expanded use of a mail system without dramatically increasing the perceived cost, downtime, or expense of an Exchange environment. Mailbox quotas began to shrink to ensure the stability and availability of mail servers and PST files were now the target of the exponential growth in data use to facilitate the increased use of email systems within an organization.
The Problem
PST files are a problem within most organizations today. When introduced to users, there were few guidelines as to how or when to use them. Some users began to route all their Exchange mail directly to their local storage, removing the administration and defensible deletion of mail content. Other users took this as an opportunity to save and store everything; began abusing the intent of the system by using it more as a file server than a mail server. Departments began using the storage containers for projects or departmental data on network shares where they were unsupported and more prone to corruption. The data stored within the files frequently fell outside of organizational compliance, retention, and discovery requirements. Data began being lost to poor management practices from the users of these files, corruption, or hardware changes. This PST file life-cycle continued for over a decade in several organizations, resulting in an enormous body of data completely unmanaged, decentralized, and with limited accessibility to those needing it most.
As time progressed, Exchange was able to restructure its storage to accommodate less expensive disks and more redundancy. The server could now accommodate the level of storage being used at a reasonable price without the same concerns of older server versions. While in the wild, PST files had become a problem. A big problem. The longer the use of PST files was permitted, the harder the problem was to resolve. Until the recent versions of Outlook, users were still permitted to store all Exchange mailbox data directly to a PST file on their system and remove all their messages from the centrally managed Exchange servers.
Manual attempts to eliminate PST files by IT departments proved to require a high level of messaging expertise, be very costly, and have an ever-expanding project duration. When things went well, a migration project was time consuming, labour intensive, and was challenging to track the progress of. The limited visibility made it challenging to accurately communicate the expected experience to the users impacted and groups intended to support them. Help desks became flooded with frustrated users wanting access to their critical data and was unable to provide any insight or accurate time-lines as to when their data would become available to them again. Engineers had to request personal passwords, be challenged with old PST file formats, duplicate items within PST files, files that were erroneous duplicated, files that were made as a backup, and file corruption of the PST files. When things went poorly, the effort frequently just failed after much expense and little progress.
The PST Flight Deck Solution
PST Flight Deck is a feature-rich, enterprise grade solution designed to reduce the time and level of effort required migrate PST files to a desired target. PST Flight Deck is designed to minimize many of the issues that plague PST migration projects. PST Flight Deck is able to discover, associate, centralize, process, ingest, and eliminate PST files automatically.
With PST Flight Deck, modules are run prior to the ingestion that are designed to prepare a PST file for quick delivery to the target system. These “pre-flight checks” are initially configured, then executed on every PST file that runs through PST Flight Deck. This process reduces the effort of an overall PST file migration by automating many of the tasks that would be performed reactively by skilled professionals, whom likely have several other responsibilities.
Architecture
PST Flight Deck is built on a modular framework that permits a nearly infinite number of solution design options. The architectural components of PST Flight Deck can be summarized as those that are required for a functional environment and those that provide expanded and supplemental functionality to the solution.
Required Components
In its most basic configuration, PST Flight Deck requires at least one server, a SQL database, and Migration Agents deployed. The actual requirements for a given project are dependent on the scope and limitations of the project. For a properly scaled environment, please consult a qualified PST Flight Deck architect to review your environment and it’s needs as it pertains to a PST Flight Deck deployment.
Below is a diagram showing a high-level architecture of the required components for PST Flight Deck and how they interact in a production environment.
Core
The Core server is responsible for several aspects required for a functional PST Flight Deck system. In a typical configuration, the Core server hosts the PST Flight Deck Core service, the Internet Information Services (IIS) instance hosting PST Flight Deck files, the Background Intelligent Transfer Service (BITS) upload directory, and any number of modules.
The Core service is responsible for communication with the PST Flight Deck system database and execution of scheduled operations. All deployed agents also check-in and update the system through the Core server. Essentially, the Core service acts as the conductor of the entire solution.
A typical deployment of PST Flight Deck begins with the installation of a Core server. In most environments, more than one PST Flight Deck server is used, however for non-production installs it is possible to install all the components on a single server. Supplemental components may also be installed on the Core server, either from the Core installer itself or independent installers.
Modules
In PST Flight Deck, a module refers to one of the components of the solution that run against successfully uploaded user data to a defined and centralized location. It is possible to categorize the PST Flight Deck modules in three categories: pre-ingestion, ingestion, and post-ingestion modules. Although workflow order is configurable, the following categorization is based on a default installation with no modification in the workflow’s order.
Pre-ingestion
Ingestion modules are typically chosen at the point of installation. These are the modules responsible for the PST Flight Deck, PST Flight Deck considers the file as “completed.”
Rate of ingestion is frequently a point of bottleneck in a migration and can be monitored in the Management Console, Admin Console, or via the PST Flight Deck Portal. Tuning an ingestion for performance requires the PST Flight Deck.
Post-ingestion
PST Flight Deck Agents
PST Flight Deck.
Supplemental Functionality
PST Flight Deck has several features that are not required for a basic configuration but provide an enhanced, more robust, or more customizable experience. The features that are required for any given deployment will vary wildly and it is encouraged to discuss your needs with a qualified PST Flight Deck Architect to determine which components are appropriate for you.
Custom Configuration
PST Flight Deck permits the flexibility to support any size of organization. Smaller organizations may be well suited in utilizing a single configuration for all its users. A larger enterprise often has several conditions which make a “one configuration fits all” approach impractical. PST Flight Deck is able to support the smallest branch offices while managing multiple large campuses within a single environment. To begin to support an enterprise’s level of complexity, PST Flight Deck permits the customization of configuration settings within its environment via profile and location designation.
Profiles
Profiles are the initial building block of the ability for a PST Flight Deck to have the level of customizable configuration needed to support today’s enterprise. Profiles are used to permit an alternate configuration for users assigned to them. Examples where profiles may be of use are:
- Support complex network topology and limitations
- Support multiple locations
- Support site specific uploads and processing
- Support alternate configurations for different user types (e.g. Executives, road warriors, factory employees)
Profiles enable the assignment of unique File Scanner, Agent, Language, and Location configurations.
Locations
Locations permit work to be assigned into geographically appropriate groupings without the overhead of deployment and management of independent migration environments. This enables an organization to centralize and process PST files in data centres close to the users and their data. Enabling the ability to support multiple locations can reduce overall environmental network consumption and the time for project completion.
Scalability
PST Flight Deck permits the ability to scale out an environment to offload the burden on any given system in the environment. For small and mid-sized projects with no special requirements, a standard deployment may be sufficient. For larger environments or projects with tight time requirements, processing modules may be distributed over several machines to scale the system to match the environment’s needs or reduce bottlenecks in the workflow. These computers with module components of PST Flight Deck installed are referred to as “nodes”.
When considering the scalability of your PST Flight Deck environment there are many factors to consider. Please consult a qualified PST Flight Deck Architect prior to scaling out an environment.
Node Functions
Part of the flexibility of PST Flight Deck is the modular approach that is taken within its architecture. The functions required to perform a migration can be run over multiple nodes. This permits many of the same modules to be run in parallel or assigned to specific locations.
Module Node
Module Nodes are servers with PST Flight Deck modules installed on them. They encompass all PST Flight Deck servers containing modules and not included in another node type. Module nodes can be used to scale processing of any module workload to permit rapid processing of items uploaded.
Most Module Nodes do not have any special considerations outside of resource availability and the ability to communicate to the Core. Extraction and repair nodes have high disk requirements and are typically located on the machine with the highspeed disks used for the BITS upload directory.
Ingest Node
Ingestion Nodes are Module Nodes that include an Ingestion module. They are typically deployed to support geographically distributed resource allocation within PST Flight Deck. Multiple ingestion nodes may also be leveraged to expedite the timeline to project completion when appropriate.
Azure Node
An Azure Node is a special node used in migrations to cloud based targets. Azure nodes can be used in a migration by itself or in support of geographically centric resource allocation for a multi-site organization. Azure Nodes are nodes deployed to Microsoft’s Azure service to act as the intended centralization target, processing, and ingestion nodes for a configured location. Azure Nodes can be used to minimize the footprint required to scale an environment out or to enable small businesses to leverage PST Flight Deck by reducing the initial hardware requirements of the solution.
Content Scanning
One of the largest challenges in PST migration projects is accurately determining ownership of a PST file. Most PST files have ownership identified within the Discovery process for PST Flight Deck. PST files identified by a Share Scanner, or by a user as not belonging to them, require some special analysis. PST Flight Deck’s Content Scanner is used to scan PST files marked as “Ownerless,” or those files located on specific fileservers. It can provide more information to help determine ownership association based on the contents of the file. Files processed by the Content Scanner can have the most common sender and most common recipient recorded and taken into consideration when determining PST file ownership. This attribute is also evaluated against a user’s proxy addresses to ensure account migration history is taken into consideration when evaluating ownership.
Centralized Uploads
The Central Upload Agent (CUA) is a component frequently utilized in a PST Flight Deck environment. The primary role of the CUA is to upload files stored on centralized repositories more rapidly without having to use a workstation’s Migration Agent.
In a traditional migration situation without a CUA, remotely stored PST files would be queued for BITS transfer using a user’s Migration Agent to facilitate the upload request.
When a CUA is deployed, and the environment is appropriately configured, files on remote file servers are uploaded by a nearby CUA rather than connecting back through the workstation.
Forced Migrations
If an environment has a CUA installed, configured, and running with sufficient permissions, it can be used to facilitate Forced Migration requests. A Forced Migration utilizes the CUA to connect to a user’s local machine and copy the eligible PST file over the administrative share on their local system. This enables a user to have their files uploaded without having to go through BITS or even a Migration Agent. This feature is useful for Migration Agents struggling to complete an upload or as a means to migrate users who are rarely connected to the domain.
Support for Application Streaming
The PST Flight Deck Application Streaming Wrapper (ASW) is designed for environments that utilize virtual application streaming to provide Outlook to users. Due to the way application streaming is performed, a typical agent approach cannot be used.
The ASW works as a wrapper for Outlook on an application streaming solution. Upon launch, it checks to see if a user is enabled for migration. If they are enabled, it quickly closes open connections to PST files associated with the user’s Outlook profile during that streaming session. Once that has completed, it launches Outlook and maintains the standard user experience without any of the PST files connected. There is no interaction required from users streaming Outlook but their experience will be impacted as PST files will no longer be attached to their streamed Outlook instance. The ASW is frequently used in conjunction with a Share Scanner and Central Upload Agent to enable discovered files to be uploaded from shared resources without the use of a traditional Migration Agent.
Remote Management
A PST Flight Deck environment can be globally distributed and centrally managed by a single Core server. Administration and operation of the environment can be performed remotely. The PST Flight Deck Management Console is available to those needing the ability to manage a PST Flight Deck migration without the need to edit the settings of the migration solution. The ability to install a console on an Operator’s workstation enables them to perform their duties without the need to grant remote access to the server hosting. The core services and Administration Console.
Interfaces
PST Flight Deck has a choice of user interfaces to interact with the product. To some extent the interface utilized depends on the role being performed. PST Flight Deck is scalable not just in processing power, but also in its ability to adapt to any size of team associated with the migration project.
Consoles
PST Flight Deck consoles are applications that run from the server or an Administrator, Architect, or Operator’s workstation. They permit the most control over a PST Flight Deck environment. There are two consoles available: The PST Flight Deck Administrator Console and the PST Flight Deck Management Console.
The PST Flight Deck Administrator Console (Admin Console) provides the most access to configure a PST Flight Deck.
The PST Flight Deck Management Console .
With the exception of the above, both interfaces are the same. The remainder of this section will refer to both as the Console.
Navigation
In general, navigation within the Console is similar no matter where you are in the Console. The core of a Console has a Ribbon Panel, Navigation panel, and the body of the console, where the selected data is displayed.
The top of the Console contains several features and is seen below:
The left side of the Console is used for navigation. The following is an example of the options available within the navigation pane of the console:
The navigation pane has menus at the bottom that act as parent containers for related options within. Selecting one of these menus will expose relevant sub-menus in the navigation pane above. The menus available are Manage, Reports, and Settings.
Managing Console Data
The Consoles have a powerful interface that permits the filtering of live data to return data you are seeking. Many areas of PST Flight Deck return a lot of data that can be too much to be useful without the ability to refine the results. The information below discusses several approaches to managing the data being returned within the grid of several options within the Console.
Columns
Additional data columns can be added to many of the grids in PST Flight Deck.
To access the additional columns of data:
- Right-click on an existing column heading
- Select “Column Chooser”
A list of available data columns will be shown. You can drag and drop from the list of columns on to the appropriate part of the user-list.
Sorting
The column headings can be clicked to change the sort order, a subsequent click on the same column heading will change the direction of the sort. Most columns support “drag and drop” functionality to change their position relative to the other data columns. This can make the data more presentable and readable. A data column can be removed if required by right-clicking on the column and choosing the “Remove this Column” option.
Filtering
It is possible to access the data filtering features in most of the Console’s grids in these ways:
- Right click on the appropriate column heading and choose the option “Filter Editor”
- Hover near the right hand corner of a column heading until the small filter icon becomes visible, and then click on it
- By default, the filter entry row is displayed underneath the column headings. A value can be entered into any of the columns in this row and the data list will automatically be filtered according to your input. Operations can be selected by clicking the small icon (filter hint). This will show this popup
Simply click on the desired operator and it will be entered.
After a filter has been applied, an option to “Edit Filter” is presented in the lower right corner of a console. This can be selected to access the advanced editor for the currently applied filter, which will permit you to perform very complex queries to return specific data within a Console.
Grouping
The list of users can be grouped by one or more columns as follows:
- Right click on an appropriate column heading and choose the option “Group By This Column”
- Right click on a column heading and choose the option “Show Group By Box”
Searching
Filtering can be used to reduce the overall list, but sometimes it is necessary to search for data within a given column. You can search for particular data within a grid by right-clicking on a column heading and selecting the option “Show Find Panel.” Previously used searches can be selected from the dropdown list.
Saving a Layout
If you identify a filter that is useful for your migration it is possible to save it for later use. To store the customized grid click on “Save” in the “Layout” ribbon. Each saved layout can be given a unique name. Previously saved layouts can be retrieved by clicking on “Load” in the “Layout” ribbon bar. The current data list can be reset to the defaults by clicking on “Reset” in the “Layout” ribbon bar. Saved layouts can be retrieved and executed by selecting “Load” on the PST Flight Deck “Layout” ribbon and selecting the name assigned to the saved layout. You can also select “Overwrite” to save over a pre-existing layout or “Delete” to remove the saved layout.
Manage
The “Manage” menu is the most commonly accessed area during a migration. Most of the functionality required for an Operator to manage the PST Flight Deck migration can be found in this section. Options are grouped into sub-menus and are described below:
Hotkeys
Some key combinations have been included to assist in the navigation of the Manage section of the console. The areas affected are frequently navigated between during a migration project. Use of these hotkeys can enable quicker daily navigation.
Additionally, there are shortcut keys associated with specific popup dialog boxes.
Progress
This view provides an overview of the project as a whole. The following are its features:
Operations
A section designed to enable an Operator to quickly be able to tell the current health of a migration in a graphical display. The following are its features:
Resources
This section permits centralized observation of resource consumption of servers running Modules within a specified location. The focus of this is to provide a point in time review of resource consumption and not a comprehensive performance monitor for the PST Flight Deck environment. Displayed is graphical output for CPU, RAM, and disk usage.
Users
This section of the Console is where most actions against users are taken. Assignment of user types, profiles, and priorities are all performed within this section. This sub-menu is where a lot of the migration monitoring and management takes place.
The grid returned under the “Users” sub-menu is comprised of two portions. Initially presented is a display of user information and general statistics about all the users in an AD environment. Each user who has discovered files can be expanded by clicking the plus sign (+) at the left of its row. The resultant sub-table has the same characteristics as the “Files” section and will be described in detail in a succeeding section of this guide. The ribbon and right-click options change depending on which section your selection is located.
The home ribbon when “Users” is selected has options impacting the selected user (with one exception). These options and a description of their functionality are as follows:
The “Action” menu has several options for selection. As with the home ribbon under the “Users” sub-menu, these options are related to the presently selected object in the grid, and changes depending on if that selection is one or many files or users. These options are also available when right-clicking on selection within the grid. The following is a list and description of the options available under the “Action” menu when a user is the selected object:
The type of user designated by the “Change User Type” option can dictate the way the features of the product affect a given user. Below is a list of available types and their description:
Files
The “Files” section of the console is also very commonly accessed while managing a migration. Like the “Users” section, it provides a lot of functionality to determine status and granular details of a migration. Actions can be executed on specific files when appropriate. The options available in the “Files” sub-menu’s home ribbon are as follows:
The “Action” menu has several options for selection. These options are also available when right-clicking on a file within the grid. The following is a list and description of the options available:
Watched items
The Watched items section shows all files and users that have been flagged for watching. This provides a consolidate view of key items. The home ribbon for this sub-menu and each action menu changes depending what is selected, a user or a file. The ribbons and action menu options are the same as their respective
Events
The Events section shows all unacknowledged events that have occurred on the system since deployed. The default view of the Events grid shows the Type, Source, Event details, a count of the occurrences of the events, the level of severity of the event, and the last logged time for the event. The home ribbon for this sub-menu has the following options:
Owners
Managing owners is a critical part of a successful migration. The Owners section shows the results of our owner identification efforts; including owner conflicts, warning levels, all ownership related metadata, the probability that ownership has been identified, who a presently is the owner of a file, and who PST Flight Deck thinks should be the owner.
In some migrations, a lot of effort is required in the Owners section to ensure the data discovered is going where it is expected to go. Upon discovery, ownership management can begin and drastically reduce the level of effort required towards the middle and end of a migration. Owner management can also be delayed until later in the migration. If file ownership is not addressed early in the project, the files could be blocked from ingestion due to not meeting the minimum ownership probability threshold.
The options on the home ribbon for the Owners sub-menu are as follows:
Schedule groups:
- Optionally, the user has an Office 365 mailbox
- Optionally, the mailbox is licensed
- Optionally, size of discovered PST’s is smaller than the free space in the mailbox
- Optionally, uploads is not in a limited or stopped state.
Office 365 mailbox information is obtained from the Office 365 user synchronization task. The options on the home ribbon for the Schedule groups sub-menu are as follows:
Reports:
Exporting
Discovery collection is an optional feature that provides legal personnel the ability to select files to be collected for a case. PST files are collected with a six-step workflow.
- Case is defined with legal and technical owners.
- Select users the PSTs are to be collected from
- Select the search criteria for the files
- Confirm the files to be collected
- Files are copied to the uploads directory
- File are moved from the uploads directory to another repository
The options on the home ribbon for the Discovery Collection sub-menu are as follows:
Settings
The Settings section is not available in the Management Console and can only be accessed by the Admin Console. The options under the Settings section are mostly designed to be configured prior to the start of a production migration and left unchanged through the migration. Changing settings in this section after Migration Agents have been deployed can negatively impact your environment. It is important to understand the impact of any change you make. If you are unfamiliar with the impact a change may have, please contact support for assistance.
The options on the home ribbon for Environment are as follows:
Settings for the environment are grouped into several submenus, described as follows:
Environment
The Environment sub-menu contains global parameters for the migration. Some of these values can be adjusted by more granular settings within the product. The Environment sub-menu consists of five sections: General, Advanced, Ownership, Groups, and Roles. These values are discussed in greater detail in the followings sections of this documentation.
General
The General tab contains three configuration settings. Below is their description and if they are required in a migration:
Advanced
The Advanced tab contains general system configuration parameters required for a functional PST Flight Deck environment. The descriptions of the options available and the sections they are contained in are found below:
Shared scanner
The shared scanner tab provides for the management of multiple central file scanner servers. This allows from a centralized location for the management of the file scanner configurations and scheduling. The options on the home ribbon for the shared scanner sub-menu are as follows:
Ownership
The Ownership tab is used to define criteria for file ownership determination. There are two main components to the Ownership tab: Weight Table and User In Path.
The Weight Table defines the weights and threshold used in determining PST file ownership. The descriptions of the criteria available and their descriptions are seen below:
Computer Owner and Custom Owner Property are values that are able to be manually imported so they may be able to be taken into consideration in determining the ownership of a file. As every organization is different, the threshold and weight of the criteria are suggested to be tuned to meet the needs of an organization and project. Generally speaking, a file should require a minimum of three criteria matching a single user until ingesting into that user’s account.
The User in Path section of the tab allows creation of a Regular Expression used to identify a common location of user account names in discovered file paths. Default values are common. Identification of a user’s name in a path can be a good indication of file ownership.
Bandwidth limitations
The Bandwidth limitations tab provides for the management of upload bandwidth limitations, which are defined by IP address. If a workstation has an IP address within a range that has a limitation, the BITS bandwidth utilization is limited to the set bandwidth. The constraint can also be limited to certain times of the day. For example, an office may require low speed BITS uploads during the day but can have higher speeds at night. The options on the home ribbon for the Bandwidth limitations sub-menu are as follows:
Role Based Web Portal
Groups and Roles enable role based configuration for access to the PST Flight Deck Portal (Portal).
The Roles tab contains two main panes: the Role settings and Current Roles.
Current Roles shows a listing of the currently configured roles in a grid showing its name, the priority assigned to that role, and a description of the role. It includes the ability to edit or delete an existing role as well as to Add a new role.
The Role settings page is where a selected role is configured. Options to configure all that is displayed for the Current role section are available as well as a list of actions that can be associated with a Role. Each action corresponds to a section within the Portal. The Actions presently available, their corresponding section in the Portal, and a description are as follows:
Any number of Roles can be created to accommodate an organization’s needs.
Groups are used to associate AD groups to roles and locations. The Groups tab contains two main sections: Group settings and Current groups.
Current groups display a grid showing the currently configured AD Group Name, the Role type associated to that group, and a description for the entry. There are options Edit or delete an existing group, or to add a new group.
The Group Settings section of the tab permits you to configure the entry under Current groups. In addition to the values displayed in the Current groups pane, you can also associate a configured group with a location, and an email for notifications.
Changing the settings in any Role or Group requires the changes to be confirmed and the appropriate section to be saved. Settings will be applied after restarting IIS.
Console Access
The Admin console has two levels of access – Expert and non-expert. Users are given access via this feature.
Windows Migration Agent
The Windows Migration Agent sub-menu provides configuration options for the settings sent out to all the workstation’s running the Migration Agent for Windows in an environment. Configuration settings are able to be applied to all Windows based Agents or can be customized to suit the needs of an organization by creating several specialized configurations. There are three tabbed menus within the Migration Agent sub menu, all containing a drop down permitting the editing of specific configurations.
File Scanner
The File Scanner tab controls the behaviour of the Discovery Agent portion of the Migration Agent. This section is where you will configure the locations you wish an agent to scan, as well as what you want it to skip. The options available in this tab are as follows:
Additional configuration options are available when creating or editing a location for scanning. Scanning location specific folder and file exclusions are available to enable a granular level of exclusions. Options to scan removable media and substituted Paths are also available on a per location basis.
Agent
The Agent tab contains configuration options related to the Agent portion of Migration Agent running on Windows workstations. This component of the Migration Agent is responsible for querying the PST Flight Deck Core service and getting information back from it. It is also responsible for managing the user experience during a migration by providing pop-up menus to engage a user in their data’s migration. The Agent also reports attached PST files that are not in a configured scanning location to the server and disconnecting PST files from profiles when appropriate.
The “General” section of this tab permits you to choose if a migration will be performed in Snapshot or in Disconnect Mode. Also in this section is the configuration for the “Polling Interval”. This value should be set as is appropriate for the number of Migration Agents deployed in an environment. The larger number of agents reporting to a server, the longer a polling interval should be set to.
The “System Tray” section contains customization on how you would like the Agent to be represented on a user’s workstation. The options available for selection/deselection are as follows:
The Advanced section of the Agent tab has additional options governing some functionality within the Migration Agent. Some of these settings aid to manage the user experience during a migration. The options available under the advanced section and a description of their function are as follows:
The Duplicate Files section of the Agent tab has settings that control processing for duplicate files. Likely duplicate files are defined as files that have the same last modified date and the same name. Options available under the advanced section and a description of their function are as follows:
The Upload Limitations section of the Agent tab has settings that aid in determining if uploads are being attempted over a slow network connection, eg, a VPN. Options available under the advanced section and a description of their function are as follows:
Language
The Language tab permits customization of the language, contents, and options contained within the prompts displayed on user workstations. This permits better support for customers with multi-lingual or compartmentalized organizations. It is recommended to copy all contents in the edit window prior to changing the defaults.
Mac OS Migration Agent
The Mac OS Migration Agent sub-menu provides configuration options for the settings sent out to all the workstation’s running the Migration Agent for Mac in an environment. Configuration settings are able to be applied to all Mac based Agents or can be customized to suit the needs of an organization by creating several specialized configurations.
The options on the home ribbon for the Mac OS Migration Agent sub-menu are as follows:
Locations
The Locations sub-menu offers the ability to manage complicated migration environments. They permit work to be assigned into geographically appropriate groupings.
The options on the home ribbon for the Locations sub-menu are as follows:
Options available during Location creation or modification are as follows:
Each location can be configured to specify a peak and limited rate for Concurrent Uploads based on disk availability. These values are designed to reduce the risk of consuming all available space on the upload location.
Locations enable an organization to centralize and process PST files in data centres close to the users, their data, and their target system.
Transparent Central Upload
Transparent Central Upload is a feature to enable better integration for the Central Upload Agent (CUA) with transparent migrations. Unlike an Agent running locally, the CUA has no ability to detach and snapshot files presently accessed by a user.
Enabling Transparent Central Upload ensures that all active Agents on the user’s workstations have validated the existence of the required registry keys or has set them prior to permitting the CUA to upload data on a remote resource. Transparent Central Upload also has the capability to schedule the period of time where it will attempt to centralize targeted files. This enables an administrator to schedule centralization of these files during a time when they will be less likely to be accessed.
Modules
Provides a list of all registered modules in the environment and the last time they checked into the server. Data is returned in a grid format with selectable and filterable results. Where appropriate, location specific configurations can be used to accommodate the needs of that location.
The options on the home ribbon for the Modules sub-menu are as follows:
Workflow
The Workflow sub-menu permits the enablement and order that the PST Flight Deck modules will be processed. The configurations can be made per location, selectable by the drop down at the top of the grid. This area permits you to enable or disable modules, permit certain modules to be skipped if a file fails successful processing by that module, how long to wait prior a file being eligible for processing by that module, or if processing should begin after all the files have been successfully queued for that task.
The options on the home ribbon for the Workflow sub-menu are as follows:
When Edit is selected a popup will appear that has the following options/settings:
Profiles
The Profiles sub-menu offers the ability to add and configure assignable user profiles. They are used to permit an alternate configuration for users associated with them and enable the assignment of unique File Scanner, Agent, Language, and Location configurations. Not all migrations require additional profiles but they are provided to enable support of complex environments where a migration project is taking place.
The options on the home ribbon for the Profiles sub-menu are as follows:
Options available during profile creation are as follows:
Module Editor
The Module Editor permits module specific configuration to be set down to a location level. Locations are specified by a drop box at the top of a tabbed interface. All modules enabled have configuration options exposed in each of the tabs. Configuration settings assigned here can be overridden using the Custom Settings option in the Module Settings sub-menu.
Common Options
There are several settings which are seen over most or all of the modules within PST Flight Deck. The following table provides a description of these configuration options:
Some modules also permit the ability to assign a cloud connection string for a storage location to support Azure nodes.
Module Specific
Some modules have configuration options that are specific to them. The following section discusses these options.
Repair
To ensure maximum efficiency of the Repair module, it is recommended that it be installed on the same machine as the Extraction module and the BITS upload location. When the Repair module is on a location local to the BITS upload location, there is special configuration required to enable the repair request to use the local path rather than a UNC path. Enabling this configuration has shown to produce a substantial performance gain for the repair module.
Extraction
The extraction module is a powerful module that performs several tasks during the extraction process. This functionality requires many additional configuration options that are specific to it. The following table provides a description of these options:
The Extraction module is also able to filter data from PST files when that content is not desired or known to be unsupported in a target system. Filtering items will remove items based on how they match the specified criteria and these items will not be migrated to the target system. Filters will be applied to the scope of the area they are placed. For example, filters placed in the “Appointment Filter” filed will only apply to message classes associated with Appointments.
Filtered content is written to the specified “Filter Location”. The following is a list and description of properties PST files can be filtered by:
Filters are case sensitive when applied. Its syntax can be built using the properties above and standard C# string methods. These methods are appended to a property and applied against the value of that property. If you wanted to find a property containing the value of “xyz” you would append .Contains(“xyz”) to the name of the property.
For more information on C# string methods and the options available for filtering, please refer to the MSDN site on the topic.
Filtering Examples
The following example on how to filter messages types by message class can be used if Enterprise Vault shortcuts are desired to be excluded from a migration:
item.MessageClass.ToLower().StartsWith(“ipm.note.enterprisevault”)
The following example shows how to exclude all items older than a year from being migrated:
item.Created < DateTime.UtcNow.AddMonths(-13)
This final example shows a filter designed to remove items larger than 25 MB from being migrated:
item.Size>26214400
Dedup
This tab contains configuration settings for the duplication function. By default, Dedup examines, and dedupes, mails items within a PST and mail items between multiple PSTs. The settings unique to this tab are described as follows:
Office 365 / Exchange
This tab contains configuration settings for the ingestion into Microsoft targets such as Office 365 and Exchange servers. PST Flight Deck uses the Advanced Ingestion Protocol (AIP) as a primary ingestion method into Microsoft targets. The settings in this tab control the behavior of this module and permit the ability to tune these settings to get the best performance possible. The settings unique to this tab are described as follows:
Enterprise Vault
This tab contains configuration settings for the ingestion into Enterprise Vault targets. PST Flight Deck uses the Enterprise Vault API as an ingestion method into Enterprise Vault targets. The settings in this tab control the behaviour of this module and permit the ability to tune these settings for the desired results. The settings Unique to this tab are described as follows:
EV Shortcut Rehydration
This tab contains configuration settings for the identification and restoration of Enterprise Vault shortcuts identified within PST files. The settings in this tab control the behaviour of this module and permit the ability to tune these settings for the desired result. There is only one unique value in this tab. The Item Ingest Parallelism is used to determine the number of items requested for rehydration per pass per thread of a running module.
Cleanup
This tab contains configuration settings for the cleanup module. The settings in this tab control the behaviour of this module and permit the ability to tune these settings for the desired result.
Leftover
This tab contains configuration settings for the Leftover module. The Leftover module copies non-ingested data to a file share or OneDrive. The settings in this tab control the behaviour of this module and permit the ability to tune these settings for the desired result.
Scheduled Tasks
Scheduled tasks sub-menu permits review and management of regular activities required in a migration. The interface shows all tasks available to the system and the time they are last run. You can execute the tasks now or you can change the configuration of a given task to run at specified times. Tasks can also be scheduled to execute on a regular interval and run continually.
Scheduled Reports
The Scheduled Reports sub-menu permits management of schedule report configurations. Scheduled reports are run periodically and are made available through the web portal to users who subscribe to the report. The reports are created by running defined SQL scripts and are saved as CSV files. Subscribed reports are emailed to the subscriber as ZIPed files.
The options on the home ribbon for the Scheduled Reports sub-menu are as follows:
The options available in the report configuration are:
Computers
The Computers sub-menu provides a grid of data showing all machines seen in the environment. The data returned also includes all computers where content was discovered. Computers can be assigned to specific locations, be designated as “Central Servers” so the Central Upload Agent uploads the content on it, and can be flagged to have the content of the files identified on that machine scanned for sender and recipient information. Should it be appropriate, comments can be added, displayed, or managed for computers identified within the environment.
The options on the home ribbon for the Computers sub-menu are as follows:
The Email sub-menu contains templates and configuration related to all email messages sent from the PST Flight Deck environment. These messages include templates for the various stages of a migration project a user goes through, as well as notification emails that are sent to PST Flight Deck team members responsible for portions of the migration. The grid seen in this section displays the template names and the type of email communication they are. All templates can be tested by selecting the applicable message, hitting the Test button on the ribbon, and providing the requested information.
There are two types of email communication in PST Flight Deck. The first type is “Batch” email messages. Batch Email messages are triggered manually by the PST Flight Deck Administrator by highlighting the applicable message and selecting the Batch Mail button on the ribbon. The resultant window will ask the email priority range that you would like to send the message to and have a button to send the messages. This will queue messages for delivery during the next scheduled send.
Automatic templates are sent when conditions for a user have been met. They can be configured to only be sent after a specific time and have a robust rules system that can be used to develop the conditions under each mail being sent. After enabling a template, the rules are evaluated regularly upon execution of a scheduled task. When the conditions match, the message is queued for delivery during the next scheduled sending of content within the queue; then is logged as having been sent. Automatic templates can also be configured to only be sent once.
Templates can be edited within the XML initially provided after launching the Edit button, or can be edited using the WYSIWYG button to launch a rich text graphical editor. If using the graphical editor and the communication requires images to be included, special consideration is required. After editing a template, it is encouraged to test the message prior to using it in production.
The options on the home ribbon for the Email sub-menu are as follows:
Leavers
The Leavers O365 sub-menu contains the configuration required to leverage PST Flight Deck to add orphaned PST files into your target system for compliance or legal search. The grid of this tab shows the licenses available for the environment and permits the designation of the number of licenses that will be used to facilitate this functionality. The options available help to define what the naming convention used to create new archives in the target will be, the usage location, email domain associated with the tenant, and the ability to set and configure legal holds against the migrated content.
Event rules engine
The Event rules engine option provides for the development and maintenance of workflows that define how various events are processed. Each event processed is given a threshold for the number of times an event happens before the action steps are taken. With this engine, you can define actions to take and notifications to be sent. Automatic actions such as agent reset can also be configured.
The steps to create an evet rule are:
- Add workflow
- Add a rule that triggers the workflow
- Add steps to be taken once the rule is satisfied
The options on the home ribbon for the Event rules engine sub-menu are as follows:
Portal
PST Flight Deck has a role based web interface designed to deliver information or decision-making authority to those in the organization who require it. The PST Flight Deck Portal (Portal) empowers the PST Flight Deck team to be able to do what they need to, when they need to, without requiring Operators, Administrators, or Architects to intervene. The result is a streamlined migration project with the proper team members being enabled and notified when appropriate for them to resolve the present issue. Throughout the Portal, selecting the Help icon can provide contextually relevant information to you when needed.
Actions, Roles, and Groups
Actions are used delegate access to specific roles. Each action correlates with a feature of the Portal. Actions can be used to grant as much access is necessary to a given role.
Roles can be created as are required and be built to suit the needs of the project. For a smaller project where limited resources are engaged throughout the entire project, one role for all the actions may be sufficient. Larger organizations may wish to create more roles for a higher degree of permission segmentation. For example, an organization may permit a help desk role to see details about a user’s progress, but not wish for them to see information related to the project as a whole. Roles can be prioritized to resolve rights assignment when a user is a member of multiple roles.
Groups are used to associate AD groups with Roles and Locations configured. This enables the ability to create project groups of users that are managed within AD and associate them with the role types created. Groups also have the ability to associate an email address for the product to email the group it is affiliated with when action items are required.
The combination of these three components creates the security configuration for the role based Portal administration.
Features
There are several Portal components associated with an Action and available for assignment. Each Portal component is created with the goal to resolve a project need. The following discusses the functions of each of the available components of the Portal.
User Search
This component provides a handy utility to search for a user or account name and return information about the current status of that user. This utility is frequently helpful for a help desk or equivalent group who receives initial calls from an organization’s users for questions concerning the migration project. The interface can provide information on the user and each of the PST files associated with the users. Initially, it provides an overview of that user’s migration. Details can be obtained by drilling into the info button against the user or any of their files. The User Search is associated with the AccessHelpdeskSection Action under Settings > Environment > Roles of the Admin Console.
Self-service Portal
The user search is also leveraged as a resource for users to see the progress of their personal migration. Appending “SelfService” to the UserSearch page will return the user page for the authenticated user. This portal may be included in project FAQ’s or communication to the user as a method to keep users informed as to the progress of their migration, while lessening the burden on teams supporting the migration. There is no Action associated with this function, but any user from the domain scanned by the product can view their migration’s status and progression by accessing the Self-service portal.
An example of the URL used to access this page is as follows:
File Action
This component of the Portal is about empowering the proper team members be able to resolve issues related to the migration without requiring the PST Flight Deck Administrator or Operator’s engagement. PST migrations are complicated projects that involve many groups of the business to get the job done. The File Action component of the Portal presents issue specific web wizards to walk users through remediation of an issue with a file.
Take the following example. A file is being migrated that is presently destined for UserA’s archive, but the product has not been able to gather sufficient information to ensure that the file belongs to that user. PST Flight Deck has a configurable ownership certainty threshold and the ability to block files from being ingested which do not meet that minimum criteria. UserA’s file was blocked from ingestion while someone manually approves the ingestion. If a file is blocked because of such an issue, the help desk could call the user to confirm that the data is theirs and they would like it migrated. Rather than communicating the decision back to the PST Flight Deck Administrator or Operator, the help desk personnel could resolve the issue themselves via the File Action feature of the portal.
The File Action component of the Portal is associated with the AccessBusinessOperationsSection Action under Settings > Environment > Roles of the Admin Console.
Support Tickets
PST Flight Deck has the ability to create, manage, track, and update project related support tickets for a specific user, file, host, or event. This feature enables Operators or other team members to be able to address the largest concerns of a migration, as well as track progress against problematic files. This product feature enables project management groups to streamline the troubleshooting and issue resolution associated with the project for a reduced impact to its end users. The Support Ticket portion of the portal is associated with the AccessIncidentsSection Action under Settings > Environment > Roles of the Admin Console.
Project Status
The project status reports are a series of reports designed to enable appropriate team members access to the data that is important to them. An Operator may wish to know what the progress day over day is while the Project Manager may want to see an overview of the entire project. There are three reports provided over the Portal to help to determine the status of a project.
The “Project Overview” report provides a high-level progress report on the number or volume of files Completed, failed, and remaining over a configurable period of time. This is a good measure to determine the current progress of the project and how far into it a migration team is.
The “Daily Progress” is a useful report to determine at a glance that everything is working as expected. It provides daily ingestion rate information for the past 24 hours. This can help you determine if more users need to be enabled and if you are maintaining the anticipated rate of ingestion through each day.
The “User Progress” report provides metrics on the number of Users with PST files identified and statistics related to them and their progress through the migration work-flow. These charts help to understand the nature of the PST problem identified and what the product is doing to resolve it.
The Project Status portal reports availability is controlled by the AccessReportsSection Action under Settings > Environment > Roles of the Admin Console.
Enabled Users
The “Enabled Users” report provides a list of all the users enabled for migration. It provides some high- level information related to their current status and progress, and a link back to the User Search page for that user. Its access is governed by the AccessProjectManagersSection Action under Settings > Environment > Roles of the Admin Console.
Import
The import feature empowers an Operator to delegate the creation of waves to project managers or other members of the business without the requirement of granting access to a Console. The feature’s accessibility is controlled by the AccessImportUsers Action under Settings > Environment > Roles of the Admin Console.
Operational Activities
The operation of a PST Flight Deck environment is something that requires daily effort and monitoring to ensure optimal throughput. Rate of centralization, rate of upload, size of available disk space, and module backlogs are just a small number of the areas of concern for a PST Flight Deck operator. A failure to perform near daily operation activities could result in a large backlog of work to be performed.
Operational activities can begin at the point of agent deployment when the Discovery process begins and information related to ownership and scope of the PST problem an organization is facing begins to become clear.
Overview
After a PST Flight Deck system is installed, there are some common steps that are taken through a project. For information related to the requirements or deployment of a PST Flight Deck system, please refer to the appropriate guide. A PST Flight Deck system must be configured and tuned appropriately. Best practices to maintain a computer operating system and a SQL database are recommended. Insufficient or contention for available resources will negatively impact the entire PST Flight Deck environment. A common area neglected in a deployment is the SQL database.
Initial Testing
In a freshly deployed environment that has a limited number of controlled agents, PST Flight Deck can be configured to expedite testing. Once appropriate, an end to end test is recommended to ensure all components of the system are properly functioning.
Typically, an initial end to end test is done with the default configuration options enabled. Once confirmed to have been successfully completed, the system should be configured to the design of the project and consecutive testing performed. Changing settings once the Migration Agents are deployed may have undesired results.
Pilots
After end to end testing is completed, it is important to evaluate and configure acceptable threshold values for extraction and ingestion failures. Doing so will prevent the review of every PST with one or two failures and permit an Operator to focus on larger areas of concern.
Once configuration and testing have been completed, the Migration Agent will need to be packaged and tested for deployment. This is frequently done on a small group of users close to the PST Flight Deck team. If the deployment installation went as expected and the workstations can be seen reporting back to the PST Flight Deck server, the system will need to be less aggressively configured to accommodate the added communication from the agents.
At this point you can begin the deployment of the agents to the remaining workstations in an environment. The business pilot will consist of the users who tested the agent package deployment and will provide an opportunity to begin to see bottlenecks within the environment. Tuning the system to a balance of resource consumption and performance is frequently initiated during the business pilot because it is the first time in a project that the system could be busy enough to maximize the performance of the system. Other pilots may wish to be performed depending on the results of initial pilot attempts.
Discovery and Owner Management
After deployment of the Agent to the remaining workstations occurs, the discovery of PST files begins. It is suggested to let this process run until the number of discovered PSTs starts to level out. In smaller projects this may take three or more weeks. Larger projects may take longer.
During the discovery process and through its completion, it is suggested that Operators focus their efforts on ownership identification through the options within Manage > Owners. Accurate and complete ownership identification is one of the challenges in PST migration projects. PST Flight Deck is able to provide suggestions as to ownerships immediately following the initial discovery results being returned. Aggressively managing ownership of files during the discovery phase of a project will drastically reduce the challenges and time needed to manage file ownership at later phases of the project.
The number and volume of PST files may continue to grow as your migration continues, but when discovery is nearing the end, consistently sustained growth can be seen week over week. At this point waves of users can start being prepared for migration. Typically, this initially involves ensuring communication has gone out to impacted users and enablement groups have been defined for the migration.
Ramp-Up
The Ramp-up phase implies growing the wave sizes until waves are keeping the system busy with available disk space and mostly completing them in the desired wave interval. Setting the Migration Priority for a wave of users enables them for migration. Commonly, a unique priority value would be set per migration wave. Daily monitoring is required at this point of the migration. Monitoring should consist of the environment, the active wave(s), and those users’ files until mostly completed. Remediation and reprocessing of some PST files or items may be required for the wave to fully complete. In addition, module tuning to achieve more performance within the limitations of the environment should also be performed during the initial monitoring and growth of migration waves.
Migration
You do not have to wait for a wave to be entirely completed prior to starting the next wave. The goal of an efficient migration project should be to minimize backlog but ensure enough work to have the environment persistently working. This typical bottleneck in a migration work flow is typically the extraction module but could be other areas if certain environmental factors dictate it. Keeping a system busy requires persistent uploads and consistent ingestion to ensure there is enough work to keep the system working at its fullest capacity without running out of available disk space.
Wrap-Up
As users complete, an Operator may choose to move their migration priority to a disabled value. This permits control over future discoveries for the user. Backup files are frequently kept for a determined period of time after a user has completed the migration. This location, and all other module’s storage locations will need to have content removed when appropriate. When a user or the migration completes, it is recommended that those users have the DisablePST registry key set on their workstations to reduce the re-introduction of PST files after a user’s source files have been removed.
Stages of a Migration
During a migration, both users and files are progressing through a workflow. Each stage of the process has specific results indicating what has happened to the file. Stages are used to describe the phases of this process so that an individual can tell the status of a user or a file at a glance.
The user workflow stage is complex and are calculated by the server on an hourly schedule. The displayed status may not correctly reflect the current status of the user. This is normally not an issue as the status of a user does not change that fast. Below is a list of stages used to describe the status of a user:
File stages are current, up to the minute, for a specific file. The following are the stages of file processing:
The primary goal of a PST Flight Deck is to move PST files from a dispersed location and ingest them into a centralized location. Files are considered “Complete” when the need to ingest the file has been satisfied. This is most commonly due to complementing a successful ingestion of a file, however can also be achieved by a file being in a status that does not get ingested (Deleting, Not a PST File…..). Files or users can be considered “Complete” but still have modules in the workflow that have not competed.
Troubleshooting
Issues do happen! When they do it is important to be able to identify that a problem did occur, the nature of the problem, and what actions need to be taken to remediate the problem. While investigating, it is encouraged to note the steps of your investigation through the PST Flight Deck comment system. This will enable you to easily see when an issue with a file is recurring and the approach being taken will need to be reviewed.
Issue Identification
There are many difficulties that can happen in PST migration projects. Being able to identify when a problem is happening at the client vs. the server or which module a failure took place in can expedite the ability to identify the problem and promptly resolve it.
Monitoring.
Events.
Point of Failure
When discussing details about an issue, a common area of interest is the “point of failure”. This is the exact point in time that the issue occurs, where it occurs, and under what conditions. The idea of determining the point of failure is to ultimately find a means by which to reproduce the issue being observed. Once any issue can be reproduced on demand, it is infinitely easier to work towards a solution. For troubleshooting, the identification of a point of failure means that we are able to reproduce the issue, collect applicable logs, sample items, and make assessments on if the issue is environmental or not. In most instances, we can collect samples that are provided to our Support team for analysis. This frequently expedites the troubleshooting of an issue and results in a drastically shortened time to resolution.
Logging
PST Flight Deck has very robust logging features that enable administrators to effectively troubleshoot problems. There are two categories of logs – agent logs and server logs. Agent logs are generated by the Migration Agent and are stored on a user’s computer. Agent logging is enabled by default. Server logs are generated by the various PST Flight Deck modules and the Core. Server logging is disabled by default. To learn how to enable logging in PST Flight Deck please visit use this on the topic.
Agent Logs
The Windows Migration Agent has two components, the discovery agent and the migration agent, and both components generate logs. The logs are stored in %temp%\Quadrotech\. Both logs can be retrieved from a workstation using the Request Client Logs feature in the Admin Console. If enabled, a user can elect to send logs to the server using the Send Log feature of the Migration Agent.
Discovery Agent
The Discovery Agent log is written to a file named FileScanner_hostname.log. The log will not be created if there are no warning levels. There is rarely a need to change the logging level but you may be directed by Quadrotech support to change it for troubleshooting.
Migration Agent
The migration log is updated every time the agent polls or every ten minutes when the agent checks for configuration changes and its name is MigrationAgent.log The Migration Agent log records items such as routine polling results, user enablement status, error conditions, work items to be processed, etc.
Server Logs
Server logs are used to log all routine work and errors conditions. They are extremely important for troubleshooting problems that occur with modules, consoles, the portal, and all client/server interactions. There is a log for every running PST Flight Deck service, the Admin console on the core server, and the web interface. Each log has a maximum size of 100MB, at which point the new log is created and the original one renamed to include “archive” and a number in its name. These logs are stored in c:\program files (x86)\Quadrotech\logs\. The various server log descriptions are in the following table. The directory where logs are stored can be changed during installation.
Getting Help
If you experience an issue while using our product, please use this guide to attempt to gather as much information as possible about the nature of the issue. Identification of the point of failure of an issue and collecting the appropriate logs when starting a ticket will aid the support process and ultimately, the resolution of the issue you are experiencing. When you have an issue that you are unable to address yourself, you may also contact a Quadrotech support representative by email: support@quadrotech-it.com
If you have a question on how to use a feature of the product, sizing, or any product related questions regarding the use of the product, then please contact your Field Enablement representative for further guidance.
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Miranda Dress
Ruffles make everything better, so we popped some on the cuffs, neck and pockets (yes, pockets) of this sleek ponte dress. Whether you're in the office or at a party, the versatile semi-fitted shape looks the part, while the full lining keeps you comfortable all day long.
- Body 39% cotton 29% polyamide 27% modal 5% elastane, Lining 100% polyester
- Machine washable
- Shift with relaxed waist to a fitted hip and straight hem
- Length finishes just above knee
- Two front pockets
- ¾ length sleeves
- Medium weight ottoman jersey
- Fully lined
- Self fabric ruffles
Miranda Ponte Dress
£80.00 £40.00
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TITLE: A question about the proof of the fact that contractible spaces are simply connected
QUESTION [2 upvotes]: In greeberg's algebraic topology, the following fact is used in the proof that contractible spaces are simply connected without justification:
Let $p:\mathbb{I}\rightarrow X$ be a continuous function such that $p(0)=p(1)$ and $X$ is contractible. Then there exists a continuous function $F:\mathbb{I}\times \mathbb{I}\rightarrow X$ such that:
1) $F(s,0)=p(0)$ for all $s\in \mathbb{I}$
2) $F(s,1)=p(s)$ for all $s\in \mathbb{I}$
3) $F(0,t)=F(1,t)$ for all $t\in \mathbb{I}$
I agree that there exists a map $F$ such that the first 2 conditions are satisfied, but what about the third ? Is there an obvious way to see that that the three conditions can be satisfied. As I mentioned the book uses this fact without justification.
Thank you
Reminder: All answers posted so far are proving the fact that contractible spaces are simply connected. Note that my original
question (found above) asks for a justification of the fact that the book stated without justification. I don't
think it s obvious.
REPLY [1 votes]: I think the question is asking about proving that for a contractible space $X$ a map $(S^1,1) \to (X,x)$ is null homotopic relative to the base point $1$. Of course a contracting homotopy of $X$ may move the base point $x$.
It is often easier to understand these matters from a more general viewpoint. The key property of $(S^1,1)$ is that it is well pointed, i.e. has the homotopy extension property. So a relevant lemma is essentially as follows, and is 7.2.11 of Topology and Groupoids. Recall an inessential map is one homotopic to a constant map.
Let $f:Y \to X$ be an inessential map, and suppose $y$ is well pointed in $Y$. Then $f$ is inessential rel $y$.
The feature of the proof is that that the null homotopy of $f$ defines a path $\alpha$ in $X$ from $x=f(y)$ to $x'$ say. Because $(Y,y)$ is well pointed, $\alpha$ defines a bijection $\alpha_*: [(Y,y), (X,x)] \to [(Y,y), (X,x')]$, where these are homotopy classes relative to the base point. Let $g: Y \to X$ be the constant map with value $x$. Then $\alpha_*[f]= \alpha_*[g]$ and so $[f]=[g]$.
These operations generalise the operations of the fundamental group on higher homotopy groups. In moving from $(S^n,1)$ to $(X,A)$ and looking at the fact that a homotopy equivalence of spaces induces an isomorphism of homotopy groups, I discovered a gluing theorem for homotopy equivalences (Section 7.4 of the above book.)
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DreamHost Wrongly Bill 1000’s of Customers Some As Much As $9’000
Webmasters that are hosted by DreamHost woke up to a nasty billing surprise this morning after DreamHost made a major mistake with their billing system, their entire customer base on Credit Cards Payments was billed a total of $7,500,000 overnight. What’s not amusing is the lack of “Taking This Seriously” attitude that has once again spilled out from DreamHost staff, maybe they should consider a rename to NightmareHost.
So we have all heard the horror stories about DreamHost and their hosting package speed issues, customer service, support etc, today it was the turn of their billing system and a member of staff “Josh” to make the biggest cock up i can remember in the hosting sector for a few years. Dreamhost had a billing glitch and billed thousands of customers for as much as 2 years in advance, with debits/charges over $400.
The particularly bad part about this error is that DreamHost customers are reporting that their credit cards or bank accounts are now overdrawn, it has been reported that a few cannot now make their mortgage payments.
Last year DreamHost wrote this post on their blog:
It’s something we like to do every day!
Yep, the secret to our crazy low prices and amazing ferraris, finally revealed:
We take your billing address and go to your homes at night to steal your jewelry, plasma TVs, and all valuable toiletries!
Not to mention all the credit card numbers we get fund our wild vegas benders (roulette is a great way to launder money) and illicit basketball leagues!
(Damn PayPal and Google Checkout, not sharing the credit card info with us!)
Yeah, overall it’s a pretty sweet scheme we’ve had these past 10 years; and now that we’ve gone into hiding you’ll never catch us, coppers!
I think those statements might come back to haunt them somehow.
Luckily LunarPages have stepped in with an unbelievable offer for DreamHost customers that have been affected, and they are taking the offer up on mass.!
You can read the blog post dreamhost made after this major error their tongue in cheek attitude to the scenario has not gone down well with customers causing a backlash, most calling it the final straw.
And if your looking for web hosting for you social CMS project please DO NOT consider using DreamHost under any circumstance’s, you have been warned. If you want to read some more info on DreamHost Services go HERE.
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\begin{document}
\title{Graded rings associated to valuations and direct limits}
\author{Silva de Souza, C. H., Novacoski, J. A. and Spivakovsky, M.}
\thanks{During the realization of this project the authors were supported by a grant from Funda\c c\~ao de Amparo \`a Pesquisa do Estado de S\~ao Paulo (process number 2017/17835-9).}
\begin{abstract} In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ in terms of the key polynomials that define $Q_n$. In order to do that, we use direct limits. In general, we describe the direct limit of a family of graded rings associated to a totally ordered set of valuations. As an example, we describe the graded ring associated to a valuation-algebraic valuation as a direct limit of graded rings associated to residue-transcendental valuations.
\end{abstract}
\subjclass[2010]{Primary 13A18}
\maketitle
\section{Introduction}
The graded ring structure associated to a valuation $\nu$, denoted by $\mathcal{G}_\nu$ (see \linebreak Definition~\ref{defAnelGrad}), has proved to be an important object on Valuation Theory. For \linebreak example, the graded ring describes information of the value group $\nu\K$ and the residue field $\K\nu$
simultaneously. It was proved in \cite{Matheus} that $\mathcal{G}_\nu$ is isomorphic to the semigroup ring $\K\nu[t^{\nu \K}]$ with a suitable multiplication. Also, this structure is related to the program developed by Teissier to prove \textit{local uniformization}, an open problem in positive characteristic with applications in resolution of singularities. This program is based on the study of the spectrum of certain graded rings (see \cite{teissier}).
\vspace{0.3cm}
Other important objects, which are also linked with programs to prove local uniformization, are key polynomials (see Definition~\ref{defiPoliChave}). These polynomials
were introduced by Mac Lane in \cite{MacLane} and generalized years later by Vaquié in \cite{Vaq}, using the structure of graded ring. We will refer to them as \textit{Mac Lane-Vaquié key polynomials}.
After that, Novacoski and Spivakovsky in \cite{josneiKeyPolyPropriedades} and Decaup, Mahboub
and Spivakovsky in \cite{spivamahboubkeypoly} introduced a new notion of key polynomials, which is the one we use in this paper. These two definitions can be well studied by using graded rings, as one can see in \cite{Andrei} and \cite{josneimonomial}.
\vspace{0.3cm}
Among key polynomials, the so called limit key polynomials are of great interest to us. Limit key polynomials were introduced in \cite{Vaq} and are one of the main aspects of the generalization of Mac Lane's original key polynomials by Vaquié. Here we use a formulation similar to the one presented in \cite{josneiKeyPolyPropriedades} (see Definition~\ref{defLKP2}). These polynomials are related with the existence of defect, which is an obstacle when dealing with valuations and local uniformization. For example, in the case where the valuation has a unique extension, the defect is the product of the \textit{relative degrees} of limit key polynomials (see \cite{Vaq2} or \cite{Nart}).
\vspace{0.3cm}
For a valuation $\nu$ on $\K[x]$, we consider the set $\Psi_n$ of all key polynomials for $\nu$ of degree $n$. In this paper, we study the structure of the graded ring associated to a limit key polynomial $Q_n$ for $\Psi_n$, denoted by $\mathcal{G}_{Q_n}$, in terms of the key polynomials $Q\in \Psi_n$. In order to do that, we describe $\mathcal{G}_{Q_n}$ as the direct limit of a direct system defined by the graded rings $\mathcal{G}_Q$ and the maps introduced in \cite{Andrei}.
\vspace{0.3cm}
Take a valuation $\nu_0$ on $\K$ with value group $\Gamma_0$. Fix a totally ordered divisible group $\Gamma$ containing $\Gamma_0$. Let
$$\mathcal{V} = \{\nu_0\}\cup \{\nu: \K[x] \rightarrow \Gamma_\infty \mid \nu \text{ is a valuation extending } \nu_0 \} $$
($\Gamma_\infty$ is defined in Section~\ref{Preli}). Consider the partial order on $\mathcal{V}$ given by $\nu_0\leq \nu$ for every $\nu\in \mathcal{V}$ and, for $\nu, \mu\in \mathcal{V}\setminus\{ \nu_0\}$, we set $\nu \leq \mu$ if and only if $\nu(f)\leq \mu(f)$ for every $f\in \K[x]$. Our first result deals with an arbitrary totally ordered subset $\mathfrak{v}= \{\nu_i\}_{i\in I}\subset \mathcal{V}$ such that there exists a valuation $\nu\in \mathcal{V}$ satisfying $\nu_i\leq \nu$ for every $i\in I$. Theorem~\ref{teoLimVmaiorRstable} will give us that $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$
is isomorphic to the additive subgroup of $\mathcal{G}_{\nu}$ generated by the set
$R=\langle \{ \inv_\nu(f)\mid f \text{ is } \mathfrak{v}\text{-stable} \} \rangle$ ($\mathfrak{v}$-stability is defined in Section~\ref{TotallyOrderLimits}).
\vspace{0.3cm}
Next, we divide the totally ordered subsets $\mathfrak{v}= \{\nu_i\}_{i\in I}\subset \mathcal{V}$ into three types: the ones with maximal element; the ones without maximal element such that every $f\in \K[x]$ is $\mathfrak{v}$-stable; the ones without maximal element such that there exists at least one polynomial that is not $\mathfrak{v}$-stable. We show that in the first and second cases there exists $\nu \in \mathcal{V}$, that we will denote by $\displaystyle\sup_{i\in I}\nu_i $, satisfying $\nu\geq \nu_i$ for every $i\in I$ and $R = \mathcal{G}_{\nu}$ (Proposition~\ref{lemVstable} and Corollary~\ref{teoLimitSeqStable}). In the third case, we show that for a polynomial $Q$ of smallest degree that is not $\mathfrak{v}$-stable we can define $\mu\in \mathcal{V}$ such that $\mu$ is equal to its truncation at $Q$ (see Definition~\ref{defTruncamento}), $\mu\geq \nu_i$ for every $i\in I$ and $R = R_Q$, where \linebreak $R_Q= \langle \{ \inv_\mu(f)\mid \deg(f)<\deg(Q) \}\rangle\subset \mathcal{G}_\mu$ (Proposition~\ref{lemamuQgammaVal} and Corollary~\ref{teoLimitSeqNotStable}).
\vspace{0.3cm}
We then give two applications of the previous results. The first one concerns limit key polynomials, our main interest. We prove that, given a limit key polynomial $Q_n$ for $\Psi_n$, the subset $\mathfrak{v}= \{\nu_Q\}_{Q\in \Psi_n}\subset \mathcal{V}$ is totally ordered without maximal element and $Q_n$ is a polynomial of smallest degree that is not $\mathfrak{v}$-stable (Corollary~\ref{corQnNotvStable}). Therefore, $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\cong R_{Q_n}$ (Corollary~\ref{corLimQn}).
\vspace{0.3cm}
The second application concerns valuation-algebraic valuations (see Definition~\ref{defValAlg}). We prove that, given a valuation-algebraic valuation $\nu$, there exists a totally ordered subset $\mathfrak{v}= \{\nu_Q\}_{Q\in \textbf{Q}}\subset \mathcal{V}$ without maximal element with each $Q$ a key polynomial for $\nu$, $\nu_Q$ a residue-transcendental valuation and $\nu = \displaystyle\sup_{Q\in \textbf{Q}}\nu_Q$ (Proposition~\ref{propValAlgStable}). Therefore, $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{Q}\cong \mathcal{G}_{\nu}$ (Corollary~\ref{corValgLim}).
\vspace{0.3cm}
This paper is organized as follows. In Section \ref{Preli}, we present the main definitions and results that will be used throughout the paper. In Section \ref{GradedRings}, we present the main results about graded rings associated to a valuation that will be useful in our discussions. In Section \ref{TotallyOrderLimits}, for a given totally ordered subset $\mathfrak{v}= \{\nu_i\}_{i\in I}\subset \mathcal{V}$, we begin presenting some properties of the direct limit of the direct system $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$ and prove Theorem~\ref{teoLimVmaiorRstable}. Then we prove Corollary~\ref{teoLimitSeqStable} and Corollary~\ref{teoLimitSeqNotStable} in Subsections \ref{FirstSecondCases} and \ref{ThirdCase}, respectively. In Section \ref{LKPLimit}, we describe the graded ring associated to a limit key polynomial via Corollary~\ref{corLimQn}. In Section \ref{ValAlgLimit}, we describe the graded ring associated to a valuation-algebraic valuation via Corollary~\ref{corValgLim}.
\section{Preliminaries}\label{Preli}
\begin{Def}
Take a commutative ring $R$ with unity. A \index{Valuation}\textbf{valuation} on $R$ is a mapping $\nu:R\lra \Gamma_\infty :=\Gamma \cup\{\infty\}$ where $\Gamma$ is a totally ordered abelian group (and the extension of addition and order to $\infty$ is done in the natural way), with the following properties:
\begin{description}
\item[(V1)] $\nu(ab)=\nu(a)+\nu(b)$ for all $a,b\in R$.
\item[(V2)] $\nu(a+b)\geq \min\{\nu(a),\nu(b)\}$ for all $a,b\in R$.
\item[(V3)] $\nu(1)=0$ and $\nu(0)=\infty$.
\end{description}
\end{Def}
Let $\nu: R \lra\Gamma_\infty$ be a valuation. The set $\supp(\nu)=\{a\in R\mid \nu(a )=\infty\}$ is called the \textbf{support of $\nu$}. A valuation $\nu$ is a \index{Valuation!Krull}\textbf{Krull valuation} if $\supp(\nu)=\{0\}$. The \textbf{value group of $\nu$} is the subgroup of $\Gamma$ generated by
$\{\nu(a)\mid a \in R\setminus \supp(\nu) \}
$ and is denoted by $\nu R$. If $R$ is a field, then we define the \textbf{valuation ring} to be $\VR_\nu:=\{ a\in R\mid \nu(a)\geq 0 \}$. The ring $\VR_\nu$ is a local ring with unique maximal ideal $\MI_\nu:=\{a\in R\mid \nu(a)>0 \}. $ We define the \textbf{residue field} of $\nu$ to be the field $\VR_\nu/\MI_\nu$ and denote it by $R\nu$. The image of $a\in \VR_\nu$ in $R\nu$ is denoted by $a\nu$.
\begin{Obs}Take a valuation $\nu$ on a field $\K$ and a valuation $\overline{\nu}$ on $\overline{\K}$, the algebraic closure of $\K$, such that $\overline{\nu}|_{\K}=\nu$. Then $\overline{\nu}\overline{\K}$ is a divisible group. Additionally, $\overline{\nu}\overline{\K}=\nu\K\otimes_\Z \Q$ (see \cite{Eng}, p.79). About the residue fields, it is known that $\overline{\K}\overline{\nu} $ is the algebraic closure of $\K\nu$ (see \cite{Eng}, p.66).
\end{Obs}
Fix a valuation $\nu$ in $\K[x]$, the ring of polynomials in one indeterminate over the field $\K$. Our main definition of key polynomial relates to the one in \cite{josneimonomial},
which is related to the one in \cite{spivamahboub}. Fix an algebraic closure $\overline{\K}$ for $\K$ and fix a valuation $\overline{\nu}$ on $\overline{\K}[x]$ such that $\overline{\nu}|_\K=\nu$.
\pagebreak
\begin{Def}
Let $f\in \overline{\K}[x]$ be a non-zero polynomial.
\begin{itemize}
\item If $\deg(f)>0$, set
$$\delta(f):=\max\{\overline{\nu}(x-a)\mid a\in \overline{\K} \text{ and }f(a)=0\}.$$
\item If $\deg(f)=0$, set $\delta(f) = -\infty$.
\end{itemize}
\end{Def}
\begin{Obs} According to \cite{josneiKeyPolyMinimalPairs}, $\delta(f)$ does not depend on the choice of the algebraic closure $\overline{\K}$ or the extension $\overline{\nu}$ of $\nu$.
\end{Obs}
\begin{Def}\label{defiPoliChave}
A monic polynomial $Q\in \K[x]$ is a \textbf{key polynomial} of level $\delta(Q)$ if, for every $f\in \K[x]$,
$$\delta(f)\geq \delta(Q) \Longrightarrow \deg(f)\geq \deg(Q). $$
\end{Def}
Let $q\in \K[x]$ be a non-constant polynomial. Then there exist uniquely determined polynomials $f_0,\ldots, f_s\in \K[x]$ with $\deg(f_i)<\deg(q)$ for every $i$, $0\leq i\leq s$, such that
$$f = f_0+f_1q+\ldots +f_sq^s.$$
We call this expression the \textbf{\textit{q}-expansion} of $f$.
\begin{Prop}\label{propequivPolichave}(Lemma 2.3 \textbf{(iii)} of \cite{josneiKeyPolyPropriedades} and Corollary 3.52 of \cite{leloup}) The following assertions are equivalent.
\begin{enumerate}[label=\rm (\roman*):]
\item $Q$ is a key polynomial for $\nu$.
\item For every $f,g\in \K[x]$ with $\deg(f)<\deg(Q)$ and $\deg(g)<\deg(Q)$, if \linebreak $fg=lQ+r$ is the $Q$-expansion of $fg$, then $\nu(fg)=\nu(r)<\nu(lQ)$.
\end{enumerate}
\end{Prop}
\begin{Def}\label{defTruncamento} Let $q\in \K[x]$ be a non-constant polynomial and $\nu$ a valuation on $\K[x]$. For a given $f\in \K[x]$, denote by $f_0, \ldots, f_s$ the coefficients of the $q$-expansion of $f$. The map
$$\nu_q(f):=\underset{0\leq i\leq s}{\min} \{ \nu(f_iq^i) \},$$
is called the \textbf{truncation} of $\nu$ at $q$.
\end{Def}
This map is not always a valuation, as we can see in Example 2.4 of \cite{josneiKeyPolyPropriedades}.
\begin{Prop}(Proposition 2.6 of \cite{josneiKeyPolyMinimalPairs}) If $Q$ is a key polynomial, then $\nu_Q$ is a valuation on $\K[x]$.
\end{Prop}
In the next lemmas, we state some properties of key polynomials and truncations. In the following, we denote by $\Gamma_\Q:=\nu(\K[x])\otimes_\Z\Q$ the divisible hull of $\nu(\K[x])$.
\begin{Lema}\label{prop3itensQQlinhaPoliChaves}(Proposition 2.10 of \cite{josneiKeyPolyPropriedades})
Let $Q,Q'\in \K[x]$ be key polynomials for $\nu$, with $\delta(Q),\delta(Q')\in \Gamma_\Q$.
\begin{enumerate}
\item If $\deg(Q)<\deg(Q')$, then $\delta(Q)<\delta(Q')$.
\item If $\delta(Q)<\delta(Q')$, then $\nu_Q(Q')<\nu(Q')$.
\item If $\deg(Q)=\deg(Q')$, then
$$\nu(Q)<\nu(Q')\Longleftrightarrow \nu_Q(Q')<\nu(Q')\Longleftrightarrow \delta(Q)<\delta(Q'). $$
\end{enumerate}
\end{Lema}
\begin{Lema}\label{lempolichaveEpsilonmenor} (Corollaries 3.9, 3.10, 3.11 and 3.13 of \cite{josneimonomial}) Let $Q,Q'\in \K[x]$ be key polynomials such that $\delta(Q)\leq \delta(Q')$.
\begin{enumerate}
\item We have $\nu_{Q'}(Q)=\nu(Q)$.
\item For every $f\in \K[x]$, we have
$\nu_Q(f)\leq \nu_{Q'}(f). $
In particular, if $\delta(Q)= \delta(Q')$ then $\nu_Q=\nu_{Q'}$.
\item For every $f\in \K[x]$, if $\nu_Q(f)=\nu(f)$, then $\nu_{Q'}(f)=\nu(f)$.
\item For every $f\in \K[x]$, if $\delta(Q)< \delta(Q')$ and $\nu_{Q'}(f)<\nu(f)$, then \linebreak $\nu_Q(f)< \nu_{Q'}(f). $
\end{enumerate}
\end{Lema}
\begin{Obs}In \cite{josneiKeyPolyPropriedades} and \cite{josneimonomial} the definition of key polynomial and the above results are stated using the notion of $\epsilon(f)$ instead of $\delta(f)$. For a non-zero polynomial $f\not \in \supp(\nu)$ with $\deg(f)>0$, one defines
$$\epsilon(f):=\underset{1\leq b\leq \deg(f)}{\max}\left\lbrace\left.\frac{\nu(f) - \nu(\partial_bf)}{b} \;\right| \nu(\partial_bf)\in\Gamma \right\rbrace\in \Gamma_\Q, $$ where $\partial_bf$ is the formal \textbf{Hasse-derivative of order} $b$ of $f$. If $f\not \in \supp(\nu)$ and $\deg(f)=0$, then we set $\epsilon(f):=-\infty$ and if $f\in\supp(\nu)$, then we set $\epsilon(f):= \infty$.
Proposition 3.1 of \cite{josneiKeyPolyMinimalPairs} shows that $\delta(f)=\epsilon(f)$ for all $f\in \K[x]$.
\end{Obs}
\section{Graded ring associated to a valuation}\label{GradedRings}
Let $\nu$ be a valuation on $\K[x]$.
For each $\gamma\in \nu(\K[x])$, we consider the abelian groups
$$\mathcal{P}_\gamma = \{ f\in \K[x]\mid \nu(f)\geq \gamma \} \text{ and } \mathcal{P}_{\gamma}^{+} = \{ f\in \K[x]\mid \nu(f)>\gamma \}. $$
\begin{Def}\label{defAnelGrad}
The \textbf{graded ring } associated to $\nu$ is defined by
$$\mathcal G_\nu={\rm gr}_{\nu}(\K[x]):= \bigoplus_{\gamma\in \nu(\K[x])}\mathcal{P}_\gamma/ \mathcal{P}_{\gamma}^{+}.$$
\end{Def}
The sum on $\mathcal G_\nu$ is defined coordinatewise and the product is given by extending the product of homogeneous elements, which is described by
$$\left( f+ \mathcal{P}_{\beta}^{+} \right) \cdot \left( g+ \mathcal{P}_{\gamma}^{+} \right): = \left( fg +\mathcal{P}_{\beta+\gamma}^{+}\right), $$
where $\beta= \nu(f)$ and $\gamma = \nu(g)$.
For $f\not \in \supp(\nu)$, we denote by $\inv_\nu(f)$ the image of $f$ in $\mathcal{P}_{\nu(f)}/ \mathcal{P}_{\nu(f)}^{+} \subseteq \mathcal{G}_\nu.$ \linebreak If $f\in \supp(\nu)$, then we define $\inv_\nu(f)=0$.
\begin{Lema}\label{lemPropinv}
Let $f,g\in \K[x]$. We have the following.
\begin{enumerate}
\item $\mathcal{G}_\nu$ is an integral domain.
\item $\inv_\nu(f)\cdot \inv_\nu(g)=\inv_\nu(fg)$.
\item $\inv_\nu(f)=\inv_\nu(g)$ if and only if $\nu(f)=\nu(g)$ and $\nu(f-g)>\nu(f)$.
\end{enumerate}
\end{Lema}
Let $\nu_i$ and $\nu_j$ be valuations on $\K[x]$ such that $\nu_i(f)\leq \nu_j(f)$ for all $f\in \K[x]$. Let $\mathcal{P}_\gamma(\K[x],\nu_i) =\{ f\in \K[x]\mid \nu_i(f)\geq \gamma \} $ (analogously we define $\mathcal{P}_\gamma(\K[x],\nu_{j}), \mathcal{P}_{\gamma}^{+}(\K[x],\nu_i)$ and $\mathcal{P}_{\gamma}^{+}(\K[x],\nu_{j})$).
We have the inclusions
$$\mathcal{P}_\gamma(\K[x],\nu_i)\subseteq \mathcal{P}_\gamma(\K[x],\nu_{j})$$
and
$$\mathcal{P}_{\gamma}^{+}(\K[x],\nu_i)\subseteq \mathcal{P}_{\gamma}^{+}(\K[x],\nu_{j}) $$
for any $\gamma\in \nu_i(\K[x])\subseteq \nu_{j}(\K[x])$.
We consider the following map defined in \cite{Andrei}:
\begin{align}\label{eqPhiij}
\phi_{ij}: \hspace{0.3cm}\mathcal G_{\nu_i}\hspace{0.2cm} &\longrightarrow \hspace{0.2cm} \mathcal G_{\nu_j}\\
\inv_{\nu_i}(f) &\longmapsto \begin{cases}
\inv_{\nu_j}(f)& \mbox{ if }\nu_i(f)=\nu_{j}(f)\\
0&\mbox{ if }\nu_i(f)<\nu_{j}(f),
\end{cases} \nonumber
\end{align}
\noindent and we extend this map naturally for an arbitrary element. This map is well-defined (Corollary 5.5 of \cite{Andrei}) and, by construction, it is a graded homomorphism.
Suppose that $q\in \K[x]$ is such that $\nu_q$ is a valuation. In this case, we denote $\mathcal{G}_{\nu_q}$ by $\mathcal{G}_q$ and $\inv_{\nu_q}(f)$ by $\inv_q(f)$.
Let $R_q$ be the additive subgroup of $\mathcal G_q$ generated by the set
$$\{ \inv_q(f)\mid f\in \K[x]_d \}, $$
where $d=\deg(q)$ and $\K[x]_d=\{ f\in \K[x]\mid \deg(f)<d \}$. We set $y_q:=\inv_q(q)$. The next two propositions say that $y_q$ can be seen as a transcendental element over $R_q$ and that $R_q$ has a subring structure if and only if $q$ is a key polynomial for $\nu$.
\begin{Prop}\label{lemAlgGraduadaAnelPoliy}(Proposition 4.5 of \cite{josneimonomial})
Suppose $q\not\in \supp(\nu)$. If
$$a_0+a_1y_q+\ldots +a_sy_q^s=0 $$
for some $a_0, \ldots, a_s \in R_q$, then $a_i =0$ for every $i$, $0\leq i \leq s$.
Moreover,
$$ \mathcal G_q= R_q[y_q].$$
\end{Prop}
\begin{Prop}\label{propequivPolichaveRq}(Theorem 5.7 of \cite{josneicaio})
Suppose $\nu_q$ is a valuation on $\K[x]$. Then the following assertions are equivalent.
\begin{enumerate}[label=\rm (\roman*):]
\item $q$ is a key polynomial for $\nu$.
\item The set $R_q$ is a subring of $\mathcal G_q$.
\end{enumerate}
\end{Prop}
\section{Totally ordered sets of valuations and direct limits}\label{TotallyOrderLimits}
Take a valuation $\nu_0$ on $\K$ and let $\Gamma_0$ denote the value group of $\nu_0$. Fix a totally ordered divisible group $\Gamma$ containing $\Gamma_0$. Let
$$\mathcal{V} = \{\nu_0\}\cup \{\nu: \K[x] \rightarrow \Gamma_\infty \mid \nu \text{ is a valuation extending } \nu_0 \}. $$
Consider the partial order on $\mathcal{V}$ given by $\nu_0\leq \nu$ for every $\nu\in \mathcal{V}$ and, for $\nu, \mu\in \mathcal{V}\setminus\{ \nu_0\}$, we set $\nu \leq \mu$ if and only if $\nu(f)\leq \mu(f)$ for every $f\in \K[x]$.
\vspace{0.3cm}
Let $\mathfrak{v}= \{\nu_i\}_{i\in I}\subset \mathcal{V}$ be a totally ordered set. We induce a total order on the index set $I$ from the order on $\mathfrak{v}$. In particular, $i<j$ implies $\nu_i<\nu_j$, that is, $\nu_i(f)\leq \nu_j(f)$ for every $f\in \K[x]$ and there exists $g\in \K[x]$ such that $\nu_i(g)<\nu_j(g)$. Since we have a total order,
$(I, \leq)$ is a directed set.\footnote{That is, $\leq$ is reflexive and transitive relation on $\mathfrak{v}$ such that, for every $\nu_i,\nu_j\in \mathfrak{v}$, there exists $\nu_k\in \mathfrak{v}$ satisfying $\nu_i\leq \nu_k$ and $\nu_j\leq \nu_k$.}
\begin{Lema}\label{lemDirectSystemValTotalOrder}
Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$. Consider the family of graded rings $\{ \mathcal{G}_{\nu_i}\}_{i\in I }$.
For $\nu_i \leq \nu_j$, let $\phi_{ij}$ be the map
\begin{align*}
\phi_{ij}:\hspace{0.3cm}\mathcal G_{\nu_i}\hspace{0.2cm} &\longrightarrow \hspace{0.2cm}\mathcal G_{\nu_j}\\
\inv_{\nu_i}(f) &\longmapsto \begin{cases}
\inv_{\nu_j}(f)& \mbox{ if }\nu_i(f)=\nu_{j}(f)\\
0&\mbox{ if }\nu_i(f)<\nu_{j}(f),
\end{cases}
\end{align*}
extended in a natural way to arbitrary (that is, not necessarily homogeneous) elements of $\mathcal G_{\nu_i}$. Then $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$ is a direct system over $I$.
\end{Lema}
\begin{proof}We need to check two properties.
\begin{itemize}
\item By definition, $\phi_{ii}(\inv_{\nu_i}(f)) = \inv_{\nu_i}(f)$ for every $f\in \K[x]$, hence $\phi_{ii}$ is the identity map on $\mathcal{G}_{\nu_i}$.
\item Take $i\leq j \leq k$,
that is,
\begin{equation}\label{eqnuinujnuk}
\nu_i(f)\leq \nu_{j}(f)\leq \nu_{k}(f)
\end{equation} for all $f\in \K[x]$.
If the strict inequality holds in some of the inequalities of \eqref{eqnuinujnuk}, then
$$(\phi_{jk}\circ \phi_{ij}) (\inv_{\nu_i}(f)) = 0 = \phi_{ik}(\inv_{\nu_i}(f)). $$
If $\nu_i(f)=\nu_{j}(f)= \nu_{k}(f)$, then
$$(\phi_{jk}\circ \phi_{ij}) (\inv_{\nu_i}(f)) = \inv_{\nu_k}(f) = \phi_{ik}(\inv_{\nu_i}(f)). $$
\end{itemize}
Therefore, $\phi_{ik}=\phi_{jk}\circ \phi_{ij}$ for all $i\leq j \leq k$.
Then $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$ is a direct system over $I$.
\end{proof}
\begin{Obs} We do not gain in generality if we suppose $\mathfrak{v}= \{\nu_i\}_{i\in I}$ simply a directed set, because every directed set in $\mathcal{V}$ is totally ordered. Indeed, if $\mathfrak{v} \subset \mathcal{V}$ is a directed set, then given $\nu, \mu\in \mathfrak{v}$ there exists $\eta\in \mathfrak{v}$ such that $\nu\leq \eta$ and $\mu \leq \eta$. By Theorem 2.4 of \cite{Nart}, the set $(-\infty, \eta) = \{\rho\in \mathcal{V}\mid \rho< \eta \}$ is well ordered. Therefore, $\nu$ and $\mu$ are comparable.
\end{Obs}
We want to describe the direct limit of the direct system $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$.
We are going to use the characterization of $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$ as a disjoint union:
the direct limit of the direct system $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$ is defined as
$$\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}:= \bigslant{\displaystyle\bigsqcup_{i\in I}\mathcal{G}_{\nu_i}}{\sim}, $$
where $\sim$ is the following equivalence relation: for $a_i\in \mathcal{G}_{\nu_i}$ and $a_j\in \mathcal{G}_{\nu_j}$ with $i\leq j$,
\begin{equation*}\label{relacaoequiv}
a_i\sim a_j \Longleftrightarrow \phi_{ij}(a_i)=a_j.
\end{equation*}
We denote by $[a_i]$ the equivalence class of $a_i$ in $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$. The operations on $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$ are induced from each $\mathcal{G}_{\nu_i}$.
Denoting by $0_i$ the additive neutral element of $\mathcal{G}_{\nu_i}$, it is easy to see that $[0_i]=[0_j]$ for all $i,j\in I$. We write only $[0]$ to denote $[0_i]$, which is the additive neutral element of $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$. Similarly, if $\inv_{\nu_i}(1)$ is the unity (i.e. the multiplicative neutral element) of $\mathcal{G}_{\nu_i}$, it is easy to see that $[\inv_{\nu_i}(1)]=[\inv_{\nu_j}(1)]$ for all $i,j\in I$. The equivalence class $[\inv_{\nu_i}(1)]$ is the unity element of $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$. We also have the following.
\begin{Lema}\label{lemClassLimite}For fixed $i\in I$ and $f\in \K[x]$, consider $[\inv_{\nu_i}(f)]\in \underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$.
\begin{enumerate}
\item We have $\nu_i(f)<\nu_j(f)$ for some $j>i$ if and only if $[\inv_{\nu_i}(f)]=[0]$.
\item For $j\geq i$, if $\nu_i(f)=\nu_j(f)$, then $[\inv_{\nu_i}(f)]=[\inv_{\nu_j}(f)]$. Moreover, if $\nu_i(f)=\nu_j(f)$ for every $j\geq i$, then $[\inv_{\nu_i}(f)]\neq [0]$.
\end{enumerate}
\end{Lema}
\begin{proof}$\,$
\begin{enumerate}
\item If $\nu_i(f)<\nu_j(f)$ for some $j>i$, then
$$\phi_{ij}(\inv_{\nu_i}(f)) = 0_j. $$
Hence, $[\inv_{\nu_i}(f)]=[0]$. On the other hand, if $[\inv_{\nu_i}(f)]=[0]$, then there exists $j\geq i$ such that $$\phi_{ij}(\inv_{\nu_i}(f)) =0_j.$$ By the definition of $\phi_{ij}$, this implies $\nu_i(f)<\nu_j(f)$ and $i<j$.
\item If $\nu_i(f)=\nu_j(f)$ for some $j\geq i$, then
$$\phi_{ij}(\inv_{\nu_i}(f)) = \inv_{\nu_j}(f). $$
Hence, we have $[\inv_{\nu_i}(f)]=[\inv_{\nu_j}(f)]$. Moreover, if $\nu_i(f)=\nu_j(f)$ for every $j\geq i$, then by the preceding item we have $[\inv_{\nu_i}(f)]\neq [0]$.
\end{enumerate}
\end{proof}
Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered subset of $\mathcal{V}$. For every $f\in \K[x]$, we say that $f$ is $\boldsymbol{\mathfrak{v}}$\textbf{-stable} if there exists $i_f\in I$ such that
\begin{equation}\label{eqVStable}
\nu_i(f)=\nu_{i_f}(f) \text{ for every } i\in I \text{ with } i\geq i_f.
\end{equation}
Let $S\subset \mathcal{G}_{\nu} $ be any subset. We denote by $\langle S \rangle$ the additive subgroup of $\mathcal{G}_{\nu}$ generated by $S$. Our main theorem states that the direct limit $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$ is isomorphic to a subgroup of $\mathcal{G}_{\nu}$ generated by a subset determined by $\mathfrak{v}$-stable elements.
\begin{Teo}\label{teoLimVmaiorRstable}Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$ and suppose there exists $\nu\in \mathcal{V}$ such that $\nu_i\leq \nu$ for every $i\in I$.
We define
$$R = \langle \{ \inv_\nu(f)\mid f \text{ is } \mathfrak{v}\text{-stable} \} \rangle\subseteq \mathcal{G}_{\nu} $$
Then $R$ is a subring of $\mathcal{G}_{\nu}$ and
$$\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\cong R.$$
\end{Teo}
\begin{proof} By construction, $R$ is an additive subgroup of $\mathcal{G}_{\nu}$. For $\inv_\nu(f), \inv_\nu(g)\in R$, we take $j = \max\{ i_f, i_g\}$. Then for every $k\geq j$, it follows that
$$\nu_j(fg) = \nu_j(f)+\nu_j(g) = \nu_k(f)+\nu_k(g) = \nu_k(fg). $$
That is, $fg$ is $\mathfrak{v}$-stable and then $\inv_\nu(fg)\in R$. This shows that $R$ is a subring of $\mathcal{G}_{\nu}$.
Consider the map given by
\begin{align*}
\tau: \hspace{0.5cm} R\hspace{0.3cm} &\longrightarrow \underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\\
\inv_{\nu}(f) & \longmapsto [\inv_{\nu_{i_f}}(f)]\\
0& \longmapsto [0],
\end{align*}
where we extend $\tau$ naturally to arbitrary (that is, not necessarily homogeneous) elements of $R$ by additivity. This map is well defined. Indeed,
take $\inv_\nu(f)\in R$. By assumption, $f$ is $\mathfrak{v}$-stable, so there exists $i_f\in I$ such that \eqref{eqVStable} is satisfied. If $j_f$ is another index such that $\nu_i(f)=\nu_{j_f}(f)$ for every $i\in I$ with $i\geq j_f$, then without loss of generality we can take $i_f\leq j_f$. Hence, by Lemma~\ref{lemClassLimite} (2), $[\inv_{\nu_{i_f}}(f)] =[\inv_{\nu_{j_f}}(f)]$. Therefore, $\tau$ is well defined.
Since we extended $\tau$ to arbitrary elements of $R$ via finite sums, this map is a group homomorphism by construction. We now check that $\tau$ is a ring isomorphism.
\begin{itemize}
\item $\tau$ is injective: since it is a group homomorphism, it is enough to check that $\ker(\tau)=\{0\}$. Indeed, given a non-zero element $\inv_\nu(f)\in R$, we know that $\nu_{i_f}(f)= \nu_j(f)$ for every $j\geq i_f$. By Lemma~\ref{lemClassLimite} (2), $\tau(\inv_\nu(f)) = [\inv_{\nu_{i_f}}(f)] \neq [0]$. Hence, $\ker(\tau)=\{0\}$ and $\tau$ is injective.
\item $\tau$ is surjective: take any $[\inv_{\nu_k}(f)]\in \underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}$. If there exists $j>k$ such that $\nu_k(f)<\nu_j(f)$, then by Lemma~\ref{lemClassLimite} (1) we have $[\inv_{\nu_{i_f}}(f)] = [0]=\tau(0)$. On the other hand, if $\nu_k(f) =\nu_j(f)$ for every $j\geq k$, then we can take $i_f=k$ and $[\inv_{\nu_k}(f)] = [\inv_{\nu_{i_f}}(f)] = \tau(\inv_\nu(f))$. Therefore, $\tau$ is surjective.
\item $\tau$ is a ring homomorphism: for any $\inv_\nu(f), \inv_\nu(g)\in R$, we can take $j\in I$ sufficiently large so that $j\geq \max\{i_{fg}, i_f, i_g\}$. We have
\begin{align*}
\tau(\inv_\nu(f) \cdot \inv_\nu(g) ) & = \tau(\inv_\nu(fg))\\
& = [\inv_{\nu_{j}}(fg)]\\
& = [\inv_{\nu_{j}}(f)\cdot \inv_{\nu_{j}}(g)]\\
& = [\inv_{\nu_{j}}(f)]\cdot [\inv_{\nu_{j}}(g)]\\
& = \tau(\inv_\nu(f))\cdot \tau(\inv_\nu(g)).
\end{align*}
Also, $\tau$ preserves the multiplicative neutral element since, by definition, $\tau(\inv_\nu(1)) = [\inv_{\nu_i}(1)]$ (for any $i\in I$), which is the unity of $\underset{\longrightarrow}{\lim}\mathcal{G}_{\nu_i}$.
\end{itemize}
Therefore, we have $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\cong R$
as commutative rings with unity.
\end{proof}
We will classify the totally ordered subsets $ \mathfrak{v}\subset \mathcal{V}$ in three classes using the following propositions.
\begin{Prop}\label{prop3valMenor} (Proposition 2.2 of \cite{novbarnabe}) Assume $\eta, \nu, \mu\in \mathcal{V}$ are such that \linebreak $\eta<\nu<\mu$. For $f\in \K[x]$, if $\eta(f)=\nu(f)$, then $\nu(f)=\mu(f)$.
\end{Prop}
\begin{Prop} (Corollary 2.3 of \cite{novbarnabe}) Let $\{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$. For every $f\in \K[x]$ we have that either $\{\nu_i(f)\}_{i\in I}$ is strictly increasing, or there exists $i_0\in I$ such that $\nu_i(f)=\nu_{i_0}(f)$ for every $i\in I$ with $i\geq i_0$.
\end{Prop}
We consider three cases:
\begin{itemize}
\item $\mathfrak{v}$ has a maximal element $\nu_m$.
\item $\mathfrak{v}$ has no maximal element and every $f\in \K[x]$ is $\mathfrak{v}$-stable.
\item $\mathfrak{v}$ has no maximal element and there is at least one polynomial $q$ that is not $\mathfrak{v}$-stable.
\end{itemize}
\subsection{First and second cases}\label{FirstSecondCases}
\begin{Prop}\label{lemVstable}Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$
such that
every $f\in \K[x]$ is $\mathfrak{v}$-stable.
Define $\nu = \displaystyle\sup_{i\in I}\nu_i: \K[x] \rightarrow \Gamma_\infty$ by $\nu(f) = \nu_{i_f}(f)$.
Then $\nu$ is a valuation on $\K[x]$ such that $\nu_i(f)\leq \nu(f)$ for every $i\in I$ and every $f\in \K[x]$. Moreover, if $\nu'\in \mathcal{V}$ is such that $\nu'\leq \nu$ and $\nu_i\leq \nu'$ for every $i\in I$, then $\nu'=\nu$.
\end{Prop}
\begin{proof}
Take $f,g\in \K[x]$. Then, by assumption, there exist $i_f,\, i_g,\, i_{f+g}$ and $i_{fg}$. Take $j=\max\{i_f,\, i_g,\, i_{f+g}, \,i_{fg}\}$. Hence, $$\nu(fg)=\nu_{j}(fg)=\nu_{j}(f)+ \nu_{j}(g) = \nu(f)+\nu(g)$$
and
$$\nu(f+g) = \nu_{j}(f+g)\geq \min\{ \nu_{j}(f), \nu_{j}(g)\}=\min\{ \nu(f), \nu(g)\}. $$
Also, $\nu(0)=\nu_{i_0}(0)=\infty$ and $\nu(1)=\nu_{i_1}(1)=0$. Therefore, $\nu$ is a valuation on $\K[x]$. In addition, for each $f\in \K[x]$ and $i\in I$, if $i<i_f$, then $\nu_i(f)\leq \nu_{i_f}(f)=\nu(f)$ and if $i\geq i_f$, then $\nu_i(f)=\nu_{i_f}(f)=\nu(f)$. Hence, $\nu_i\leq \nu$.
Moreover, suppose $\nu'\in \mathcal{V}$ is such that $\nu'\leq \nu$ and $\nu_i\leq \nu'$ for every $i\in I$. Thus, for every $f\in \K[x]$ we have
$$\nu(f)\geq \nu'(f)\geq \nu_{i_f}(f)=\nu(f). $$
Therefore, $\nu'=\nu$.
\end{proof}
\begin{Obs}If $\mathfrak{v}= \{\nu_i\}_{i\in I}$ has a maximal element $\nu_{m}$, then every $f$ is $\mathfrak{v}$-stable (take $i_f=m$). Hence, $\nu$ in Proposition~\ref{lemVstable} coincides with $\nu_{m}$.
\end{Obs}
We have the following corollary, which covers the first and second cases.
\begin{Cor}\label{teoLimitSeqStable} Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$ such that
every $f\in \K[x]$ is $\mathfrak{v}$-stable. Consider the direct system $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$. Take $\nu = \displaystyle\sup_{i\in I}\nu_i$ as in Proposition~\ref{lemVstable}. Then $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\cong \mathcal{G}_{\nu}$
as commutative rings with unity.
\end{Cor}
\begin{proof} It follows from Theorem~\ref{teoLimVmaiorRstable} because $R = \mathcal{G}_{\nu}$.
\end{proof}
\subsection{Third case}\label{ThirdCase}
Now we treat the third case. Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$ such that $\mathfrak{v}$ has no maximal element and there is at least one polynomial that is not $\mathfrak{v}$-stable.
Consider the set
$$C(\mathfrak{v}):=\{ f\in \K[x]\mid f \text{ is } \mathfrak{v}\text{-stable }\}.$$
For every $f\in C(\mathfrak{v})$ we set $\mathfrak{v}(f) = \nu_{i_f}(f)$. Let $Q$ be a monic polynomial of smallest degree $d$ not $\mathfrak{v}$-stable and take $\gamma\in \Gamma_\infty$ such that $\gamma>\nu_i(Q)$ for every $i\in I$.
\begin{Prop}\label{lemamuQgammaVal} We have the following.
\begin{enumerate}
\item Consider the map
$$\mu(f_0+f_1Q+\ldots+f_rQ^r)=\underset{0\leq j \leq r}{\min}\{ \mathfrak{v}(f_j)+j\gamma \}, $$
where $f_0+f_1Q+\ldots+f_rQ^r$ is the $Q$-expansion of $f$. Then $\mu$ is a valuation such that $\mu|_\K = \nu_0$.
\item We have $\nu_i<\mu$ for every $i\in I$.
\item We have $\mu = \mu_Q$ and $Q$ is a key polynomial for $\mu$.
\end{enumerate}
\end{Prop}
\begin{proof}$\,$
\begin{enumerate}
\item For the first part, see Theorem 2.4 of \cite{novbarnabe}. We easily see that $\mu\in \mathcal{V}$.
\item Using Proposition 1.21 of \cite{vaquieextension} and Theorem 5.1 of \cite{MacLane}, one can prove that $ \nu_i(f)\leq \mu(f)$ for every $f\in \K[x]$. Also, $\nu_i(Q)<\gamma=\mu(Q)$ for every $i\in I$.
Hence, $\nu_i<\mu$ for every $i\in I$.
\item It follows immediately from the definition of $\mu$ that $\mu = \mu_Q$.
We now prove that $Q$ is a key polynomial for $\mu$. Take $f,g\in \K[x]$ with $\deg(f)<\deg(Q)$ and $\deg(g)<\deg(Q)$ and suppose $fg = lQ+r$ is the $Q$-extension of $fg$. We will prove that $\mu(fg) = \mu(r)<\mu(lQ)$. First we see that $\mu(fg) = \mu(r)$. Suppose, aiming for a contradiction, that $\mu(fg) \neq \mu(r)$. If $fg = lQ+r$ is the $Q$-expansion of $fg$, then $\deg(l),\deg(r)<\deg(Q)$. By the minimality of $\deg(Q)$, all $f,g,l$ and $r$ are $\mathfrak{v}$-stable. Take $i>\max\{i_{f}, i_{g}, i_{l}, i_{r}\}$. Hence, $\nu_i(fg) = \mu(fg)\neq \mu(r) =\nu_i(r)$. We have
$$\nu_i(l)+\nu_i(Q) = \nu_i(lQ)=\nu_i(fg-r)=\min\{ \nu_i(fg), \nu_i(r)\}. $$
However, for $j>i$ we have $\nu_j(l) = \nu_i(l)$, $\nu_j(fg)= \nu_i(fg)$ and $\nu_j(r)=\nu_i(r)$. Hence, $\nu_j(Q) = \min\{ \nu_i(fg), \nu_i(r)\}-\nu_i(l)$ for all $j>i$, contradicting the fact that $Q$ is not $\mathfrak{v}$-stable. Thus, $\mu(fg) = \mu(r)$. Therefore,
$$\mu(lQ)=\mu(l)+\mu(Q) > \nu_i(l)+ \nu_i(Q) \geq \min\{\nu_i(r), \nu_i(fg) \} = \mu(fg)=\mu(r). $$
By Proposition~\ref{propequivPolichave}, $Q$ is a key polynomial for $\mu$.
\end{enumerate}
\end{proof}
\begin{Lema}\label{lemMenorIgualMu} Fix $i\in I$. We have the following.
\begin{enumerate}
\item If $\nu_i(f)<\mu(f)$, then $\nu_i(f)<\nu_j(f)$ for every $j>i$.
\item We have $\nu_i(f) = \mu(f)$ if and only if $\nu_i(f)=\nu_j(f)$ for every $j\geq i$. In particular, $\nu_i(f)=\mu(f)$ implies that $f$ is $\mathfrak{v}$-stable.
\end{enumerate}
\end{Lema}
\begin{proof}$\,$
\begin{enumerate}
\item If there existed $j> i$ such that $\nu_i(f)=\nu_j(f)$, then we would have \linebreak $\nu_i(f)=\nu_j(f)<\mu(f)$ and $\nu_i<\nu_j<\mu$, contradicting Proposition~\ref{prop3valMenor}.
\item Suppose $\nu_i(f)=\mu(f)$. If there existed $j>i$ such that $\nu_i(f)<\nu_j(f)$, then we would have $\mu(f)=\nu_i(f)<\nu_j(f)$, which is a contradiction. Conversely, suppose $\nu_i(f)=\nu_j(f)$ for every $j\geq i$. If $\nu_i(f)<\mu(f)$, then by the preceding item we would have $\nu_i(f)<\nu_j(f)$ for every $j>i$, contradicting our assumption.
\end{enumerate}
\end{proof}
Let $R_Q$ be the additive subgroup of $\mathcal{G}_\mu$ generated by the set $\{ \inv_\mu(f)\mid f\in \K[x]_d\}$. Since $Q$ is a key polynomial for $\mu$, Proposition~\ref{propequivPolichaveRq} guarantees that $R_Q$ is a subring of $\mathcal{G}_\mu$.
\begin{Cor}\label{teoLimitSeqNotStable} Let $\mathfrak{v}= \{\nu_i\}_{i\in I}$ be a totally ordered set in $\mathcal{V}$ such that $\mathfrak{v}$ has no maximal element and there is at least one polynomial that is not $\mathfrak{v}$-stable.
Let $Q$ be a polynomial of smallest degree $d$ that is not $\mathfrak{v}$-stable and take $\gamma\in \Gamma_\infty$ such that $\gamma>\nu_i(Q)$ for every $i\in I$. Take $\mu$ as in Proposition~\ref{lemamuQgammaVal} and $R_Q$ as in the above paragraph. Consider the direct system $\{( \mathcal{G}_{\nu_i}, \phi_{ij})\}^{i,j\in I }_{i\leq j}$. Then $\underset{\longrightarrow}{\lim\,} \mathcal{G}_{\nu_i}\cong R_{Q}$
as commutative rings with unity.
\end{Cor}
\begin{proof} By Theorem~\ref{teoLimVmaiorRstable}, we have $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{\nu_i}\cong R$. We need to show that $R=R_Q$. Since $\deg(f)<d$ implies $f$ is $\mathfrak{v}$-stable (by the minimality of $d=\deg(Q)$), it follows that $R_Q\subseteq R$.
Take $\inv_\mu(f)\in R$, that is, $f$ is $\mathfrak{v}$-stable. Let $f = gQ+f_0$ the euclidean division of $f$ by $Q$. Hence, $\deg(f_0)<d$ and $f_0$ is also $\mathfrak{v}$-stable. We first see that $gQ$ is not $\mathfrak{v}$-stable. Indeed, if $gQ$ was $\mathfrak{v}$-stable, then by Lemma~\ref{lemMenorIgualMu} (2) we would have, for some $i\in I$, $\nu_i(gQ) = \mu(gQ)$, that is, $\nu_i(g)+\nu_i(Q) = \mu(g)+\mu(Q)$. However, we know that $\nu_i(Q)<\mu(Q)$ and $\nu_i(g)\leq \mu(g)$, implying that $\nu_i(g)+\nu_i(Q) < \mu(g)+\mu(Q)$, which is a contradiction. Therefore, $gQ$ is not $\mathfrak{v}$-stable.
Take $i = \max\{i_f, i_{f_0}\}$. Hence,
$$\nu_i(f) = \nu_j(f) \text{ and } \nu_i(f_0) = \nu_j(f_0). $$
for every $j\geq i$.
By Lemma~\ref{lemMenorIgualMu}, for $j\geq i$ we have $$\mu(gQ)> \nu_j(gQ) \text{ and }\mu(f) = \nu_j(f)= \nu_j(f_0)= \mu(f_0).$$
Thus, for $j >i$, we have
$$\mu(f-f_0)=\mu(gQ)>\nu_j(gQ)>\nu_i(qQ) \geq \min\{ \nu_i(f), \nu_i(f_0)\}=\min\{ \mu(f), \mu(f_0)\}.$$
Hence, $\mu(f-f_0)>\mu(f) = \mu(f_0)$, that is, $\inv_{\mu}(f) = \inv_{\mu}(f_0)\in R_Q$. Therefore, $R=R_Q$.
\end{proof}
\section{Limit key polynomials and direct limits}\label{LKPLimit}
Let $\nu$ be a valuation on $\K[x]$.
Take $n\in \N$ and let $\Psi_n$ denote the set of all key polynomials for $\nu$ with degree $n$. Assume that $\Psi_n\neq \emptyset$ and that $\{ \nu(Q)\mid Q\in \Psi_n \}$ does not have a maximal element.
Consider the set
$$\mathcal{K}_n:=\{ f\in K[x]\mid \nu_Q(f)<\nu(f) \text{ for all } Q\in \Psi_n \}.$$
\begin{Def}\label{defLKP2}
A monic polynomial $Q_n\in K[x]$ is called a \textbf{limit key polynomial} for $\Psi_n$
if $Q_n\in \mathcal{K}_n$ and $Q_n$ has the least degree among polynomials in $\mathcal{K}_n$.
\end{Def}
\begin{Obs} If a polynomial $Q_n$ satisfies Definition~\ref{defLKP2}, then it follows from the minimality of $\deg(Q_n)$, the non-existence of maximal element for $\{ \nu(Q)\mid Q\in \Psi_n \}$ and Proposition~\ref{propequivPolichave} (ii) that $Q_n$ is indeed a key polynomial for $\nu$.
\end{Obs}
Consider the following relation on $\Psi_n$:
\begin{equation}\label{QmenorQlinha}
Q\preceq Q' \Leftrightarrow \nu_Q\leq \nu_{Q'} \text{ and } Q\prec Q' \Leftrightarrow \nu_Q<\nu_{Q'}.
\end{equation}
We note that if we take $Q,Q'\in \Psi_n$, then either $Q\preceq Q'$ or $Q'\preceq Q$. Indeed, considering $\delta(Q), \delta(Q')\in \Gamma_\Q$, since they belong to a totally ordered group, we have $\delta(Q)\leq \delta(Q')$ or $\delta(Q')\leq \delta(Q)$. By Lemma~\ref{lempolichaveEpsilonmenor} (2), $\nu_{Q}(f)\leq \nu_{Q'}(f)$ or $\nu_{Q'}(f)\leq \nu_{Q}(f)$, that is, $Q\preceq Q'$ or $Q'\preceq Q$.
Therefore,
$(\Psi_n, \preceq)$ is a directed set.
\begin{Cor}
Consider the family of graded rings $\{ \mathcal{G}_Q\}_{Q\in \Psi_n}$ and, for $Q \preceq Q'$, let $\phi_{QQ'}$ be the map
\begin{align*}
\phi_{QQ'}: \hspace{0.5cm} \mathcal G_Q\hspace{0.3cm} &\longrightarrow \mathcal G_{Q'}\\
\inv_Q(f) &\longmapsto \begin{cases}
\inv_{Q'}(f)& \mbox{ if }\nu_Q(f)=\nu_{Q'}(f)\\
0&\mbox{ if }\nu_Q(f)<\nu_{Q'}(f),
\end{cases}
\end{align*}
extended naturally to arbitrary (that is, not necessarily homogeneous) elements of $\mathcal G_Q$. Then $\{( \mathcal{G}_Q, \phi_{QQ'})\}^{Q,Q'\in \Psi_n}_{Q\preceq Q'}$ is a direct system over $\Psi_n$.
\end{Cor}
\begin{proof} This follows from Lemma~\ref{lemDirectSystemValTotalOrder} because $\{ \nu_Q\}_{Q\in \Psi_n}$ is a totally ordered set.
\end{proof}
In the next lemma we gather some properties of limit key polynomials for $\Psi_n$. These properties will allow us to prove that $\mathfrak{v} = \{\nu_Q \}_{Q\in \Psi_n}$ is a totally ordered subset of $\mathcal{V}$ with no maximal element.
\begin{Lema}\label{lemPropQn}Let $Q_n$ be a limit key polynomial for $\Psi_n$.
\begin{enumerate}
\item We have $\delta(Q)<\delta(Q_n)$ for every $Q\in \Psi_n$. Hence, $\deg(Q_n)\geq n$.
\item For every $Q\in \Psi_n$, we have $\nu_Q(f)\leq \nu_{Q_n}(f)$ for all $f\in \K[x]$ and \linebreak $\nu_Q(Q_n)<\nu_{Q_n}(Q_n)=\nu(Q_n)$. Also, $\nu_{Q_i}(Q_n)<\nu_{Q_j}(Q_n)$ for every $Q_i\prec Q_j$ in $\Psi_n$.
\item If $\deg(f)< \deg(Q_n)$, then there exists $Q_f\in \Psi_n$ such that $\nu_{Q_f}(f)=\nu_{Q_n}(f)=\nu(f)$. Moreover, $\nu_{Q}(f)=\nu_{Q_f}(f)$ for every $Q\in \Psi_n$ with $Q_f \preceq Q$.
\end{enumerate}
\end{Lema}
\begin{proof}
\noindent \begin{enumerate}
\item Suppose $\delta(Q_n)\leq \delta(Q)$. Hence, by Lemma~\ref{lempolichaveEpsilonmenor} (2), $\nu_{Q_n}(f)\leq \nu_Q(f)$ for every $f\in \K[x]$. In particular, $\nu(Q_n)=\nu_{Q_n}(Q_n)\leq \nu_Q(Q_n)$. However, this contradicts $Q_n\in \mathcal{K}_n$. Therefore, $\delta(Q)<\delta(Q_n)$ for every $Q\in \Psi_n$. Now suppose $\deg(Q_n)<n=\deg(Q)$. By Lemma~\ref{prop3itensQQlinhaPoliChaves} (1), we would have $\delta(Q_n)<\delta(Q)$, a contradiction. Thus, $\deg(Q_n)\geq n$.
\item By Lemma~\ref{lempolichaveEpsilonmenor}, Lemma~\ref{prop3itensQQlinhaPoliChaves} and the preceding item, we have \linebreak $\nu_Q(f)\leq \nu_{Q_n}(f)$ for all $f\in \K[x]$ and $\nu_Q(Q_n)<\nu_{Q_n}(Q_n)=\nu(Q_n)$ for every $Q\in \Psi_n$.
Also, take $Q_i\prec Q_j$, that is, there exists $g\in \K[x]$ such that $\nu_{Q_i}(g)<\nu_{Q_j}(g)$. Thus, by Lemma~\ref{lempolichaveEpsilonmenor} (2), we must have $\delta(Q_i)<\delta(Q_j)$. Since $\nu_{Q_j}(Q_n)<\nu(Q_n)$, we see by Lemma~\ref{lempolichaveEpsilonmenor} (4) that $\nu_{Q_i}(Q_n)<\nu_{Q_j}(Q_n)$.
\item Take $f\in \K[x]$ such that $\deg(f)<\deg(Q_n)$. By the preceding item, \linebreak $\nu_Q(f)\leq \nu_{Q_n}(f)=\nu(f)$ for every $Q\in \Psi_n$. Since $Q_n$ has the minimal degree among polynomials in $\mathcal{K}_n$, there exists $Q_f\in \Psi_n$ such that $\nu_{Q_f}(f)= \nu_{Q_n}(f)=\nu(f)$. Now, if $Q\in \Psi_n$ is such that $Q_f\preceq Q$, then
$$\nu_{Q_n}(f)\geq \nu_Q(f)\geq\nu_{Q_f}(f) = \nu_{Q_n}(f). $$
Hence, $\nu_Q(f)=\nu_{Q_f}(f)$.
\end{enumerate}
\end{proof}
\begin{Cor}\label{corQnNotvStable} Let $\nu$ be a valuation on $\K[x]$ extending $\nu_0$. Let $Q_n$ be a limit key polynomial for $\Psi_n$. Then $\mathfrak{v} = \{\nu_Q \}_{Q\in \Psi_n}$ is a totally ordered subset of $\mathcal{V}$ such that $\mathfrak{v}$ has no maximal element. Also, $Q_n$ is a polynomial of least degree that is not $\mathfrak{v}$-stable. If we take $\gamma=\nu(Q_n)$, then $\nu_{Q_n}(f)=\underset{0\leq j \leq r}{\min}\{ \mathfrak{v}(f_j)+j\gamma \}$.
\end{Cor}
\begin{proof} We see that $\mathfrak{v} = \{\nu_Q \}_{Q\in \Psi_n}$ is a totally ordered set since,
given $Q_i,Q_j\in \Psi_n$ we must have $Q_i \preceq Q_j$ or $Q_j \preceq Q_i$ and then $\nu_{Q_i}\leq \nu_{Q_j}$ or $\nu_{Q_j}\leq \nu_{Q_i}$.
\vspace{0.3cm}
Suppose, aiming for a contradiction, that $\mathfrak{v} $ has a maximal element $\nu_{Q_k}$. Then $\nu_{Q}(f)\leq \nu_{Q_k}(f)$ for all $Q\in \Psi_n$ and $f\in \K[x]$. Since $\{ \nu(Q)\mid Q\in \Psi_n \}$ does not have a maximal element, there exists $Q_i\in \Psi_n$ such that $\delta(Q_k)<\delta(Q_i)$ (Lemma~\ref{prop3itensQQlinhaPoliChaves}). By Lemma~\ref{lempolichaveEpsilonmenor} (2), $\nu_{Q_k}(f)\leq \nu_{Q_i}(f)$, which implies $\nu_{Q_k}(f)=\nu_{Q_i}(f)$ for all $f\in \K[x]$. However, by Lemma~\ref{lemPropQn} (2), we have $\nu_{Q_i}(Q_n)<\nu(Q_n)$ and this, together with Lemma~\ref{lempolichaveEpsilonmenor} (4), implies $\nu_{Q_k}(Q_n)<\nu_{Q_i}(Q_n)$, a contradiction to the maximality of $\nu_{Q_k}$. Therefore, $\mathfrak{v}$ has no maximal element.
\vspace{0.3cm}
By Lemma~\ref{lemPropQn} (2), we have $\nu_{Q_i}(Q_n)<\nu_{Q_j}(Q_n)$ for $Q_i\prec Q_j$, that is, $Q_n$ is not $\mathfrak{v}$-stable. Also, we have $\nu_{Q}(Q_n)<\nu(Q_n)=\gamma$ for all $Q\in \Psi_n$.
Moreover, by Lemma~\ref{lemPropQn} (3), if $\deg(g)<\deg(Q_n)$, then there exists $Q_g\in \Psi_n$ such that $\nu_{Q_g}(g)= \nu_{Q}(g) $ for every $Q\in \Psi_n$ with $Q_g\preceq Q$. That is, $g$ is $\mathfrak{v}$-stable and $\nu(g) = \nu_{Q_g}(g)=\mathfrak{v}(g)$.
Thus, for every $f\in \K[x]$, we can write $f = f_0+f_1Q_n+\ldots+f_rQ_{n}^r$ and conclude that
$$\nu_{Q_n}(f) = \underset{0\leq j \leq r}{\min}\{ \nu(f_j)+j\nu(Q_n) \} = \underset{0\leq j \leq r}{\min}\{ \mathfrak{v}(f_j)+j\gamma \}. $$
\end{proof}
\begin{Cor}\label{corLimQn} We have $\underset{\longrightarrow}{\lim\,}\mathcal{G}_Q\cong R_{Q_n}$ as commutative rings with unity.
\end{Cor}
\begin{proof}
It follows from Corollary~\ref{teoLimitSeqNotStable}.
\end{proof}
\section{Valuation-algebraic valuations and direct limits}\label{ValAlgLimit}
In this last section, we give an application for Corollary~\ref{teoLimitSeqStable}. We start defining the concepts of valuation-transcendental and valuation-algebraic valuations.
\begin{Def} A valuation $\nu$ on $\K[x]$ extending $\nu_0$ is called \linebreak \textbf{value-transcendental} if either it is not Krull or the quotient group $\nu(\K[x])/\nu_0\K$ is not a torsion group. We say that $\nu$ is \textbf{residue-transcendental} if it is Krull and the field extension $\K(x)\nu\mid \K\nu_0$ is transcendental.
\end{Def}
\begin{Def}\label{defValAlg} A valuation $\nu$ on $\K[x]$ extending $\nu_0$ is called \textbf{valuation-transcendental} if it is value-transcendental or residue-transcendental. We say that $\nu$ is \textbf{valuation-algebraic} if it is not valuation-transcendental.
\end{Def}
\begin{Obs}\label{obsAbhyankar} By Abhyankar inequality (see \cite{zar}, p.330), we see that a valuation cannot be value-transcendental and residue-transcendental at the same time.
\end{Obs}
\begin{Obs}Explicitly, a valuation $\nu$ on $\K[x]$ extending $\nu_0$ is valuation-algebraic if it is a Krull valuation, $\nu(\K[x])/\nu_0\K$ is a torsion group and $\K(x)\nu\mid \K\nu_0$ is an algebraic field extension.
\end{Obs}
\begin{Lema}\label{lemValuationAlgTrunc} Let $\nu$ be a valuation-algebraic valuation on $\K[x]$ extending $\nu_0$ on $\K$. Suppose that $q$ is a polynomial such that $\nu_q$ is a valuation. Then $\nu_q$ is residue-transcendental.
\end{Lema}
\begin{proof} By Theorem 3.1 of \cite{josneiKeyPolyMinimalPairs}, we have that $\nu_q$ is valuation-transcendental. Given $f\in \K[x]$, $f\neq 0$, we know that $$\nu_q(f) = \min_{0\leq i \leq r} \{ \nu(f_i)+i\nu(q) \}\in \nu(\K[x]),$$ where $f_0, \ldots, f_r$ are the coefficients of the $q$-expansion of $f$. Since $\nu$ is valuation-algebraic, $\nu(\K[x])/\nu_0\K$ is a torsion group. Hence, $\nu_q(f)$ is a torsion element in $\nu_q(\K[x])/\nu_0\K$ for every $f\in \K[x]$, $f\neq 0$. Therefore, $\nu_q$ is not value-transcendental and, due to Remark~\ref{obsAbhyankar}, we see that $\nu_q$ is a residue-transcendental valuation.
\end{proof}
To a given valuation-algebraic valuation $\nu$, we are going to associate a totally ordered subset of $\mathcal{V}$. In order to do that, we use the results of \cite{josneiKeyPolyPropriedades} on complete sets, which we define bellow.
\begin{Def} Let $\nu$ be a valuation on $\K[x]$. A set $\textbf{Q}\subset \K[x]$ is called a \textbf{complete set} for $\nu$ if for every $f\in \K[x]$ there exists $Q\in \textbf{Q}$ with $\deg(Q)\leq \deg(f)$ such that $\nu_Q(f)=\nu(f)$.
\end{Def}
\begin{Prop}\label{propcompleteset}(Theorem 1.1 of \cite{josneiKeyPolyPropriedades})
Every valuation $\nu$ on $\K[x]$ admits a complete set \textbf{Q} of key polynomials.
\end{Prop}
\begin{Obs} As remarked in \cite{josneimonomial}, the definition of complete set in Theorem 1.1 of \cite{josneiKeyPolyPropriedades} does not require that $\deg(Q)\leq \deg(f)$. However, the proof of the Theorem shows that this inequality always holds.
\end{Obs}
\begin{Prop}\label{propValAlgStable} Let $\nu\in \mathcal{V}$ be a valuation-algebraic valuation. Then there exists a totally ordered subset $\mathfrak{v} = \{ \nu_i \}_{i\in I}\subset \mathcal{V}$ without maximal element such that every $f\in \K[x]$ is $\mathfrak{v}$-stable, $\nu = \displaystyle \sup_{i\in I}\nu_i$ and each $\nu_i$ is residue-transcendental.
\end{Prop}
\begin{proof}
By Proposition~\ref{propcompleteset}, there exists a complete set \textbf{Q} of key polynomials for $\nu$. Consider $\mathfrak{v} = \{ \nu_{Q} \}_{Q\in \textbf{Q}}\subset \mathcal{V}$, which is totally ordered due to Lemma~\ref{lempolichaveEpsilonmenor} (2).
We order the set $\textbf{Q}$ by posing $Q\preceq Q'$ if and only if $\nu_Q\leq \nu_{Q'}$. By Lemma~\ref{lemValuationAlgTrunc}, each $\nu_Q$ is residue-transcendental.
We now show that every $f\in \K[x]$ is $\mathfrak{v}$-stable. Indeed, for every $f\in \K[x]$, there exists $Q\in \textbf{Q}$ such that $\deg(Q)\leq \deg(f)$ and $\nu_Q(f)=\nu(f)$. If $Q\preceq Q'$, then we have the following:
\begin{itemize}
\item if $\delta(Q')\leq \delta(Q)$, then $\nu_{Q'}(f)\leq \nu_Q(f)$ (Lemma~\ref{lempolichaveEpsilonmenor} (2)), that is, $\nu_{Q'}(f) = \nu_Q(f)=\nu(f)$;
\item if $\delta(Q)< \delta(Q')$, then by Lemma~\ref{lempolichaveEpsilonmenor} (3) we have that $\nu_Q(f)=\nu(f)$ implies $\nu_{Q'}(f)=\nu(f)$.
\end{itemize}
Hence, $Q\preceq Q'$ implies $\nu_{Q'}(f)=\nu_{Q}(f)=\nu(f)$. That is, $f$ is $\mathfrak{v}$-stable and $\nu=\underset{Q\in \textbf{Q}}{\sup} \nu_Q$ as in Proposition~\ref{lemVstable}.
Moreover, suppose $\{ \nu_{Q} \}_{Q\in \textbf{Q}}$ has a maximal element. Then $\nu =\underset{Q\in \textbf{Q}}{\sup} \nu_Q = \nu_{Q_m}$ for some $Q_m\in \textbf{Q}$, which is a contradiction since $\nu$ is valuation-algebraic and $\nu_{Q_m}$ is residue-transcendental. Therefore, $\{ \nu_{Q} \}_{Q\in \textbf{Q}}$ does not have a maximal element.
\end{proof}
\begin{Cor}\label{corValgLim} Let $\nu\in \mathcal{V}$ be a valuation-algebraic valuation and take
$\mathfrak{v} = \{ \nu_{Q} \}_{Q\in \textbf{Q}}\subset \mathcal{V}$ as in Proposition~\ref{propValAlgStable}. Then $\underset{\longrightarrow}{\lim\,}\mathcal{G}_{Q}\cong \mathcal{G}_{\nu}$
as commutative rings with unity.
\end{Cor}
\begin{proof}
It follows from Corollary~\ref{teoLimitSeqStable}.
\end{proof}
| 61,285
|
TITLE: Closest hermitian matrix in operator norm induced metric
QUESTION [2 upvotes]: Let $M (n)$ be the set of $n \times n$ complex matrices and $\| \cdot \|$ an
operator norm on $M (n)$ arising from some inner-product on $\mathbb{C}^n$.
Note the Cartesian Decomposition of $A \in M (n)$ is
$$ A = \left( \frac{A + A^{\ast}}{2} \right) + i \left( \frac{A -
A^{\ast}}{2 i} \right) . $$
I'd like to show $\frac{A + A^{\ast}}{2}$ is the closest hermitian to $A$ in
the $\| \cdot \|$ induced metric. That is, how to prove
$$ \left\| A - \frac{A + A^{\ast}}{2} \right\| \leqslant \| A - H \| $$
for all hermitian matrices $H$ in $M(n)$?
For the Frobenius norm, I proved the result using $\| H+iK \|_F^2 = \| H\|_F^2+\|K\|_F^2$ (for hermitian $H$ and $K$). Unfortunately, this strategy doesn't work here, and I'm unsure what else to try.
REPLY [1 votes]: Yes, this is true. Note that:
$$\left\|A - \frac{A + A^*}{2}\right\| = \left\|\frac{A - A^*}{2}\right\| = \left\|\frac{(A - H) - (A - H)^*}{2}\right\|,$$
for all Hermitian $H$. Given $A - H$ ranges over all $M(n)$ for $A \in M(n)$ and Hermitian $H$, the conjecture is equivalent to
$$\left\|\frac{A - A^*}{2}\right\| \le \|A\|$$
for all $A \in M(n)$.
Recall that $\|A\|$ is the greatest singular value of $A$, and since $A^*A$ and $AA^*$ share the same (non-zero) eigenvalues, we therefore have $\|A^*\| = \|A\|$. Therefore,
$$\left\|\frac{A - A^*}{2}\right\| \le \frac{\|A\| + \|A^*\|}{2} = \|A\|,$$
as required.
| 102,722
|
\begin{document}
\title{Eigenfunction expansions associated with operator differential equation depending on spectral parameter nonlinearly}
\author{Volodymyr Khrabustovskyi}
\address{Ukrainian State Academy of Railway Transport, Kharkiv, Ukraine}
\email{v{\_}khrabustovskyi@ukr.net}
\subjclass[2000]{Primary 34B05, 34B07, 34L10}
\keywords{Relation generated by pair of differential expressions
one of which depends on spectral parameter in nonlinear manner,
non-injective resolvent, generalized resolvent}
\begin{center}
{\Large \textbf{Eigenfunction expansions associated with operator differential equation depending on spectral parameter nonlinearly}}\\\medskip {\large Volodymyr Khrabustovskyi}\\\medskip
{\small Ukrainian
State Academy of Railway Transport, Kharkiv, Ukraine}\\
{\small v{\_}khrabustovskyi@ukr.net}
\end{center}\vspace{13px}
\begin{flushright}
\noindent \textit{Dedicated to my teacher Professor F.S. Rofe-Beketov\\ on the occasion of his jubilee}\bigskip
\end{flushright}
{\small \noindent \textbf{Abstract.} For operator differential equation which depends
on the spectral parameter in the Nevanlinna manner we obtain the expansions in eigenfunctions.\bigskip
\noindent \textbf{Keywords and phrases:} Relation generated by
pair of differential expressions one of which depends on spectral
parameter in nonlinear manner, non-injective resolvent,
generalized resolvent, Weyl type operator function and solution, eigenfunction expansion.\medskip
\noindent \textbf{2000 MSC:} 34B05, 34B07, 34L10}\bigskip
\section*{Introduction}
We consider either on finite or infinite interval operator
differential equation of arbitrary order
\begin{gather}
\label{GEQ__1_} l_\lambda[y]=m[f],\ t\in\bar{\mathcal{I}},\
\mathcal{I}=(a,b)\subseteq\mathbb{R}^1
\end{gather}
in the space of vector-functions with values in the separable
Hilbert space $\mathcal{H}$, where
\begin{gather}
\label{GEQ__2_} l_\lambda[y]=l[y]-\lambda m[y]-n_\lambda[y],
\end{gather}
$l[y],m[y]$ are symmetric operator differential expression. The
order of $l_\lambda[y]$ is equal to $r>0$. For the expression
$m[y]$ the subintegral quadratic form $m\{y,y\}$ of its Dirichlet
integral $m[y,y]=\int_{\mathcal{I}}m\{y,y\}dt$ is nonnegative for
$t\in\bar{\mathcal{I}}$. The leading coefficient of the expression
$m[y]$ may lack the inverse from $B(\mathcal{H})$ for any
$t\in\bar{\mathcal{I}}$ and even it may vanish on some intervals.
For the operator differential expression $n_\lambda[y]$ the form
$n_\lambda\{y,y\}$ depends on $\lambda$ in the Nevanlinna manner
for $t\in\bar{\mathcal{I}}$. Therefore the order $s\geq 0$ of
$m[y]$ is even and $\leq r$.
In paper \cite{KhrabMAG} (see also \cite{KhrabArxiv}) in the Hilbert space $L^2_m(\mathcal{I})$ with metrics generated
by the form $m[y,y]$ for equation
(\ref{GEQ__1_})-(\ref{GEQ__2_}) we constructed analogs
$R(\lambda)$ of the generalized resolvents which in general are
non-injective and which possess the following representation:
\begin{gather}\label{GEQ__3_}
R(\lambda)=\int_{\mathbb{R}^1}{dE_\mu\over \mu-\lambda}
\end{gather}
where $E_\mu$ is a generalized spectral family for which
$E_\infty$ is less or equal to the identity operator.
This analogue in an integro-differential operator depending on the characteristic operator of
the equation
\begin{gather}\label{GEQ__5_}
l_\lambda[y]=-{(\Im l_\lambda)[f]\over \Im\lambda},\
t\in\bar{\mathcal{I}},
\end{gather}
where $(\Im l_\lambda)[f]={1\over 2i}(l[f]-l^*[f])$. This characteristic operator was defined in \cite{KhrabArxiv,KhrabMAG}. It is an analogue of the characteristic matrix of scalar differential operator \cite{Shtraus1} (see also \cite[p. 280]{Naimark}). The operator $R(\lambda)$ in the case $n_\lambda[y]\equiv 0$ is the generalized resolvent of the minimal relation corresponding to equation \eqref{GEQ__1_} (see the details in \cite{Khrab6,KhrabMAG,KhrabArxiv}).
In this paper we calculate $E_{\Delta } $ and derive an inequality of Bessel type. In the case when the expression $n_{\lambda } \left[y\right]$ submits in a special way to the expression $m\left[y\right]$ we obtain the inversion formulae and the Parseval equality. The general results obtained in the work are illustrated on the example of equation \eqref{GEQ__1_} with coefficients which are periodic on the semi-axes. We remark that in the case $n_{\lambda } \left[y\right]\equiv 0$ it follows from \cite{Khrab3,Khrab6} that if $\mathcal{I}={\mathbb R}^{1} ,\, r>s$ and $\dim \mathcal{H}<\infty $ then $E_{\mu } $ for equation \eqref{GEQ__1_} with periodic coefficients has no jumps. (For $r=s$ in the described case $E_{\mu } $ may have jump (see e.g. \cite{Khrab6})). We show that in contrast to the case $n_{\lambda } \left[y\right]\equiv 0$ if $r=1,\, \dim \mathcal{H}=2$ then $E_{\mu } $ for equation \eqref{GEQ__1_} with periodic coefficients on the axis may have jump.
In the case $n_{\lambda } \left[y\right]\equiv 0$ the eigenfunction expansion results above are obtained in paper \cite{Khrab6}, which contains its comparison with results, which was obtained earlier for this case.
Eigenfunction expansions for differential operators and relations are considered in the monographs \cite{CodLev,DS,Atkinson,Berez1,Berez2,LyaSto,Marchenko,RBKholkin,Sakhno}. Let us notice that for infinite systems first eigenfunction expansion results are obtained in \cite{RB60} for operator Sturm-Liouvill equation. (Later it was done in \cite{Gor} in another way). The expansion in eigenfunctions of operator equation of highest order (analogous to scalar case \cite{Shtraus1}) was obtained in \cite{Bruk1}).
Also for the case of the half-axis we obtain for equation \eqref{GEQ__1_} a generalization of the result from \cite{Shtraus2} on the expansion in solutions of scalar Sturm-Liouville equation which satisfy in the regular end-point the boundary condition depending on a spectral parameter. To do this we introduce for equation \eqref{GEQ__1_} Weyl type functons and solutions. Such solutions for operator equation of first order containing spectral parameter in Nevanlinna manner was constructed in \cite{Khrab5}. For finite canonical systems
the parametrisation of Weyl functions for such equation was obtained in \cite{Orlov} with the help of $\mathcal{J}$-theory. For these systems depending on spectral parameter in linear manner, such parametrization in another form was obtained in \cite{0} with the help of abstract Weyl function. The method of studying of differential operators and relations based on use of the abstract Weyl function and its generalization (Weyl family) was proposed in \cite{DerMalamud,DHMdS1,DHMdS2}.
A part of the results of this paper is contained in the preliminary form in preprint \cite{KhrabArxiv}.
We denote by $(\ .\ )$ and $\|\cdot\|$ the scalar product and the
norm in various spaces with special indices if it is necessary. For differential operation $l$ we denote
$\Re l={1\over 2}(l+l^*)$, $\Im l={1\over 2i}(l-l^*)$.
Let an interval $\Delta \subseteq \mathbb{R}^1 ,\,
f\left(t\right)\, \left(t\in \Delta \right)$ be a function with
values in some Banach space $B$. The notation $f\left(t\right)\in
C^{k} \left(\Delta,B\right),\ k=0,\, 1,\, ...$ (we omit the index
$k$ if $k=0$) means, that in any point of $\Delta$
$f\left(t\right)$ has continuous in the norm $\left\| \, \cdot \,
\right\| _{B}$ derivatives of order up to and including $l$ that
are taken in the norm $\left\| \, \cdot \,\right\| _{B} $; if
$\Delta$ is either semi-open or closed interval then on its ends
belonging to $\Delta$ the one-side continuous derivatives exist.
The notation $f\left(t\right)\in C_{0}^{k} \left(\Delta,B\right)$
means that $f\left(t\right)\in C^{k} \left(\Delta,B \right)$ and
$f\left(t\right)=0$ in the neighbourhoods of the ends of $\Delta$.
\section{Characteristic operator. Weyl type operator function and solution}
In order to formulate the eigenfunction expansion results we present in this section several results from \cite{KhrabMAG} (see also \cite{KhrabArxiv}). Comparing with \cite{KhrabMAG,KhrabArxiv} some of these results are given here in either more general or more weak form. Lemmas \ref{lm11}, \ref{lm12}, Theorem \ref{th12} and Corollary \ref{cor11} are new.
We consider an operator differential equation in separable Hilbert
space $\mathcal{H}_{1} $:
\begin{equation} \label{GEQ__46_}
\frac{i}{2} \left(\left(Q\left(t\right)x\left(t\right)\right)^{{'}
} +Q^{*} \left(t\right)x'\left(t\right)\right)-H_{\lambda }
\left(t\right)x\left(t\right)=W_{\lambda }
\left(t\right)F\left(t\right),\quad t\in \bar{\mathcal{I}},
\end{equation}
where $Q\left(t\right),\, \left[\Re \, Q\left(t\right)\right]^{-1}
,\, H_{\lambda } \left(t\right)\in B\left(\mathcal{H}_{1}
\right),\, Q\left(t\right)\in C^{1}
\left(\bar{\mathcal{I}},B\left(\mathcal{H}_{1} \right)\right)$;
the operator function $H_{\lambda } \left(t\right)$ is continuous
in $t$ and is Nevanlinna's in $\lambda $. Namely the following
condition holds:
(\textbf{A}) The set $\mathcal{A}\supseteq \mathbb{C}\setminus
\mathbb{R}^1$ exists, any its point have a neighbourhood
independent of $t\in \bar{\mathcal{I}}$, in this neighbourhood
$H_{\lambda } \left(t\right)$ is analytic $\forall t\in
\bar{\mathcal{I}};\, \forall \lambda \in \mathcal{A}\, H_{\lambda
} \left(t\right)=H^*_{\bar \lambda}(t)\in
C\left(\bar{\mathcal{I}},B\left(\mathcal{H}_1\right)\right)$; the
weight $W_{\lambda } \left(t\right)=\Im H_{\lambda }
\left(t\right)/\Im \lambda \ge 0\left(\Im \lambda \ne 0\right)$.
In view of \cite{Khrab5} $\forall \mu \in \mathcal{A}\bigcap
\mathbb{R}^1:\, W_{\mu } \left(t\right)=\left.\partial H_{\lambda}
\left(t\right)/\partial \lambda\right|_{\lambda=\mu} $ is Bochner
locally integrable in the uniform operator topology.
For convenience we suppose that $0\in \bar{\mathcal{I}}$ and we
denote $\Re \, Q\left(0\right)=G$.
Let $X_{\lambda } \left(t\right)$ be the operator solution of
homogeneous equation \eqref{GEQ__46_} satisfying the initial
condition $X_{\lambda } \left(0\right)=I$, where $I$ is an
identity operator in $\mathcal{H}_{1} $.
For any $\alpha ,\, \beta \in \bar{\mathcal{I}},\, \alpha \le
\beta $ we denote $\Delta _{\lambda } \left(\alpha ,\beta
\right)=\int _{\alpha }^{\beta }X_{\lambda }^{*}
\left(t\right)W_{\lambda } \left(t\right)X_{\lambda }
\left(t\right) dt$,\\ $N=\left\{h\in \mathcal{H}_{1} \left|h\in
Ker\Delta _{\lambda } \left(\alpha ,\beta \right)\right. \forall
\alpha ,\beta \right\},P$ is the ortho-projection onto $N^{\bot }
$. $N$ is independent of $\lambda \in \mathcal{A}$ \cite{Khrab5}.
For $x\left(t\right)\in \mathcal{H}_{1} $ we denote
$U\left[x\left(t\right)\right]=\left(\left[\Re \,
Q\left(t\right)\right]x\left(t\right),x\left(t\right)\right)$.
\begin{definition}\cite{Khrab4,Khrab5}
An analytic operator-function $M\left(\lambda \right)=M^{*}
\left(\bar{\lambda }\right)\in B\left(\mathcal{H}_{1} \right)$ of
non-real $\lambda $ is called a characteristic operator of
equation \eqref{GEQ__46_} on $\mathcal{I}$,
if for $\Im \lambda \ne 0$ and for any $\mathcal{H}_{1} $ -
valued vector-function $F\left(t\right)\in L_{W_{\lambda } }^{2}
\left(\mathcal{I}\right)$ with compact support the corresponding
solution $x_{\lambda } \left(t\right)$ of equation
\eqref{GEQ__46_} of the form
\begin{equation} \label{GEQ__47_}
x_{\lambda } \left(t,F\right)=\mathcal{R}_\lambda F=\int
_{\mathcal{I}}X_{\lambda } \left(t\right) \left\{M\left(\lambda
\right)-\frac{1}{2} sgn\left(s-t\right)\left(iG\right)^{-1}
\right\}X_{\bar{\lambda }}^{*} \left(s\right)W_{\lambda }
\left(s\right)F\left(s\right)ds
\end{equation}
satisfies the condition
\begin{gather}\label{GEQ__47++_}
\left(\Im \lambda \right)\mathop{\lim }\limits_{\left(\alpha
,\beta \right)\uparrow \mathcal{I}} \left(U\left[x_{\lambda }
\left(\beta ,F\right)\right]-U\left[x_{\lambda } \left(\alpha
,F\right)\right]\right)\le 0,\ \Im \lambda \ne
0.
\end{gather}
\end{definition}
Let us note that in \cite{Khrab5} characteristic operator was defined if
$Q(t)=Q^*(t)$. Our case is equivalent to this one since equation
\eqref{GEQ__46_} coincides with equation of
\eqref{GEQ__46_} type with $\Re Q(t)$ instead of $Q(t)$ and
with $H_\lambda(t)-{1\over 2}\Im Q'(t)$ instead of $H_\lambda(t)$.
The properties of characteristic operator and sufficient conditions of the characteristic operators
existence are obtained in \cite{Khrab4,Khrab5}.
In the case $\mathrm{dim}\mathcal{H}_1<\infty$,
$Q(t)=\mathcal{J}=\mathcal{J}^*=\mathcal{J}^{-1}$, $-\infty<a=0$
the description of characteristic operators was obtained in \cite{Orlov} (the
results of \cite{Orlov} were specified and supplemented in
\cite{Khrab+}). In the case $\mathrm{dim} \mathcal{H}_1=\infty$
and $\mathcal{I}$ is finite the description of characteristic operators was obtained
in \cite{Khrab5}. These descriptions are obtained under the
condition that
\begin{gather}\label{star8}
\exists\lambda_0\in \mathcal{A},\ [\alpha,\beta]\subseteq
\overline{\mathcal{I}}:\ \Delta_{\lambda_0}(\alpha,\beta)\gg 0.
\end{gather}
\begin{definition}\label{def+}
\cite{Khrab4,Khrab5} Let $M\left(\lambda \right)$ be the characteristic operator of
equation (\ref{GEQ__46_}) on $\mathcal{I} $. We say that the
corresponding condition (\ref{GEQ__47++_}) is separated for
nonreal $\lambda =\mu _{0} $ if for any $\mathcal{H}_1$-valued
vector function $f\left(t\right)\in L_{W_{\mu_{0} }(t) }^{2}
\left(\mathcal{I} \right)$ with compact support the following
inequalities holds simultaneously for the solution $x_{\mu _{0} }
\left(t\right)$ (\ref{GEQ__47_}) of equation
(\ref{GEQ__46_}):
\begin{gather}
\label{12} \displaystyle \lim\limits_{\alpha\downarrow a}\Im\mu
_{0} U\left[x_{\mu _{0} } \left(\alpha\right)\right]\ge 0,\quad
\mathop{\lim }\limits_{\beta \uparrow b} \Im\mu _{0} U\left[x_{\mu
_{0} } \left(\beta \right)\right]\le 0.
\end{gather}
\end{definition}
\begin{theorem}\cite{Khrab4,Khrab5}\label{th++}
Let $P=I$, $M\left(\lambda \right)$ be the characteristic operator of equation
(\ref{GEQ__46_}), $\mathcal{P}(\lambda)=iM(\lambda)G+{1\over 2}I$, so that we have the following representation
\begin{gather}
\label{13} \displaystyle M\left(\lambda \right)=\left(\mathcal{P}
\left(\lambda \right)-\frac{1}{2} I\right)\left(iG\right)^{-1}.
\end{gather}
Then the condition (\ref{GEQ__47++_}) corresponding to
$M\left(\lambda \right)$ is separated for $\lambda =\mu _{0} $ if
and only if the operator $\mathcal{P} \left(\mu_{0} \right)$ is
the projection, i.e.
\begin{gather*}
\displaystyle \mathcal{P} \left(\mu _{0}
\right)=\mathcal{P} ^{2} \left(\mu _{0} \right).
\end{gather*}
\end{theorem}
\begin{definition}\cite{Khrab4,Khrab5}\label{def++}
If the operator-function $M\left(\lambda \right)$ of the form
(\ref{13}) is the characteristic operator of equation (\ref{GEQ__46_}) on
$\mathcal{I} $ and, moreover, $\mathcal{P} \left(\lambda
\right)=\mathcal{P}^{2} \left(\lambda \right)$, then
$\mathcal{P}\left(\lambda \right)$ is called a characteristic
projection of equation (\ref{GEQ__46_}) on $\mathcal{I}$.
\end{definition}
The properties of characteristic projections and sufficient conditions for their
existence are obtained in \cite{Khrab5}. Also \cite{Khrab5}
contains the description of characteristic projections and abstract analogue of
Theorem \ref{th++}. Necessary and sufficient conditions for existence of characteristic operator, which corresponds to such separated boundary conditions that corresponding boundary condition in
regular point is self-adjoint, are obtained in \cite{KhrabMAG} (see also \cite{KhrabArxiv}) with the help of Theorem \ref{th++}. In the case of self-adjoint boundary conditions the analogue of
this result for regular differential operators in space of
vector-functions was proved in \cite{RB} (see also
\cite{RBKholkin}). For finite canonical systems depending on
spectral parameter in a linear manner such analogue was proved in
\cite{Mogil}. These analogues were obtained in a different way
comparing with the proof in \cite{KhrabMAG,KhrabArxiv}.
{
From this point and till the end of Corollary \ref{cor11} we suppose that $\mathcal{H}_{1} =\mathcal{H}^{2n}$,
\begin{gather}\label{Q}
Q(t)=\left(\begin{matrix}0&iI_n\\-iI_n& 0\end{matrix}\right)=J/i,
\end{gather}
where $I_n$ is the identity operator in $\mathcal{H}^{n}$,
$\mathcal{I}=(0,b)$, $b\leq\infty$ and condition \eqref{star8} holds. Let condition \eqref{GEQ__47++_} be separated and $\mathcal{P}(\lambda)$ be a corresponding characteristic projection. In view of \cite[p. 469]{Khrab5} the Nevanlinna pair $\left\{-a\left(\lambda \right),\, b\left(\lambda \right)\right\},\, a(\lambda),b(\lambda)\in B\left(\mathcal{H}^{n} \right)$ (see for example \cite{DHMdS2}) and Weyl function $m\left(\lambda \right)\in B\left(\mathcal{H}^{n} \right)$ of equation \eqref{GEQ__46_} on $\mathcal{I}$ \cite{Khrab5} exist such that
\begin{gather} \label{GEQ__64ad1_}
\mathcal{P}\left(\lambda \right)=\left(\begin{array}{c} {I_{n} } \\ {m\left(\lambda \right)} \end{array}\right)\left(b^{*} \left(\bar{\lambda }\right)-a^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)^{-1} \left(a_{2}^{*} \left(\bar{\lambda }\right),\, -a_{1}^{*} \left(\bar{\lambda }\right)\right),
\\ \label{GEQ__65ad1_}
I-\mathcal{P}\left(\lambda \right)=\left(\begin{array}{c} {a\left(\lambda \right)} \\ {b\left(\lambda \right)} \end{array}\right)\left(b\left(\lambda \right)-m\left(\lambda \right)a\left(\lambda \right)\right)^{-1} \left(-m\left(\lambda \right),I_{n} \right),\\\label{13+}
\left(b^{*} \left(\bar{\lambda }\right)-a^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)^{-1} ,\, \, \left(b\left(\lambda \right)-m\left(\lambda \right)a\left(\lambda \right)\right)^{-1} \in B\left(\mathcal{H}^{n} \right).
\end{gather}
(Conversely \cite{Khrab5} $\mathcal{P}\left(\lambda \right)$ \eqref{GEQ__64ad1_} is a characteristic projection for any Nevanlinna pair $\left(-a\left(\lambda \right),\, b\left(\lambda \right)\right)$ and any Weyl function $m\left(\lambda \right)$ of equation \eqref{GEQ__46_} on $\mathcal{I}$.)
Let also domain ${D}\subseteq {\mathbb C}_{+} $ be such that $\forall \lambda \in {D}:\, \, 0\in \rho \left(a\left(\lambda \right)-ib(\lambda)\right)$ (for example ${D}={\mathbb C}_{+} $ if $\exists\lambda_\pm\in\mathbb{C}_\pm$ such that $a^{*} \left(\lambda_\pm \right)b\left(\lambda_\pm \right)=b^{*} \left(\lambda_\pm\right)a\left(\lambda_\pm \right)$ ). Let domain ${ D}_{1} $ be symmetric to ${D}$ with respect to real axis. Then the operator $\mathcal{R}_{\lambda } F$ \eqref{GEQ__47_} for $\lambda \, \in { D}\bigcup { D}_{1} $ can be represented in the following form with using the operator solulion $U_{\lambda } \left(t\right)\in B\left(\mathcal{H}^{n} ,\, \mathcal{H}^{2n} \right)$ of equation \eqref{GEQ__46_}, ($F=0$) satisfying accumulative (or dissipative) initial condition and operator solution $V_{\lambda } \left(t\right)\in B\left(\mathcal{H}^{n} ,\, \mathcal{H}^{2n} \right)$ of Weyl type of the same equation. More precisely the following proposition holds.
\begin{proposition}\label{rm21}Let $\lambda \in {D}\bigcup {D}_{1} $, $\mathcal{H}_1$-valued $F(t)\in L^2_{W_\lambda}(\mathcal{I})$\footnote{Norms $\|\cdot\|_{L^2_{W_\lambda}(\mathcal{I})}$ are equivalent for $\lambda\in \mathcal{A}$ \cite{Khrab5}.}. Then solution \eqref{GEQ__47_} of equation \eqref{GEQ__46_} is equal to
\begin{gather}\label{rlambdaf}
\mathcal{R}_{\lambda } F=\int _{0}^{t}V_{\lambda } \left(t\right)U_{\bar{\lambda }}^{*} \left(s\right)W_{\lambda } \left(s\right)F\left(s\right)ds +\int _{t}^{b}U_{\lambda } \left(t\right)V_{\bar{\lambda }}^{*} \left(s\right)W_{\lambda } \left(s\right)F\left(s\right)ds,
\end{gather}
where the integrals converge strongly if the interval of integration is infinite.
Here
\begin{equation} \label{GEQ__66ad1_}
U_{\lambda } \left(t\right)=X_{\lambda } \left(t\right)\left(\begin{array}{c} {a\left(\lambda \right)} \\ {b\left(\lambda \right)} \end{array}\right),\, \, V_{\lambda } \left(t\right)=X_{\lambda } \left(t\right)\left(\begin{array}{c} {b\left(\lambda \right)} \\ {-a\left(\lambda \right)} \end{array}\right){K} ^{-1} \left(\lambda \right)+U_{\lambda } \left(t\right)m_{a,b} \left(\lambda \right),
\end{equation}
where
\begin{gather} \label{GEQ__67ad1_}
{K} \left(\lambda \right)=a^{*} \left(\bar{\lambda }\right)a\left(\lambda \right)+b^{*} \left(\bar{\lambda }\right)b\left(\lambda \right),\, \, {K} ^{-1} \left(\lambda \right)\in B\left(\mathcal{H}^{n} \right),
\\\label{GEQ__68ad1_}
m_{a,b} \left(\lambda \right)=m_{a,b}^{*} \left(\bar{\lambda }\right)={K} ^{-1} \left(\lambda \right)\left(a^{*} \left(\bar{\lambda }\right)+b^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)\left(b^{*} \left(\bar{\lambda }\right)-a^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)^{-1} ,
\\ \label{add_ineq}
\int_{0}^\beta V_{\lambda }^{*} \left(t\right)W_{\lambda } \left(t\right)V_\lambda\left(t\right)dt\leq
{\left(b(\bar\lambda)-m^*(\lambda)a(\bar\lambda)\right)^{-1}(\Im m(\lambda))(b^*(\bar{\lambda})-a^*(\bar\lambda)m(\lambda))^{-1}\over\Im\lambda}
\end{gather}
$\forall[0,\beta]\subseteq\bar{\mathcal{I}}$ and therefore
\begin{gather}\label{add_ineq+}
V_{\lambda } \left(t\right)h\in L_{W_{\lambda } \left(t\right)}^{2} \left(\mathcal{I}\right)\forall h\in \mathcal{H}^{n}.
\end{gather}
Moreover if $a\left(\lambda \right)=a\left(\bar{\lambda }\right),\, b\left(\lambda \right)=b\left(\bar{\lambda }\right)$ as $\Im\lambda\not= 0$ then we can set $D=\mathbb{C}_+$ and
\begin{gather}\label{vlambda}
\int_{0}^\beta V_{\lambda }^{*} \left(t\right)W_{\lambda } \left(t\right)V_\lambda\left(t\right)dt \le \frac{\Im m_{a,b} \left(\lambda \right)}{\Im\lambda },\ \Im\lambda \ne 0.
\end{gather}
\end{proposition}
Proposition \ref{rm21} contains in \cite{KhrabMAG} in less complete form. Therefore we prove it here.
\begin{proof}
In view of \eqref{13}, \eqref{GEQ__64ad1_}, \eqref{GEQ__65ad1_} $\mathcal{R}_{\lambda } F$ has a representation (\ref{rlambdaf}) where
\begin{gather}
\label{prop_add1}
V_{\lambda } \left(t\right)=X_{\lambda } \left(t\right)\left(\begin{array}{c} {{I} _{n} } \\ {m\left(\lambda \right)} \end{array}\right)\left(a_{2}^{*} \left(\bar{\lambda }\right)-a_{1}^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)^{-1}.
\end{gather}
Due to Lemma 1.2 from \cite{KhrabArxiv, KhrabMAG} the integrals in \eqref{rlambdaf} converge strongly if the interval of integration is infinite.
In view of \cite{GorGor, Khrab5} and the fact that $\mathcal{P}^{*} \left(\bar{\lambda }\right)G\mathcal{P}\left(\lambda \right)=0$ \cite{Khrab5} one has
\begin{gather}
\label{prop_add2}
a\left(\lambda \right)=\mp i \left(u\left(\lambda \right)+I_{n} \right)S\left(\lambda \right),\, b\left(\lambda \right)=\left(u\left(\lambda \right)-I_{n} \right)S\left(\lambda \right),\, \, \lambda \in {\mathbb C}_{\pm }
\end{gather}
where $u\left(\lambda \right)=u^{*} \left(\bar{\lambda }\right)\in B(\mathcal{H}^n)$ is some contraction, $S\left(\lambda \right),S^{-1} \left(\lambda \right)\in B\left(\mathcal{ H}^{n} \right);\, u\left(\lambda \right),\, S\left(\lambda \right)$ analytically depend on $\lambda \in \mathbb{C}\setminus\mathbb{R}^1$.
In view of \eqref{prop_add2}
\begin{gather}\label{prop_add2+}
K\left(\lambda \right)=-4S^{*} \left(\bar{\lambda }\right)u\left(\lambda \right)S\left(\lambda \right)
\end{gather}
and so $K^{-1} \left(\lambda \right)\in B\left(\mathcal{H}^n\right),\, \lambda \in {D}\bigcup {D}_{1} $ since $u^{-1} \left(\lambda \right)=-2iS\left(\lambda \right)\left(a\left(\lambda \right)-ib\left(\lambda \right)\right)^{-1} \in B\left(\mathcal{H}^{n} \right)\, \lambda \in {D}\cup D_1$.
Using (\ref{prop_add2}), (\ref{prop_add2+}) it can be directly shown that initial conditions in point $t=0$ for solutions $V_{\lambda } \left(t\right)$ \eqref{GEQ__66ad1_} and $V_{\lambda } $ (\ref{prop_add1}) coincides and that $m_{a,b} \left(\lambda \right)=m_{a,b}^{*} \left(\bar{\lambda }\right)$. So (\ref{rlambdaf})-(\ref{GEQ__68ad1_}) is proved.
In view of \cite[p. 450]{Khrab5} one has $\forall [0,\beta]\subseteq\bar{\mathcal{I}}$
\begin{gather}
\label{prop_add3}
\mathcal{P}^{*} \left(\lambda \right)\Delta _{\lambda } \left(0,\, \beta\right)\mathcal{P}\left(\lambda \right)\le \frac{1}{2{\Im}\lambda } \mathcal{P}^{*} \left(\lambda \right)G\mathcal{P}\left(\lambda \right).
\end{gather}
Now inequality (\ref{add_ineq}) (and therefore (\ref{add_ineq+})) follows from (\ref{prop_add3}) in view of (\ref{GEQ__64ad1_}), (\ref{GEQ__67ad1_}), (\ref{prop_add1}).
If $a\left(\lambda \right)=a\left(\bar{\lambda }\right),\, b\left(\lambda \right)=b\left(\bar{\lambda }\right)$ as $\Im\lambda\not= 0$, then operator $u\left(\lambda \right)$ is unitary and independent in $\lambda $ (cf. \cite{Khrab5}). Now in formulae \eqref{GEQ__67ad1_}, \eqref{GEQ__68ad1_} and the right-hand-side of \eqref{add_ineq} we substitute $a\left(\bar{\lambda }\right)$, $b\left(\bar{\lambda }\right)$ by $a\left({\lambda }\right)$, $b\left({\lambda }\right)$ and by direct calculations and with the help of (\ref{prop_add2}) we prove that right hand sides of inequalities (\ref{add_ineq}), (\ref{vlambda}) coincides. The proposition is proved.
\end{proof}
For an arbitrary Nevanlinna pair $\left\{-a\left(\lambda \right),\, b\left(\lambda \right)\right\}$ Weyl solution $V_{\lambda } \left(t\right)$ ({\ref{GEQ__66ad1_}}) does not satisfy in general to inequality (\ref{vlambda}) for $\lambda \in {D}\bigcup {D}_{1} $, and corresponding Weyl function $m_{a,\, b} \left(\lambda \right)$ (\ref{GEQ__68ad1_}) does not satisfy the condition
\begin{gather}
\label{mab}
\frac{{\Im}m_{a,b} \left(\lambda \right)}{{\Im}\lambda } \ge 0,\, \, \lambda \in {D}\bigcup {D}_1.
\end{gather}
But if we choose pair $\left\{-a\left(\lambda \right),\, b\left(\lambda \right)\right\}$ "in canonical way" then corresponding Weyl solution $V_{\lambda } $ ({\ref{GEQ__66ad1_}}) satisfies (\ref{vlambda}).
Namely let $v\left(\lambda \right)\in B\left(\mathcal{H}^{n} \right)$ is a contraction analically depending on $\lambda $ in domain ${D}\subseteq {\mathbb C}_{+} $ and let $v^{-1} \left(\lambda \right)\in B\left(\mathcal{ H}^{n} \right),\, \lambda \in {D}$. Let us consider the following pair $\left\{a\left(\lambda \right),\, b\left(\lambda \right)\right\}$, where
\begin{gather}\label{alambda}
a\left(\lambda \right)=-i\left(v\left(\lambda \right)+I_{n} \right),\, b\left(\lambda \right)=v\left(\lambda \right)-I_{n},\ \lambda \in {D}.
\end{gather}
Let us extend the pair $\left\{a\left(\lambda \right),b\left(\lambda \right)\right\}$ (\ref{alambda}) to the domain ${D}_{1} $ which is symmetric to ${D}$ with the respect to real axis in the following way
\begin{gather}\label{barlambda}
v\left(\lambda \right)=\left(v^{*} \left(\bar{\lambda} \right)\right)^{-1} ,\, \, \, \lambda \in {D}_1
\end{gather}
(and therefore $v^*\left(\bar{\lambda }\right)$ is stretching as $\lambda \in {D}$). As a result we obtain the pair of \eqref{prop_add2} type with $D$ (respectively $D_1$) instead of $\mathbb{C}_+$ (respectively $\mathbb{C}_-$) and
$$u(\lambda)=\begin{cases}v(\lambda),&\lambda\in D\\v^*(\bar{\lambda}),&\lambda\in D_1\end{cases},\quad S(\lambda)=\begin{cases}I_n,&\lambda\in D\\-v(\lambda),&\lambda\in D_1\end{cases}.$$
Therefore if $\lambda \in {D}\bigcup {D}_{1} $ then for pair $\left\{a\left(\lambda \right),b\left(\lambda \right)\right\}$ \eqref{alambda}, \eqref{barlambda} the projections (\ref{GEQ__64ad1_}), (\ref{GEQ__65ad1_}) exist and therefore for operator $M\left(\lambda \right)$ (\ref{13}), (\ref{GEQ__64ad1_}) condition (\ref{GEQ__47++_}) holds and is separated.
\begin{lemma}\label{lm11}
The operator Weyl function $m_{a,b} \left(\lambda \right)$ ({\ref{GEQ__68ad1_}}) corresponding to the pair $\left\{a\left(\lambda \right),\, b\left(\lambda \right)\right\}$ (\ref{alambda}), (\ref{barlambda}) satisfies for any $h\in \mathcal{H}^{n} $ the identity
\begin{gather}\label{e_lm11}
\Im\left( m_{a,b} \left(\lambda \right)h,h\right)=\frac{1}{4} \left\| \sqrt{v\left(\bar{\lambda }\right)v^{*} \left(\bar{\lambda }\right)-I_{n} } \left(I_{n} -im\left(\lambda \right)\right)g\right\| ^{2} +\Im\left(\, m\left(\lambda \right)g,g\right),\, \, \lambda \in {D},
\end{gather}
where $g=\left(\left(I_{n} -v^{*} \left(\bar{\lambda }\right)\right)+i\left(I_{n} +v^{*} \left(\bar{\lambda }\right)m\left(\lambda \right)\right)\right)^{-1} h$, $(\dots)^{-1}\in B(\mathcal{H}^n)$.
\end{lemma}
\begin{proof} The Weyl function $m_{a,b} \left(\lambda \right)$ ({\ref{GEQ__68ad1_}}), (\ref{alambda}), (\ref{barlambda}) is equal to
\begin{gather}\label{mab1} m_{a,b} (\lambda )=\frac{-1}{4} \left(i
(I_{n} +v^{*} (\bar{\lambda }))-(I_{n} -v^{*} (\bar{\lambda }))m(\lambda )\right)\left(I_{n} -v^{*} (\bar{\lambda })+i(I_{n} +v^{*} (\bar{\lambda }))m(\lambda )\right)^{-1} ,\, \lambda \in {D},
\end{gather}
where $(\dots)^{-1}\in B(\mathcal{H}^n)$ in view of \eqref{13+}, \eqref{alambda}, \eqref{barlambda}.
Now identity (\ref{e_lm11}) follows from (\ref{mab1}) by direct calculation.
\end{proof}
In view of the fact that $m_{a,b}\left(\bar{\lambda }\right)=m_{a,b}^{*} \left(\lambda \right)$, inequality (\ref{add_ineq}), Lemma \ref{lm11}, condition (\ref{star8}) and formula (\ref{prop_add1}) the following theorem is valid.
\begin{theorem}\label{th12}
The solution $V_{\lambda } \left(t\right)$ (\ref{GEQ__66ad1_})-(\ref{GEQ__68ad1_}), (\ref{alambda}), (\ref{barlambda}) satisfies inequality (\ref{vlambda}) for $\lambda \in {D}\bigcup {D}_{1} $ (and therefore $m_{a,b} \left(\lambda \right)$ (\ref{GEQ__68ad1_}), (\ref{alambda}), (\ref{barlambda}) satisfies inequality (\ref{mab}) with $"\gg"$ instead of $"\geq"$).
\end{theorem}
\begin{lemma}\label{lm12} Let $v\left(\lambda \right)\in B\left(\mathcal{ H}^{n} \right)$ be a contraction analytically depending on $\lambda \in {\mathbb C}_{+} $. Let limit points of the set $S=\left\{\lambda \in {\mathbb C}_{+} :\, v^{-1} \left(\lambda \right)\notin B\left(\mathcal{H}^{n} \right)\right\}$ that belong to ${\mathbb C}_{+} $ be isolated. Let ${D}={\mathbb C}_{+} \backslash S$. Let us consider operator-function $m_{ab} \left(\lambda \right)$ (\ref{GEQ__68ad1_}), (\ref{alambda}), (\ref{barlambda}) as $\lambda \in {D}\bigcup { D}_{1} $. Then the points of the set $S$ are the removable singular points of this function. If $m_{a,b} \left(\lambda \right)$ is extended on the set $S$ in a proper way then we obtain the Nevanlinna operator-function $m_{a.b}(\lambda)=m_{a,b}^*(\bar\lambda)$.
\end{lemma}
\begin{proof}
Let $\lambda _{0} \in S$ be not the limit point of $S$. Then $\lambda _{0} $ is a removable singular point of scalar function $\left(m_{a,b} \left(\lambda \right)f,f\right)\, \forall f\in \mathcal{H}^{n} $ in view of (\ref{e_lm11}). Hence $\exists\, m_{0}\in B\left(\mathcal{H}^{n} \right):\, \mathop{\lim }\limits_{\lambda \to \lambda _{0} } \left(m_{a,b} \left(\lambda \right)f,f\right)=\left(m_{0} f,f\right)\forall f\in \mathcal{H}^{n} $ in view of principle of uniform boundedness \cite[p. 164]{Halmos}, \cite[p. 322]{Kato}. If we define $m_{a,b} \left(\lambda \right)$ in point $\lambda _{0} $ as $m_{a,b} \left(\lambda _{0} \right)=m_{0} $ then we obtain the operator-function which is analytic in point $\lambda _{0} $ in view of \cite[p. 195]{Kato}. The analicity of $m_{a,b} \left(\lambda \right)$ in limit points of ${S}$ belonging to $\mathbb{C}_+$ is proved analogously.
\end{proof}
\begin{corollary}\label{cor11}
Let the construction $v(\lambda)\in B(\mathcal{H}^n)$ satisfy condition of Lemma \ref{lm12}. Then corresponding solution $V_\lambda(t)$ (\ref{GEQ__66ad1_})-(\ref{GEQ__68ad1_}), (\ref{alambda}), (\ref{barlambda}) satisfies inequality (\ref{vlambda}) ($V_\lambda(t)\overset{def}{=} (\ref{prop_add1}),\ \lambda\notin D\cup D_1$).
\end{corollary}
For the construction of solutions of Weyl type and descriptions of Weyl function in various situation see \cite{0,Khrab5} and references in \cite{0}.
}
We consider in the separable Hilbert space $\mathcal{H}$
differential expression $l_\lambda\left[y\right]$ of order $r>0$ with coefficients from
$B\left(\mathcal{H}\right)$. This
expression is presented in the divergent form, namely
\begin{equation} \label{GEQ__51+_}
l_\lambda\left[y\right]=\sum\limits _{k=0}^{r}i^{k} l_{k}(\lambda)\left[y\right] ,
\end{equation}
where $l_{2j}(\lambda)=D^{j} p_{j} \left(t,\lambda\right)D^{j} $, $l_{2j-1}(\lambda)
=\frac{1}{2} D^{j-1} \left\{Dq_{j} \left(t,\lambda\right)+s_{j}
\left(t,\lambda\right)D\right\}D^{j-1}$, $D={d\over dt}$.
Let $-l_{\lambda } $ depend on $\lambda $ in Nevanlinna
manner. Namely, from now on the following condition holds:
(\textbf{B}) The set $\mathcal{B}\supseteq {\mathbb{C}\setminus
\mathbb{R}}^1 $ exists, any its points have a neighbourhood
independent on $t\in \bar{\mathcal{I}}$, in this neighbourhood
coefficients $p_{j} =p_{j} \left(t,\lambda \right),\, \, q_{j}
=q_{j} \left(t,\lambda \right),\, \, s=s_{j} \left(t,\lambda
\right)$ of the expression $l_{\lambda } $ are analytic $\forall
t\in \bar{\mathcal{I}}$; $\forall \lambda \in \mathcal{B}{\rm ,}\,
\, p_{j} \left(t,\lambda \right)$, $q_{j} \left(t,\lambda
\right)$, $s_{j} \left(t,\lambda \right)\in C^{j}
\left(\bar{\mathcal{I}},B\left(\mathcal{H}\right)\right)$ and
\begin{equation} \label{GEQ__48_}
p_{n}^{-1} \left(t,\lambda \right)\in
B\left(\mathcal{H}\right)\left(r=2n\right),\, \left(q_{n+1}
\left(t,\lambda \right)+s_{n+1} \left(t,\lambda
\right)\right)^{-1} \in B\left(\mathcal{H}\right)\,
\left(r=2n+1\right),\ t\in\bar{\mathcal{I}};
\end{equation}
these coefficients satisfy the following conditions
\begin{gather} \label{GEQ__49_}
p_{j} \left(t,\lambda \right)=p_{j}^{*} \left(t,\bar{\lambda
}\right),\, q_{j} \left(t,\lambda \right)=s_{j}^{*}
\left(t,\bar{\lambda }\right),\ \lambda \in\mathcal{B}\ (\Longleftrightarrow\ l_{\lambda }
=l_{\bar{\lambda }}^{*},\ \lambda\in\mathcal{B}),
\end{gather}
\begin{multline} \label{GEQ__50_}
\forall h_{0} ,\ldots ,h_{\left[\frac{r+1}{2} \right]} \in
\mathcal{H}:\\ \frac{\Im \left(\sum\limits
_{j=0}^{\left[r/2\right]}\left(p_{j} \left(t,\lambda \right)h_{j}
,h_{j} \right) +\frac{i}{2} \sum\limits
_{j=1}^{\left[\frac{r+1}{2} \right]}\left\{\left(s_{j}
\left(t,\lambda \right)h_{j} ,h_{j-1} \right)-\left(q_{j}
\left(t,\lambda \right)h_{j-1} ,h_{j} \right)\right\} \right)}{\Im
\lambda } \le 0,\\ t\in \bar{\mathcal{I}},\, \, \, \Im \lambda \ne
0.
\end{multline}
Therefore the order of expression $\Im l_{\lambda } $ is even and
therefore if $r=2n+1$ is odd, then $q_{m+1} ,\, s_{m+1} $ are
independent on $\lambda $ and $s_{n+1} =q_{n+1}^{*} $.
Condition \eqref{GEQ__50_} is equivalent to the condition:
${\left(\Im l_{\lambda } \right)\left\{f,f\right\}
\mathord{\left/{\vphantom{\left(\Im l_{\lambda }
\right)\left\{f,f\right\} \Im \lambda
}}\right.\kern-\nulldelimiterspace} \Im \lambda } \le 0,\, \, t\in
\bar{\mathcal{I}},\, \, \Im \lambda \ne 0$. Here for differential expression
$L[y]=\sum\limits _{k=0}^{R}i^{k} L_{k}[y]
$ with sufficiently smooth coefficients from $B(\mathcal{H})$, where $L_{2j} =D^{j} P_{j} \left(t\right)D^{j}$,\ $L_{2j-1}
=\frac{1}{2} D^{j-1} \left\{DQ_{j} \left(t\right)+S_{j}
\left(t\right)\, D\right\}D^{j-1} $, we denote by
\begin{multline} \label{GEQ__28_}
L\left\{f,g\right\}=\sum\limits _{j=0}^{\left[{R
\mathord{\left/{\vphantom{R 2}}\right.\kern-\nulldelimiterspace}
2} \right]}\left(P_{j} \left(t\right)f^{\left(j\right)}
\left(t\right),\, g^{\left(j\right)} \left(t\right)\right)+\\+
\frac{i}{2} \sum\limits _{j=1}^{\left[\frac{R+1}{2}
\right]}\left(S_{j} \left(t\right)f^{\left(j\right)}
\left(t\right),\, g^{\left(j-1\right)}
\left(t\right)\right)-\left(Q_{j}
\left(t\right)f^{\left(j-1\right)}
\left(t\right),g^{\left(j\right)} \left(t\right)\right)
\end{multline}
the bilinear form which corresponds to subintegral expression of the Dirichlet integral for
expression $L[y]$.
Let $m[y]$ be the same as $l_\lambda[y]$ differential expression
of even order $s\le r$ with operator coefficients $\tilde{p}_{j}
\left(t\right)=\tilde{p}_{j}^{*} \left(t\right),\, \,
\tilde{q}_{j} \left(t\right),\, \tilde{s}_{j}
\left(t\right)=\tilde{q}_{j}^{*} \left(t\right)\in C^j(\bar{\mathcal{I}},B(\mathcal{H}))$ that are
independent on $\lambda $. Let
\begin{multline}
\label{GEQ__52_} \forall h_{0} ,\, \ldots ,\,
h_{\left[\frac{r+1}{2} \right]} \in \mathcal{H}:\, \, 0\le
\sum\limits _{j=0}^{{s \mathord{\left/{\vphantom{s
2}}\right.\kern-\nulldelimiterspace} 2} }\left(\tilde{p}_{j}
\left(t\right)h_{j} ,h_{j} \right) +{\Im}\sum\limits _{j=1}^{{s
\mathord{\left/{\vphantom{s 2}}\right.\kern-\nulldelimiterspace}
2} }\left(\tilde{q}_{j}
\left(t\right)h_{j-1} ,\, h_{j} \right) \le\\
\le -\frac{\Im \left(\sum\limits
_{j=0}^{\left[{r \mathord{\left/{\vphantom{r
2}}\right.\kern-\nulldelimiterspace} 2} \right]}\left(p_{j}
\left(t,\lambda \right)h_{j} ,h_{j} \right) +\frac{i}{2}
\sum\limits _{j=1}^{\left[\frac{r+1}{2} \right]}\left(\left(s_{j}
\left(t,\lambda \right)h_{j} ,h_{j-1} \right)-\left(q_{j}
\left(t,\lambda \right)h_{j-1} ,h_{j} \right)\right) \right)}{\Im
\lambda },\\ t\in \bar{\mathcal{I}},\, \, \, \Im \lambda \ne 0.
\end{multline}
Condition \eqref{GEQ__52_} is equivalent to the condition:
$0\le m\left\{f,f\right\}\le -({\Im l_{\lambda })\left\{f,f\right\}/\Im \lambda } $, $t\in
\bar{\mathcal{I}}$, $\Im \lambda \ne 0$.
In the case of even $r=2n\ge s$ we denote
\begin{gather}\label{GEQ__6_}
Q\left(t,l_\lambda\right)={J}/{i} ,\, \, \,
S\left(t,l_\lambda\right)=Q\left(t,l_\lambda\right), \\ \label{GEQ__7_}
H\left(t,\, l_\lambda\right)=\left\| h_{\alpha \beta } \right\| _{\alpha
,\, \beta =1}^{2} ,\, \, h_{\alpha \beta } \in
B\left(\mathcal{H}^n \right),
\end{gather}
where $h_{11} $ is a three diagonal operator matrix whose
elements under the main diagonal are equal to $\left(\frac{i}{2}
q_{1} ,\, \ldots ,\, \frac{i}{2} q_{n-1} \right)$, the elements
over the main diagonal are equal to $\left(-\frac{i}{2} s_{1} ,\,
\, \ldots ,\, \, -\frac{i}{2} s_{n-1} \right)$, the elements on
the main diagonal are equal to $\left(-p_{0} ,\, \, \ldots ,\, \,
-p_{n-2} ,\, \, \frac{1}{4} s_{n} p_{n}^{-1} q_{n} -p_{n-1}
\right)$; $h_{12} $ is an operator matrix with the identity
operators $I_{1} $ under the main diagonal, the elements on the
main diagonal are equal to $\left(0,\, \, \ldots ,\, \, 0,\, \,
-\frac{i}{2} s_{n} p_{n}^{-1} \right)$, the rest elements are
equal to zero; $h_{21} $ is an operator matrix with identity
operators $I_{1} $ over the main diagonal, the elements on the
main diagonal are equal to $\left(0,\, \, \ldots ,\, \, 0,\, \,
\frac{i}{2} p_{n}^{-1} q_{n} \right)$, the rest elements are equal
to zero; $h_{22} =\mathrm{diag}\left(0,\, \, \ldots ,\, \, 0,\, \,
p_{n}^{-1} \right)$.
Also in this case we denote
\footnote{\label{foot1}$W\left(t,l_\lambda,m\right)$ is given for the case
$s=2n$ . If $s<2n$ one have set the corresponding elements of
operator matrices $m_{\alpha \beta } $ be equal to zero. In
particular if $s<2n$ then $m_{12} =m_{21} =m_{22} =0$ and
therefore $W\left(t,l_\lambda,m\right)=\mathrm{diag}\left(m_{11}
,0\right)$ in view of (14) from \cite{KhrabArxiv,KhrabMAG}.}
\begin{equation} \label{GEQ__8_}
W\left(t,\, l_\lambda,\, m\right)=C^{*-1} \left(t,l_\lambda\right)\left\{\left\|
m_{\alpha \beta } \right\| _{\alpha ,\, \beta =1}^{2}
\right\}C^{-1} \left(t,l_\lambda\right), m_{\alpha \beta } \in
B\left(\mathcal{H}^{n} \right),
\end{equation}
where $m_{11} $ is a tree diagonal operator matrix whose elements
under the main diagonal are equal to $\left(-\frac{i}{2}
\tilde{q}_{1} ,\, \ldots ,\, -\frac{i}{2} \tilde{q}_{n-1}
\right)$, the elements over the main diagonal are equal to
$\left(\frac{i}{2} \tilde{s}_{1} ,\, \ldots ,\, \frac{i}{2}
\tilde{s}_{n-1} \right)$, the elements on the main diagonal are
equal to $\left(\tilde{p}_{0} ,\, \ldots ,\, \tilde{p}_{n-1}
\right)$; $m_{12} =\mathrm{diag}\left(0,\, \ldots ,\, 0,\,
\frac{i}{2} \tilde{s}_{n} \right)$, $m_{21}
=\mathrm{diag}\left(0,\, \ldots ,\, 0,\, -\frac{i}{2}
\tilde{q}_{n} \right)$, $m_{22} =\mathrm{diag}\left(0,\, \ldots
,\, 0,\, \tilde{p}_{n} \right)$.
The operator matrix $C\left(t,l_\lambda\right)$ is defined by the condition
\begin{multline}
\label{GEQ__9_} C\left(t,l_\lambda\right)col\left\{f\left(t\right),\,
f'\left(t\right),\, \ldots ,\, f^{\left(n-1\right)}
\left(t\right),\, f^{\left(2n-1\right)} \left(t\right),\, \ldots
,\, f^{\left(n\right)} \left(t\right)\right\}=\\=
col\, \left\{f^{\left[0\right]}
\left(t|l_\lambda\right),\, f^{\left[1\right]} \left(t|l_\lambda\right),\, \ldots
,\, f^{\left[n-1\right]} \left(t|l_\lambda\right),\, f^{\left[2n-1\right]}
\left(t\left|l_\lambda\right. \right),\, \ldots ,\, f^{\left[n\right]}
\left(t\left|l_\lambda\right. \right)\right\},
\end{multline}
where $f^{\left[k\right]} \left(t\left|L\right. \right)$ are
quasi-derivatives of vector-function $f\left(t\right)$ that
correspond to differential expression $L[y]$; $C^{-1}(t,l_\lambda)\in B(\mathcal{H}^r)$, $t\in\bar{\mathcal{I}}$, $\lambda\in\mathcal{B}$ in view of (14) from \cite{KhrabArxiv,KhrabMAG}.
The quasi-derivatives corresponding to $l_\lambda[y]$ are equal (cf.
\cite{RBUpsala}) to
\begin{gather} \label{GEQ__10_}
y^{\left[j\right]} \left(t\left|l_\lambda\right. \right)=y^{\left({
j}\right)} \left(t\right),\, \, \, { j}=0,\, \, ...,\, \,
\left[\frac{r}{2} \right]-1, \\ \label{GEQ__11_}
y^{\left[n\right]} \left(t\left|l_\lambda\right.
\right)=\left\{\begin{array}{l} {p_{n} y^{\left(n\right)}
-\frac{i}{2} q_{n} y^{\left(n-1\right)} ,\, \, r=2n} \\
{-\frac{i}{2} q_{n+1} y^{\left(n\right)} ,\, \, r=2n+1}
\end{array}\right.,
\\\label{GEQ__12_}
y^{\left[r-j\right]} \left(t\left|l_\lambda\right.
\right)=-Dy^{\left[r-j-1\right]} \left(t\left|l_\lambda\right.
\right)+p_{j} y^{\left(j\right)} +\frac{i}{2} \left[s_{j+1}
y^{\left(j+1\right)} -q_{j} y^{\left(j-1\right)} \right],\, j=0,\,
...,\, \left[\frac{r-1}{2} \right],\, q_{0} \equiv 0.
\end{gather}
At that $l_\lambda\left[y\right]=y^{\left[r\right]} \left(t\left|l_\lambda\right.
\right)$. The quasi-derivatires $y^{\left[k\right]}
\left(t\left|m\right. \right)$ corresponding to $m[y]$ are defined in
the same way with even $s$ instead of $r$ and $\tilde{p}_{j},
\tilde{q}_{j} ,\tilde{s}_{j} $ instead of $p_{j} ,q_{j} ,s_{j} $.
In the case of odd $r=2n+1>s$ we denote
\begin{gather} \label{GEQ__14_}
Q\left(t,l_\lambda\right)=\begin{cases}J/i\oplus q_{n+1}\\
q_{1}\end{cases},\quad S\left(t,l_\lambda\right)=\begin{cases} J/i\oplus
s_{n+1},& n>0 \\ s_{1},& n=0\end{cases},
\\
\label{GEQ__15_} H\left(t,\, l_\lambda\right)=\begin{cases}\left\|
h_{\alpha \,\beta } \right\|_{\alpha ,\, \beta =1}^{2},& n>0 \\
p_{0},& n=0
\end{cases},
\end{gather}
where $B\left(\mathcal{H}^{n} \right)\ni h_{11} $ is a
three-diagonal operator matrix whose elements under the main
diagonal are equal to $\left(\frac{i}{2} q_{1} ,\, \ldots ,\,
\frac{i}{2} q_{n-1} \right)$, the elements over the main diagonal
are equal to $\left(-\frac{i}{2} s_{1} ,\, \ldots ,\, -\frac{i}{2}
s_{n-1} \right)$, the elements on the main diagonal are equal to
$\left(-p_{0} ,\, \ldots ,\, -p_{n-1} \right)$, the rest elements
are equal to zero. $B\, \left(\mathcal{ H}^{n+1} ,\, \mathcal{
H}^{n} \right)\ni h_{12} $ is an operator matrix whose elements
with numbers $j,\, j-1$ are equal to $I_{1} ,\, j=2,\, \ldots ,\,
n$, the element with number $n,\, n+1$ is equal to $\frac{1}{2}
s_{n} $, the rest elements are equal to zero. $B\,
\left(\mathcal{H}^{n} ,\, \mathcal{H}^{n+1} \right)\ni h_{21} $ is
an operator matrix whose elements with numbers $j-1,\, j$ are
equal to $I_{1} ,\, j=2,\, \ldots ,\, n$, the element with number
$n+1,\, n$ is equal to $\frac{1}{2} q_{n} $, the rest elements are
equal to zero. $B\, \left(\mathcal{H}^{n+1} \right)\ni h_{22} $ is
an operator matrix whose last row is equal to $\left(0,\, \ldots
,\, 0,\, -iI_{1} ,\, -p_{n} \right)$, last column is equal to
$col\, \left(0,\, \ldots ,\, 0,\, iI_{1} ,\, -p_{n} \right)$, the
rest elements are equal to zero.
Also in this case we denote \footnote{See the previous footnote}
\begin{equation} \label{GEQ__16_}
W\left(t,\, l_\lambda,\, m\right)=\left\| m_{\alpha \beta } \right\| _{\alpha ,\, \beta =1}^{2} ,
\end{equation}
where $m_{11} $ is defined in the same way as $m_{11} $
\eqref{GEQ__8_}. $B\left(\mathcal{H}^{n+1} ,\, \mathcal{H}^{n}
\right)\ni m_{12} $ is an operator matrix whose element with
number $n,\, n+1$ is equal to $-\frac{1}{2} \tilde{s}_{n} $, the
rest elements are equal to zero. $B\, \left(\mathcal{H}^{n} ,\,
\mathcal{H}^{n+1} \right)\ni m_{21} $ is an operator matrix whose
element with number $n+1,\, n$ is equal to $-\frac{1}{2}
\tilde{q}_{n} $, the rest elements are equal to zero. $B\,
\left(\mathcal{H}^{n+1} \right)\ni m_{22} =\mathrm{diag}\left(0,\,
\ldots ,\, 0,\, \tilde{p}_{n} \right)$.
Obviously in view of \eqref{GEQ__49_}, \eqref{GEQ__52_} for $H\left(t,l_\lambda\right)$ \eqref{GEQ__7_},
\eqref{GEQ__15_} and $W\left(t,l_\lambda,m\right)$
\eqref{GEQ__8_}, \eqref{GEQ__16_} one has
\begin{equation} \label{GEQ__17_}
H^{*} \left(t,l_\lambda\right)=H\left(t,l_{\bar\lambda} \right), W^{*} \left(t,l_\lambda,\, m\right)=W\left(t,l_\lambda,\, m \right),\ t\in\bar{\mathcal{I}},\lambda\in\mathcal{B}.
\end{equation}
\begin{lemma}\cite{KhrabMAG} (see also \cite{KhrabArxiv})\label{lm1}
Let the order of $\Im l_\lambda$ is even. Then
\begin{equation} \label{GEQ__18_}
\Im H\left(t,l_\lambda\right)=W\left(t,l_\lambda,\, -\Im l_\lambda\right)=W\left(t,l_{\bar\lambda}
,\, -\Im l_\lambda\right),\ t\in\bar{\mathcal{I}},\lambda\in\mathcal{B}.
\end{equation}
\end{lemma}
For sufficiently smooth vector-function $f\left(t\right)$ by
corresponding capital letter we denote (if $f(t)$ has a subscript
then we add the same subscript to $F$)
\begin{multline} \label{GEQ__23_}
\mathcal{H}^{r} \ni F\left(t,\, l_\lambda,m\right)=\\=\begin{cases}
\left(\sum\limits _{j=0}^{s/2}\oplus f^{\left(j\right)}
\left(t\right) \right)\oplus
0 \oplus ...\oplus 0 ,\quad r=2n,&r=2n+1>1,s<2n\\
\left(\sum\limits _{j=0}^{n-1}\oplus f^{\left(j\right)}
\left(t\right) \right)\oplus 0 \oplus ...\oplus 0 \oplus \,
\left(-if^{\left(n\right)} \left(t\right)\right),&r=2n+1>1,
s=2n\\ f\left(t\right),&r=1 \\ \left(\sum\limits
_{j=0}^{n-1}\oplus f^{\left(j\right)} \left(t\right) \right)\oplus
\left(\sum\limits _{j=1}^{n}\oplus f^{\left[r-j\right]}
\left(t\left|l_\lambda\right. \right) \right),& r=s=2n.\end{cases}
\end{multline}
\begin{theorem}\cite{KhrabMAG} (see also \cite{KhrabArxiv})\label{th1}
Equation \eqref{GEQ__1_} is equivalent to the following first
order system
\begin{equation} \label{GEQ__54_}
\frac{i}{2} \left(\left(Q\left(t,l_{\lambda }
\right)\vec{y}\left(t\right)\right)^{{'} } +Q^{*}
\left(t,l_{\lambda }
\right)\vec{y}\hspace{2px}'\left(t\right)\right)-H\left(t,l_{\lambda
} \right)\vec{y}\left(t\right)=W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }} ,m\right),
\end{equation}
where $Q\left(t,l_{\lambda } \right),\, H\left(t,l_{\lambda }
\right)$ are defined by \eqref{GEQ__6_}, \eqref{GEQ__7_},
\eqref{GEQ__14_}, \eqref{GEQ__15_} and $W\left(t,l_{\bar{\lambda }} ,m\right)$,
$F\left(t,l_{\bar{\lambda }} ,m\right)$ are defined by
\eqref{GEQ__8_}, \eqref{GEQ__16_}, \eqref{GEQ__23_} with $l_{\bar\lambda } $ instead of
$l_\lambda$.
Namely if $y\left(t\right)$ is a solution of equation
\eqref{GEQ__1_} then
\begin{multline} \label{GEQ__25_}
\vec{y}\left(t\right)=\vec{y}\left(t,\, l_\lambda,\, m,\,
f\right)=\\=\begin{cases}\left(\sum\limits _{j=0}^{n-1}\oplus
y^{\left(j\right)} \left(t\right) \right)\oplus \left(\sum\limits
_{j=1}^{n}\oplus \left(y^{\left[r-j\right]} \left(t\left|l_\lambda\right.
\right)-f^{\left[s-j\right]}
\left(t\left|m\right. \right)\right) \right),& r=2n\\
\left(\sum\limits _{j=0}^{n-1}\oplus y^{\left(j\right)}
\left(t\right) \right)\oplus \left(\sum\limits _{j=1}^{n}\oplus
\left(y^{\left[r-j\right]} \left(t\left|l_\lambda\right.
\right)-f^{\left[s-j\right]} \left(t\left|m\right. \right)\right)
\right)\oplus \left(-iy^{\left(n\right)} \left(t\right)\right),&
r=2n+1>1 \\\quad \text{{\rm (here }}f^{\left[k\right]}
\left(t\left|m\right.
\right)\equiv 0\text{ \rm as }k< \frac{s}{2}\text{\rm)}\\
\\y\left(t\right),& r=1\end{cases}
\end{multline}
is a solution of \eqref{GEQ__54_}. Any solution of
equation \eqref{GEQ__54_} is equal to \eqref{GEQ__25_}, where
$y\left(t\right)$ is some solution of equation
\eqref{GEQ__1_}.
\end{theorem}
Let us notice that different vector-functions $f(t)$ can generate
different right-hand-sides of equation \eqref{GEQ__54_} but unique right-hand-side of
equation \eqref{GEQ__1_}.
Due to Theorem 1.2 from \cite{KhrabMAG,KhrabArxiv} and Lemma \ref{lm1} we have $\frac{\Im H\left(t,l_{\lambda } \right)}{\Im
\lambda }=W\left(t,l_{\lambda } ,-\frac{\Im l_{\lambda } }{\Im
\lambda } \right) \ge 0,\, \, t\in \bar{\mathcal{I}},\, \, \, \Im \lambda
\ne 0$ and therefore
$H\left(t,l_{\lambda } \right)$ satisfy condition \textbf{(A)}
with $\mathcal{A}=\mathcal{B}$. Therefore $\forall\mu\in
\mathcal{B}\cap \mathbb{R}^1$ $W(t,l_\mu,-{\Im l_\mu\over\Im\mu
})=\left.{\partial
H(t,l_\lambda)\over\partial\lambda}\right|_{\lambda=\mu}$ is
Bochner locally integrable in uniform operator topology. Here in
view of \eqref{GEQ__7_}, \eqref{GEQ__15_} $\forall \mu \in
\mathcal{B}\bigcap \mathbb{R}^{1}\ \exists \frac{\Im l_{\mu }
}{\Im \mu } \mathop{=}\limits^{def} \frac{\Im l_{\mu +i0} }{\Im
\left(\mu +i0\right)} =\left.\frac{\partial l_{\lambda }
}{\partial \lambda }\right|_{\lambda=\mu} $, where the
coefficients ${\partial p_j(t,\mu)\over\partial\lambda}$,
${\partial q_j(t,\mu)\over\partial\lambda}$, ${\partial
s_j(t,\mu)\over\partial\lambda}$ of expression ${\partial l_{\mu }
\mathord{\left/{\vphantom{\partial l_{\mu }
\partial \mu }}\right.\kern-\nulldelimiterspace} \partial \mu } $
are Bochner locally integrable in the uniform operator topology.
Also in view of Theorem 1.2 from \cite{KhrabMAG,KhrabArxiv} and Lemma \ref{lm1} one has
\begin{equation} \label{GEQ__53_}
0\le W\left(t,l_{\lambda } ,m\right)\le W\left(t,l_{\lambda }
,-\frac{\Im l_{\lambda } }{\Im \lambda } \right)=\frac{\Im
H\left(t,l_{\lambda } \right)}{\Im \lambda } \quad t\in
\bar{\mathcal{I}},\, \, \, \Im \lambda \ne 0
\end{equation}
Let us consider in $\mathcal{H}_1=\mathcal{H}^r$ the equation
\begin{equation} \label{GEQ__51_}
\frac{i}{2} \left(\left(Q\left(t,l_{\lambda }
\right)\vec{y}\left(t\right)\right)^{{'} } +Q^{*}
\left(t,l_{\lambda }
\right)\vec{y}\hspace{2px}'\left(t\right)\right)-H\left(t,l_{\lambda
} \right)\vec{y}\left(t\right)=W\left(t,l_{\lambda } -\frac{\Im
l_{\lambda } }{\Im \lambda } \right)F\left(t\right).
\end{equation}
This equation is an equation of \eqref{GEQ__46_} type due to
\eqref{GEQ__17_}, \eqref{GEQ__53_}. Equation
\eqref{GEQ__5_} is equivalent to equation \eqref{GEQ__51_}
with $F\left(t\right)=F\left(t,l_{\bar{\lambda }} ,-\frac{\Im
l_{\lambda } }{\Im \lambda } \right)$ due to Theorem \ref{th1} and \eqref{GEQ__18_}.
\begin{definition}\cite{KhrabMAG,KhrabArxiv} Every characteristic operator of equation
\eqref{GEQ__51_} corresponding to the equation
\eqref{GEQ__5_} is said to be a characteristic operator of
equation \eqref{GEQ__5_} on $\mathcal{I}$.
\end{definition}
In some cases we will suppose additionally that
\begin{multline}\exists \lambda _{0} \in \mathcal{B};\, \, \alpha
,\beta \in \bar{\mathcal{I}},\, \, 0\in \left[\alpha ,\beta
\right]\text{, the number }\delta >0:\\
\label{GEQ__55_}
-\int _{\alpha }^{\beta }\left(\frac{\Im l_{\lambda _{0} } }{\Im
\lambda _{0} } \right)\left\{y\left(t,\lambda _{0}
\right),y\left(t,\lambda _{0} \right)\right\} \, dt\ge \delta
\left\| P\vec{y}\left(0,l_{\lambda _{0} } ,m,0\right)\right\| ^{2}
\end{multline}
for any solution $y\left(t,\lambda _{0} \right)$ of
\eqref{GEQ__1_} as $\lambda =\lambda _{0} ,\, \, f=0$, where $P\in B(\mathcal{H}^r)$ is the orthoprojection onto subspace $N^\perp$ which corresponds to equation (\ref{GEQ__51_}). In view
of Theorem 1.2 from \cite{KhrabMAG,KhrabArxiv} this condition is equivalent to the fact that
for the equation \eqref{GEQ__51_}
\begin{gather*}\exists \lambda _{0} \in \mathcal{A}=\mathcal{B};\, \, \alpha
,\beta \in \bar{\mathcal{I}},\, \, 0\in \left[\alpha ,\beta
\right]\text{, the number }\delta >0:
\ \left(\Delta _{\lambda _{0} } \left(\alpha ,\beta
\right)g,g\right)\ge \delta \left\| Pg\right\| ^{2} ,\quad g\in
\mathcal{H}^{r} .
\end{gather*}
Therefore in view of \cite{Khrab5} the fulfillment of
\eqref{GEQ__55_} implies its fulfillment with $\delta
\left(\lambda \right)>0$ instead of $\delta $ for all $\lambda \in
\mathcal{B}$.
Let us notice what in view of \eqref{GEQ__52_} $l_{\lambda } $
can be a represented in form \eqref{GEQ__2_} where
\begin{equation} \label{GEQ__561_}
l=\Re l_{i} , n_{\lambda } =l_{\lambda } -l-\lambda m; {\Im
n_{\lambda } \left\{f,f\right\}
\mathord{\left/{\vphantom{n_{\lambda } \left\{f,f\right\}
\mathcal{I}\lambda \ge 0}}\right.\kern-\nulldelimiterspace}
\Im\lambda \ge 0} ,t\in \bar{\mathcal{I}},\, {\Im}\lambda \ne 0.
\end{equation}
From now on we suppose that $l_{\lambda } $ has a representation
\eqref{GEQ__2_}, \eqref{GEQ__561_} and therefore the
order of $n_\lambda$ is even.
We consider pre-Hilbert spaces $\mathop{H}\limits^{\circ } $ and
$H$ of vector-functions $y\left(t\right)\in C_{0}^{s}
\left(\bar{\mathcal{I}},\mathcal{H}\right)$ and
$y\left(t\right)\in C^{s}
\left(\bar{\mathcal{I}},\mathcal{H}\right),\,
m\left[y\left(t\right),\, y\left(t\right)\right]<\infty $
correspondingly with a scalar product
\[\left(f\left(t\right),\, g\left(t\right)\right)_{m} =m\left[f\left(t\right),\, g\left(t\right)\right],\]
where
\begin{gather}
m\left[f,\, g\right]=\int\limits_{\mathcal{I}}m\left\{f,\, g\right\}dt,
\end{gather}
Here $m\left\{f,\, g\right\}$ is defined by \eqref{GEQ__28_}
with expression $m[y]$ from condition \eqref{GEQ__52_} instead of
$L[y]$. Namely, $$m\left\{f,\, g\right\}=\sum\limits _{j=0}^{s/2} (\tilde{p}_{j}
\left(t\right)f^{(j)}(t) ,g^{(j)}(t) ) +{i\over 2}\sum\limits _{j=1}^{s/2}
\left((\tilde{q}^*_{j}
\left(t\right)f^{(j)}(t) ,\, g^{(j-1)}(t) )-(\tilde{q}_{j}
\left(t\right)f^{(j-1)}(t) ,\, g^{(j)}(t) )\right).$$
By $\mathop{L_{m}^{2} }\limits^{\circ } \left(\mathcal{I}\right)$
and $L_{m}^{2} \left(\mathcal{I}\right)$ we denote the completions
of spaces $\mathop{H}\limits^{\circ } $ and $H$ in the norm
$\left\| \, \bullet \, \right\| _{m} =\sqrt{\left(\, \bullet ,\,
\bullet \right)_{m} } $ correspondingly. By
$\mathop{P}\limits^{\circ } $ we denote the orthoprojection in
$\mathop{L_{m}^{2} }\limits^{} \left(\mathcal{I}\right)$ onto
$\mathop{L_{m}^{2} }\limits^{\circ } \left(\mathcal{I}\right)$.
\begin{theorem} \cite{KhrabMAG} (see also \cite{KhrabArxiv})\label{th4}
Let $M\left(\lambda \right)$ be a characteristic operator of equation
\eqref{GEQ__5_}, for which the condition \eqref{GEQ__55_}
with $P=I_{r} $ holds if $\mathcal{I}$ is infinite. Let
$\Im\lambda\not= 0$, $f\left(t\right)\in H$ and
\begin{multline} \label{GEQ__62_}
col\left\{y_{j} \left(t,\lambda ,f\right)\right\}=\\=\int
_{\mathcal{I}}X_{\lambda } \left(t\right) \left\{M\left(\lambda
\right)-\frac{1}{2} sgn\left(s-t\right)\left(iG\right)^{-1}
\right\}X_{\bar{\lambda }}^{*}
\left(s\right)W\left(s,l_{\bar{\lambda }}
,m\right)F\left(s,l_{\bar{\lambda }} ,m\right)\, ds,\, y_{j} \in
\mathcal{H}
\end{multline}
be a solution of equation \eqref{GEQ__54_}, that corresponds
to equation \eqref{GEQ__1_}, where $X_{\lambda }
\left(t\right)$ is the operator solution of homogeneons equation
\eqref{GEQ__54_} such that $X_{\lambda } \left(0\right)=I_{r}
;\, \, G=\Re Q\left(0,l_{\lambda } \right)$ (if $\mathcal{I}$ is
infinite integral \eqref{GEQ__62_} converges strongly). Then
the first component of vector function \eqref{GEQ__62_} is a
solution of equation \eqref{GEQ__1_}. It defines densely
defined in $L_{m}^{2} \left(\mathcal{I}\right)$
integro-differential operator
\begin{equation} \label{GEQ__63_}
R\left(\lambda \right)f=y_{1} \left(t,\lambda ,f\right),\quad f\in H
\end{equation}
which has the following properties after closing
\noindent 1${}^\circ$
\begin{gather}\label{GEQ__64_}
R^{*} \left(\lambda \right)=R\left(\bar{\lambda }\right),\, \, \,
{\Im}\lambda \ne 0
\end{gather}
\noindent 2${}^\circ$
\begin{gather}\label{GEQ__65_}
R\left(\lambda \right)\text{ is holomorphic on
}\mathbb{C}\setminus \mathbb{R}^1
\end{gather}
\noindent 3${}^\circ$
\begin{gather}\label{GEQ__66_}
\left\| R\left(\lambda \right)f\right\| _{L_{m}^{2}
\left(\mathcal{I}\right)}^{2} \le \frac{\Im \left(R\left(\lambda
\right)f,f\right)_{L_{m}^{2} \left(\mathcal{I}\right)} }{\Im
\lambda } ,\, \, \, {\Im}\lambda \ne 0,\, \, \, f\in L_{m}^{2}
\left(\mathcal{I}\right)
\end{gather}
\end{theorem}
Let us notice that the definition of the operator $R\left(\lambda
\right)$ is correct. Indeed if $f\left(t\right)\in H$,
$m\left[f,f\right]=0$, then $R\left(\lambda \right)f\equiv 0$
since $W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }},m \right)\equiv 0$ due to
\eqref{GEQ__53_} and Theorem 1.2 from \cite{KhrabMAG,KhrabArxiv}.
Also let us notice that if $L^2_m(\mathcal{I})=\overset{\circ\ }{L^2_m}(\mathcal{I})$ then Theorem \ref{th4} is valid with $f(t)\in \overset{\circ}H$ instead of $f(t)\in H$ and without condition \eqref{GEQ__55_} with $P=I_r$ if $\mathcal{I}$ is infinite.
The resolvent $R(\lambda)$ can be represented in another forms. (In \eqref{prop12formula}, \eqref{prop13formula} integrals converge strongly if the interval of integration is infinite.)
\begin{proposition}\cite{KhrabMAG}\label{rm31} Let us represent characteristic operator $M\left(\lambda \right)$ from Theorem \ref{th4} in the form \eqref{13}. Then $R(\lambda)f$ \eqref{GEQ__63_} can be represented in the form
\begin{multline}\label{prop12formula}
R\left(\lambda \right)f=\int _{a}^{t}\sum _{j=1}^{r}y_{j} \left(t,\lambda \right) \sum _{k=0}^{{s\mathord{\left/ {\vphantom {s 2}} \right. \kern-\nulldelimiterspace} 2} }\left(x_{j}^{\left(k\right)} \left(s,\bar{\lambda }\right)\right)^{*} \mathrm{m}_{k} \left[f\left(s\right)\right]ds+\\+
\int _{t}^{b}\sum _{j=1}^{r}x_{j} \left(t,\lambda \right) \sum _{k=0}^{{s\mathord{\left/ {\vphantom {s 2}} \right. \kern-\nulldelimiterspace} 2} }\left(y_{j}^{\left(k\right)} \left(s,\bar{\lambda }\right)\right)^{*} \mathrm{m}_{k} \left[f\left(s\right)\right]ds
\end{multline}
where $x_{j} \left(t,\lambda \right),y_{j} \left(t,\lambda \right)\in B\left(\mathcal{H}\right)$ are operator solutions of equation \eqref{GEQ__1_} as $f=0$, such that $\left(x_{1} \left(t,\lambda \right),\, \ldots ,x_{r} \left(t,\lambda \right)\right)$ is the first row $\left[X_{\lambda } \left(t\right)\right]_{1}\in B(\mathcal{H}^r,\mathcal{H}) $ of operator matrix $X_{\lambda } \left(t\right),\, \left(y_{1} \left(t,\lambda \right),\, \ldots ,y_{r} \left(t,\lambda \right)\right)=\left[X_{\lambda } \left(t\right)\right]_{1} \mathcal{P}\left(\lambda \right)\left(iG\right)^{-1} $,
\begin{gather}\label{mk}
\mathrm{m}_{k} \left[f\left(s\right)\right]=\tilde{p}_{k} \left(s\right)f^{\left(k\right)} \left(s\right)+\frac{i}{2} \left(\tilde{q}_{k}^{*} \left(s\right)f^{\left(k+1\right)} \left(s\right)-\tilde{q}_{k} \left(s\right)f^{\left(k-1\right)} \left(s\right)\right)\left(\tilde{q}_{0} \equiv 0,\, \tilde{q}_{\frac{s}{2} +1} \equiv 0\right).
\end{gather}
\end{proposition}
In view of Proposition \ref{rm21}, Corollary \ref{cor11}, Theorem \ref{th4} and also Theorem 1.2 from \cite{KhrabArxiv,KhrabMAG} the following proposition is valid.
\begin{proposition}\cite{KhrabMAG}\label{rm32} Let $r=2n,\ \mathcal{I}=(0,b)$, $b\leq\infty$, condition \eqref{GEQ__55_} hold with $P=I_r$. (Therefore for equation \eqref{GEQ__51_} condition \eqref{star8} holds.) Let for characteristic operator $M\left(\lambda \right)$ of equation \eqref{GEQ__5_} condition \eqref{GEQ__47++_} be separated. (Therefore $M\left(\lambda \right)$ has representation \eqref{13} where characteristic projection $\mathcal{ P}\left(\lambda \right)$ can be represented in the form \eqref{GEQ__64ad1_}, \eqref{GEQ__65ad1_} with the help of some Nevanlinna pair $\{-a(\lambda),b(\lambda)\}$ and some Weyl function $m(\lambda)$ of equation \eqref{GEQ__51_}; this equation with $F(t)=0$ has an operator solutions $U_{\lambda } \left(t\right),\, V_{\lambda } \left(t\right)$ \eqref{GEQ__66ad1_}-\eqref{GEQ__68ad1_}). Let domains ${D,}\, {D}_{1} $ be the same as in Proposition \ref{rm21}. Then $R(\lambda)f$ \eqref{GEQ__63_} for $\lambda \in {D}\bigcup {D}_{1}$ can be represented in the form
\begin{multline}\label{prop13formula}
R\left(\lambda \right)f=\int _{0}^{t}\sum _{j=1}^{n}v_{j} \left(t,\lambda \right) \sum _{k=0}^{s/2}\left(u_{j}^{\left(k\right)} \left(s,\bar{\lambda }\right)\right)^{*} \mathrm{m}_{k} \left[f\left(s\right)\right]ds +\\
+\int _{t}^{b}\sum _{j=1}^{n}u_{j} \left(t,\lambda \right) \sum _{k=0}^{s/2}\left(v_{j}^{\left(k\right)} \left(s,\bar{\lambda }\right)\right)^{*} \mathrm{m}_{k} \left[f\left(s\right)\right]ds ,
\end{multline}
where $u_{j} \left(t,\lambda \right),\, v_{j} \left(t,\lambda \right)\in B\left(\mathcal{H}\right)$ are operator solutions of equation \eqref{GEQ__1_} as $f=0$, such that, $\left(u_{1} \left(t,\lambda \right),\ldots u_{n} \left(t,\lambda \right)\right)=\left[X_{\lambda } \left(t\right)\right]_{1} \left(\begin{array}{c} {a\left(\lambda \right)} \\ {b\left(\lambda \right)} \end{array}\right)$,
\begin{equation} \label{GEQ__115_}
\left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)=\left[X_{\lambda } \left(t\right)\right]_{1} \left(\begin{array}{c} {b\left(\lambda \right)} \\ {-a\left(\lambda \right)} \end{array}\right)K^{-1} \left(\lambda \right)+\left(u_{1} \left(t,\lambda \right),\ldots ,u_{n} \left(t,\lambda \right)\right)m_{a,b} \left(\lambda \right),
\end{equation}
$K\left(\lambda \right),\, m_{a,b} \left(\lambda \right)$ see \eqref{GEQ__67ad1_}, \eqref{GEQ__68ad1_};
$$\|\left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)h\|^2_{m}\leq {\Im(m(\lambda)g,g)\over \Im\lambda},\ \Im\lambda\not= 0,$$
where $g=(b^*(\bar\lambda)-a^*(\bar\lambda)m(\lambda))^{-1}h$, $h\in\mathcal{H}^n$ and therefore $$\left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)h\in L_{m}^{2} \left(\mathcal{I}\right)\, \, \forall h\in \mathcal{H}^{n}. $$
Moreover if $a\left(\lambda \right)=a\left(\bar{\lambda }\right),\, b\left(\lambda \right)=b\left(\bar{\lambda }\right)$ as $\Im\lambda\not= 0$ then we can set $D=\mathbb{C}_+$ and
\begin{gather}\label{norm}
\left\| \left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)h\right\| _{m}^{2} \le \frac{\Im\left(m_{a,b} \left(\lambda \right)h,h\right)}{\Im\lambda },\ \Im\lambda \ne 0.
\end{gather}
Let contraction $v(\lambda)\in B(\mathcal{H}^n)$ satisfy the conditions of Lemma \ref{lm12} and domains $D$, $D_1$ be the same as in Lemma \ref{lm12}. Then corresponding solution $\left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)$ (\ref{GEQ__115_}),(\ref{GEQ__67ad1_}),(\ref{GEQ__68ad1_}),(\ref{alambda}),(\ref{barlambda}) satisfies inequality \eqref{norm} ($\left(v_{1} \left(t,\lambda \right),\ldots ,v_{n} \left(t,\lambda \right)\right)\overset{def}{=}[V_\lambda(t)]_1$, $\lambda\notin D\cup D_1$, where $[V_\lambda(t)]_1\in B(\mathcal{H}^n,\mathcal{H})$ is an analogue of $[X_\lambda(t)]_1$ for $V_\lambda(t)$ \eqref{prop_add1}).
\end{proposition}
Comparison of Theorem \ref{th4}, Propositions \ref{rm31} , \ref{rm32} with results for various particular cases see in \cite{KhrabMAG}.
\section{Eigenfunction expansions}
It is known \cite{DSnoo1} that \eqref{GEQ__64_} -
\eqref{GEQ__66_} implies \eqref{GEQ__3_}, where
$E_{\mu}\in B\left(L_{m}^{2} \left(\mathcal{I}\right)\right)$,
$E_{\mu }=E_{\mu -0}$,
\begin{equation} \label{GEQ__83_}
0\le E_{\mu _{1} } \le E_{\mu _{2} } \le \mathbf{I} ,\, \, \, \mu
_{1} <\mu _{2} ;\, \, \, E_{-\infty } =0.
\end{equation}
Here $\mathbf{I}$ is the identity operator in $L_{m}^{2}
\left(\mathcal{I}\right)$. We denote $E_{\alpha \beta }
=\frac{1}{2} \left[E_{\beta +0} +E_{\beta } -E_{\alpha +0}
-E_{\alpha } \right]$.
\begin{theorem}\label{th7}
Let $M\left(\lambda \right)$ be the characteristic operator of
equation \eqref{GEQ__5_} (and therefore by
\cite[p.162]{Khrab5} $\Im {P} M\left(\lambda \right){P} /\Im
\lambda \ge 0$ as $\Im \lambda \ne 0$) and $\sigma \left(\mu
\right)=w-\mathop{\lim }\limits_{\varepsilon \downarrow 0}
\frac{1}{\pi } \int _{0}^{\mu }\Im { P} M\left(\mu +i\varepsilon
\right) {P} d\mu $ be the spectral operator-function that
corresponds to ${P} M\left(\lambda \right){P} $.
Let the condition
\eqref{GEQ__55_} with $P=I_r$ hold if $\mathcal{I}$ is
infinite. Let $E_{\mu } $ be generalized spectral family
\eqref{GEQ__83_} corresponding by \eqref{GEQ__3_} to the
resolvent $R\left(\lambda \right)$ from Theorem \ref{th4} which is
constructed with the help of characteristic operator $M\left(\lambda \right)$. Let $\mathcal{B}^1=\mathcal{B}\cap\mathbb{R}^1$. Then
for any $\left[\alpha ,\beta \right]\subset \mathcal{B}^1$ the
equalities
\begin{equation} \label{GEQ__84_}
\begin{matrix}
{\mathop{P}\limits^{\circ } E_{\alpha ,\beta } f\left(t\right)=
\mathop{P}\limits^{\circ } \int _{\alpha }^{\beta }\left[X_{\mu }
\left(t\right)\right]_{1} d\sigma \left(\mu \right)\varphi
\left(\mu ,\, f\right),\text{ if }f\left(t\right)\in
\mathop{H}\limits^{\circ },\ \mathcal{I}\text{ is infinite},}
\\ {E_{\alpha ,\beta } f\left(t\right)=\int _{\alpha }^{\beta }\left[X_{\mu } \left(t\right)\right]_{1} d\sigma \left(\mu \right)\varphi \left(\mu ,f\right) ,\, \, if\, \, f\left(t\right)\in H,\, \mathcal{I}\text{ is finite}} \end{matrix}
\end{equation}
are valid in $L_{m}^{2} \left(\mathcal{I}\right)$, where
$\left[X_{\lambda } \left(t\right)\right]_{1} \in
B\left(\mathcal{H}^{r} ,\, \mathcal{H}\right)$ is the first row of
the operator solution $X_{\lambda } \left(t\right)$ of homogeneous
equation \eqref{GEQ__54_} which is written in the matrix form and such
that $X_{\lambda } \left(0\right)=I_{r} $,
\begin{equation} \label{GEQ__85_}
\varphi \left(\mu ,f\right)=\left\{\begin{array}{l} {\int
_{\mathcal{I}}\left(\left[X_{\mu } \left(t\right)\right]_{1}
\right)^{*} m\left[f\right]dt\text{ if }f\left(t\right)\in
\mathop{H}\limits^{\circ } } \\ {\int
_{\mathcal{I}}\left(\left[X_{\mu } \left(t\right)\right]_{1}
\right)^{*} W\left(t,l_{\mu } ,m\right)F\left(t,l_{\mu }
,m\right)dt,\text{ if }f\left(t\right)\in H,\, \mathcal{I}\text{ is finite}}\text{ or }f(t)\in \overset{\circ}H
\end{array}\right.,
\end{equation}
$\mu \in \left[\alpha ,\beta \right]$.
Moreover, if vector-function $f\left(t\right)$ satisfy the
following conditions
\begin{gather}\label{star3}
\begin{matrix}
\mathop{{ P} }\limits^{\circ } E_{\infty } f=f,\, \, \mathop{{ P}
}\limits^{\circ } \int _{\mathbb{R}^{1}\setminus
\mathcal{B}^1}dE_{\mu } f =0\text{ if } f\in
\mathop{H}\limits^{{}^\circ },\ \mathcal{I}\text{ is infinite}
\\E_{\infty }f=f,\
\int_{\mathbb{R}^1\setminus \mathcal{B}^1}dE_{\mu } f =0\text{ if
}f\in H,\, \, \mathcal{I}\text{ is finite }
\end{matrix}
\end{gather}
then the inversion formulae in $L_{m}^{2} \left(\mathcal{I}\right)$
\begin{equation} \label{GEQ__87_}
\begin{matrix} {f\left(t\right)=\mathop{P}\limits^{\circ }
\int_{\mathcal{B}^1}\left[X_{\mu } \left(t\right)\right]_{1} d\sigma
\left(\mu \right)\varphi \left(\mu
,f\right)\text{ if }f\left(t\right)\in \mathop{H}\limits^{\circ } },\mathcal{I}\text{ is infinite}, \\
{f\left(t\right)=\int _{\mathcal{B}^1}^{}\left[X_{\mu }
\left(t\right)\right]_{1} d\sigma \left(\mu \right)\varphi
\left(\mu ,f\right),\, \, if\, f\left(t\right)\in H,\mathcal{I}\,
\, is\, finite} \end{matrix}
\end{equation}
and Parceval's equality
\begin{equation} \label{GEQ__88_}
m\left[f,g\right]=\int _{\mathcal{B}^1}\left(d\sigma \left(\mu
\right)\varphi \left(\mu ,f\right),\varphi \left(\mu
,g\right)\right) ,
\end{equation}
are valid, where $g\left(t\right)\in \mathop{H}\limits^{\circ } $
if $\mathcal{I}$ is infinite or $g\left(t\right)\in H$, if
$\mathcal{I}$ is finite.
In general case for $f\left(t\right),\, g\left(t\right)\in
\mathop{H}\limits^{\circ }$ if $\mathcal{I}$ is infinite or
$f\left(t\right),g\left(t\right)\in {H}$ if $\mathcal{I}$ is
finite, the inequality of Bessel type
\begin{equation} \label{GEQ__89_}
m\left[f\left(t\right),g\left(t\right)\right]\le \int
_{\mathcal{B}^1}\left(d\sigma \left(\mu \right)\varphi \left(\mu
,f\right),\, \varphi \left(\mu ,g\right)\right)
\end{equation}
is valid.
\end{theorem}
Let us notice that $\mathcal{B}^1=\cup_k (a_k,b_k)$, $(a_j,b_j)\cap
(a_k,b_k)=\varnothing$, $k\not= j$ since $\mathcal{B}^1$ is an open
set. In \eqref{GEQ__87_}
$\mathop{P}\limits^{\circ}\int_{\mathcal{B}^1}=\sum_k\lim\limits_{\alpha_k\downarrow
a_k,\beta_k\uparrow
b_k}\mathop{P}\limits^{\circ}\int_{\alpha_k}^{\beta_k}$. In
\eqref{GEQ__87_}-\eqref{GEQ__89_} we understand
$\int_{\mathcal{B}^1}$ similarly.
\begin{proof}
Let for definiteness $r=s=2n$, $\mathcal{I}$ is infinite (for
another cases the proof becomes simpler). Let the vector-functions
$f\left(t\right),\, g\left(t\right)\in \mathop{H}\limits^{\circ }
,\, \lambda =\mu +i\varepsilon ,G_{\lambda } \left(t,l_{\lambda }
,m\right)$ be defined by \eqref{GEQ__23_} with
$g\left(t\right)$ instead of $f\left(t\right)$. In view of the
Stieltjes inversion formula, we have
\begin{multline} \label{GEQ__90_}
\left(E_{\alpha ,\beta } f,g\right)_{L_{m}^{2}
\left(\mathcal{I}\right)} =\mathop{\lim }\limits_{\varepsilon
\downarrow 0} \frac{1}{2\pi i} \int _{\alpha }^{\beta
}\left(\left[y_{1} \left(t,\lambda ,f\right)-y_{1}
\left(t,\bar{\lambda },f\right)\right],g\right)_{m} d\mu =\\
=\mathop{\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{2\pi i}
\int_{\alpha }^{\beta }\bigg[\left(\vec{y}_{1} \left(t,l_{\lambda
} ,m,f\right),G\left(t,l_{\lambda }
,m\right)\right)_{L_{W\left(t,l_{\lambda } ,m\right)}^{2}
\left(\mathcal{I}\right)} -\left(\vec{y}_{1}
\left(t,l_{\bar{\lambda }} ,m,f\right),G\left(t,l_{\bar{\lambda }}
,m\right)\right)_{L_{W\left(t,l_{\bar{\lambda }} ,m\right)}^{2}
\left(\mathcal{I}\right)} +\\
+2i\int _{\mathcal{I}}\left(\left(\Im {p}_{n}^{ -1}
\left(t,\lambda \right)\right)f^{\left[n\right]}
\left(t\left|m\right. \right),g^{\left[n\right]}
\left(t\left|m\right. \right)\right)dt\bigg]d\mu =\\
=\mathop{\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{2\pi i}
\int _{\alpha }^{\beta }\left[\left( M\left(\lambda \right) \int
_{\mathcal{I}}X_{\bar{\lambda }}^{*}
\left(t\right)W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }} ,m\right)dt, \, \int
_{\mathcal{I}}X_{\lambda }^{*} \left(t\right)W\left(t,l_{\lambda }
,m\right)G\left(t,l_{\lambda } ,m\right)dt \right)\right. -
\\
\left. -\left( M^{*} \left(\lambda \right) \int
_{\mathcal{I}}X_{\lambda }^{*} \left(t\right)W\left(t,l_{\lambda }
,m\right)F\left(t,l_{\lambda } ,m\right)dt ,\, \int
_{\mathcal{I}}X_{\bar{\lambda }}^{*}
\left(t\right)W\left(t,l_{\bar{\lambda} }
,m\right)G\left(t,l_{\bar{\lambda }} ,m\right)dt
\right)\right]d\mu=\\
=\int _{\alpha }^{\beta }\left(d\sigma \left(\mu \right)\int
_{\mathcal{I}}X_{\mu }^{*} \left(t\right)W\left(t,l_{\mu }
\right)F\left(t,l_{\mu } ,m\right)dt ,\, \int _{\mathcal{I}}X_{\mu
}^{*} \left(t\right)W\left(t,l_{\mu } \right)G_{\mu }
\left(t,l_{\mu } ,m\right)dt \right),
\end{multline}
where the second equality is a corollary of formula (40) from \cite{KhrabMAG,KhrabArxiv},
the next to last is a corollary of \eqref{GEQ__62_} and the
last one follows from the well-known generalization of the
Stieltjes inversion formula \cite[p.
803]{Shtraus1}, \cite[p. 952]{Bruk1}. (In the case of
finite $\mathcal{I}$ we have to substitute in \eqref{GEQ__90_}
$M(\lambda)$ by $P M(\lambda) P$ and then when passing to the next
to the last equality in \eqref{GEQ__90_} we have to use the
remark after the proof of Lemma 2.1 from \cite{KhrabMAG,KhrabArxiv}.) But for $\lambda \in
\mathcal{B}$
\begin{equation} \label{GEQ__91_}
\int_{\mathcal{I}}X_{\bar{\lambda }}^{*}
\left(t\right)W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }} \right)dt =\int
_{\mathcal{I}}\left(\left[X_{\bar\lambda } \left(t\right)\right]_{1}
\right)^{*} m\left[f\right]dt ,
\end{equation}
because in view of Theorem 2.1 from \cite{KhrabMAG,KhrabArxiv}
\begin{multline*}
\forall h\in \mathcal{H}^{r} :(\int _{\mathcal{I}}X_{\bar{\lambda
}}^{*} \left(t\right)W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }} \right)dt,h )=\\
=\int _{\mathcal{I}}\left(W\left(t,l_{\bar{\lambda }}
,m\right)F\left(t,l_{\bar{\lambda }} \right),X_{\bar{\lambda }}
\left(t\right)h\right)dt =(\int
_{\mathcal{I}}\left(\left[X_{\bar{\lambda }} \right]_{1}
\right)^{*} m\left[f\right],h)dt.
\end{multline*}
Due to \eqref{GEQ__90_}, \eqref{GEQ__91_},
\eqref{GEQ__85_}
\begin{equation} \label{GEQ__92_}
\left(E_{\alpha ,\beta } f,g\right)_{L_{m}^{2}
\left(\mathcal{I}\right)} =\int _{\alpha }^{\beta }\left(d\sigma
\left(\mu \right)\varphi \left(\mu ,f\right),\varphi \left(\mu
,g\right)\right) .
\end{equation}
The equality \eqref{GEQ__88_} and inequality
\eqref{GEQ__89_} are the corollaries of \eqref{GEQ__92_}.
Representing $\varphi \left(\mu ,g\right)$ in \eqref{GEQ__92_}
by the second variant of \eqref{GEQ__85_}, changing in
\eqref{GEQ__92_} the order of integration and replacing
$\int_{\alpha}^\beta$ by integral sum and using
Theorem 2.1 from \cite{KhrabMAG,KhrabArxiv} we obtain that
\begin{multline*}
\left(E_{\alpha ,\beta } f,g\right)_{L_{m}^{2}
\left(\mathcal{I}\right)} =(\int _{\alpha }^{\beta }\left[X_{\mu }
\left(t\right)\right]_{1} d\sigma \left(\mu \right)\varphi
\left(\mu ,f\right),g\left(t\right) )_{L_{m}^{2}
\left(\mathcal{I}\right)} =\\
=\int _{\alpha }^{\beta }\left[X_{\mu } \left(t\right)\right]_{1}
d\sigma \left(\mu \right)\varphi \left(\mu
,f\right),g\left(t\right) )_{L_{m}^{2} \left(\mathcal{I}\right)}
\end{multline*}
and \eqref{GEQ__84_} is proved since
$g(t)\in\overset{\circ}{H}$. Equalities \eqref{GEQ__87_} are
the corollary of \eqref{GEQ__84_}, \eqref{star3}. Theorem
\ref{th7} is proved.
\end{proof}
Let us notice that if
$L_m^2(\mathcal{I})=\overset{\circ\ }{L_m^2}(\mathcal{I})$ then
Theorem \ref{th7} is valid without condition \eqref{GEQ__55_}
with $P=I_r$ if $\mathcal{I}$ is infinite.
Formulae (\ref{GEQ__84_}), (\ref{GEQ__87_}), (\ref{GEQ__88_}) are similar to corresponding formulas for scalar differential operators from \cite{Shtraus1} (operator case see \cite{Bruk1}). For such operators the formulas that corresponds to (\ref{GEQ__84_}), (\ref{GEQ__87_}), (\ref{GEQ__88_}) are represented in \cite[p. 251, 255]{DS}, \cite[p. 516]{Naimark} in another form. Let us represent for example inversion formula (\ref{GEQ__87_}), in the form analogues to \cite{DS,Naimark}.
\begin{proposition}\label{prop21}
Let all conditions of Theorem \ref{th7} hold. Let us represent spectral operator-function $\sigma \left(\mu \right)$ in matrix form: $\sigma \left(\mu \right)=\left\| \sigma _{ij} \left(\mu \right)\right\| _{i,j=1}^{r},\ \sigma _{ij} \left(\mu \right)\in B\left(\mathcal{H}\right)$. Then the following inversion formulae in $L_{m}^{2} \left(\mathcal{I}\right)$
\begin{gather}
\label{prop21_1}
f\left(t\right)=\mathop{P}\limits^{\circ } \int _{\mathcal{B}^1}\sum _{i,j=1}^{r}x_{i} \left(t,\mu \right)d\sigma _{ij} \left(\mu \right) \int _{\mathcal{I}}x_{j}^{*} \left(s,\mu \right)m\left[f\left(s\right)\right] ds,\\\label{prop21_2}
f\left(t\right)=\mathop{P}\limits^{\circ } \int _{\mathcal{B}^1}\sum _{i,j=1}^{r}x_{i} \left(t,\mu \right) d\sigma _{ij} \left(\mu \right) \int _{\mathcal{I}}\sum _{k=0}^{s/2}\left(x_{j}^{\left(k\right)} \left(s,\mu \right)\right)^{*} \mathrm{m}_{k} \left[f\left(s\right)\right]ds
\end{gather}
are valid if $\mathcal{I}$ is infinite, vector function $f\left(t\right)\in \mathop{H}\limits^{\circ } $ satisfies \eqref{star3}. Let $\mathcal{I}$ is finite. Then $\mathop{P}\limits^{\circ } $ in (\ref{prop21_1})-(\ref{prop21_2}) disappears, and (\ref{prop21_1}) (respectively, (\ref{prop21_2})) is valid for vector functions $f\left(t\right)\in \mathop{H}\limits^{\circ } $ (respectively, $f\left(t\right)\in H$) satisfying (\ref{star3}).
In formulae (\ref{prop21_1}), (\ref{prop21_2}): $x_i(t,\mu)\in B(\mathcal{H})$,
$(x_1(t,\mu),\dots,x_r(t,\mu))=[X_\mu(t)]_1$, $\mathrm{m}_k[f(s)]$ see \eqref{mk}.
\end{proposition}
The proof of this proposition is carried out in the same way as the proof of (\ref{GEQ__87_}) taking into account the proof of Remark 3.1 from \cite{KhrabMAG}.
Further we present several statements which allow to check the fulfilment of conditions (\ref{star3}) of Theorem \ref{th7} in various situations.
It is known (see for example \cite{Khrab2,Khrab3} or \cite[Ex. 3.2]{KhrabMAG,KhrabArxiv})
that even in the case $n_{\lambda } \left[y\right]\equiv 0$ in
\eqref{GEQ__1_}, \eqref{GEQ__2_} there is such $E_{\mu} $
satisfying \eqref{GEQ__3_}, \eqref{GEQ__62_}-\eqref{GEQ__66_}, \eqref{GEQ__83_} that ${E} _{\infty }
\ne \mathbf{I}$.
On the other hand if $n_\lambda[y]\equiv 0$ then $R(\lambda)$ is a generalized resolvent of relation $\mathcal{L}_0$ and $\forall f\in
D\left(\mathcal{L}_{0} \right){E} _{\infty } f=f$ in view of
\cite{Khrab1,Khrab3}. Here $\mathcal{L}_{0}$ is the minimal relation generated in $L^2_m(\mathcal{I})$ by the pair of expressions $l[y]$ and $m[y]$; in particular $\mathcal{L}_0\supset \{\{y(t),f(t)\}:\ y(t)\in C_0^r(\mathcal{I}),f(t)\in \overset{\, \circ}H,l[y]=m[f]\}$ (see \cite{Khrab6,KhrabArxiv,KhrabMAG}).
Let expression $n_{\lambda } $ in representation
\eqref{GEQ__2_}, \eqref{GEQ__561_} have a divergent form
with coefficients $\tilde{\tilde{p}}_{j} =\tilde{\tilde{p}}_{j}
\left(t,\lambda \right),\tilde{\tilde{q}}_{j}
=\tilde{\tilde{q}}_{j} \left(t,\lambda
\right),\tilde{\tilde{s}}_{j} =\tilde{\tilde{s}}_{j}
\left(t,\lambda \right)$.
We denote $m\left(t\right)$ three-diagonal $\left(n+1\right)\times
\left(n+1\right)$ operator matrix, whose elements under main
diagonal are equal to $\left(-\frac{i}{2} \tilde{q}_{1} ,\, \ldots
,\, -\frac{i}{2} \tilde{q}_{n} \right)$, the elements over the
main diagonal are equal to $\left(\frac{i}{2} \tilde{s}_{1} ,\,
\ldots ,\, \frac{i}{2} \tilde{s}_{n} \right)$, the elements on the
main diagonal are equal to $\left(\tilde{p}_{0} ,\, \ldots ,\,
\tilde{p}_{n} \right)$, where $\tilde{p}_{j} ,\tilde{q}_{j}
,\tilde{s}_{j} =\tilde{q}_{j}^{*} $ are the coefficients of
expressions $m$. (Here either $2n$ or $2n+1$ is equal to the order
$r$ of $l_\lambda$). If order of $n_{\lambda } $ is less or equal
to $2n$, we denote $n\left(t,\lambda \right)$ the analogues
$\left(n+1\right)\times \left(n+1\right)$ operator matrix with
$\tilde{\tilde{p}}_{j} ,\tilde{\tilde{q}}_{j}
,\tilde{\tilde{s}}_{j} $ instead of $\tilde{p}_{j} ,\tilde{q}_{j}
,\tilde{s}_{j} $. If order $m$ or order $n_{\lambda } $ is less
than $2n$, we set the correspondent elements of $m\left(t\right)$
or $n\left(t,\lambda \right)$ be equal to zero.
\begin{theorem}\label{th8}
Let in \eqref{GEQ__1_}, \eqref{GEQ__2_} the order of the
expression $n_{\lambda }[y] $ is less or equal to the order of the
expression $(l-\lambda m)[y]$ (and therefore in view of
(\ref{GEQ__561_}) the order of $l-\lambda m$ is equal to $r$;
so $Q\left(t,l_{\lambda } \right)=Q\left(t,l-\lambda m\right)$).
Let $y=R_{\lambda } f,\, f\in H$ be the generalized resolvent of
the relation $\mathcal{L}_{0} $ and $y$ satisfy equation
\eqref{GEQ__1_}. Let $y_{1} =R\left(\lambda \right)f,\, f\in
H$ be the operator \eqref{GEQ__62_}, \eqref{GEQ__63_} from
Theorem \ref{th4}.
Let the following conditions hold for $\tau >0$ large enough:
1$^\circ$.
{\small
\begin{multline} \label{GEQ__94_}
\left.\mathop{\lim }\limits_{\alpha \downarrow a,\, \beta \uparrow
b} \frac{(\Re Q\left(t,l_{\lambda } \right)\left(\vec{y}_{1}
\left(t,l_{\lambda } ,m,f\right)-\vec{y}\left(t,l-\lambda
m,m,f\right)),\left(\vec{y}_{1} \left(t,l_{\lambda }
,m,f\right)\right)-\vec{y}\left(t,l-\lambda
m,m,f\right)\right)}{\Im \lambda }\right|_\alpha^\beta\le\\\le 0,\
\lambda =i\tau
\end{multline}
}
2$^\circ$.
\begin{equation} \label{GEQ__95_}
\Im{n}\left({\rm t,}\lambda \right)\le c\left(t,\tau
\right)m\left(t\right),\, t\in \bar{\mathcal{I}},\, \, \lambda
=i\tau ,
\end{equation}
where the scalar function $c\left(t,\tau \right)$ satisfles the
following condition:
\begin{equation} \label{GEQ__96_}
\mathop{\sup }\limits_{t\in \bar{\mathcal{I}}} c\left(t,\tau
\right)=o\left(\tau \right),\, \, \, \tau \to +\infty .
\end{equation}
Then for generalized spectral family ${E}_{\mu } $
\eqref{GEQ__83_} corresponding by \eqref{GEQ__3_} to the
resolvent $R\left(\lambda \right)$
\eqref{GEQ__62_}-\eqref{GEQ__63_} from Theorem \ref{th4}
and for generalized spectral family $\mathcal{E}_{\mu } $
corresponding to the generalized resolvent $R_{\lambda } $ one has
${E} _{\infty } =\mathcal{E} _{\infty } $.
\end{theorem}
Let us notice that in view of \eqref{GEQ__95_} the coefficient
at the highest derivative in the expression $l-\lambda m$ has
inverse from $B\left(\mathcal{H}\right)$ if $t\in
\bar{\mathcal{I}}$, $\Im \lambda \ne 0$.
\begin{proof}
Let $f\left(t\right)\in H$, $y_{1} =R\left(\lambda \right)f$,
$y=R_{\lambda } f$. Then $z=y_{1} -y$ satisfies the following
equation
\begin{equation} \label{GEQ__97_}
l\left[z\right]-\lambda m\left[z\right]=n_{\lambda } \left[y_{1} \right].
\end{equation}
Applying to the equation \eqref{GEQ__97_} the Green formula
\cite[Theorem 1.3]{KhrabMAG,KhrabArxiv} one has
$$
\left.\int_{\alpha }^{\beta }\Im \left(n_{\lambda } \left\{y_{1}
,z\right\}\right)dt +\int _{\alpha }^{\beta }m\left\{z,z\right\}dt
=\frac{1}{2} \frac{\Re \left(Q\left(t,l_{\lambda }
\right)\vec{z},\vec{z}\right)}{\Im \lambda } \right|_\alpha^\beta
,
$$
where $\vec{z}=\vec{z}\left(t,l-\lambda m,n_{\lambda } ,y_{1}
\right)=\vec{y}_1\left(t,l_{\lambda }
,m,f\right)-\vec{y}\left(t,l-\lambda m,m,f\right)$ in view Lemma 1.2 from \cite{KhrabMAG,KhrabArxiv} and of
\eqref{GEQ__25_}. Hence for $\tau >0$
large enough
\begin{multline} \label{GEQ__98_}
m\left[z,z\right]\le -\int_{\mathcal{I}}\Im\left(n_{\lambda }
\left[y_{1} ,z\right]\right)dt/\tau
\leq\\
\leq{\int _{\mathcal{I}}\left|\left({ n}\left({\rm t,}\lambda
\right)col\left\{y_{1} ,y'_{1} ,\ldots ,y_{1}^{\left(n\right)}
\right\},col\left\{z,z',\ldots ,z^{\left(n\right)}
\right\}\right)\right| dt \mathord{\left/{\vphantom{\int
_{\mathcal{I}}\left|\left({\rm n}\left({\rm t,}\lambda
\right)col\left\{y_{1} ,y'_{1} ,\ldots ,y_{1}^{\left(n\right)}
\right\},col\left\{z,z',\ldots ,z^{\left(n\right)}
\right\}\right)\right| dt \tau }}\right.\kern-\nulldelimiterspace}
\tau } ,\quad \lambda =i\tau
\end{multline}
in view of \eqref{GEQ__94_}. But due to the inequality of the
Cauchy type for dissipative operators \cite[p. 199]{Nagy} and
\eqref{GEQ__95_}, \eqref{GEQ__96_}: subintegral function
in the last integral in \eqref{GEQ__98_} is less or equal to
$\left(m\left\{z,z\right\}\right)^{1/2}
\left(m\left\{y_1,y_1\right\}\right)^{1/2} o\left(1\right)$ with
$\lambda =i\tau ,\, \tau \to +\infty $. Therefore $\left\|
z\right\|_m \le o\left({1/\tau}\right)\left\| f\right\| _m $ since
$\left\| R_{\lambda } \right\| \le {1 \mathord{\left/{\vphantom{1
\left|\Im \lambda \right|}}\right.\kern-\nulldelimiterspace}
\left|\Im \lambda \right|} $. Hence
$$\left\| R\left(\lambda
\right)-R_{\lambda } \right\|\le o\left({1
\mathord{\left/{\vphantom{1 \tau
}}\right.\kern-\nulldelimiterspace} \tau } \right),\quad \lambda
=i\tau ,\quad \tau \to +\infty$$ To complete the proof of the
theorem it remains to prove the following
\begin{lemma}\label{lm10+}
Let $R_{k} \left(\lambda \right)=\int
_{\mathbb{R}^1}\frac{dE^k_{\mu } }{\mu -\lambda } ,\, k=1,2$,
where $E_{\mu }^{k} $ are the generalized spectral families the
type \eqref{GEQ__83_} in Hilbert space $\mathbf{H}$. If
$\left\| R_{1} \left(\lambda \right)-R_{2} \left(\lambda
\right)\right\| \le o\left({1 \mathord{\left/{\vphantom{1 \tau
}}\right.\kern-\nulldelimiterspace} \tau } \right),\, \, \,
\lambda =i\tau ,\, \, \tau \to +\infty $, then $E_{\infty }^{1}
=E_{\infty }^{2} $.
\end{lemma}
\begin{proof}
Let $f\in $\textbf{H} $\, \sigma \left(\mu
\right)=\left(\left(E_{\mu }^{1} -E_{\mu }^{2} \right)f,f\right)$.
One has
\begin{multline*}
\left|\left(\left[R_{1} \left(\lambda \right)-R_{2} \left(\lambda
\right)\right]f,f\right)\right|=\\=\frac{1}{\tau } \left|-\int
_{\Delta }d\sigma \left(\mu \right) +\int _{\Delta }\frac{\mu
d\sigma \left(\mu \right)}{\mu -\lambda } +\int _{R^{1}
\backslash \Delta }\frac{\lambda d\sigma \left(\mu \right)}{\mu
-\lambda } \right|\le o\left({1 \mathord{\left/{\vphantom{1 \tau
}}\right.\kern-\nulldelimiterspace} \tau } \right)\left\|
f\right\| ^{2} ,\, \, \lambda =i\tau ,\, \, \tau \to +\infty
\end{multline*}
Therefore
\begin{equation} \label{GEQ__99_}
\left|-\int _{\Delta }d\sigma \left(\mu \right)+\int _{\Delta
}\frac{\mu d\sigma \left(\mu \right)}{\mu -\lambda } +\int
_{R^{1} \backslash \Delta }\frac{\lambda d\sigma \left(\mu
\right)}{\mu -\lambda } \right|\le o\left(1\right),\, \, \,
\lambda =i\tau ,\, \, \tau \to +\infty .
\end{equation}
For an arbitrarily small $\varepsilon >0$ we choose such finite
interval $\Delta \left(\varepsilon \right)$ that for any finite
interval $\Delta \supseteq \Delta \left(\varepsilon
\right):\left|\int _{R^{1} \backslash \Delta }\frac{\lambda
d\sigma \left(\mu \right)}{\mu -\lambda }
\right|<\frac{\varepsilon }{2} ,\, \, \lambda =i\tau $. But for
any finite interval $\Delta \supseteq \Delta \left(\varepsilon
\right)\, \exists N=N\left(\Delta \right):\forall \tau
>N:\left|\int _{\Delta }\frac{\mu d\sigma \left(\mu \right)}{\mu
-\lambda } \right|<\frac{\varepsilon }{2} ,\, \, \lambda =i\tau
$. Therefore $\forall \varepsilon >0,\Delta \supseteq \Delta
\left(\varepsilon \right):\left|\int _{\Delta }d\sigma \left(\mu
\right) \right|<\varepsilon $ in view of \eqref{GEQ__99_}.
Hence $\forall f\in $\textbf{H}$:\left(E_{\infty }^{1}
f,f\right)=\left(E_{\infty }^{2} f,f\right)\Rightarrow E_{\infty
}^{1} =E_{\infty }^{2} $.
Lemma \ref{lm10+} and Theorem \ref{th8} are proved.
\end{proof}
\end{proof}
\begin{corollary}\label{cor3}
Let the conditions of Theorems \ref{th7}, \ref{th8} hold. Then for
generalized spectral family $E_{\mu } $ from Theorem \ref{th7}
$\forall f\left(t\right)\in D\left(\mathcal{L}_{0}
\right):E_{\infty } f=f$.
\end{corollary}
\begin{remark}
\label{rem3} If $L_{m}^{2}
\left(\mathcal{I}\right)=\mathop{L_{m}^{2} }\limits^{\circ }
\left(\mathcal{I}\right)$, then it is sufficient to verify
condition \eqref{GEQ__94_} in Theorem \ref{th8} for $f\in
\mathop{H}\limits^{\circ } $.
\end{remark}
\begin{proposition}
\label{prop2} Let the order of expression $n_{\lambda } $ be less
or equal to the order of expression $l-\lambda m$ and the
coefficient of $l-\lambda m$ at the highest derivative has the
inverse from $B\left(\mathcal{H}\right)$ for $t\in
\bar{\mathcal{I}},\, \lambda \in \mathcal{B} \left(l-\lambda
{m}\right)$, where $\mathcal{B}\left(l-\lambda m\right)$ is an
analogue of the set $\mathcal{B} =\mathcal{B}\left(l_\lambda
\right)$. Let interval $\mathcal{I}$ be finite and for equation
\eqref{GEQ__1_}, \eqref{GEQ__2_} with $n_{\lambda }
\left[y\right]\equiv 0$ condition \eqref{GEQ__55_} holds with
${P} ={I} _{r}$.
Then for equation \eqref{GEQ__1_}-\eqref{GEQ__2_} this condition also holds with $P=I_r$.
The boundary value problem which is obtained by adding to equation \eqref{GEQ__1_}-\eqref{GEQ__2_} with $n_\lambda[y]\equiv 0$ (respectively, \eqref{GEQ__1_}-\eqref{GEQ__2_}) boundary conditions
\begin{gather}\label{bk1}
\exists h=h\left(\lambda ,f\right)\in \mathcal{H}^{r} :\, \,
\vec{y}\left(a,l_\lambda-\lambda m ,m,f\right)=\mathcal{M}_{\lambda } h,\, \,
\, \vec{y}\left(b,l_\lambda-\lambda m ,m,f\right)=\mathcal{N}_{\lambda }
h\\
\label{bk2}
\text{(respectively, }\exists h_1=h_1\left(\lambda ,f\right)\in \mathcal{H}^{r} :\, \,
\vec{y}\left(a,l_\lambda ,m,f\right)=\mathcal{M}_{\lambda } h_1,\, \,
\, \vec{y}\left(b,l_\lambda ,m,f\right)=\mathcal{N}_{\lambda }
h_1\text{)},\end{gather}
has the unique solutions $y=R_{\lambda } f$ (respectively, $y_{1} =R\left(\lambda \right)f$) for an arbitrary $f(t)\in H$.
Here the operator-functions $\mathcal{M}_{\lambda },
\mathcal{N}_{\lambda} \in B\left(\mathcal{H}^{r} \right)$ depend
analytically on the non-real $\lambda $,
\begin{equation*}
\mathcal{M}_{\bar{\lambda }}^{*} \left[\Re Q\left(a,l_{\lambda }
\right)\right]\mathcal{M}_{\lambda } =\mathcal{N}_{\bar{\lambda
}}^{*} \left[\Re Q\left(b,l_{\lambda }
\right)\right]\mathcal{N}_{\lambda },\ \Im \lambda \ne
0,
\end{equation*}
where $Q\left(t,l_{\lambda } \right)$ is the coefficient of
equation \eqref{GEQ__54_} corresponding by Theorem \ref{th1}
to equation \eqref{GEQ__1_},
\begin{equation*}
\left\| \mathcal{M}_{\lambda } h\right\| +\left\|
\mathcal{N}_{\lambda } h\right\| >0,\ 0\ne h\in
\mathcal{H}^{r} ,\, \Im \lambda \ne 0,
\end{equation*}
the lineal $\left\{\mathcal{M}_{\lambda } h\oplus
\mathcal{N}_{\lambda } h\left|h\in \mathcal{H}^{r} \right.
\right\}\subset \mathcal{H}^{2r} $ is a maximal
$\mathcal{Q}$-nonnegative subspace if $\Im \lambda \ne 0$, where
$\mathcal{Q}=\left(\Im \lambda \right)\mathrm{diag} \left(\Re
Q\left(a,l_{\lambda } \right),\, -\Re Q\left(b,l_{\lambda }
\right)\right)$ (and therefore
\begin{equation*}
\Im \lambda \left(\mathcal{N}_{\lambda }^{*} \left[\Re
Q\left(b,l_{\lambda } \right)\right]\mathcal{N}_{\lambda }
-\mathcal{M}_{\lambda }^{*} \left[\Re Q\left(a,l_{\lambda }
\right)\right]\mathcal{M}_{\lambda } \right)\le 0,\ \left.
\Im \lambda \ne 0\right).
\end{equation*}
Operator $R_\lambda f$ (respectively, $R(\lambda)f$) is a generalized resolvent of
$\mathcal{L}_0$ (respectively, a resolvent of \eqref{GEQ__63_} type). This resolvent is constructed by applying of Theorem \ref{th4} with the characteristic operator
\begin{gather}\label{mlambda+}
M(\lambda)=-{1\over 2}\left(X_\lambda^{-1}(a)\mathcal{M}_\lambda+X_\lambda^{-1}(b)\mathcal{N}_\lambda\right)\left(X_\lambda^{-1}(a)\mathcal{M}_\lambda-X_\lambda^{-1}(b)\mathcal{N}_\lambda\right)^{-1}(iG)^{-1},
\end{gather}
to equation \eqref{GEQ__1_}, \eqref{GEQ__2_} with $n_\lambda[y]\equiv 0$ (respectively, \eqref{GEQ__1_}-\eqref{GEQ__2_}). Here $\left(X_\lambda^{-1}(a)\mathcal{M}_\lambda-X_\lambda^{-1}(b)\mathcal{N}_\lambda\right)^{-1}\in B(\mathcal{H}^r)$, $\Im\lambda\not= 0$, $X_\lambda(t)$ is an operator solution from Theorem \ref{th4} which corresponds to equation $(l-\lambda m)[y]=0$ (respectively, $l_\lambda[y]=0$).
The resolvents $y=R_{\lambda } f$ and $y_{1} =R\left(\lambda \right)f$ satisfy condition \eqref{GEQ__94_} of Theorem \ref{th8}.
\end{proposition}
Let us notice that if $\mathcal{I}$ is finite and condition \eqref{GEQ__55_} holds with $P=I_r$ then $M(\lambda)$ \eqref{mlambda+} is a characteristic operator of equation \eqref{GEQ__5_} and any characteristic operator of equation \eqref{GEQ__5_} has representation (\ref{mlambda+}). Also we notice that if $\mathcal{I}$ is finite, $n_\lambda[y]\equiv 0$ and condition \eqref{GEQ__55_} holds with $P=I_r$ then any generalized resolvent of $\mathcal{L}_0$ can be constructed as an operator $R(\lambda)$ from Theorem \ref{th4} in view of \cite{KhrabMAG,KhrabArxiv}.
\begin{proof}
In view of Theorems 3.2, 3.3 from \cite{KhrabMAG,KhrabArxiv} it is sufficient to prove only
proposition about condition \eqref{GEQ__55_}.
Let for definiteness order $l=$ order $m=$ order $n_{\lambda }
=2n$.
Let for equation \eqref{GEQ__1_}-\eqref{GEQ__2_} with
$n_{\lambda } \left[y\right]\equiv 0$ condition
\eqref{GEQ__55_} with ${P} ={ I}_{r}$ hold, but for equation
\eqref{GEQ__1_}, \eqref{GEQ__2_} that is not true. Then in
view of \cite{Khrab5} the solutions $y_k\left(t\right)$ of
equation \eqref{GEQ__1_}-\eqref{GEQ__2_} with $f(t)=0$, $\lambda
=i$ exist for which
\begin{equation} \label{GEQ__100_}
\int _{\alpha }^{\beta }\left(m+\Im n_{i} \right)\left\{y_{k}
,y_{k} \right\} dt\to 0,\quad \vec{y}_{k} \left(0,l_{i}
,m,0\right)=f_{k} ,\; \left\| f_{k} \right\| =1,
\end{equation}
where $i\Im n_i =n_{i} $ in view of \eqref{GEQ__561_}. Hence
in view of Theorem 1.2 from \cite{KhrabMAG,KhrabArxiv})
\begin{equation} \label{GEQ__101_}
\int _{\alpha }^{\beta }\left(W_{i} \left(t,l+im,n_{i} \right)
Y_{k} \left(t,l+im,n_i\right),Y_{k}\left(t,l+im,n_i\right)
\right)dt = \int _{\alpha }^{\beta }n_{i} \left\{y_{k} ,y_{k}
\right\}dt \to 0.
\end{equation}
On the other hand
\begin{equation} \label{GEQ__102_}
X_{i} \left(t\right)f_{k} =\tilde X_{i} \left(t\right)f_{k}
+\tilde X_{i} \left(t\right)\int _{o}^{t}\tilde X_{i}^{-1}
\left(s\right)J^{-1} W\left(s,l+im,n_{i} \right)Y_{k}
\left(s,l+im,n_i\right) ds .
\end{equation}
in view of Theorem \ref{th1} and the fact that
$\vec{y}_k(t,l-im,n_i,y_k)=\vec{y}_k(t,l_i,m,0)$, where $\tilde
X_{\lambda } \left(t\right)$ is an analogue of $X_{\lambda }
\left(t\right)$ for the case $n_{\lambda } \left[y\right]\equiv
0$.
Comparing \eqref{GEQ__101_}, \eqref{GEQ__102_} we see that
\begin{equation} \label{GEQ__103_}
\left\|X_{i} \left(t\right)f_{k} -\tilde X_{i} \left(t\right)f_{k}
\right\| \to 0
\end{equation}
uniformly in $t\in \left[\alpha ,\beta \right]$.
In view of \eqref{GEQ__100_} subsequence $y_{k_{q} } $ exist
such that
\begin{equation} \label{GEQ__104_}
m\left\{y_{k_{q} } ,y_{k_{q} } \right\}\mathop{\to }\limits^{a.a.} 0,\quad n_{i} \left\{y_{k_{q} } ,y_{k_{q} } \right\}\mathop{\to }\limits^{a.a.} 0.
\end{equation}
Due to second proposition in \eqref{GEQ__104_} and the arguments
as in the proof of Proposition 3.1 from \cite{KhrabMAG} one has
\begin{equation} \label{GEQ__105_}
y_{k_{q} }^{\left[j\right]} \left(t\left|n_{i} \right.
\right)\mathop{\to }\limits^{a.a.} 0\quad j=n,\ldots ,2n.
\end{equation}
Let us denote $\tilde y_{k_{q} } \left(t\right)=\tilde X_{i}
\left(t\right)f_{k_{q} } $. In view of Theorem \ref{th1} and
\eqref{GEQ__103_}
\begin{equation} \label{GEQ__106_}
\left\| y_{k_{q} }^{\left(j\right)} \left(t\right)-\tilde y_{k_{q}
}^{\left(j\right)} \left(t\right)\right\| \to 0,\quad j=1,\ldots
,n-1,
\end{equation}
\begin{multline} \label{GEQ__107_}
\left\| \left(p_{n} \left(t\right)-i\tilde{p}_{n}
\left(t\right)\right)\left[y_{k_{q} }^{\left(n\right)}
\left(t\right)-\tilde y_{k_{q} }^{\left(n\right)}
\left(t\right)\right]-\right.\\\left.-{i\over 2}(q_n(t)-i\tilde
q_n(t))\left[y_{k_q}^{(n-1)}(t)-\tilde y_{k_q}^{(n-1)}(t)\right]
-y_{k_{q} }^{\left[n\right]} \left(t\left|n_{i} \right.
\right)\right\| =\\=\left\| y_{k_{q} }^{\left[n\right]}
\left(t|l_i\right)-\tilde y_{k_{q} }^{\left[n\right]}
\left(t|l-im\right)\right\| \to 0
\end{multline}
uniformly in $t\in \left[\alpha ,\beta \right]$. Comparing
\eqref{GEQ__104_}, \eqref{GEQ__105_},
\eqref{GEQ__107_} and using $\left(p_{n}
\left(t\right)-i\tilde{p}_{n} \left(t\right)\right)^{-1} \in
B\left(\mathcal{H}\right)$ we have
\begin{equation} \label{GEQ__108_}
\left(\tilde{p}_{n} \left(t\right)\tilde y_{k_{q}
}^{\left(n\right)} \left(t\right),\tilde y_{k_{q}
}^{\left(n\right)} \left(t\right)\right)\mathop{\to
}\limits^{a.a.} 0.
\end{equation}
In view of \eqref{GEQ__106_}, \eqref{GEQ__108_},
\eqref{GEQ__104_}
$$
m\left\{\tilde y_{k_{q} } ,\tilde y_{k_{q} } \right\}\mathop{\to
}\limits^{a.a.} 0,\quad \vec{y}_{k_q}(0,l-im, m,0)=f_{k_q},
$$
that contradicts to the condition \eqref{GEQ__55_} with
$P=I_{r} $ for equation \eqref{GEQ__1_}, \eqref{GEQ__2_}
with $n_{\lambda } \left[y\right]\equiv 0$. Proposition
\ref{prop2} is proved.
\end{proof}
If the set ${\mathbb R}^{1} \backslash \mathcal{B}^1$ has no finite limit points then to verify the condition $\mathop{P}\limits^{\circ } \int \ldots =0 $ or $\int \ldots =0$ in \eqref{star3} we can use the following proposition which is a corollary of Lemma from \cite[p. 789]{Shtraus2}.
\begin{proposition}\label{prop_str1}
Let $R\left(\lambda \right)=\int _{{\mathbb R}^{1} }\frac{dE_{\mu } }{\mu -\lambda } $, where $E_{\mu } $ is generalized spectral family in Hilbert space $\mathbf{H} ;\, g\in \mathbf{H} $. If $\sigma $ is not a point of continuity of $E_{\mu} g$, then $\exists c\left(\sigma,g\right)>0$: $\left\| R\left(\sigma +i\tau \right)g\right\| \sim \frac{c\left(\sigma ,g\right)}{\left|\tau \right|} ,\, \, \, \tau \to 0$.
\end{proposition}
\begin{proof} Let $\Delta $ be a jump of $E_{\mu } $ in the point $\sigma $. Then $\Delta g\ne 0$,
\begin{gather}\label{prop_str1_1}
R\left(\sigma +i\tau \right)g=i\frac{\Delta }{\tau } g+\int _{{\mathbb R}^{1} }\frac{d\tilde{E}_{\mu } g}{\mu -\lambda } \text{ where }\tilde{E}_{\mu } =\begin{cases} E_{\mu } ,&\mu \leq\sigma \\ E_{\mu } -\Delta ,&\mu >\sigma \end{cases}
\end{gather}
Since the second term in (\ref{prop_str1_1}) is $o\left({\raise0.7ex\hbox{$ 1 $}\!\mathord{\left/ {\vphantom {1 \left|\tau \right|}} \right. \kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$ \left|\tau \right| $}} \right)$ in view of \cite[p. 789]{Shtraus2}\footnote{ Lemma from \cite[789]{Shtraus2} is proved for families $E_{\mu } $ with $E_{\infty } =$identity operator. But analysis of its proof shows that it is valid in general case.} proposition in proved.
\end{proof}
In the next theorem $\mathcal{I}=\mathbb{R}^1$ and condition
\eqref{GEQ__55_} hold with $P=I_{r} $ both on the negative
semi-axis $\mathbb{R}^1_{-} $ (i.e. as $\mathcal{I}=\mathbb{R}^1_{-} $) and on the
positive semi-axis $\mathbb{R}^1_{+} $ (i.e. as $\mathcal{I}=\mathbb{R}^1_{+} $).
\begin{theorem}\label{th9}
Let $\mathcal{I}=\mathbb{R}^1$, the coefficient of the expression
$l_\lambda[y]$ \eqref{GEQ__2_} be periodic on each of the
semi-axes $\mathbb{R}^1_{-} $ and $\mathbb{R}^1_{+} $ with periods $T_{-} >0$ and $T_{+}
>0$ correspondingly. Then the spectrums of the monodromy operators
$X_{\lambda } \left(\pm T_{\pm } \right)$ ($X_{\lambda }
\left(t\right)$ is from Theorem \ref{th4}) do not intersect the
unit circle as $\Im \lambda \ne 0$, the characteristic operator $M\left(\lambda
\right)$ of the equation \eqref{GEQ__5_} is unique and equal
to
\begin{equation} \label{GEQ__109_}
M\left(\lambda \right)=\left(\mathcal{P}\left(\lambda
\right)-\frac{1}{2} I_{r} \right)\, \left(iG\right)^{-1} \quad
\left(\Im \lambda \ne 0\right),
\end{equation}
where the projection $\mathcal{P}\left(\lambda \right)=P_{+}
\left(\lambda \right)\left(P_{+} \left(\lambda \right)+P_{-}
\left(\lambda \right)\right)^{-1} ,\, P_{\pm } \left(\lambda
\right)$ are Riesz projections of the monodromy operators
$X_{\lambda } \left(\pm T_{\pm } \right)$ that correspond to their
spectrums lying inside the unit circle, $\left(P_{+} \left(\lambda
\right)+P_{-} \left(\lambda \right)\right)^{-1} \in
B\left(\mathcal{H}^{r} \right)$ as $\Im \lambda \ne 0$.
Also let $\dim \mathcal{H}<\infty $, a finite interval $\Delta
\subseteq \mathcal{B}^1 $. Then in Theorem \ref{th7} $d\sigma
\left(\mu \right)=d\sigma _{ac} \left(\mu \right)+d\sigma _{d}
\left(\mu \right),\mu \in \Delta $. Here $\sigma _{ac} \left(\mu
\right)\in AC\left(\Delta \right)$ and, for $\mu \in \Delta $,
$$\sigma '_{ac} \left(\mu \right)=\frac{1}{2\pi } G^{-1}
\left(Q_{-}^{*} \left(\mu \right)GQ_{-} \left(\mu
\right)-Q_{+}^{*} \left(\mu \right)GQ_{+} \left(\mu
\right)\right)G^{-1}, $$ where the projections $Q_{\pm } \left(\mu
\right)=q_{\pm } \left(\mu \right)\left(P_{+} \left(\mu
\right)+P_{-} \left(\mu \right)\right)^{-1} ,\, q_{\pm } \left(\mu
\right)$ are Riesz projections of the monodromy matrixes $X_{\mu }
\left(\pm T_{\pm } \right)$ corresponding to the multiplicators
belonging to the unit circle and such that they are shifted inside
the unit circle as $\mu $ is shifted to the upper half plane,
$P_{\pm } \left(\mu \right)=P_{\pm } \left(\mu +i0\right);\,
\sigma _{d} \left(\mu \right)$ is a step function.
\end{theorem}
Let us notice that the sets on which $q_{\pm } \left(\mu
\right),P_{\pm } \left(\mu \right),\left(P_{+} \left(\mu
\right)+P_{-} \left(\mu \right)\right)^{-1} $ are not infinitely
differentiable do not have finite limit points $\in \mathcal{B}^1 $
as well as the set of points of increase of $\sigma _{d} \left(\mu
\right)$.
\begin{proof}
The proof of Theorem \ref{th9} is similar to that on in the case
$n_{\lambda } \left[y\right]\equiv 0$ \cite{Khrab6}.
\end{proof}
The following examples demonstrate effects that are the results of
appearance in $l_{\lambda } $ \eqref{GEQ__2_} of perturbation
$n_{\lambda } $ depending nonlinearly on $\lambda $.
In Examples \ref{ex3}, \ref{ex4} nonlinear in $\lambda $
perturbation does not change the type of the spectrum. In this
examples $\dim \mathcal{H}=1,\, \, m\left[y\right]=-y''+y$.
$L_{m}^{2} \left(\mathcal{I}\right)=\mathop{L_{m}^{2}
}\limits^{\circ } \left(\mathcal{I}\right)=W_{2}^{1,2}
\left(\mathbb{R}^{1} \right)$. In Example \ref{ex5} such
perturbation implies an appearance of spectral gap with
"eigenvalue" in this gap.
\begin{example}\label{ex3}
Let
$$
l_{\lambda } \left[y\right]=iy'-\lambda
\left(-y''+y\right)-\left(-\frac{h}{\lambda } y\right),\, \, \,
h\ge 0.$$ Here $\mathcal{B} =\mathbb{C}\setminus 0,\, \, E_{0}
=E_{+0} $, spectral matrix $\sigma \left(\mu \right)\in AC_{loc}
$,
\\
$\sigma '\left(\mu \right)=\frac{1}{2\pi } \left(\begin{array}{cc}
{\frac{2}{\sqrt{4h+1-4\mu ^{2} } } } & {0} \\ {0} & {\frac{1}{2}
\sqrt{4h+1-4\mu ^{2} } } \end{array}\right)$, as $\left|\mu
\right|<\sqrt{h+{1 \mathord{\left/{\vphantom{1
4}}\right.\kern-\nulldelimiterspace} 4} } $,
\\
$\sigma '\left(\mu \right)=0$, as $\left|\mu \right|>\sqrt{h+{1
\mathord{\left/{\vphantom{1 4}}\right.\kern-\nulldelimiterspace}
4} } $.
In Example \ref{ex3} nonlinear in $\lambda $ perturbation change
edges of spectral band.
\end{example}
\begin{example}\label{ex4}
Let
$$
l_{\lambda } \left[y\right]=y^{\left(IV\right)} -\lambda
\left(-y''+y\right)-\left(-\frac{h}{\lambda } y\right),\, \, \,
h\ge 0.$$ Here $\mathcal{B} =\begin{cases}
\mathbb{C}\setminus\{0\},&h\ne 0 \\ \mathbb{C},&h=0
\end{cases},\, \, \, E_{0} =E_{+0} $, spectral matrix
$\sigma \left(\mu \right)\in AC_{loc} $,
\begin{gather*}
\sigma '\left(\mu \right)=\begin{cases} \frac{1}{2\pi }
\sqrt{\frac{\lambda +\sqrt{D} }{D} } \left(\begin{array}{cccc}
{\frac{2}{\lambda +\sqrt{D} } } & {0} & {0} & {-1} \\ {0} & {1} &
{\frac{-\lambda +\sqrt{D} }{2} } & {0} \\ {0} & {\frac{-\lambda
+\sqrt{D} }{2} } & {\left(\frac{\lambda -\sqrt{D} }{2} \right)^{2}
} & {0} \\ {-1} & {0} & {0} & {\frac{\lambda +\sqrt{D} }{2} }
\end{array}\right),&\text{ as }-\sqrt{h} <\mu <0,\, \mu
>\sqrt{h} \\ \frac{1}{2\pi } \cdot \frac{1}{\sqrt{\lambda
-2\sqrt{q} } } \left(\begin{array}{cccc} {\frac{1}{\sqrt{q} } } &
{0} & {0} & {-1} \\ {0} & {1} & {-\sqrt{q} } & {0} \\ {0} &
{-\sqrt{q} } & {\sqrt{q} \left(\lambda -\sqrt{q} \right)} & {0} \\
{-1} & {0} & {0} & {\lambda -\sqrt{q} }
\end{array}\right),&\text{ as }\mu ^{*} <\mu <\sqrt{h},
\end{cases},
\end{gather*}
where $D=\mu ^{2} -4q,\, q={h/\mu} -\mu ,\,
\mu ^{*} =\mu ^{*} \left(h\right)$ - nonnegative root of equation
$D=0$. $\sigma '\left(\mu \right)=0$, as $\mu \notin
\left[-\sqrt{h} ,0\right]\bigcup \left[\mu ^{*} ,\infty \right)$.
In Example \ref{ex4} nonlinear in $\lambda $ perturbation implies
an appearance of additional spectral band $\left[-h,0\right]$,
variation of edge of semi-infinite spectral band and appearance
of interval $\left(\mu ^{*} ,\sqrt{h} \right)$ of fourfold
spectrum.
\end{example}
\begin{example}\label{ex5}
Let $\dim \mathcal{H}=2$,
$$l_{\lambda } \left[y\right]=\left(\begin{array}{cc} {0} & {-1} \\
{1} & {0}
\end{array}\right)y'-\lambda y-\left(\begin{array}{cc} {-{h
\mathord{\left/{\vphantom{h \lambda
}}\right.\kern-\nulldelimiterspace} \lambda } } & {0} \\ {0} & {0}
\end{array}\right)y,\quad h\ge 0.$$
Here $\mathcal{B} =\begin{cases} \mathbb{C}\setminus\{0\},& h\ne 0 \\
\mathbb{C},& h=0 \end{cases},$ spectral matrix $\sigma \left(\mu
\right)=\sigma _{ac} \left(\mu \right)+\sigma _{d} \left(\mu
\right)$, $\sigma_{ac}(\mu)\in AC_{loc}, \sigma '_{ac} \left(\mu
\right)\ne 0$, as $\left|\mu \right|>\sqrt{h} ,\, \, \sigma '_{ac}
\left(\mu \right)=0$, as $\left|\mu \right|<\sqrt{h} $,
step-function $\sigma _{d} \left(\mu \right)$ has only one jump
$\left(\begin{array}{cc} {0} & {0} \\ {0} & {{\sqrt{h}
\mathord{\left/{\vphantom{\sqrt{h}
2}}\right.\kern-\nulldelimiterspace} 2} } \end{array}\right)$ in
point $\mu =0$ (inside of spectral gap). In this point
$$\left(E_{+0} -E_{0} \right)f=\left(\begin{array}{cc} {0} & {0} \\
{0} & {{\sqrt{h} \mathord{\left/{\vphantom{\sqrt{h}
2}}\right.\kern-\nulldelimiterspace} 2} } \end{array}\right)\int
_{-\infty }^{\infty }e^{-\sqrt{h} \left|t-s\right|}
f\left(s\right) ds,\, \, \, f\left(t\right)\in L^{2}
\left(\mathbb{R}^{1} \right).$$
\end{example}
Let us explain that in Examples \ref{ex3}, \ref{ex4}: 1) Spectral matices are locally absolutely continions in view of Theorem 3.6 and estimates of the type $\left\| M\left(\lambda \right)\right\| \sim \frac{c}{\left|\lambda \right|^{\alpha } } \, \left(\lambda \to i0\right),\, \alpha <1$ for corresponding characteristic operators (cf. \cite{Khrab1}) that follows from \eqref{GEQ__109_}; 2) Equalities $E_{0} =E_{+0} $ follow from Proposition \ref{prop_str1}, equality $L_{m}^{2} \left(\mathbb{R}^{1} \right)=\mathop{L_{m}^{2} }\limits^{\circ } \left(\mathbb{R}^{1} \right)$
and estimates of the type $\left\| R\left(i\tau \right)g\right\| _{m} \le \frac{c\left(g\right)}{\left|\tau \right|^{\beta } } ,\, \left(\tau \to 0\right),\, \beta <1,\, g\in \mathop{{\rm H} }\limits^{\circ } $ that follows from Theorems \ref{th4}, \ref{th9} and Floquet Theorem.
Let us notice that in view of Floquet theorem conditions of
Theorem \ref{th8} ((\ref{GEQ__94_}) with account of Remark
\ref{rem3}) hold for all Examples \ref{ex3}-\ref{ex5}.
The following theorem is a generalization of results from \cite{Shtraus2} on the expansion in solutions of scalar Sturm-Louville equation which satisfy in regular end point the boudary condition depending on spectral parameter.
\begin{theorem} \label{th24}
Let $r=2n,\ \mathcal{I}=\left(0,\infty \right)$, condition (\ref{GEQ__55_}) with $P=\mathcal{I}_{2n} $ hold. Let contraction $v\left(\lambda \right)\in B\left(\mathcal{H}^{n} \right)$ satisfy the conditions of Lemma \ref{lm12}. Let $v\left(\lambda \right)$ analytically depend in $\lambda $ in any points of $\mathcal{B}^1=\mathbb{R}^{1} \bigcap \mathcal{B}$ and be unitary in this points.
Let $R\left(\lambda \right)$ (\ref{GEQ__63_}) correspond to characteristic operator $M\left(\lambda \right)$ (\ref{13}), (\ref{GEQ__64ad1_}) of equation (\ref{GEQ__5_}), where characteristic projection (\ref{GEQ__64ad1_}) corresponds to some Weyl function $m(\lambda)$ of equation (\ref{GEQ__51_}) and to pair (\ref{alambda}), (\ref{barlambda}) which is constructed with the help of this $v\left(\lambda \right)$. Let the generalized spectral family $E_\mu$ correspond to $R\left(\lambda \right)$ by (\ref{GEQ__3_}).
Let $m_{a,b} \left(\lambda \right)$ be Nevanlinna operator-function from Lemma \ref{lm12} corresponding by (\ref{GEQ__64ad1_}), (\ref{GEQ__68ad1_}), (\ref{alambda}), (\ref{barlambda}) to this $v\left(\lambda \right)$. Let $\sigma _{a,b} \left(\mu \right)=w-\mathop{\lim }\limits_{\varepsilon \downarrow 0} \frac{1}{\pi } \int_{0}^{\mu }{\Im}m_{a,b} \left(\mu +i\varepsilon \right) d\mu $ be the spectral operator-function that corresponds to $m_{a,b} \left(\lambda \right)$.
Then every proposition of Theorem \ref{th7} is valid with $\sigma _{a,b} \left(\mu \right)$ instead of $\sigma \left(\mu \right)$, $\left(u_{1} \left(t,\lambda \right),\ldots ,u_{n} \left(t,\lambda \right)\right)$ instead of $\left[X_{\lambda } \left(t\right)\right]_{1} $ and
\begin{multline*}\varphi \left(\mu ,f\right)=\int _{\mathcal{I}}\left(u_{1} \left(t,\mu \right),\ldots ,u_{n} \left(t,\mu \right)\right)^{*} m\left[f\left(t\right)\right] dt=\\=\int _{\mathcal{I}}\sum _{k=0}^{{s\mathord{\left/ {\vphantom {s 2}} \right. \kern-\nulldelimiterspace} 2} }\left(u_{1}^{\left(k\right)} \left(t,\mu \right),\ldots ,u_{n}^{\left(k\right)} \left(t,\mu \right)\right)^{*} \mathrm{m}_{k} \left[f\left(t\right)\right]dt
\end{multline*}
instead of $\varphi(\mu,f)$ \eqref{GEQ__85_}, where $u_{j} \left(t,\lambda \right)$ see (\eqref{GEQ__115_}), $\mathrm{m}_{k} \left[f\left(t\right)\right]$ see \eqref{mk}.
Therefore if we represent spectral operator-function $\sigma _{ab} \left(\mu \right)$ in matrix form: $\sigma _{a,b} \left(\mu \right)=\left\| \left(\sigma _{ab} \left(\mu \right)\right)_{ij} \right\| _{i,j=1}^{n} ,\, \left(\sigma _{a,b} \left(\mu \right)\right)_{ij} \in B\left(\mathcal{H}\right)$ then, for example\footnote{Also \eqref{GEQ__84_}, Parseval equality \eqref{GEQ__88_}, Bessel inequality \eqref{GEQ__89_} can be represented in a similar way}, the following inversion formula is valid in $L_{m}^{2} \left(0,\infty \right)$ for any vector-function $f(t)\in \overset{\, \circ}H$ satisfying \eqref{star3} :
\begin{gather*}
f\left(t\right)=
\mathop{P}\limits^{\circ } \int _{\mathcal{B}^1}\sum _{i,j=1}^{n}u_{i} \left(t,\mu \right) d\left(\sigma _{a,b} \left(\mu \right)\right)_{ij} \int _{\mathcal{I}}u_{j}^{*} \left(s,\mu \right)m\left[f\left(s\right)\right] ds=\\=\mathop{P}\limits^{\circ } \int _{\mathcal{B}^1}\sum _{i,j=1}^{n}u_{i} \left(t,\mu \right) d\left(\sigma _{a,b} \left(\mu \right)\right)_{ij} \int _{\mathcal{I}}\sum\limits_{k=0}^{s/2}\left(u_{j}^{\left(k\right)} \left(s,\mu \right)\right)^*\mathrm{m}_{k} \left[f\left(s\right)\right] ds.
\end{gather*}
\end{theorem}
The proof is carried out in the same way as the proofs of Theorem \ref{th7} and Proposition \ref{prop21} with the help of Proposition \ref{rm21} and Lemma \ref{lm12}.
Let us notice that in contrast to operator spectral function $\sigma_{a,b}(\mu)$ from Theorem \ref{th24} the scalar spectral function in \cite{Shtraus2} was constructed with the help of different formulae that corresponds to such intervals of real axis where $v\left(\mu \right)\ne -1$ or $v\left(\mu \right)\ne 1$. But already in matrix case it is impossible to construct the spectral matrix according to \cite{Shtraus2} since here for some real $\lambda$ (and even for any real $\lambda$) $(v(\lambda)+I_n)^{-1}$ and $(v(\lambda)-I_n)^{-1}$ may simultaneously do not belong to $B(\mathcal{H}^n)$.
\begin{remark}
Let contraction $v\left(\lambda \right)\in B\left(\mathcal{H}^{n} \right)$ is analytical in any point $\lambda \in \bar{{\mathbb C}}_{+} \bigcap \mathcal{B}$ and is unitary in any point $\lambda \in \mathcal{B}^1\not=\varnothing $. Let $\dim \mathcal{H}<\infty $. Then $v\left(\lambda \right)$ satisfy conditions of Lemma \ref{lm12}.
If $\dim \mathcal{H}=\infty $ in general it is not valid. Namely let domain $D\subset {\mathbb C}_{+} $, $dist\left\{\bar{D},\, \left[-a,a\right]\right\}>0\, \, \forall a\in {\mathbb R}_{+}^{1} $; $set\left\{\lambda _{k} \right\}_{k=1}^{\infty } \subset D$ in dense in $D$. Let us consider in $\mathcal{H}^{n} =\left(l^{2} \right)^{n} $ the following operator $v\left(\lambda \right)=v_1\left(\lambda \right)\oplus I_{n-1} $, where $v_1\left(\lambda \right)=diag\left\{\frac{\lambda -\lambda _{k} }{\lambda -\bar{\lambda }_{k} } \right\}_{k=1}^{\infty } $. This operator is analytical in any point $\lambda \in \bar{{\mathbb C}}_{+} $, is a contraction for $\lambda \in {{\mathbb C}}_{+} $ and is unitary for $\lambda \in {\mathbb R}^{1} $. But for this operator the set $S=\left\{\lambda \in {\mathbb C}_{+} :v^{-1} \left(\lambda \right)\notin B\left(\mathcal{H}^n\right)\right\}=\bar{D}$.
\end{remark}
Finally, we note that, obviously, an analogue of Theorem \ref{th24} is valid for $\mathcal{I}=(0,b)$, $b<\infty$.
| 33,664
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“Tragedy does that to us. Sometimes it will force us to engage life differently as we did before. This isn’t a bad thing. It’s just the way life works, and there is nothing we can do about it.”-Jeff Huxford, M.D.
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Optimal Health
15606 S Gilbert rd Lot 61 Chandler, Arizona 85225(480) 814-7317
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AboutOptimal Health is located at the address 15606 S Gilbert rd Lot 61 in Chandler, Arizona 85225. They can be contacted via phone at (480) 814-7317 for pricing, hours and directions. Optimal Health specializes in Factories, Supplemental Health, Employer-Sponsored Pensions.
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The Aims and Principles of The Association of Private Pet Cemeteries and CrematoriaThe following aims and principles of the Association are designed to foster trust and understanding for all involved with the death of companion animals.
- Promote & offer guidance on funeral facilities and services for companion animals to pet owners and pet professionals.
- Set a Code of Practice and define services for cremations and burials which together ensure quality, clarity and protection for pet owners.
- Offer help and advice to pet crematoria, cemeteries and those providing pet funeral services.
- Consult on legislation affecting the pet funeral sector.
The over riding principle of the Association is to provide services to you, the pet owner, that are clearlydescribed so there can be no doubt you are getting the service you expect. Everything is carried out in accordance with the Consumer Protection Against Unfair Trading Regulations which states that everything must be detailed so there is no confusion.
An acceptable cremation to one person may looked on as nothing more than a waste disposal to another. Unfortunately mis-selling is rife in the sector which makes it all the more important that you check everything carefully by using our Codes of Practice. We also provide other detailed advice to help you.
It is hoped that one day all reputable establishments will join and a uniform set of standards be applicable to the whole country. In this way the public will be protected and businesses will be able to compete on a level playing field with a strong representative voice over new legislation.
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Mailbag: Anti-Vaccine Myths, Pharma Shill Gambit and Vaccine Court
Recently, an anti-vaccine activist by the name of “peter” posted a comment on my article Irrefutable Evidence Shows That Anti-Vaccine Activists Still Have No Clue (were I destroyed some anti-vaccine propaganda written by Dave Mihalovic). It did not address any arguments and attacked me directly, yet I think it can still be useful to unravel and debunk his claims about scientific skeptics, vaccines and the U.S. legal system. This exercise also highlights some of the common rhetorical devices that anti-vaccine activists make use of in their efforts to undermine modern medicine.
The basics flaws in his approach are (1) the use of the “pharma shill” gambit, (2) the gross ignorance about the Office of Special Masters of the U.S. Court of Federal Claims (also known as “vaccine court”) and the Vaccine Injury Compensation Program (VICP) and (3) the deployment of anti-vaccine crankery about the alleged negative health effects of vaccines.
As we shall see, the pharma shill gambit is just a vacuous rationalization to psychologically shield anti-vaccine beliefs, vaccine court / VICP are more beneficial to individuals who have suffered a genuine adverse effect of vaccines (faster, cheaper and requires a much lighter burden of evidence to be met) and the peer-review literature as well as scientific reports by the Institute of Medicine (IOM) and others show that vaccines are not linked to the conditions he claims.
Here is what Peter has to say about me and vaccines:
Peter accuses me of being a moronic pharma shill and that I support criminal drug companies. Further, he triumphantly states the rhetorical question why the U. S. government does not let people who claim to have gotten a vaccine injury if vaccines are safe. Finally, the spouts the same old canards about how vaccines allegedly cause a great number of disease in children and adults.
Let us examine these three categories of claims.
The pharma shill gambit
The pharma shill gambit consists of condescendingly dismissing arguments by asserting that the person making those arguments are somehow paid off by the pharmaceutical industry (Gorski, 2009). This is based on the preconceived notion that no independent thinker could possibly come to the conclusion that vaccines are safe and effective. When an anti-vaccine activist come across such individuals, cognitive dissonance strikes and the only way to make their worldview consistent against is to dismiss the science-based arguments and evidence in favor of vaccines by investing and attacking the motives of the person presenting those arguments and that evidence.
Throughout my time as a skeptical debunker of pseudoscience, I have been accused of being a shill for companies producing vaccines (because I point out that vaccines are safe and effective), companies producing antiretroviral medication against HIV/AIDS (because I criticize HIV/AIDS denialists), companies producing psychiatric medication (because I refute anti-psychiatry cranks), Al Gore, the EPA and Greenpeace (because I debunk climate change denialists), Israel and the Anti-Defamation League (because I reject Holocaust denial), NASA (because I explain that the evidence for humans having been to the moon is very strong), Monsanto (because I find the assertions by anti-GMO zealots to be highly flawed and unpersuasive) and NSA (for dismissing crank 9/11 conspiracy theories).
So I just have one question for the people constantly spouting various shill gambits: where is my money!? Surely, I should be a multi-billionaire by now?
Vaccine court is cheaper, faster and requires a lighter burden of evidence for compensation.
Another problem was that pharmaceutical companies could spend a lot of money on lawyers and drag out the legal proceedings to the detriment of families. This mean that families with children who had gotten actual vaccine side-effects had to wait a long time for the trials to finish and the entire process was presumably quite expensive for them, both in terms of time and money. Furthermore, some (but not all) legal claims required evidence of fault from the pharmaceutical company. This meant that the burden of evidence was fairly high.
The government understood these problems and established the Vaccine Court to handle (U. S. Court of Federal Claims, 2004).
Vaccines do not cause autism
Around a dozen independent researcher studies published in the peer-reviewed scientific literature has shown that there is no link between vaccines and autism. This research is reviewed in Gerber and Offit (2009) and the fact that it converges on the same conclusion means that the evidence strongly favors a rejection of a causal link between the MMR vaccine and autism. The same conclusion was reached by the independent Institute of Medicine in their 2004 report entitled Immunization Safety Review: Vaccines and Autism (Institute of Medicine, 2004): source behind the claim that the MMR vaccine cause autism was a paper published in the Lancet back in 1998 by Andrew Wakefield. However, the interpretation section was retracted in 2004 (and most co-authors distanced themselves from the paper) and the entire paper retracted in 2010. Investigations (see Deer, 2011 for an overview) revealed that Wakefield had performed painful and medically unnecessary tests on children with autism without approval of an ethical review board, he manipulated and altered research data (thus committing scientific fraud) and received over 400 000 GBP from lawyer Richard Barr (who was working on an anti-vaccine lawsuit against GlaxoSmithKline). In addition, the study did not have a control group and many of the children subjects of the study had developed problems prior to getting th MMR vaccine. Wakefield was later struck off the Medical Register and banned from practicing medicine in the U. K.
As a final note in this section, it is ironic for anti-vaccine cranks to oppose the MMR vaccine if the fear autism. This is because one of the complications of rubella infection in pregnant women is the child may develop autism (Muhle, Trentacoste and Rapin, 2004). So by vaccinating against rubella, some future children are actually being protected against autism.
Vaccines do not cause asthma or allergies
Allergies are thought to develop by insufficient stimulation of Th1 cells (and thus too little inhibition of Th2 cells that can contribute to allergies) as well as insufficient stimulation of regulatory T cells by infection. Yet no vaccines are given for the most common childhood infection (upper-respiratory viral infections) and children get over half a dozen infections per year on average (Offit and Hackett, 2003). This means that vaccines are probably not a factor for the development of allergies.
Many studies have been performed that also show no link between vaccines and allergies (reviewed in Offit and Hackett, 2003). This paper also discusses the lack of association between vaccines and asthma:
One well-controlled study was performed using the computerized records of children born between 1991 and 1997 who were enrolled in 4 large health maintenance organizations (HMOs). This cohort was used to identify 18 407 children with asthma. Relative risks of asthma in vaccinated children were determined by comparison with children who did not receive vaccines. The relative risk for asthma was 0.92 for the combination diphtheria-tetanus-whole-cell pertussis vaccine, 1.09 for the oral polio vaccine, and 0.97 for the combination measles-mumps-rubella vaccine. In children who had at least 2 medical encounters during their first year of life, the relative risk for asthma was 1.07 after receipt of the Haemophilus influenzae type b (Hib) vaccine and 1.09 for the hepatitis B vaccine.
Thus, the risk for the vaccinated group was not clinically significantly larger than the risk for the unvaccinated group.
Vaccines do not cause Sudden Infant death syndrome (SIDS)
An Institute of Medicine immunization safety review from 2003 concluded the following after reviewing the existing literature (Institute of Medicine, 2003):
Based on this review, the committee concluded that the evidence favors rejection of a causal relationship between some vaccines and SIDS; and that the evidence is inadequate to accept or reject a causal relationship between other vaccines and SIDS, SUDI, or neonatal death.
Although the precise cause(s) of SIDS has not yet been identified, researchers have shown that risk factors for SIDS include brain abnormalities, low birth weight, respiratory infections, sleeping on stomach or side, sleeping on a soft surface, sleeping in the same bed as parents, having a mother below the age of 20, having a mother that smokes and a family history of SIDS (Mayo Clinic, 2011).
No evidence that vaccines cause mitochondrial dysfunction
Here is what the CDC has to say about the claim that vaccines cause mitochondrial disorders (CDC, 2010):
Do vaccines cause or worsen mitochondrial diseases?.
The remaining claims (neurological disorders, childhood gastrointestinal disorders and cancers) are too broad vaguely phrased to be subject to a proper refutation, but Peter has not provided a shred of evidence that vaccines are linked with these categories of diseases. Thus, there is no reason to take what he says seriously.
Conclusion:
This is a the debating rhetorics of a typical anti-vaccine activist: never provide any real scientific evidence, angrily accuse defenders of science-based medicine to be pharma shills, claim that vaccines cause all sorts of diseases and promote conspiracy theories about a solution that is actually beneficial both for vaccine production and those that have had real vaccine side-effects.
References and further reading:
CDC. (2010). Mitochondrial Disease – Frequently Asked Questions. Accessed: 2013-07-03.
Deer, Brian. (2011). How the case against the MMR vaccine was fixed. BMJ, 342.
Gerber, J. S., & Offit, P. A. (2009). Vaccines and Autism: A Tale of Shifting Hypotheses. Clinical Infectious Diseases, 48(4), 456-461.
Gorski, D. (2009). The Pharma Shill Gambit. Science-Based Medicine. Accessed: 2013-07-03.
Institute of Medicine. (2003). Immunization Safety Review: Vaccinations and Sudden Unexpected Death in Infancy. Washington, D. C: National Academies Press.
Institute of Medicine. (2004). Immunization Safety Review: Vaccines and Autism. Washington, D. C: National Academies Press.
Mayo Clinic. (2011). Sudden infant death syndrome (SIDS). Accessed: 2013-07-03.
Muhle, Rebecca, Trentacoste, Stephanie V., & Rapin, Isabelle. (2004). The Genetics of Autism. Pediatrics, 113(5), e472-e486.
Offit, Paul A., & Hackett, Charles J. (2003). Addressing Parents’ Concerns: Do Vaccines Cause Allergic or Autoimmune Diseases? Pediatrics, 111(3), 653-659. doi: 10.1542/peds.111.3.653
U. S. Court of Federal Claims. (2004). Vaccine Program/Office of Special Masters. Accessed: 2013-07-03.
One thought on “Mailbag: Anti-Vaccine Myths, Pharma Shill Gambit and Vaccine Court”
Pingback: Mailbag: Countering Miscellaneous Pseudoscientific Nonsense | Debunking Denialism
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BER?”
At this moment when dark clouds loom over Silicon Valley, VR “weds the nerdy thing with the hippie mystic thing,” high-tech but like a dream and “an elixir of unbounded experience.”
But he’s well aware of the “Matrix” dangers. He realized early on, he writes, that “it could turn out to be the evilest invention of all time.”
It’s a pretty simple proposition, he tells me: “If you control the person’s reality, you control the person.” Or as he writes in the book, “Never has a medium been so potent for beauty and so vulnerable to creepiness.”
Recently, the creepiness has been on display.
Mark Zuckerberg stumbled into more trouble for tone-deafness when he used his cartoon avatar to take a disaster adventure trip, a “magical” 360-degree virtual reality tour of hurricane devastation in Puerto Rico to promote his new Facebook Spaces app.
“Oh, God,”.
We sit down at his dining room table amid a wild cornucopia of stuff, including a lamp with a hot pink feathered shade and black cats lounging on chairs and hanging, Cirque du Soleil-style, from carpeted staircases. There are also musical instruments — a golden Wurlitzer pedal harp; a rare pre-Depression Mason & Hamlin piano that. Lanier can play.
Like his house, his new book is crammed full of strange and mesmerizing stuff.. There, he had to confront more than his share of bullies growing up, once by swinging a baritone horn at them.
His mother died when he was about 9, when her car flipped over on the freeway as she was coming back from getting her driver’s license. His father, who worked for a time as the science editor of “Amazing,” “Fantastic” and “Astounding” pulp science-fiction magazines, then let his 11-year-old son design their new house in New Mexico: a geodesic dome.
Ther. Lanier bids a loving good night to his wife, Lena, a child psychologist, and their 11-year old daughter, Lilibell. Then he brings out his Microsoft HoloLens headsets and a big mug of chocolate milk. “I’m more like the child than the parent, I’m afraid,” he says.
I spend some time wearing the headset painting graffiti in the air with my hand, and Mr. Lanier explains why the brain can see more than the eyes.
I ask about social-media sites getting hijacked by Russians pushing propaganda aimed at putting Donald Trump in the White House. Vanity Fair has compared this juncture, with anxious lawmakers demanding accountability from the resistant tech companies, to the moment when we all had to start taking off our shoes at airports.
“Expect some smelly feet,” Mr. Lanier says.
Unlike many here, he does not think of humans as ants in his experiments.
“Hopefully, in this period, when we’re dealing with this really crude and early stuff like Facebook feeds, Instagram, Snapchat,” he says, “we’ll be able to get the politics straight and find a path for people to have dignity and autonomy before the hard-core stuff comes. Unless we all kill ourselves through this other stuff, which is a possibility, too... And for those who are not giving Facebook money, the only — and I want to emphasize, the only, underlined and in bold and italics — reward they can get or positive feedback is just getting attention. And if you have a system where the only possible prize is getting more attention, then you call that system Christmas for Asses, right? It’s a creep-amplification device.
.”
He continues: .”
I remark that Facebook’s Sheryl Sandberg expressed surprise that their fiendish little invention could do something so nefarious.
“People in the community knew,” Mr. Lanier says, adding that he wrote essays and participated in debates in the early 1990s about how easy it would be the create unreality and manipulate society, how you could put out a feed of information that would put people in illusory worlds where they thought they had sought out the information but actually they had been guided “the way a magician forces a card.”
“So for somebody to say they didn’t know the algorithms could do that,” Mr. Lanier says in a disbelieving tone. “If somebody didn’t know, they should’ve known.”
So what happens when fake news marries virtual reality?
“It could be much more significant,” Mr. Lanier says. “When you look at all the ways of manipulating people that you can do with just a crude thing like a Facebook feed — when people are just looking at images and text on their phones and they’re not really inside synthetic worlds yet — when you can do it with virtual reality, it’s like total control of the person. So what I’m hoping is that we’re going to figure this stuff out so we don’t make ourselves insane before virtual reality becomes mature.”. Lanier about the sexual harassment and gender inequity problems roiling the Valley.
“Well, sometimes, I think there’s a kind of emerging new male jerk persona of the digital age, which would be some kind of a cross between the Uber guy and the pharma bro and maybe Milo Yiannopoulos and maybe Palmer Luckey and maybe Steve Bannon,” he says. “Because, there’s this sort of smug, superior, ‘I’ve got the levers of power, and I know better than you.’ It’s sort of this weird combination of a lot of power and a lot of insecurity at the same time.” He believes that Gamergate led to the alt right. “It was one of the feeders,” he says.
He talks about another personality that is emerging from the digital age.
.”.
“I think we know that Facebook is turning us into trained dogs,” he says. “We know we’re being trained. We can feel ourselves being turned into trained circus animals. And we long for that independence that cats show. So when you look at a cat video, what you’re really seeing is this receding identity that you want to cling to and find again.”
RELATED: Jaron Lanier submits to a round of Confirm or Deny, here.
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I started drawing dog cartoons about a year ago, most of them include my pooch Gizmo, a rescue doodle, you can find out more about his furriness if you go to this link.
Last year I was preparing for an exhibition ‘Rescue Me’. I wanted to continue to include dogs in my artwork. So in collaboration with my writer husband Richard Attree, we decided to start on our third dog book which will include my cartoons and illustrations – The title of the book will be ‘Shaggy Dog Tails’ – I woof Therefore I am. A collection of short stories.
You can find our first two novels ‘Nobody’s Poodle’ and Somebody’s Doodle on Amazon.
Now as we are in #lockdown the cartoons have helped me cope with the strict isolation that the Spanish Government has imposed on us ( I live in Tenerife). I certainly don’t want to produce any art that is too depressing. I have had positive feeback from twitter and facebook so I thought I would start to put all my cartoons onto one website wooftasticcartoons.com .
I hope my cartoons bring a smile to your face 🙂
You can join me on facebook or wooftasticbooks twitter page
The Cartoon below is of Betty who belongs to Beverley Cuddy the editor of ‘Dogs Today Magazine’. Betty is keeping update with the current doggy news.
‘Betty likes nothing better than relaxing on the sofa with a copy of Dogs Today Magazine – Leave the rushing around and bouncing off walls to more energetic dogs’
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Sabia Wade
- Full Spectrum Doula, Doula Business Coach, Creator & Executive Director of For The Village & Birthing Advocacy Doula Trainings.
- From 2015-2017, I was a volunteer full spectrum doula for The Prison Birth Project.
- In February 2018, I quit my job and became a full time doula in San Diego, CA
- In April 2018, I started For The Village, which provides low income and marginalized people access to free and low cost doula services.
- In 2019, I launched Birthing Advocacy Doula Training and here we are now!
Venice Cotton
- Executive Assistant & Program Manager with For The Village.
- Trained as a Full Spectrum Doula in 2018
- A traumatic birthing experience at 17 led to a passionate career in birth work
- Left corporate world in 2019 to pursue reproductive justice full-time.
Hope Jackson
- Intern with For The Village
- Hart Fellow Program at the Sanford School of Public Policy @Duke University
- Research entitled “Understanding Racial Disparities in Reproductive Health Outcomes: Black Doula-Patient- Provider Relationship”
- Full spectrum doula since December of 2021
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TITLE: What is an vacuum in QFT in curved spacetime?
QUESTION [4 upvotes]: Sometimes in some public lectures about General Relativity (GR) and Quantum Mechanics, in college, the professors dealt with the vacuum concept, precisely in the context of Quantum Field Theory (QFT), like: Minkowski Vacuum, Rindler Vacuum.
I'm understanding this concept as the background spacetime, like for instance, in Schwarschild geometry the trajectory of light-like geodesics and time-like geodesics are profoundly different from flat geometries like Minkowski spacetime.
From this point of view we have then a Schwarschild Vacuum, but I'm not sure if this is the right way to see this QFT picture.
So what is a Vacuum in GR and QFT?
REPLY [4 votes]: A vacuum isn't a terribly well-defined concept in general, but in Minkowski space for free field theories bounded from below, we have the good luck of all the definitions lining up : there's a unique state $\Omega \in \mathcal{H}$ that is invariant under Poincaré transformations and such that $\langle \hat{H} \rangle_\Omega$ minimizes the energy.
This definition doesn't work out in general due to the fact that both no state may be invariant under Poincaré transformation (ie any spacetime that isn't maximally symmetric) and that states of lowest energy may fail to be unique (for instance the Higgs vacua). If we allow for arbitrary coordinate transformations, the notion of a state of lowest energy may even fail to be invariant under such.
In the broadest sense, a vacuum state is basically any state we wish to pick up to build the theory from. Given some operator algebra $\mathscr{A}$, you pick a cyclic state $\psi \in \mathcal{H}$ (so that applying $\mathscr{A}$ on $\psi$ is dense in $\mathcal{H}$). As there are no prefered vacuum in general theory, all we can really ask is that it relates to every other state.
For convenience (and because it is called a vacuum after all), it's typically a state which obeys some properties. It's often going to be a state of lowest energy in a given coordinate system (maybe for some observer), with some arbitrary choice if there is more than one such state (for instance pick a Higgs vacua with a certain phase). Different observers will see different vacua as is known from the Unruh and Hawking radiation, so we just require conditions for specific observers.
For instance you will have three commonly used vacuum in Schwarzschild spacetime. The Boulware vacuum is roughly equivalent to a vacuum state from the perspective of an observer at infinity, the Unruh vacuum is one for a free-falling observer, while the Hartle-Hawking vacuum has the benefit of being well-behaved on both sides of the horizon (this isn't so much the case for Boulware or Unruh) but isn't really a vacuum state as it is never empty of radiation.
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16
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Wednesday, August 19, 2009
Pecan Sticky Buns
Ages and ages ago, I posted a picture of these Sticky Buns. I've made them to sell at the bakery in Colorado and I've made them for myself and now I've made them again to sell at the farmers market. They're super delicious. It should be said that Carole Walters, whose recipe I've poached and simplified, has never lead me wrong. She is, however, quite a complicated recipe developer, liking the multi-stepped, overnight-type things. They're often worth the time or effort, but they're not for the shy of heart. Nor for dieters. But if you've neither of those frightful qualities, you'll be glad of this recipe.
It is one of the best things to pop these out of the pan all gooey gorgeous and burn your tongue on a hot pecan. When your eyes roll back in your head, you'll know it was worthwhile. Just plan a day ahead for this recipe and you're set.
Simple Sweet Dough
Makes 2#
1 T sugar
2-1/4 tsp (1 pkg) active dry yeast
1/4 cup warm water
3 T sugar
1 tsp salt
3 cups flour + more for kneading
1 cup (2 sticks) cold butter
1/2 cup milk
3 egg yolks
1 tsp vanilla
Combine the yeast with the tablespoon of sugar and the warm water in a small bowl. Let rest 5 minutes, until yeast is dissolved and the mixture is bubbly.
In a separate bowl, cut the butter into 3 T sugar, salt, and flour until the mixture looks like sand, very finely textured. Make a well in the dry ingredients and add the milk, egg yolks and vanilla. Use a fork to blend the wet ingredients lightly together and then add in the yeast. Mix into the dry ingredients a little at a time until a rough dough forms. Knead lightly, adding up to 1/4 cup of extra flour, until the dough is smooth. Dough will be really soft and sticky. Butter a bowl and place dough into it. Smooth more butter over the top of the dough. Cover tightly and refrigerate overnight.
Sticky Buns
Makes 12 muffin-sized buns
Topping
1/4 cup soft butter
3/4 cup brown sugar, packed
2 T honey
2 T light corn syrup
2/3 cup pecans, roughly chopped
Mix until smooth. Prepare the muffin tins by buttering each one generously, then smear a spoonful of this topping into each muffin cup, as evenly as possible.
Filling
2 T soft butter
2 T sugar
2 T brown sugar
1/2 tsp cinnamon
1/2 cup pecans, roughly chopped
Heat oven to 350º. Roll out 1# of the sweet dough* about 1/2" thick or into a rectangle. Mix all ingredients and spread over the dough. Roll up the dough on the widest side, pressing to seal the end, and slice into 12 equal pieces. Press one piece lightly into each muffin cup. Bake 20-25 minutes, turning once midway through, or until the tops are golden brown and a toothpick inserted through the center comes out clean. Let rest for 5 minutes, then invert — carefully! hot + gooey = burns — onto a platter. Eat as soon as you can safely touch the pecans without singeing a finger. They're also great the next day, if they last.
*Note that the dough recipe will make twice as much as you'll need for the sticky buns. Either double your sticky bun recipe or, as I like to do, make one half into cinnamon rolls instead. Super easy filling: roll out the dough to 1/2" thick, smear with butter, sprinkle with 2 T sugar and 1 tsp cinnamon, then roll them up and slice into 12 again. Bake in a buttered 9" square dish, about 20-25 minutes. Ice them if you like, but they're delicious plain.
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Tyrese Gibson Joins Black Nativity
Tyrese Gibson has signed on to star in Black Nativity for writer-director Kasi Lemmons. We reported in November that Forest Whitaker, Jennifer Hudson, and Angela Bassett have joined the cast.
The project is based on the Langston Hughes libretto that was adapted into a Broadway musical in 1961. The story follows a young boy from Baltimore, who spends Christmas in Harlem with the grandparents he has never met. The boy learns about the importance of family through his grandfather's Christmas Eve sermon. Forest Whitaker and Angela Bassett are playing the boy's grandparents, with Jennifer Hudson portraying his mother, who had a falling out with her father years ago. No details were given regarding Tyrese Gibson's character, but it is believed that this will be the first time the actor sings on film.
Kasi Lemmons is directing from her own screenplay, adapted from the Langston Hughes libretto. William Horberg, Celine Rattray, Galt Niederhoffer, and Daniela Taplin Lundberg are producing. Shooting is currently under way in New York<<
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\begin{document}
\maketitle
\renewcommand{\thefootnote}{}
\footnote{The second author was supported by a National Science Foundation Graduate Research Fellowship.}
\footnote{{\bf\noindent Key words:} aggregation, recurrence, regular tree, rotor-router model, sandpile group, transience}
\footnote{{\bf\noindent 2000
Mathematics Subject Classifications:} Primary 05C05; Secondary 05C25, 60G50}
\renewcommand{\thefootnote}{\arabic{footnote}}
\begin{abstract}
The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a perfect ball whenever it can be, provided the initial rotor configuration is acyclic (that is, no two neighboring vertices have rotors pointing to one another). This is proved by defining the {\it rotor-router group} of a graph, which we show is isomorphic to the sandpile group. We also address the question of recurrence and transience: We give two rotor configurations on the infinite ternary tree, one for which chips exactly alternate escaping to infinity with returning to the origin, and one for which every chip returns to the origin. Further, we characterize the possible ``escape sequences" for the ternary tree, that is, binary words $a_1 \ldots a_n$ for which there exists a rotor configuration so that the $k$-th chip escapes to infinity if and only if $a_k=1$.
\end{abstract}
\section{Introduction}
The rotor-router model is a deterministic analogue of random walk, first defined by Priezzhev et al.\ under the name ``Eulerian walkers'' \cite{PDDK} and popularized more recently by Jim Propp \cite{Kleber}. To define rotor-router walk on a tree $T$, for each vertex of $T$ we choose a cyclic ordering of its neighbors. Each vertex is assigned a ``rotor'' which points to one of the neighboring vertices. A chip walks on the vertices of $T$ according to the following rule: when the chip reaches a vertex $v$, the rotor at $v$ rotates to point to the next neighbor in the ordering, and the chip steps in direction of the newly rotated rotor. In {\it rotor-router aggregation}, we grow a cluster of points in $T$ by repeatedly starting chips at a fixed vertex $o$ and letting them walk until they exit the cluster. Beginning with $A_1 = \{o\}$, define the cluster $A_n$ inductively by
\[ A_n = A_{n-1} \cup \{x_n\}, \qquad n >1 \]
where $x_n \in T$ is the endpoint of a rotor-router walk started at $o$ and stopped on first exiting $A_{n-1}$. We do not change the positions of the rotors when adding a new chip. Thus the sequence $(A_n)_{n \geq 1}$ depends only on the choice of the initial rotor configuration.
Recent interest has focused on rotor-router aggregation in the integer lattice $\Z^d$. Jim Propp noticed from simulations in $\Z^2$ that the shape $A_n$ is extremely close to circular, and asked why this was so \cite{Kleber}. The spherical shape of $A_n$ in $\Z^d$ is proved in \cite{LP1,LP2}. Here we prove an analogous result for rotor-router aggregation on the infinite $d$-regular tree. We say that a rotor configuration is \emph{acyclic} if the rotors form no oriented cycles. On a tree, this condition is equivalent to forbidding oriented cycles of length~2: there is no pair of neighboring vertices $x,y$ such that both the rotor at $x$ points to $y$ and the rotor at $y$ points to $x$. As the following result shows, provided we start with an acyclic rotor configuration, the occupied cluster $A_n$ is a perfect ball for suitable values of $n$.
\begin{theorem}
\label{aggregintro}
Let $T$ be the infinite $d$-regular tree, $d \geq 3$, and let
\[ B_r = \{x \in T \,:\, |x| \leq r\} \]
be the ball of radius $r$ centered at the origin $o \in T$, where $|x|$ is the number of edges in the path from $o$ to $x$. Write
\[ b_r = \# B_r = 1 + d \frac{(d-1)^r-1}{d-2}. \]
Let $A_n$ be the region formed by rotor-router aggregation in $T$, starting from $n$ chips at $o$. If the initial rotor configuration is acyclic, then
\[ A_{b_r} = B_r. \]
\end{theorem}
The proof of Theorem~\ref{aggregintro} uses the {\it sandpile group} of a wired regular tree (that is, a finite regular tree with the leaves collapsed to a single vertex, and an edge added from the root to this vertex), whose structure was found in \cite{sandpiletree}. In section~2 we define the {\it rotor-router group} of a graph and show that it is isomorphic to the sandpile group. We then use this isomorphism in section~3 to prove Theorem~\ref{aggregintro}.
Much previous work on the rotor-router model has taken the form of comparing the behavior of rotor-router walk with the expected behavior of random walk. For example, Cooper and Spencer \cite{CS} show that for any configuration of chips on even lattice sites in $\Z^d$, letting each chip perform rotor-router walk for $n$ steps results in a configuration that differs by only constant error at each point from the expected configuration had the chips performed independent random walks. In section~4, we continue in this vein by investigating the recurrence and transience of rotor-router walk on trees. A walk which never returns to the origin visits each vertex only finitely many times, so the positions of the rotors after a walk has escaped to infinity are well-defined. We construct two ``extremal'' rotor configurations on the infinite ternary tree, one for which walks exactly alternate returning to the origin with escaping to infinity, and one for which every walk returns to the origin. The latter behavior is something of a surprise: to our knowledge it represents the first example of rotor-router walk behaving fundamentally differently from the expected behavior of random walk.
In between these two extreme cases, a variety of intermediate behaviors are possible. We say that a binary word $a_1 \ldots a_n$ is an {\it escape sequence} for the infinite ternary tree if there exists an initial rotor configuration on the tree so that the $k$-th chip escapes to infinity if and only if $a_k=1$. The following result characterizes all possible escape sequences on the ternary tree.
\begin{theorem}
Let $a = a_1 \ldots a_n$ be a binary word. For $j \in \{1,2,3\}$ write $a^{(j)} = a_{j} a_{j+3} a_{j+6} \ldots$. Then $a$ is an escape sequence for some rotor configuration on the infinite ternary tree if and only if for each $j$ and each $k \geq 2$, every subword of $a^{(j)}$ of length $2^k-1$ contains at most $2^{k-1}$ ones.
\end{theorem}
We conclude in section~5 with an open question about the transience of rotor-router walk in $\Z^d$ for $d\geq 3$.
\section{The Rotor-Router Group}
In this section we define the {\it rotor-router group} of a graph and show it is isomorphic to the sandpile group. The definition of the sandpile group is recalled below. In the next section we use this isomorphism together with the results of \cite{sandpiletree} to study the rotor-router aggregation model on a regular tree.
The isomorphism between the rotor-router and sandpile groups, Theorem~\ref{groupisom}, is mentioned in the physics literature; see \cite{PDDK, PPS}. To our knowledge the details of the proof are not written down anywhere. While our main focus is on the tree, the isomorphism is just as easily proved for general graphs, and it seems to us worthwhile to record the general proof here.
Let $G$ be a strongly connected finite directed graph, which may have multiple edges but not loops. Fix a vertex $s$ in $G$ and call it the sink. To define rotor-router walk on $G$, for each vertex $x\neq s$ we fix a cyclic ordering of the edges emanating from $x$. A \emph{rotor configuration} $T$ on $G$ assigns to each non-sink vertex $x$ an edge $T(x)$ emanating from $x$. Each step of the walk then consists of two parts: If the chip is located at $x$, we first increment the rotor $T(x)$ to the next edge in the ordering of the edges emanating from~$x$, and then move the chip along this new edge.
Given a rotor configuration~$T$, write $e_x(T)$ for the rotor configuration resulting from starting a chip at $x$ and letting it walk according to the rotor-router rule until it reaches the sink. (Note that if the chip visits a vertex infinitely often, it visits all of its outbound neighbors infinitely often; since $G$ is strongly connected, the chip eventually reaches the sink.)
The set of edges $\{T(x)\}_{x\neq s}$ in a rotor configuration forms a spanning subgraph of $G$ in which every vertex except the sink has out-degree one. If this subgraph contains no directed cycles (equivalently, no cycles), we call it an \emph{oriented spanning tree} of $G$. Write $Rec(G)$ for the set of oriented spanning trees of $G$. Note that as we have defined them, oriented spanning trees are always rooted at the sink (i.e., all paths in the tree lead to the sink).
\begin{lemma}
If $T \in Rec(G)$, then $e_x(T) \in Rec(G)$.
\end{lemma}
\begin{proof}
Let $Y$ be any collection of vertices of $G$. If the chip started at $x$ reaches the sink without ever visiting $Y$, then the rotors at vertices in $Y$ point the same way in $e_x(T)$ as they do in $T$, so they do not form an oriented cycle. If the chip does visit $Y$, let $y\in Y$ be the last vertex it visits. Then either $y=s$, or the rotor at $y$ points to a vertex not in $Y$; in either case, the rotors at vertices in $Y$ do not form an oriented cycle.
\end{proof}
We will need slightly more refined information about the intermediate states that occur before the chip falls into the sink. These states may contain oriented cycles, but only of a very restricted form. For a vertex $x$ we write $Cyc_x(G)$ for the set of rotor configurations $U$ such that
\begin{itemize}
\item[(i)] $U$ contains an oriented cycle; and
\item[(ii)] If the rotor $U(x)$ is deleted, the resulting subgraph contains no oriented cycles.
\end{itemize}
\begin{lemma}
\label{cyclecriterion}
Starting from a rotor configuration $T_0 \in Rec(G)$ with a chip at $x_0$, let $T_k$ and $x_k$ be the rotor configuration and chip location after $k$ steps of rotor-router walk. Then
\begin{itemize}
\item[(i)] If $T_k \notin Rec(G)$, then $T_k \in Cyc_{x_k}(G)$.
\item[(ii)] If $T_k \in Rec(G)$, then $x_k \notin \{x_0,\ldots,x_{k-1}\}$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) It suffices to show that any oriented cycle in~$T_k$ contains~$x_k$. Let~$Y$ be any set of vertices of~$G$ not containing~$x_k$. If~$Y$ is disjoint from $\{x_0, \ldots, x_{k-1}\}$, then the rotors at vertices in~$Y$ point the same way in~$T_k$ as they do in~$T_0$, so they do not form an oriented cycle. Otherwise, let $y\in Y$ be the vertex visited latest before time~$k$. The rotor~$T_k(y)$ points to a vertex not in~$Y$, so the rotors at vertices in~$Y$ do not form an oriented cycle.
(ii) Suppose $x_k \in \{x_0,\ldots,x_{k-1}\}$. Let~$y_0=x_k$, and
for $i=0,1,\ldots$ let~$y_{i+1}$ be the target of the rotor~$T_k(y_i)$. Then the last exit from~$x_k$ before time~$k$ was to~$y_1$, and by induction if $y_1, \ldots, y_{i-1}$ are different from~$x_k$, then~$y_{i-1}$ was visited before time~$k$, and the last exit from~$y_{i-1}$ before time~$k$ was to~$y_i$. It follows that~$y_i=x_k$ for some~$i\geq 1$, and hence~$T_k$ contains an oriented cycle.
\end{proof}
\begin{lemma}
If $T_1, T_2 \in Rec(G)$ and $e_x(T_1)=e_x(T_2)$, then $T_1=T_2$.
\end{lemma}
\begin{proof}
We will show that any $T\in Rec(G)$ can be recovered from $e_x(T)$ by reversing one rotor step at a time.
Given rotor configurations $U,U'$ and vertices $y,y'$, we say that $(U',y')$ is a predecessor of $(U,y)$ if a chip at $y'$ with rotors configured according to $U'$ would move to $y$ in a single step with resulting rotors configured according to $U$. Given $U$ and $y$, for each neighbor $z$ of $y$ whose rotor $U(z)$ points to $y$, there is a unique predecessor of the form $(U',z)$, which we will denote $P_z(U,y)$.
Suppose $(U,y)$ is an intermediate state in the evolution from $T$ to $e_x(T)$. If $U \notin Rec(G)$, then by case (i) of Lemma~\ref{cyclecriterion} there is a cycle of rotors
$y \rightarrow y_1 \rightarrow y_2 \rightarrow \ldots \rightarrow y_n \rightarrow y$ in $U$.
If $z$ is a vertex different from $y_n$ whose rotor $U(z)$ points to $y$, then $z$ is not in this cycle, so the predecessor $P_z(U,y)$ has a cycle disjoint from its chip location. Thus $P_z(U,y)$ does not belong to $Rec(G)$ or to $Cyc_z(G)$, so by Lemma~\ref{cyclecriterion} it cannot be an intermediate state in the evolution from $T$ to $e_x(T)$. The state immediately preceding $(U,y)$ in the evolution from $T$ to $e_x(T)$ must therefore be $P_{y_n}(U,y)$.
Now suppose $U \in Rec(G)$. By case (ii) of Lemma~\ref{cyclecriterion}, $U$ is the rotor configuration when $y$ is first visited. If $y=x$, then $U=T$. Otherwise, let $x=x_0 \rightarrow x_1 \rightarrow \ldots \rightarrow x_k=s$ be the path in $U$ from $x$ to the sink. Then the last exit from $x$ before visiting $y$ was to $x_1$. By induction, if $x_1, \ldots, x_{j-1}$ are different from $y$, then $x_{j-1}$ was visited before $y$ and the last exit from $x_{j-1}$ before visiting $y$ was to $x_j$. It follows that $x_j=y$ for some $j\geq 1$, and the state immediately preceding $(U,y)$ must be $P_{x_{j-1}}(U,y)$.
\end{proof}
Thus for any vertex $x$ of $G$, the operation $e_x$ of adding a chip at $x$ and routing it to the sink acts invertibly on the set of states $Rec(G)$ whose rotors form oriented spanning trees rooted at the sink. It is for this reason that we call these states recurrent. We define the {\it rotor-router group} $RR(G)$ as the subgroup of the permutation group of $Rec(G)$ generated by $\{e_x\}_{x\neq s}$. For any two vertices $x$ and $y$, the operators $e_x$ and $e_y$ commute; this commutativity is proved in \cite{DF} for a broad class of models encompassing both the abelian sandpile and the rotor-router.
Hence the group $RR(G)$ is abelian.
\begin{lemma}
\label{transitivity}
$RR(G)$ acts transitively on $Rec(G)$.
\end{lemma}
\begin{proof}
Given $T_1, T_2 \in Rec(G)$, for each vertex $x\neq s$ let $u(x)$ be the number of rotor turns needed to get from $T_1(x)$ to $T_2(x)$. Let $v(x)$ be the number of chips ending up at~$x$ if $u(y)$ chips start at each vertex~$y$, with rotors starting in configuration~$T_1$, and each chip takes a single step. After each chip has taken a single step, the rotors are in configuration $T_2$, hence
\[ \left( \prod_{x \neq s} e_x^{u(x)} \right) T_1 = \left( \prod_{x \neq s} e_x^{v(x)}
\right) T_2. \]
Letting $g = \prod_{x \neq s} e_x^{u(x)-v(x)}$ we obtain $T_2 = gT_1$.
\end{proof}
Given vertices $x$ and $y$, write $d_{xy}$ for the number of edges in $G$ from $x$ to $y$, and write
\[ d_x = \sum_y d_{xy} \]
for the outdegree of $x$.
\begin{theorem}
\label{groupisom}
Let $G$ be a strongly connected finite directed graph without loops, let $RR(G)$ be its rotor-router group, and $SP(G)$ its sandpile group. Then $RR(G) \simeq SP(G)$.
\end{theorem}
\begin{proof}
Let $V$ be the vertex set of $G$. The sandpile group of $G$ \cite{CR,Dhar}
is the quotient
\[ SP(G) = \Z^V \big/ (s,\Delta_x)_{x\in V} \]
where $s \in V$ is the sink and
\[ \Delta_x = \sum_{y\in V} d_{xy} y - d_x x. \]
Define $\phi : \Z^V \rightarrow RR(G)$ by
\[ \phi \left(\sum_{x\in V} u_x x\right) = \prod_{x \in V} e_x^{u_x}. \]
Starting with $d_x$ chips at a vertex $x$ and letting each chip take one rotor-router step results in $d_{xy}$ chips at each vertex $y$, with the rotors unchanged, hence
\[ e_x^{d_x} = \prod_{y\in V} e_y^{d_{xy}}. \]
Thus $\phi(\Delta_x)=Id$. Since also $\phi(s)=e_s=Id$, the map $\phi$ descends to a map $\bar{\phi}:SP(G) \rightarrow RR(G)$. This latter map is surjective since $\phi$ is surjective; to show that $\bar{\phi}$ is injective, by Lemma~\ref{transitivity} we have
\[ \# RR(G) \geq \# Rec(G) = \# SP(G), \]
where the equality on the right is the matrix-tree theorem \cite[5.6.8]{Stanley}.
\end{proof}
\section{Aggregation on the Tree}
Fix $d\geq 3$, and let $T$ be the infinite $d$-regular tree. Fix an origin vertex~$o$ in~$T$. In {\it rotor-router aggregation}, we grow a cluster of points in~$T$ by repeatedly starting chips at the origin and letting them walk until they exit the cluster. Beginning with $A_1 = \{o\}$, define the cluster~$A_n$ inductively by
\[ A_n = A_{n-1} \cup \{x_n\}, \qquad n >1. \]
where $x_n \in T$ is the endpoint of a rotor-router walk started at~$o$ and stopped on first exiting~$A_{n-1}$. We do not change the positions of the rotors when adding a new chip. In this section we use the group isomorphism in Theorem~\ref{groupisom} to show that $A_n$ is a perfect ball for suitable values of $n$ (Theorem~\ref{treecirc}).
A function $H$ on the vertices of a directed graph $G$ is {\it harmonic} at a vertex $x$ if
\[ d_x H(x) = \sum_{y \in V} d_{xy} H(y), \]
where $d_{xy}$ is the number of edges from~$x$ to~$y$, and~$d_x$ is the outdegree of~$x$.
\begin{lemma}
\label{HPinvariant}
Let $G=(V,E)$ be a finite directed graph without loops.
Suppose chips on $G$ can be moved by a sequence of rotor-router steps, starting with $u(x)$ chips at each vertex $x$ and ending with $v(x)$ chips at each vertex $x$, in such a way that the initial and final rotor configurations are the same. If $H$ is a function on $V$ that is harmonic at all vertices which emitted chips, then
\[ \sum_{x \in V} H(x) u(x) = \sum_{x \in V} H(x) v(x). \]
\end{lemma}
\begin{proof}
Let $u=u_0, u_1, \ldots, u_k=v$ be the intermediate configurations. If $u_{i+1}$ is obtained from $u_i$ by routing a chip from $x_i$ to $y_i$, then
\begin{equation}\label{stepbystep} \sum_{x \in V} H(x) (u(x)-v(x)) = \sum_i H(x_i)-H(y_i). \end{equation}
If the initial and final rotor configurations are the same, then each rotor makes an integer number of full turns, so the sum in (\ref{stepbystep}) can be written
\[ \sum_i H(x_i)-H(y_i) = \sum_{x \in V} N(x) \sum_{y \in V} d_{xy} (H(x)-H(y)) \]
where $N(x)\in \Z_{\geq 0}$ is the number of full turns performed by the rotor at~$x$. By the harmonicity of $H$, the inner sum on the right vanishes whenever $N(x)>0$.
\end{proof}
Next we describe our choice of graph $G$ and harmonic function $H$.
By the \emph{$d$-regular tree of height~$n$} we will mean the finite rooted tree in which each non-leaf vertex has $d-1$ children, and the path from each leaf to the root has $n-1$ edges. We denote this tree by~$T_n$. Let $\hat{T}_n$ be the graph obtained from $T_n$ by adding a single additional leaf $o$ whose parent is the root $r$ of $T_n$. This is an undirected graph; when applying the results above, which are phrased in terms of directed graphs for maximum generality, we think of it as \emph{bidirected}: each edge is replaced by a pair of directed edges pointing in opposite directions.
Denote by $(X_t)_{t\geq 0}$ the simple random walk on $\hat{T}_n$, and let $\tau \geq 0$ be the first hitting time of the set of leaves. Fix a leaf $z\neq o$, and let
\begin{equation} \label{ourharmonicfunction} H(x) = \PP_x(X_\tau = z) \end{equation}
be the probability that random walk started at $x$ and stopped at time $\tau$ stops at $z$. This function is harmonic at all non-leaf vertices.
We briefly recall the well-known martingale argument from gambler's ruin used to find the value of $H(r)$. The process
\[ M_t = a^{-|X_t|} \]
is a martingale, where $a=d-1$ and $|x|$ denotes the number of edges in the path from $o$ to $x$. Since $M_t$ has bounded increments and $\EE_r \tau < \infty$, we obtain from optional stopping
\[ a^{-1} = \EE_r M_0 = \EE_r M_\tau = p + (1-p) a^{-n} \]
where $p = \PP_r(X_\tau = o)$. Solving for $p$ we obtain
\begin{equation} \label{gamblersruin} \PP_r(X_\tau=o) = \frac{a^{n-1}-1}{a^n-1}. \end{equation}
In the event that the walk stops at a leaf $z\neq o$, by symmetry it is equally likely to stop at any such leaf. Since there are $a^{n-1}$ such leaves, we obtain from (\ref{gamblersruin})
\begin{equation} \label{hittingprobofleaf} H(r) = \frac{1 - \PP_r(X_\tau = o)}{a^{n-1}}
= \frac{a-1}{a^n-1}. \end{equation}
The \emph{wired $d$-regular tree} of height $n$ is the graph~$\bar{T}_n$ obtained from~$\hat{T}_n$ by collapsing all the leaves to a single vertex~$s$, the {\it sink}. We do not collapse edges; thus each neighbor of the sink except for $r$ has $a=d-1$ edges to the sink. The proof of Theorem~\ref{treecirc} will use the following fact about the sandpile group of the wired regular tree.
\begin{lemma}
\label{degreeofroot}
The root $r$ of $\bar{T}_n$ has order $\frac{a^n-1}{a-1}$ in the sandpile group $SP\big(\bar{T}_n\big)$.
\end{lemma}
\begin{proof}
See \cite{sandpiletree}, Proposition~4.2.
\end{proof}
The next lemma concerns rotor-router walk on $\hat{T}_n$ stopped on hitting the leaves. The leaves play the role of sinks, and the dynamics are the same as for rotor-router walk on the wired tree $\bar{T}_n$. However, we are interested in counting how many chips stop at each leaf, which is why we preserve the distinction between $\hat{T}_n$ and $\bar{T}_n$. Since the rotors at the leaves play no role, we view our rotor configuration as living on $\bar{T}_n$. Such a configuration is \emph{acyclic} if no two neighboring vertices have rotors pointing to one another; in the notation of the previous section, the acyclic configurations are precisely those in $Rec(\bar{T}_n)$.
\begin{lemma}
\label{exitmeasure}
Let $a=d-1$. Given an acyclic rotor configuration on $\bar{T}_n$, starting with $\frac{a^n-1}{a-1}$ chips at the root $r$ of $\hat{T}_n$, and stopping each chip when it reaches a leaf, exactly one chip stops at each leaf $z\neq o$, and the remaining $\frac{a^{n-1}-1}{a-1}$ chips stop at $o$. Moreover, the starting and ending rotor configurations are identical.
\end{lemma}
\begin{proof}
By Theorem~\ref{groupisom} and Lemma~\ref{degreeofroot}, the element $e_r \in RR(\bar{T}_n)$ has order $m=\frac{a^n-1}{a-1}$, so $e_r^m$ is the identity permutation of $Rec(\bar{T}_n)$, hence the starting and ending rotor configurations are identical. Fix a leaf $z\neq o$ of $\hat{T}_n$ and let~$H$ be the function on vertices of $\hat{T}_n$ given by (\ref{ourharmonicfunction}). Since $H$ is harmonic on the non-leaf vertices, by Lemma~\ref{HPinvariant} and (\ref{hittingprobofleaf}), the number of chips stopping at $z$ is
\[ \sum H(x) v(x) = \sum H(x) u(x) = \frac{a^n-1}{a-1} H(r) = 1. \]
Since there are $a^{n-1}$ leaves $z\neq o$, the remaining $\frac{a^n-1}{a-1}-a^{n-1} = \frac{a^{n-1}-1}{a-1}$ chips stop at o.
\end{proof}
The \emph{principal branches} of the infinite $d$-regular tree $T$ are the $d$ subtrees rooted at the neighbors of the origin. The ball of radius $\rho$ centered at the origin in $o \in T$ is
\[ B_\rho = \{x \in T \,:\, |x| \leq \rho \} \]
where $|x|$ is the number of edges in the path from $o$ to $x$. Write
\[ b_\rho = \# B_\rho = 1 + (a+1) \frac{a^\rho-1}{a-1}. \]
As the following result shows, provided we start with an acyclic configuration of rotors, the rotor-router aggregation cluster $A_n$ is a perfect ball at those times when an appropriate number of chips have aggregated. It follows that at all other times, the cluster is as close as possible to a ball: if $b_\rho<n<b_{\rho+1}$ then $B_\rho \subset A_n \subset B_{\rho+1}$.
\begin{theorem}
\label{treecirc}
Let $A_n$ be the region formed by rotor-router aggregation on the infinite $d$-regular tree, starting from $n$ chips at the origin. If the initial rotor configuration is acyclic, then $A_{b_\rho} = B_\rho$ for all $\rho \geq 0$.
\end{theorem}
\begin{proof}
Define a modified aggregation process $A'_n$ as follows. Stop the $n$-th chip when it either exits the occupied cluster $A'_{n-1}$ or returns to $o$, and let
\[ A'_n = A'_{n-1} \cup \{x'_n\} \]
where $x'_n$ is the point where the $n$-th chip stops. By relabeling the chips, this yields a time change of the original process, i.e.\ $A'_n = A_{f(n)}$ for some sequence $f(1), f(2), \ldots$. Thus it suffices to show $A'_{c_\rho}=B_\rho$ for some sequence $c_1, c_2, \ldots$. We will show by induction on $\rho$ that this is the case for
\[ c_\rho = 1 + (a+1) \sum_{t=1}^{\rho} \frac{a^t-1}{a-1}, \]
and that after $c_\rho$ chips have stopped, the rotors are in their initial state. For the base case $\rho=1$, we have $c_1 = a+2=d+1$. The first chip stops at $o$, and the next $d$ stop at each of the neighbors of $o$, so $A'_{d+1}=B_1$. Since the rotor at $o$ has performed one full turn, it is back in its initial state.
Assume now that $A'_{c_{\rho-1}} = B_{\rho-1}$ and that the rotors are in their initial acyclic state. Starting with $c_\rho - c_{\rho-1}$ chips at $o$, let each chip in turn perform rotor-router walk until either returning to $o$ or exiting the ball $B_{\rho-1}$. Then each chip is confined to a single principal branch of the tree, and each branch receives~$\frac{a^\rho-1}{a-1}$ chips.
By Lemma~\ref{exitmeasure}, exactly one chip will stop at each leaf $z \in B_{\rho}-B_{\rho-1}$, and the remainder will stop at $o$. Thus $A'_{c_\rho} = B_\rho$. Moreover, by Lemma~\ref{exitmeasure}, once all chips have stopped, the rotors are once again in their initial state, completing the inductive step.
\end{proof}
\section{Recurrence and Transience}
\begin{figure}
\centering
\includegraphics[scale=.7]{Tree1.eps}
\\ ~ \\ ~ \\
\includegraphics[scale=.7]{Tree2.eps}
\caption{The ball $B_n$ in the regular ternary tree (top), the branch $Y_n$ (left), and its sub-branches $L$ and $R$.}
\label{fig:Tree1}
\end{figure}
In this section we explore questions of recurrence and transience for the rotor-router walk on regular trees. We aim to study to what extent the rotor-router walk behaves as a deterministic analogue of random walk. We find that the behavior depends quite dramatically on the initial configuration of rotors.
A chip performing rotor-router walk starting at the origin $o$ in the infinite $d$-regular tree either returns to the origin or escapes to infinity within a single principal branch of the tree, leaving the rotors in the other branches unchanged.
Therefore, as shown in Figure~1, we focus on a single branch~$Y_n$ of the ball~$B_n$ in the $d$-regular tree. In the notation of the previous section,~$Y_n$ is the graph obtained from $\hat{T}_n$ by collapsing all the leaves except for $o$ to a single vertex, which we label $b$ for boundary. Starting chips at the root~$r$ of~$Y_n$, and stopping them either when they reach~$b$ or return to~$o$, we will compare the hitting rates of~$o$ and~$b$ for rotor-router walk with the expected hitting rates for random walk.
To each rotor direction we associate an index from $\{1,\ldots,d\}$, with direction $d$ corresponding to a rotor pointing to the parent vertex. Rotors cycle through the indices in order. In the ternary tree ($d=3$) we will sometimes refer to the three rotor directions as left (direction $1$), right (direction $2$) and up (direction $3$).
\begin{lemma}
\label{ternarycase}
Suppose $d=3$.
If all rotors in $Y_n$ initially point in direction~$1$, then the first $2^{n}-1$ chips started at~$r$ alternate, the first stopping at~$b$, the next stopping at~$o$, the next at~$b$, and so on. After this sequence of $2^n-1$ walks, all rotors again point in direction~$1$.
\end{lemma}
\begin{proof}
Induct on $n$. In the base case $n=2$, there is only one rotor, which sends the first chip in direction $2$ to $b$, the next chip up in direction $3$ to $o$, and the third chip in direction $1$ to $b$, at which point the rotor is again in its initial state.
Now suppose that the lemma holds for $Y_{n-1}$. Let $L$ and $R$ be the two principal branches of $Y_n$. We think of $L$ and $R$ as each having a rotor that points either to $b$ or back up to $r$. The initial state of these rotors is pointing to $r$. The first
chip is sent from the root to $R$, which by induction sends it to $b$. Note that the root rotor is now pointing towards $R$, the $R$-rotor is pointing to $b$, and the $L$-rotor is pointing to $r$ (Figure~\ref{fig:FourSteps}a). We now observe that the next four chips form a
pattern that will be repeated. The second chip is sent directly to $o$ (Figure~\ref{fig:FourSteps}b) and
the third chip is sent to $L$ which sends it to $b$ (Figure~\ref{fig:FourSteps}c). The fourth chip
is sent to $R$, but by induction this chip is returned and then it is sent
to $o$ (Figure~\ref{fig:FourSteps}d). Finally, the fifth chip is sent to $L$, returned, sent to $R$, and
through to~$b$ (Figure~\ref{fig:FourSteps}e). Note that the root rotor is now again pointing towards $R$,
the $R$-rotor is again pointing to~$b$, and the $L$-rotor is again pointing to $r$. In this cycle of four chips, the two branches $R$ and $L$ see two chips apiece. This cycle repeats $2^{ n - 2 } - 1$
times, and each subtree sees $2^{ n - 1 } - 2$ chips.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{FourSteps.eps}
\caption{The four-chip cycle, which begins after the first chip has been routed to $b$.}
\label{fig:FourSteps}
\end{figure}
Recall that the first chip was sent to $R$, so $R$ it has seen a total of $2^{ n - 1 }
- 1$ chips. By induction, all the rotors in $R$ are in their initial
configuration. We have sent a total of $2^n - 3$ chips. The next chip
is sent to $o$, and the last to $L$, which sends it to $b$. Now $L$ has seen
$2^{ n - 1 } - 1$ chips, so by induction all of its rotors are in their
initial configuration. The root rotor is pointing towards $L$, its initial
configuration. We have sent a total of $2^n - 1$ chips, alternating
between $b$ and $o$, and all of the rotors of $Y_n$ are in the initial
configuration, so the inductive step is complete.
\end{proof}
We remark that the obvious generalization of Lemma~\ref{ternarycase} to trees of degree $d > 3$ fails; indeed, we do not know of a starting rotor configuration on trees of higher degree which results in a single chip stopping at $o$ alternating with a string of $d-1$ chips stopping at $b$.
Consider now the case of the infinite ternary tree $T$. A chip performing rotor-router walk started at the origin $o\in T$ must either return to $o$ or escape to infinity visiting each vertex only finitely many times.
Thus the state of the rotors after a chip has escaped to infinity is well-defined. We can therefore run a sequence of~$m$ rotor-router walks and count the number~$R(m)$ that return to the origin. The following result shows that there is an initial rotor configuration on the tree for which the rotor-router walk behaves as an exact quasirandom analogue to the random walk, in which chips exactly alternate returning to the origin with escaping to infinity.
\begin{prop}
\label{ternarythm}
Let $T$ be the infinite ternary tree, with principal branches labeled $Y^{(1)}$, $Y^{(2)}$, and $Y^{(3)}$ in correspondence with the direction indexing of the rotor at the origin. Set the rotors along the rightmost path to infinity in $Y^{(3)}$ initially pointing in direction~$2$, and all remaining rotors initially pointing in direction~$1$. Let $E(m)$ be the expected number of chips that return to the origin if $m$ chips perform independent random walks on~$T$. Let $R(m)$ be the number of chips that return to the origin if $m$ chips sequentially perform rotor-router walks on~$T$. Then $|E(m) - R(m)| \leq \frac{1}{2}$ for all $m$.
\end{prop}
\begin{proof}
Lemma~\ref{ternarycase} implies that for the branches $Y^{(1)}$ and $Y^{(2)}$, the chips sent to a given branch alternate indefinitely with the first escaping to infinity, the next returning to $o$, and so on. Likewise, chips sent to $Y^{(3)}$ will alternate indefinitely with the first returning to $o$, the next escaping to infinity, and so on. Since chips on the full tree $T$ are routed cyclically through the branches beginning with $Y^{(2)}$, we see that the chips too will alternate indefinitely between escaping to infinity and returning to the origin, with the first escaping to infinity. Thus $R(m) = \left\lfloor \frac{m}{2} \right\rfloor$. Taking $n \rightarrow \infty$ in (\ref{gamblersruin}) we obtain $E(m) = \frac{m}{2}$, and the result follows.
\end{proof}
\begin{lemma}
\label{finitetreesreturn}
For any $d\geq 3$, if all rotors in~$Y_n$ initially point in direction $d-1$, then the first $n-1$ chips started at~$r$ all hit~$o$ before hitting~$b$. After these $n-1$ chips have stopped at~$o$, the final rotors all point in direction~$d$.
\end{lemma}
\begin{proof}
Induct on $n$. In the base case $n=2$, the first chip steps directly from $r$ to $o$, leaving the single rotor pointing in direction $d$.
Now suppose the lemma holds for $Y_{n-1}$. Let $Z_1, \ldots, Z_{d-1}$ be the principal branches of $Y_n$. The first chip placed at~$r$ is sent directly to~$o$. By the inductive hypothesis, the first $n-2$ chips that are sent to each branch $Z_i$ are returned to~$r$ before hitting~$b$. Thus each of the next $n-2$ chips started at~$r$ is sent to $Z_1$, returned to~$r$, sent to $Z_2$, and so on until it is sent to $Z_{d-1}$, returned to~$r$ and then routed to~$o$. The root rotor now points in direction~$d$, and since each branch~$Z_i$ received exactly $n-2$ chips, its final rotors all point in direction~$d$ by the inductive hypothesis.
\end{proof}
Our next result shows that, perhaps surprisingly, the initial rotors can be set up so as to make rotor-router walk on the $d$-regular tree recurrent.
\begin{prop}
\label{infinitetreesreturn}
On the infinite $d$-regular tree $T$, if all rotors initially point in direction $d-1$, then every chip in an infinite succession of chips started at the origin eventually returns to the origin.
\end{prop}
\begin{proof}
By Lemma~\ref{finitetreesreturn}, for each $n$, the $n$-th chip sent to each principal branch~$Y$ returns to the origin before hitting height $n+1$ of~$T$.
\end{proof}
Note also that if all the rotors in the first $n-1$ levels of~$T$ initially point in direction $d-1$, and all remaining rotors initially point in direction~$d$, then after $n-1$ chips have been sent to a given branch $Y$ and returned to the origin, by Lemma~\ref{finitetreesreturn} all rotors in $Y$ point in direction~$d$, so the next chip sent to~$Y$ escapes to infinity.
We continue our exploration of recurrence and transience on the infinite ternary tree~$T$, allowing now for arbitrary rotor configurations. We focus on a single principal branch~$Y$ of the infinite tree, rooted at a neighbor~$r$ of the origin $o\in T$. We include the edge~$(o,r)$ in~$Y$, so that~$r$ has degree~$d$ in~$Y$, and~$o$ has degree one. Thus each chip started at the origin will move to $r$ on its first step. Given a rotor configuration on~$Y$, we define the {\it escape sequence} for the first~$n$ chips to be the binary word $a = a_{1}\ldots a_{n}$, where for each~$j$,
\[a_{j} = \begin{cases} 0, & \text{if the $j^{th}$ chip returns to the origin;} \\
1, & \text{if the $j^{th}$ chip escapes to infinity.} \end{cases} \]
As noted previously, a chip cannot stay within a finite height indefinitely without returning to the origin, so $a$ is well-defined.
We define a map $\psi$ associating to an escape sequence $a = a_{1} \ldots a_{n}$ a pair of shorter sequences. First, we rewrite $a$ as the concatenation of subwords $b_{1} \cdots b_{m}$ where each $b_{j} \in \{0,10,110\}$. Since at least one of any three consecutive chips entering~$Y$ is routed back to the origin by the rotor at the root~$r$ of~$Y$, at most two of any three consecutive letters in an escape sequence~$a$ can be ones. Therefore, any escape sequence can be factored in this way up to the possible concatenation of an extra $0$. Now we define $\psi(a) = (c,d)$ by
\begin{equation} \label{psidef} (c_{j},d_{j}) = \begin{cases} (0,0), & \text{if} \; b_{j} = 0\\
(1,1), & \text{if} \; b_{j} = 110\\
(0,1), & \text{if} \; b_{j} = 10 \; \text{and} \; \#\{i<j|b_{i}=10\} \; \text{is odd}\\
(1,0), & \text{if} \; b_{j} = 10 \; \text{and} \; \#\{i<j|b_{i}=10\} \; \text{is even.} \end{cases}
\end{equation}
In the other direction, given a pair of binary words $c$ and $d$, each of length~$m$, define $\phi(c,d) = b_{1} \cdots b_{m}$, where
\[b_{j}
= \begin{cases} 0,& \text{if}\; (c_{j},d_{j}) = (0,0)\\
10, & \text{if}\; (c_{j},d_{j}) = (1,0) \;\text{or}\; (0,1)\\
110, & \text{if} \;(c_{j},d_{j}) = (1,1). \end{cases} \]
Note that $\phi$ is a left inverse of $\psi$, i.e.\ $\phi \circ \psi(a) = a$, up to possible concatenation of an extra $0$.
\begin{lemma}
\label{phi}
Let $Y$ be a principal branch of the infinite ternary tree. Fix a rotor configuration on $Y$ with the root rotor pointing to $o$. Let $c$ and $d$ be the escape sequences for the configurations on the left and right sub-branches of $Y$, respectively. Then $\phi(c,d)$ is the escape sequence for the full branch $Y$.
\end{lemma}
\begin{proof}
We claim that each word $b_j$ is the escape sequence for the $j^{th}$ full rotation of the root rotor. Note that after the root rotor has performed $j-1$ full rotations, each of the sub-branches $L$ and $R$ of $Y$ has seen exactly $j-1$ chips, so the next chip sent to $L$ (resp.\ R) will either return to $r$ or escape to infinity accordingly as $c_j=0$ or $c_j=1$ (resp.\ $d_j=0$ or $d_j=1$).
Consider first the case $(c_j,d_j) = (0,0)$. After $j-1$ full rotations of the root rotor, the next chip that enters $Y$ will be routed first to $L$, then returned to $r$, sent to $R$, returned to $r$, and finally routed back up to the origin. The root rotor has now performed a full turn, with corresponding escape sequence $b_j=0$.
If $(c_j,d_j) = (1,0)$, the next chip entering $Y$ will be routed to $L$, where it escapes to infinity. The following chip will be routed to $R$ and then back up to the origin, completing a full rotation of the root rotor. In this case we have escape sequence $b_j = 10$. If $(c_j,d_j) = (0,1)$, the next chip entering $Y$ will be routed to $L$, back up to $r$, and then to $R$ where it escapes to infinity. The following chip will be routed directly up to the origin leaving the root rotor pointing up once again. Again, in this case $b_j = 10$. Finally, if $(c_j,d_j) = (1,1)$, the next two chips entering $Y$ will escape to infinity, the first through $L$ and the second through $R$. The following chip will be routed directly up to the origin, once again leaving the root rotor pointing up. In this case we have $b_j = 110$.
\end{proof}
To adapt Lemma~\ref{phi} to the case when the root rotor is not pointing up, we define \emph{extended escape sequences} $c'$ and $d'$ associated to the two sub-branches.
If the root rotor initially points to $L$, let $c' = 0c$ and $d' = d$. If the root rotor initially points to $R$, let $c' = 0c$ and $d' = 0d$. Then $a = \phi(c',d')$ is the escape sequence of the full branch $Y$.
We now introduce the condition that is central to characterizing which words can be escape sequences:
\renewcommand{\theequation}{$P_k$}
\begin{equation}
\text{any subword of length $2^{k}-1$ contains at most $2^{k-1}$ ones}
\end{equation}
We next show that the map $\psi$ preserves this requirement.
\renewcommand{\theequation}{\arabic{equation}}
\begin{lemma}
\label{psi}
Let $a$ be a binary word satisfying $(P_{k})$ and let $\psi(a) = (c,d)$ as defined in (\ref{psidef}). Then $c$ and $d$ each satisfy $(P_{k-1})$.
\end{lemma}
\begin{proof}
Let $c'$ be a subword of $c$ of length $2^{k-1}-1$ and let $d'$ be the corresponding subword of $d$. Let $a' = \phi(c',d')$, which is a subword of $a0$. The formula for $\phi$ guarantees that $a'$ has one zero for each letter of $c'$, so $a'$ has $2^{k-1}-1$ zeros. Since the last letter of $a'$ is zero, and $a$ satisfies $(P_k)$, it follows that $a'$ has at most $2^{k-1}$ ones (else after truncating the final zero, the suffix of $a'$ of length $2^k-1$ has at most $2^{k-1}-2$ zeros, hence at least $2^{k-1}+1$ ones).
Let $m$ be the number of ones in $c'$. Since the instances of $(0,1)$ and $(1,0)$ alternate in the formula for $\psi(a)=(c,d)$, it follows that $d'$ must have at least $m-1$ ones. Since the number of ones in $c'$ and $d'$ combined equals the number of ones in $a'$, we obtain $2m-1 \leq 2^{k-1}$, hence $m \leq 2^{k-2}$. The same argument with the roles of $c$ and $d$ reversed shows that $d$ has at most $2^{k-2}$ ones.
\end{proof}
\begin{lemma}
\label{branch-escape}
Let $a=a_1\ldots a_n$ be a binary word of length $n$. Then $a$ is an escape sequence for some rotor configuration on the infinite branch $Y$ if and only if $a$ satisfies $(P_{k})$ for all $k$.
\end{lemma}
\begin{proof}
Suppose $a$ is an escape sequence. We prove that $a$ satisfies $(P_{k})$ for each $k$ by induction on $k$. That $a$ satisfies $(P_{1})$ is trivial. Now suppose that every escape sequence satisfies $(P_{k-1})$ and let $c$ and $d$ be the extended escape sequences of the left and right sub-branches respectively. Then $a = \phi(c,d)$ up to the possible concatenation of an extra zero. Let $a'$ be a subword of $a$ of length $2^{k}-1$, and let $\psi(a')=(c',d')$. Then there are words $c''$ and $d''$ each of which is a subword of $c$ or $d$, and which are equal to $c'$ and $d'$, respectively, except possibly in the first letter; moreover the first letters satisfy $c'_1 \leq c''_1$ and $d'_1 \leq d''_1$.
By the formula for $\psi$, the number of ones in $a'$ is the sum of the number of ones in $c'$ and $d'$. If $c'$ has length at most $2^{k-1}-1$, then since $c$ and $d$ satisfy $(P_{k-1})$, each of $c'$ and $d'$ has at most $2^{k-2}$ ones, and therefore $a'$ has at most $2^{k-1}$ ones. On the other hand, if $c'$ has length at least $2^{k-1}$, then the number of zeros in $a'$ is at least $2^{k-1}-1$. Thus $a'$ has at most $2^{k-1}$ ones, so $a$ satisfies $(P_{k})$.
The proof of the converse is by induction on $n$. For $n=1$ the statement is trivial. Suppose that every binary word of length $n-1$ satisfying $(P_{k})$ for each $k$ is an escape sequence. Then by Lemma~\ref{psi}, $\psi(a) = (c,d)$ gives a pair of binary words each satisfying $(P_{k})$ for all $k$. If $c$ and $d$ have length $n-1$ or less, then they are escape sequences by induction, hence $a$ is an escape sequence by Lemma~\ref{phi}. If $c$ and $d$ are of length~$n$, then the definition of~$\psi$ implies that $a_j=0$ for all $j<n$, in which case $a$ is an escape sequence by the remark following Proposition~\ref{infinitetreesreturn}.
\end{proof}
We can now establish our main result characterizing all possible escape sequences on the infinite ternary tree.
\begin{theorem}
\label{escape}
Let $a=a_1\ldots a_n$ be a binary word. For $j \in \{1,2,3\}$ write $a^{(j)} = a_{j} a_{j+3} a_{j+6} \ldots$. Then $a$ is an escape sequence for some rotor configuration on the infinite ternary tree $T$ if and only if each $a^{(j)}$ satisfies $(P_{k})$ for all $k$.
\end{theorem}
\begin{proof}
Let $Y^{(1)}$, $Y^{(2)}$, and $Y^{(3)}$ be the three principal branches of $T$ assigned so that the rotor at the origin initially points to $Y^{(3)}$. Then $a$ is the escape sequence for $T$ if and only if $a^{(j)} = a_{j} a_{j+3} a_{j+6} \ldots $ is the escape sequence for $Y^{(j)}$. The result now follows from Lemma~\ref{branch-escape}.
\end{proof}
\section{Concluding Remark}
We conclude with an open question. While Theorem~\ref{escape} completely characterizes the possible escape sequences for rotor-router walk on the infinite ternary tree, we know nothing about the possible escape sequences for rotor-router walk on another natural class of transient graphs, namely $\Z^d$ for $d\geq 3$. The open question is this: does there exist a rotor configuration on $\Z^d$ for $d\geq 3$, analogous to the configuration on the tree described in Proposition~\ref{infinitetreesreturn}, so that every chip in an infinite sequence of chips started at the origin eventually returns to the origin? We remark that Jim Propp has found such a configuration on $\Z^2$.
\section*{Acknowledgments}
The authors thank Yuval Peres, Jim Propp and Parran Vanniasegaram for useful discussions.
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\subsection{Generation of Measures via Given ``Preintegrable'' Functions}
\label{SS:PRE}
We describe a way to obtain a measure on a set $\OM$,
\NOT{which will be used in the sequel.}
which we use in \S\ref{SS:EnhInf}. This method seems convenient when a
measure has to be constructed by some ``integration'' of a family of given
measures.
The starting point is a set $\Pre$ of $[0,\I]$-valued \fns\ on $\OM$, called
{\EM preintegrable}, with a functional ({\EM integral}) $\cI:\Pre\to[0,\I[$,
satisfying the following assumptions:
\begin{enumerate}
\item
$\Pre$ is a cone, i.e.\ $\Pre$ contains $0$ and is stable w.r.t.\ addition
and multiplication by finite nonnegative real constants, and $\cI$ is
additive and non-negatively linear.
\item
If $f,g:\OM\to\BAR{\bR^+}$ s.t.\ $f,f+g\in\Pre$ then $g\in\Pre$.
(Consequently, $f,g\in\Pre,\:f\ge g\Rightarrow\cI(f)\ge\cI(g)$.)
\item
If $f_n\in\Pre,\:n\in\bN$, $f_n\uparrow$ and $\cI(f_n)$ is bounded, then
$\lim f_n\in\Pre$ and $\cI(\lim f_n)=\lim(\cI(f_n))$.
\item \label{it:comp}
If $f_0\in\Pre$ and $\cI(f_0)=0$ then any \fn\ $f\le f_0$ belongs to $\Pre$.
\end{enumerate}
Now say that a set $E\subset\OM$ is {\EM measurable} if
$$\forall f\in\Pre\; f\cdot 1_E\in\Pre.$$
One proves easily, using the above assumptions, that the measurable sets
form a $\sigma$-algebra. {\EM Measurable \fns} will be \fns\ measurable
w.r.t.\ this $\sigma$-algebra. Note that if $g$ is measurable and bounded,
then
$$\forall f\in\Pre\; f\cdot g\in\Pre$$.
To define the measure $\mu$ on this $\sigma$-algebra, a measurable set $E$
will have finite measure iff $1_E\in\Pre$ and then $\mu(E):=\cI(1_E)$.
Otherwise $\mu(E)=\I$. The assumptions on $\Pre$ and $\cI$ imply readily
that $\mu$ is $\sigma$-additive.
(Moreover, by \ref{it:comp}.\ $\mu$ is complete, i.e.\ every subset of a set
of measure $0$ is measurable.)
Thus $\int\:d\mu$ is defined. As usual, a \fn\ $f\ge0$
is {\EM integrable} if it is measurable and has finite integral.
An important fact is that any integrable \fn\ $f\ge0$ is preintegrable, and
any measurable preintegrable \fn\ $f\ge0$ is integrable, and then $\cI(f)$
and $\int f\,d\mu$ coincide. (Thus, to find the integral of a measurable
\fn\ $f\ge0$ one just checks if $f$ is in $\Pre$. If it is, its integral is
$\cI(f)$, otherwise $\int f\,d\mu=\I$).
Indeed, Note first that any $\{0,\I\}$-valued $f\in\Pre$ must have
$\cI(f)=0$, since $\cI(f)$ is finite and $2\cI(f)=\cI(2f)=\cI(f)$.
Therefore if $f\ge0$ is measurable preintegrable, then $\{f=+\I\}$ is
null.
Now an integrable $f\ge0$ can be obtained from characteristic \fns\ of
sets of finite measure by addition and increasing limits with bounded
integral, hence it is in $\Pre$. On the other hand, if $f\ge0$ is measurable
preintegrable, and $a>0$, then $E=\{\I>f>a\}$ is measurable. $1_E$ is of the
form $f\cdot g$ where $g$ is measurable bounded, hence $1_E$ is
preintegrable, implying $\mu(E)=\cI(1_E)<\I$. From that one easily deduces
$f$ integrable and $\int f\,d\mu=\cI(f)$.
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Indian Women: Changing The 'Game'
Indian Women: Changing The 'Game'
In the lead up to the Commonwealth Games in Gold Coast NDTV and WINS - The Australian Broadcasting Corporation discuss why women's sport in India at elite and grassroots have a yawning gap and how can it be bridged. Injeti Srinivas, Sports Secretary, believes the lack of a sporting culture in the country in the main problem. He also added that there is disparity between male and female athletes at the grassroots level but not at the elite level.
Show Comments
| 197,922
|
TITLE: Quick question about solution to Chapter 14 Question 5(i) in Spivak's Calculus: Find a function $g$ such that $\int_0^x t g(t)dt=x+x^2$
QUESTION [0 upvotes]: Question 5(i) in Chapter 14 of Spivak's Calculus reads as follows:
Find a function $g$ such that$\displaystyle \int_0^x t g(t)dt=x+x^2$
The solution manual proposes the following function as a viable possibility:
$g(t)=\begin{cases}\frac{1}{t}+2 \quad &\text{ if t $\gt 0$} \\0 \quad &\text{ if $t=0$} \end{cases}$
I just wanted to make sure that the specification of the value that $g$ takes on when $t=0$ is completely arbitrary. Any value $M \in\mathbb R$ would have worked, right?
In fact, the integral is completely agnostic to the value that $g$ takes on at $0$ because, by definition, $\int_0^0=0$ and at $x=0$, we obviously have that $x+x^2=0+0^2=0$. This will always be the case regardless of the finite value that $g(0)$ is assigned. Is this correct?
REPLY [1 votes]: Indeed, you could have taken any value for $g(0)$, you would not change the value of the integral, and the function $g$ would not be continuous in any case. In fact, you could change the values of the integrand at countably many points and still get the same result for the Riemann integral. In full generality, the set of discontinuities has to have Lebesgue measure zero. This is known as the Lebesgue-Vitali theorem.
| 218,021
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Red Roof Inn Boston Southborough/Worcester
- Free WiFi and free parking
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Red Roof Inn Boston Southborough/Worcester
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| 381,719
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\section{Summary and conclusion}
\label{sec:7}
The main purpose of this paper is to describe a method for extracting Hamiltonian systems of finite dimension
from a class of Hamiltonian field theories with Poisson brackets of the form of \eqref{eq:2.2}, as described
in section \ref{sec:2}. The method was exemplified by considering a four-wave truncation of Euler's equation
for two-dimensional vortex dynamics. In section \ref{sec:3} we described a direct method of truncation, one
that produces equations that are energy conserving but not guaranteed to be Hamiltonian. Sections \ref{sec:4}
and \ref{sec:5} contain the main results of the paper, the description of the method of beatification followed
by truncation. This was applied to Euler's equation to produce our Hamiltonian four-wave example. Lastly,
in section \ref{sec:6} we briefly used numerics and recurrence plots to compare our Hamiltonian four-wave model
with the non-Hamiltonian version.
Clearly there are many applications possible for our methodology developed here, since the class of systems
of section \ref{sec:2} includes many models from geophysical fluid dynamics and plasma physics. Moreover, it
is clear that the ideas pertain to more complicated Hamiltonian models such as those with more field variables,
as are common in plasma physics modeling (see e.g.\ Ref.~\onlinecite{thiffeault}), three-dimensional magnetofluid
models (see e.g.\ Ref.~\onlinecite{remarkable}), and sophisticated kinetic theories (see e.g.\ Ref.~\onlinecite{burby}).
In addition, one could retain more waves in the truncation, use an alternative basis other than Fourier, and proceed to
higher order in the beatification procedure in order to capture higher degree of nonlinearity and more complete dynamics.
Because beatification yields a Poisson bracket that is independent of the dynamical variable, conventional structure
preserving numerical methods, such as symplectic integrators, could be implemented.
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Are you enjoying Giveaway Week here at KCW?! We are only halfway through and have so much more for you to win! Today’s giveaway comes from an adorable shop in California, Sew Modern. They also have an online shop and run a popular blog as well. So I’m sure this isn’t the first time you’ve heard of them:)
I’ll let Lauren tell you a bit about their shop:
Sew Modern is an online and brick and mortar shop in Los Angeles. We carry every color of Robert Kaufman Kona Cotton as well as other solids including Art Gallery’s gorgeous Pure Elements. We offer free shipping on US orders over $100 dollars and most orders ship the same day. We have children’s patterns from Oliver & S, Figgy’s, Green Bee Designs and more.
And they really do have spectacular fabric. I spent a ton of time happily digging around in their [online] stacks and came up with some perfect fabrics you can use for this season’s STORYBOOK theme. Smiling piggies, sleeping unicorns, vikings, and storybook brownstones! I think any kid would be super psyched to have any one of these fabric made into an outfit.
1. ed emberbly, happy drawing too! 2. alexander henry, the vikings 3. good night moon, mouse 4. leah duncan, brownstones 5. cotton and steel, tiger stripes 6. heather ross, far far away
While poking around the Sew Modern shop I also came across a bunch of fantastic flannel prints. My local fabric stores carry a lot of flannel, but the selection is limited to baby pink, baby blue, or plaid. Not the biggest variety. My kids love cozy, flannel pjs and they would flip out if I made them out of flannel with glow-in-the-dark stars. I mean come on! How could they not?!
1. outer space stars flannel–glows in the dark! 2. happy camper flannel 3. polka dot, gray 4.miss kate flannel 5. super softly flannel, gold 6. alphabet, cocoa
1. SOAK nail polish set– Lizzy House 2. We Love to Sew by Annabel Wrigley 3. Green Bee, Dylan 4. Paper, Fabric, Thread by Kristen Sutcliffe 5. cat snip scissors
Sew Modern is giving away this gorgeous Patchwork City Bundle curated by Elizabeth Hartman (of Oh Fransson!) for Robert Kaufman Fabrics. Use the Rafflecopter widget below to enter by October 22. This giveaway is open to international entries, void where prohibited by law. Winner will receive the Patchwork City Bundle from Sew Modern!
Good Luck & Happy KCW!
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\begin{document}
\title{\Large Instability of an integrable nonlocal NLS}
\author{Fran\c cois Genoud \\
\small\emph{Delft Institute of Applied Mathematics, Delft University of Technology}\\
\small\emph{Mekelweg~4, 2628 CD Delft, The Netherlands}\\
\small s.f.genoud@tudelft.nl}
\date{}
\maketitle
\begin{abstract}
In this note we discuss the global dynamics of an integrable nonlocal NLS
on $\R$, which has been the object of recent investigation by integrable systems methods.
We prove two results which are in striking contrast with the case of the local cubic focusing NLS on $\R$.
First, finite time blow-up solutions exist with arbitrarily small initial data in $H^s(\R)$, for any $s\ge0$.
On the other hand, the solitons of the local NLS, which are also solutions of the nonlocal equation, are unstable
by blow-up for the latter.
\end{abstract}
\section{Introduction}
The nonlocal nonlinear Schr\"{o}dinger equation
\begin{equation}\label{nls}
iu_t(t,x)+u_{xx}(t,x)+u^2(t,x)\wb{u}(t,-x)=0, \qquad u(t,x):\R\times\R \to \C,
\end{equation}
has recently been shown to be a completely integrable system, with infinitely many conservation laws \cite{mus1,mus2}.
The equation is related to two different areas of physics: gain/loss systems in optics and so-called
$PT$-symmetric quantum mechanics, see \cite{mus0,yang,ben,mos} and references therein.
Mathematically, the feature connecting \eqref{nls} to these areas
is the $PT$-symmetry of the `nonlinear potential' $u(t,x)\wb{u}(t,-x)$. Namely,
this quantity is invariant under the joint transformation $x\to -x$ and $i\to -i$ (parity and time reversal).
The inverse-scattering transform was applied in \cite{mus1,mus2} to produce a variety of solutions to \eqref{nls}.
In particular, a `one-soliton solution' is obtained, which blows up in finite time (actually, up to rescaling,
at all times $t=2m+1, \ m\in\Z$). The purpose of this note is to use this peculiar solution to prove some results
about the global dynamics of \eqref{nls}, which are in striking contrast with the case of
the local focusing cubic equation
\begin{equation}\label{local_nls}
iu_t(t,x)+u_{xx}(t,x)+|u(t,x)|^2u(t,x)=0, \qquad u(t,x):\R\times\R \to \C.
\end{equation}
We will first show that \eqref{nls} is locally well-posed (in $H^1(\R)$) but
then we prove that there exist solutions which blow up in finite time (in $L^\infty(\R)$),
with arbitrarily small initial data in $H^s(\R)$, for any $s\ge0$. This
shows in particular that the trivial solution, $u\equiv0$, is unstable by blow-up.
Let us now observe that \eqref{nls} reduces to \eqref{local_nls}
provided the discussion is restricted to even solutions.
Hence, the well-known solitons
$u_\omega(t,x)=\e^{i\omega t}\ffi_\omega(x)$ of \eqref{local_nls},
where
\begin{equation}\label{sech}
\ffi_\omega(x)=\frac{2\sqrt{2\omega}}{\e^{\sqrt{w}x}+\e^{-\sqrt{w}x}}, \qquad \omega>0,
\end{equation}
are also solutions of \eqref{nls}. These standing waves are orbitally stable with respect to \eqref{local_nls} but
we show that they are unstable by blow-up with respect to \eqref{nls}.
The rest of the paper is organised as follows. In Section~\ref{well} we prove the local well-posedness
and the blow-up instability of the zero solution. In Section~\ref{blow} we prove the blow-up instability of the solitons
\eqref{sech}. We conclude in Section~\ref{defoc} with some remarks on the `defocusing' equation
\begin{equation}\label{defoc_nls}
iu_t(t,x)+u_{xx}(t,x)-u^2(t,x)\wb{u}(t,-x)=0, \qquad u(t,x):\R\times\R \to \C.
\end{equation}
\subsection*{Notation} For non-negative quantities $A,B$ we write $A \lesssim B$ if $A\le CB$ for some
constant $C>0$, whose exact value is not essential to the analysis.
\subsection*{Acknowledgement} I am grateful to Stefan Le Coz for an interesting discussion about
\eqref{nls}, and to the anonymous referee, whose remarks helped improve the presentation of the paper.
\section{Instability of the trivial solution}\label{well}
We start with a local well-posedness result.
\begin{theorem}\label{local.thm}
Given any initial data $u_0\in H^1(\R)$, there exists a unique maximal solution
$u\in C\big([0,T_\mathrm{max}),H^1(\R)\big)$ of \eqref{nls} such that $u(0,\cdot)=u_0$, where
$T_\mathrm{max}=T_\mathrm{max}(\norm{u_0}_{H^1(\R)})$.
\end{theorem}
\begin{proof}
The theorem is proved by a fixed point argument, similar to the
case of the local equation \eqref{local_nls}. However, some calculations are different due to the
nonlocal nonlinearity, so we give the proof here for completeness.
Fix $u_0\in H^1(\R)$, define $F(u)(x)=u^2(x)\wb{u}(-x)$ and a map
$\tau: X_T\to X_T$ by
\[
\tau(u)(t)=S(t)u_0+i\int_0^tS(t-s)F(u)(s) \diff s,
\]
where $X_T=L^\infty\big((0,T);H^1(\R)\big)$ for some $T>0$ and $S(t)$ is the free Schr\"odinger group.
We shall prove the existence of a unique fixed point of $\tau$ in the ball
\[
B_R=\{u\in X_T : \norm{u}_{X_T}<R\},
\]
for suitable values of $T,R>0$. That this fixed point can be extended to a maximal solution
$u\in C\big([0,T_\mathrm{max}),H^1(\R)\big)$ of \eqref{nls} then follows by standard arguments.
First observe that, for any $p\ge2$, the Sobolev embedding theorem yields
\begin{align*}
\norm{F(u)}_{L^p}^p &= \int_\R |u(x)|^{2p}|u(-x)|^p \diff x \\
&\le \Big\{\int_\R |u(x)|^{2pr}\diff x\Big\}^{1/r}\Big\{\int_\R |u(-x)|^{ps}\diff x\Big\}^{1/s} \\
&= \norm{u}_{L^{2pr}}^{2p}\norm{u}_{L^{ps}}^{p}
\lesssim \norm{u}_{H^1}^{3p},
\end{align*}
where $r,s\ge1$ are arbitrary H\"older conjugate exponents. It follows that
\begin{equation}\label{Festimate}
\norm{F(u)}_{L^p} \lesssim \norm{u}_{H^1}^{3} \quad \ \text{for any} \ p\ge 2,
\end{equation}
and a similar estimate yields
\begin{equation}\label{Fxestimate}
\norm{F(u)_x}_{L^p} \lesssim \norm{u}_{H^1}^{3} \quad \text{for any} \ p\ge 2,
\end{equation}
where
\begin{equation}\label{F_x}
F(u)_x=\big[u^2(x)\wb{u}(-x)\big]_x=2u(x)u_x(x)\wb{u}(-x)-u^2(x)\wb{u}_x(-x).
\end{equation}
By Strichartz's estimate and \eqref{Festimate}--\eqref{Fxestimate} with $p=2$,
we see in particular that $\tau$ indeed maps $X_T$ into $X_T$. Furthermore,
there exist constants $C_1,C_2>0$ such that
\begin{align*}\label{tauestimate}
\norm{\tau(u)}_{X_T} &\le C_1\norm{u_0}_{H^1}+T\norm{F(u)}_{L^\infty(0,T;H^1)} \\
&\le C_1\norm{u_0}_{H^1}+TC_2\norm{u}^3_{X_T} .
\end{align*}
Choosing $R=2C_1\norm{u_0}_{H^1}$ and $T>0$ such that $C_1TR^2=1/2$, it follows that, for any
$u\in B_R$,
\[
\norm{\tau(u)}_{X_T} \le \frac{R}{2}+TC_1 \norm{u}^2_{X_T}\norm{u}_{X_T}
\le \frac{R}{2}+\frac12 \norm{u}_{X_T}<R.
\]
Hence, for these values of $T,R>0$, $\tau$ maps the ball $B_R$ into itself.
We now show that, if $T>0$ is small enough, then $\tau$ is a contraction in $B_R$. We have
\begin{equation}\label{taudiff}
\norm{\tau(u)-\tau(v)}_{X_T}\le T\norm{F(u)-F(v)}_{L^\infty(0,T;H^1)}, \quad u,v\in X_T.
\end{equation}
Writing
$|F(u)-F(v)|=|\int_0^1 \frac{\dif}{\dif\theta} F(\theta u +(1-\theta) v) \diff\theta|$,
we obtain
\[
|F(u)-F(v)|\lesssim |u(x)+v(x)||u(x)-v(x)||u(-x)+v(-x)|+|u(x)+v(x)|^2|u(-x)-v(-x)|
\]
and it follows that
\begin{equation}\label{FL2}
\norm{F(u)-F(v)}_{L^2}\lesssim \big(\Vert{u}\Vert_{H^1}^2+\Vert{v}\Vert_{H^1}^2\big)\norm{u-v}_{L^2}.
\end{equation}
On the other hand, in view of \eqref{F_x}, letting
\[
G(u)(x)=2u(x)u_x(x)\wb{u}(-x) \quad\text{and}\quad H(u)(x)=u^2(x)\wb{u}_x(-x)
\]
we have
\[
|[F(u)-F(v)]_x| \le |G(u)-G(v)|+|H(u)-H(v)|,
\]
where
\begin{align*}
|G(u)-G(v)| &\lesssim |u(x)-v(x)||u_x(x)+v_x(x)||u(-x)+v(-x)| \\
&\phantom{\lesssim}+|u(x)+v(x)||u_x(x)-v_x(x)||u(-x)+v(-x)| \\
&\phantom{\lesssim}+|u(x)+v(x)||u_x(x)+v_x(x)||u(-x)-v(-x)|
\end{align*}
and
\begin{align*}
|H(u)-H(v)| &\lesssim |u(x)+v(x)||u(x)-v(x)||u_x(-x)+v_x(-x)| \\
&\phantom{\lesssim}+|u(x)+v(x)|^2|u_x(-x)+v_x(-x)|.
\end{align*}
It follows that
\begin{equation}\label{F_xL2}
\norm{[F(u)-F(v)]_x}_{L^2}\lesssim \big(\Vert{u}\Vert_{H^1}^2+\Vert{v}\Vert_{H^1}^2\big)\norm{u-v}_{H^1}.
\end{equation}
By \eqref{taudiff}, \eqref{FL2} and \eqref{F_xL2}, there is a constant $C>0$ such that
\[
\norm{\tau(u)-\tau(v)}_{X_T}\le CT\big(\Vert{u}\Vert_{X_T}^2+\Vert{v}\Vert_{X_T}^2\big)\norm{u-v}_{X_T}.
\]
Hence, if $u,v\in B_R$ we have
\[
\norm{\tau(u)-\tau(v)}_{X_T}\le 2CTR^2\norm{u-v}_{X_T},
\]
showing that $\tau$ is a contraction in $B_R$ provided $T<(2CR^2)^{-1}$.
The contraction mapping principle now yields a unique fixed point of $\tau$ in $B_R$,
which concludes the proof.
\end{proof}
\medskip
For the local equation \eqref{local_nls}, the next chapter of the story is well known.
One proves that, for any $u_0\in H^1(\R)$, the maximal solution is global, i.e.~that
$T_\mathrm{max}=\infty$. This is usually done by means of the energy and charge functionals
\[
E(u)=\frac12\int_\R |u_x|^2\diff x -\frac14\int_\R|u|^4 \diff x, \qquad
Q(u)=\frac12\int_\R |u|^2\diff x.
\]
Using the conservation of these quantities along the flow and
the Gagliardo--Nirenberg inequality, one shows that
the first term in $E$ is controlled by the second one, and must remain bounded. Hence, global existence in
$H^1(\R)$ is ensured by the blow-up alternative.
The corresponding conservation laws for \eqref{nls} are \cite{mus1,mus2}
\[
E(u)=\frac12\int_\R u_x(x)\wb{u}_x(-x)\diff x -\frac14\int_\R u^2(x)\wb{u}^2(-x) \diff x \quad\text{and}\quad
Q(u)=\frac12\int_\R u(x)\wb{u}(-x)\diff x.
\]
Even though each of these integrals is real, in general none of the three terms has a definite sign, unless
$u$ is even (or odd), in which case we recover the energy and charge of the local equation \eqref{local_nls}.
This predicament wipes away any hope of proving a global well-posedness result for \eqref{nls},
even for small initial data. In fact, we have the following result.
\begin{theorem}\label{global.thm}
For any $0<\alpha<1$, there exists a solution $u^{\alpha}(t,x)$ of \eqref{nls}, defined on $[0,T_\alpha)\times\R$,
where $T_\alpha=\pi/3\alpha^2$, with the following properties:
\begin{itemize}
\item[(i)] $u^{\alpha}$ blows up in $L^\infty(\R)$ as $t\to T_\alpha$, with
$\lim_{t\to T_\alpha}|u^{\alpha}(t,0)|=\infty$;
\item[(ii)] $u^{\alpha}_0=u^{\alpha}(0,\cdot)$ satisfies
$\norm{u^{\alpha}_0}_{H^k(\R)} \lesssim \alpha^{1/2}$, for all $k\in \N$.
\end{itemize}
\end{theorem}
\begin{proof}
The result is obtained from the explicit solution
\begin{equation}\label{2param}
u^{\alpha,\beta}(t,x)=\frac{2\sqrt{2}(\alpha+\beta)}{\e^{-4i\alpha^2t}\e^{2\alpha x}+\e^{-4i\beta^2t}\e^{-2\beta x}}.
\end{equation}
For any $\alpha,\beta>0, \alpha\neq\beta$, this function blows up at all times
\[
T_m=\frac{(2m+1)\pi}{4(\alpha^2-\beta^2)}, \qquad m\in \Z,
\]
with $\lim_{t\to T_m}|u^{\alpha,\beta}(t,0)|=\infty$, and is a solution of \eqref{nls}
in the sense of Theorem~\ref{local.thm} between these times,
i.e.~$u^{\alpha,\beta}\in C\big((T_m,T_{m+1}),H^1(\R)\big), \ m\in \Z$.
To simplify the analysis, we choose $\beta=\alpha/2$, so that $u^{\alpha,\beta}$ reduces to
\begin{equation}\label{1param}
u^{\alpha}(t,x)=\frac{3\sqrt{2}\alpha}{\e^{-4i\alpha^2t}\e^{2\alpha x}+\e^{-4i\beta^2t}\e^{-\alpha x}},
\end{equation}
and the first blow-up time to the right of $t=0$ becomes
\[
T_\alpha=\frac{\pi}{3\alpha^2}.
\]
For the initial condition $u^{\alpha}_0=u^{\alpha}(0,\cdot)$, direct calculations then show that
\begin{equation}\label{norms}
\norm{u^{\alpha}_0}_{L^2}^2=\frac{4\pi\alpha}{3}, \quad
\norm{(u^{\alpha}_0)_x}_{L^2}^2=\frac{8\pi\alpha^3}{3\sqrt{3}}, \quad
\norm{(u^{\alpha}_0)_{xx}}_{L^2}^2=\frac{8\pi\alpha^5}{\sqrt{3}}.
\end{equation}
Upon inspection of the integrals involved, one easily sees that for all $k\in\N$,
there is a constant $C_k>0$, independent of $\alpha$, such that
\begin{equation}\label{sobolev_norms}
\norm{\frac{\dif^ku^{\alpha}_0}{\dif x^k}}_{L^2}^2 = C_k \alpha^{2k+1}.
\end{equation}
For $\alpha\in(0,1)$, this completes the proof.
\end{proof}
\begin{remark}\label{rem}
\rm
(a) If $\alpha=\beta=\sqrt{\omega}/2$, then $u^{\alpha,\beta}(t,x)$ reduces to the usual soliton
$\e^{i\omega t}\ffi_\omega(x)$, with $\ffi_\omega$ defined in \eqref{sech}.
(b) A direct verification shows that the solution $u^{\alpha,\beta}(t,x)$ only blows up at $x=0$, i.e.~the
denominator in \eqref{2param} never vanishes if $x\neq0$.
(c) The particular choice $\beta=\alpha/2$ enables one to compute explicitly the norms in \eqref{norms}.
In fact, the relations \eqref{sobolev_norms} are easily derived by choosing $\beta=\gamma\alpha$ with,
say, $\gamma\in(0,1)$, and using the change of variables $y=\alpha x$ in the integrals.
\end{remark}
\section{Instability of the solitons}\label{blow}
The blow-up instability of the solitons \eqref{sech} is now a consequence of Remark~\ref{rem}~(a). More precisely, fixing
$\alpha=\sqrt{\omega}/2$ and letting $\beta=\sqrt{\omega+\delta}/2$ with $0<\delta\ll1$, we obtain finite time
blow-up solutions $u^{\alpha,\beta}$ as close as we want to $\e^{i\omega t}\ffi_\omega(x)$.
\begin{theorem}
Fix $\omega>0$. For any $\ep>0$ there exists $q_{\omega,\ep}\in H^1(\R)$ such that
\[
\norm{\ffi_\omega-q_{\omega,\ep}}_{H^1(\R)}<\ep
\]
and the solution with initial data
$u(0,\cdot)=q_{\omega,\ep}$ blows up in finite time.
\end{theorem}
\begin{proof}
Define $q_{\omega,\delta}(x)$ as $u^{\alpha,\beta}(0,x)$, with $\alpha=\sqrt{\omega}/2$ and
$\beta=\sqrt{\omega+\delta}/2, \ \delta>0$, namely
\[
q_{\omega,\delta}(x)=\frac{\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+\delta})}{\e^{\sqrt{w}x}+\e^{-\sqrt{w+\delta}x}}.
\]
We only need to check that
\begin{equation}
\norm{\ffi_\omega-q_{\omega,\delta}}_{H^1} \to 0 \quad\text{as} \ \delta\to0.
\end{equation}
To show that
\begin{equation}\label{L2}
\int_\R |\ffi_\omega(x) - q_{\omega,\delta}(x)|^2\diff x
\to 0 \quad\text{as} \ \delta\to0,
\end{equation}
we first observe that $|\ffi_\omega(x) - q_{\omega,\delta}(x)|\to 0$ as $\delta\to0$ for all $x\in\R$.
Furthermore if $0<\delta<1$, we have, for $-\infty<x\le0$,
\[
\frac{\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+\delta})}{\e^{\sqrt{w}x}+\e^{-\sqrt{w+\delta}x}}
\le \frac{\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+1})}{\e^{\sqrt{w}x}+\e^{-\sqrt{w}x}},
\]
while, for $0<x<\infty$,
\[
\frac{\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+\delta})}{\e^{\sqrt{w}x}+\e^{-\sqrt{w+\delta}x}}
\le \frac{\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+1})}{\e^{\sqrt{w}x}+\e^{-\sqrt{w+1}x}},
\]
and so \eqref{L2} follows by dominated convergence. Applying similar estimates to
the derivative
\[
(q_{\omega,\delta})_x(x)=
\sqrt{2}(\sqrt{\omega}+\sqrt{\omega+\delta})
\frac{\sqrt{w+\delta}\,\e^{-\sqrt{w+\delta}x}-\sqrt{w}\,\e^{\sqrt{w}x}}{(\e^{\sqrt{w}x}+\e^{-\sqrt{w+\delta}x})^2}
\]
and using again dominated convergence, we also have
\begin{equation*}
\int_\R |(\ffi_\omega)_x(x) - (q_{\omega,\delta})_x(x)|^2\diff x
\to 0 \quad\text{as} \ \delta\to0,
\end{equation*}
from which the conclusion follows.
\end{proof}
\section{Remarks on the defocusing case}\label{defoc}
The `defocusing' equation \eqref{defoc_nls}
has also been considered in \cite{mus1,mus2}.
Our local well-posedness result, Theorem~\ref{local.thm},
carries over to \eqref{defoc_nls}, with an identical proof.
On the other hand, it is shown in \cite[p.~936]{mus2} that `one-soliton' solutions of the type \eqref{2param}
are not available in the defocusing case.
Global well-posedness for \eqref{defoc_nls} seems to be an open problem.
| 108,126
|
\section{Appendix setup}
\label{sec:setup}
We will first provide a note on the organization of the appendix and follow that up with introducing the notations.
\subsection{Organization}
\label{ssec:org}
\begin{itemize}
\item In subsection~\ref{ssec:notations}, we will recall notation from the main paper and introduce some new notation that will be used across the appendix.
\item In section~\ref{sec:tailAverageIterateCovariance}, we will write out expressions that characterize the generalization error of the proposed accelerated SGD method. In order to bound the generalization error, we require developing an understanding of two terms namely the bias error and the variance error.
\item In section~\ref{sec:commonLemmas}, we prove lemmas that will be used in subsequent sections to prove bounds on the bias and variance error.
\item In section~\ref{sec:biasContraction}, we will bound the bias error of the proposed accelerated stochastic gradient method. In particular, lemma~\ref{lem:main-bias} is the key lemma that provides a new potential function with which this paper achieves acceleration. Further, lemma~\ref{lem:bound-bias} is the lemma that bounds all the terms of the bias error.
\item In section~\ref{sec:varianceContraction}, we will bound the variance error of the proposed accelerated stochastic gradient method. In particular, lemma~\ref{lem:main-variance} is the key lemma that considers a stochastic process view of the proposed accelerated stochastic gradient method and provides a sharp bound on the covariance of the stationary distribution of the iterates. Furthermore, lemma~\ref{lem:bound-variance} bounds all terms of the variance error.
\item Section~\ref{sec:proofMainTheorem} presents the proof of Theorem~\ref{thm:main}. In particular, this section aggregates the result of lemma~\ref{lem:bound-bias} (which bounds all terms of the bias error) and lemma~\ref{lem:bound-variance} (which bounds all terms of the variance error) to present the guarantees of Algorithm~\ref{algo:TAASGD}.
\end{itemize}
\subsection{Notations}
\label{ssec:notations}
We begin by introducing $\M$, which is the fourth moment tensor of the input $\a\sim\D$, i.e.:
\begin{align*}
\M\defeq\Eover{\distr}{\a\otimes\a\otimes\a\otimes\a}
\end{align*}
Applying the fourth moment tensor $\M$ to any matrix $\S\in\R^{d\times d}$ produces another matrix in $\R^{d\times d}$ that is expressed as:
\begin{align*}
\M\S \defeq \E{(\a\T\S\a)\a\a\T}.
\end{align*}
With this definition in place, we recall $\infbound$ as the smallest number, such that $\M$ applied to the identity matrix $\eye$ satisfies:
\begin{align*}
\M\eye=\E{\twonorm{\a}^2\a\a\T}\preceq\infbound\ \Cov
\end{align*}
Moreover, we recall that the condition number of the distribution $\cnH = \infbound/\mu$, where $\mu$ is the smallest eigenvalue of $\Cov$. Furthermore, the definition of the statistical condition number $\cnS$ of the distribution follows by applying the fourth moment tensor $\M$ to $\Covinv$, i.e.:
\begin{align*}
\M\Covinv&=\E{(\a\T\Covinv\a)\cdot\a\a\T}\preceq\cnS\ \H
\end{align*}
We denote by $\tensor{A}_{\mathcal{L}}$ and $\tensor{A}_{\mathcal{R}}$ the left and right multiplication operator of any matrix $\A\in\R^{d\times d}$, i.e. for any matrix $\S\in\R^{d\times d}$, $\tensor{A}_{\mathcal{L}}\S=\A\S$ and $\tensor{A}_{\mathcal{R}}\S=\S\A$.
\underline{\bf Parameter choices:} In all of appendix we choose the parameters in Algorithm~\ref{algo:TAASGD} as
\begin{align*}
\alpha = \frac{\sqrt{\cnH\cnHh}}{\ctwo\sqrt{2\cone-\cone^2}+\sqrt{\cnH\cnHh}},\ \ \beta = \cthree\frac{\ctwo\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnHh}},\ \ \gamma = \ctwo\frac{\sqrt{2\cone-\cone^2}}{\mu\sqrt{\cnH\cnHh}}, \ \ \delta=\frac{\cone}{\infbound}
\end{align*}
where $\cone$ is an arbitrary constant satisfying $0 < \cone < \frac{1}{2}$. Furthermore, we note that $\cthree=\frac{\ctwo\sqrt{2\cone-\cone^2}}{\cone}$, $\ctwo^2=\frac{\cfour}{2-\cone}$ and $\cfour< 1/6$.
Note that we recover Theorem~\ref{thm:main} by choosing $\cone = 1/5, \ctwo = \sqrt{5}/9, \cthree=\sqrt{5}/3, \cfour = 1/9$. We denote
\begin{align*}
c\defeq\alpha(1-\beta) \text{ and, } \g\defeq\alpha\delta+(1-\alpha)\gamma.
\end{align*}
Recall that $\xs$ denotes unique minimizer of $P(\x)$, i.e. $\xs=\arg\min_{\x \in \R^d} \Eover{\distr}{(b-\iprod{\x}{\a})^2}$. We track $\thetav_k=\begin{bmatrix}\x_k-\xs\\ \yt[k]-\xs\end{bmatrix}$. The following equation captures the updates of Algorithm~\ref{algo:TAASGD}:
\begin{align}
\label{eq:mainRec}
\thetav_{k+1}&=\begin{bmatrix}0&\eye-\delta\widehat{\H}_{k+1}\\-c\cdot\eye&(1+c)\cdot\eye-\g\cdot\widehat{\H}_{k+1}\end{bmatrix}\thetav_k
+\begin{bmatrix}\delta\cdot\epsilon_{k+1}\av_{k+1}\\\g\cdot\epsilon_{k+1}\av_{k+1}\end{bmatrix}\nonumber\\
&\defeq\Ah_{k+1}\thetav_{k}+\zetav_{k+1},
\end{align}
where, $\widehat{\H}_{k+1} \defeq \av_{k+1} \av_{k+1}^\top$, $\widehat{\A}_{k+1} \defeq \begin{bmatrix}0&\eye-\delta\widehat{\H}_{k+1}\\-c\cdot\eye&(1+c)\cdot\eye-\g\cdot\widehat{\H}_{k+1}\end{bmatrix}$
and $\zetav_{k+1} \defeq \begin{bmatrix}\delta\cdot\epsilon_{k+1}\av_{k+1}\\\g\cdot\epsilon_{k+1}\av_{k+1}\end{bmatrix}$.
\noindent Furthermore, we denote by $\phiv_k$ the expected covariance of $\thetav_k$, i.e.:
\begin{align*}
\phiv_k\defeq\E{\thetav_k\otimes\thetav_k}.
\end{align*}
Next, let $\mathcal{F}_k$ denote the filtration generated by samples $\{(\a_1,b_1),\cdots, (\a_k,b_k)\}$. Then,
\begin{align*}
\A&\eqdef \E{\Ah_{k+1}|\mathcal{F}_{k}}=\begin{bmatrix}
\zero & \eye - \delta \Cov \\ -c\eye & (1+c)\eye - \g\Cov
\end{bmatrix}.
\end{align*}
By iterated conditioning, we also have
\begin{align}\label{eqn:theta-det}
\E{\thetav_{k+1}\middle \vert \mathcal{F}_{k}} = \A \thetav_k.
\end{align}
Without loss of generality, we assume that $\Cov$ is a diagonal matrix. We now note that we can rearrange the coordinates through an eigenvalue decomposition so that $\A$ becomes a block-diagonal matrix with $2\times2$ blocks. We denote the $j^{\textrm{th}}$ block by $\A_j$:
\begin{align*}
\A_j \eqdef \begin{bmatrix}
0 & 1 - \delta \lambda_j \\ -c & 1+c - \g \lambda_j
\end{bmatrix},
\end{align*}
where $\lambda_j$ denotes the $j^{\textrm{th}}$ eigenvalue of $\Cov$.
Next,
\begin{align*}
\BT&\eqdef \E{\Ah_{k+1}\otimes\Ah_{k+1}|\mathcal{F}_{k}}, \mbox{ and }\\
\Sigh&\eqdef \E{\zetav_{k+1}\otimes\zetav_{k+1}|\mathcal{F}_k} = \begin{bmatrix}\delta^2&\delta\cdot \g\\\delta\cdot \g&\g^2\end{bmatrix}\otimes\Sig\preceq\sigma^2\cdot\begin{bmatrix}\delta^2&\delta\cdot \g\\\delta\cdot \g&\g^2\end{bmatrix}\otimes\H.
\end{align*}
Finally, we observe the following:
\begin{align*}
\E{(\A-\Ah_{k+1})\otimes(\A-\Ah_{k+1})|\mathcal{F}_k}&=\A\otimes\A-\E{\Ah_{k+1}\otimes\A|\mathcal{F}_k}\\&\quad\quad-\E{\Ah_{k+1}\otimes\A|\mathcal{F}_k}+\E{\Ah_{k+1}\otimes\Ah_{k+1}|\mathcal{F}_k}\\
&=-\A\otimes\A+\E{\Ah_{k+1}\otimes\Ah_{k+1}|\mathcal{F}_k}\\
\implies \E{\Ah_{k+1}\otimes\Ah_{k+1}|\mathcal{F}_k}&=\E{(\A-\Ah_{k+1})\otimes(\A-\Ah_{k+1})|\mathcal{F}_k}+\A\otimes\A
\end{align*}
We now define:
\begin{align*}
\RT&\defeq\E{(\A-\Ah_{k+1})\otimes(\A-\Ah_{k+1})|\mathcal{F}_k}, \mbox{ and }\\
\DT&\defeq\A\otimes\A.
\end{align*}
Thus implying the following relation between the operators $\BT,\DT$ and $\RT$:
\begin{align*}
\BT=\DT+\RT.
\end{align*}
\section{The Tail-Average Iterate: Covariance and bias-variance decomposition}
\label{sec:tailAverageIterateCovariance}
\input{files/10.1.tail-average-iterate-covariance.tex}
\section{Useful lemmas}
\label{sec:commonLemmas}
In this section, we will state and prove some useful lemmas that will be helpful in the later sections.
\input{files/10.2.usefulLemmas.tex}
\section{Lemmas and proofs for bias contraction}
\label{sec:biasContraction}
\input{files/10.3.biasBounds.tex}
\input{files/10.4.biasLemmas.tex}
The following lemma bounds the total error of $\thetavb^{\textrm{bias}}$.
\begin{lemma}\label{lem:bound-bias}
\iffalse
\begin{align*}
&\iprod{\begin{bmatrix}
\Cov & \zero \\ \zero & \zero
\end{bmatrix}}{\E{\thetavb^{\textrm{bias}} \otimes \thetavb^{\text{bias}}}} \\ &\leq \frac{1792}{(n-t)^2(\cone\cfour)^{5/4}}\cdot\frac{(\cnH\cnS)^{9/4}d}{\delta\cfour}\cdot\exp\bigg(-(t+1)\frac{\ctwo\cthree\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnS}}\bigg)\norm{\thetat[0]}^2 +\\&\qquad\qquad \frac{5376}{(\cone\cfour)^{1/4}}\frac{(\cnH\cnS)^{5/4}d}{\delta\cfour}\exp\left(\frac{-n \ctwo \cthree \sqrt{2\cone-\cone^2} }{\sqrt{\cnH\cnS}}\right) \cdot \norm{\thetat[0]}^2.
\end{align*}
\fi
\iffalse
\begin{align*}
&\iprod{\begin{bmatrix}
\Cov & \zero \\ \zero & \zero
\end{bmatrix} }{\E{\thetavb^{\textrm{bias}} \otimes \thetavb^{\text{bias}}}} \leq \UC\cdot\frac{(\cnH\cnS)^{9/4}d}{\delta}\cdot\exp\bigg(-(t+1)\frac{\ctwo\cthree\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnS}}\bigg)\norm{\thetat[0]}^2 \nonumber\\&\qquad\qquad\qquad+ \UC\cdot\frac{(\cnH\cnS)^{5/4}d}{\delta}(n-t)\exp\left(\frac{-n \ctwo \cthree \sqrt{2\cone-\cone^2} }{\sqrt{\cnH\cnS}}\right) \cdot \norm{\thetat[0]}^2
\end{align*}
\fi
\begin{align*}
&\iprod{\begin{bmatrix}
\Cov & \zero \\ \zero & \zero
\end{bmatrix} }{\E{\thetavb^{\textrm{bias}} \otimes \thetavb^{\text{bias}}}} \leq \UC\cdot\frac{(\cnH\cnS)^{9/4}d\cnH}{(n-t)^2}\cdot\exp\bigg(-(t+1)\frac{\ctwo\cthree\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnS}}\bigg)\cdot \big(P(\x_0)-P(\xs)\big) \nonumber\\&\qquad\qquad\qquad+ \UC\cdot(\cnH\cnS)^{5/4}d\cnH\cdot\exp\left(\frac{-n \ctwo \cthree \sqrt{2\cone-\cone^2} }{\sqrt{\cnH\cnS}}\right) \cdot \big(P(\x_0)-P(\xs)\big)
\end{align*}
Where, $\UC$ is a universal constant.
\end{lemma}
\begin{proof}
\input{files/10.5.biasActBound.tex}
\end{proof}
\section{Lemmas and proofs for Bounding variance error}
\label{sec:varianceContraction}
\input{files/10.6.varianceBounds.tex}
\input{files/10.7.varianceLemmas.tex}
\begin{lemma}
\label{lem:var1N2bound}
\iffalse
\begin{align*}
&\bigg\vert\iprod{\begin{bmatrix}\Cov&0\\0&0\end{bmatrix}}{ \bigg((\eyeT-\AL)^{-2}\AL+(\eyeT-\AR\T)^{-2}\AR\T\bigg)\phiv_{\infty}}\bigg\vert\\&\leq4\sigma^2 \cdot d \cdot\bigg(\ \frac{2}{\cfour}\cdot\bigg(1+\big(\frac{1+\sqrt{\cone\cfour}}{1-\cfour}\big)^2\bigg) + 3 \cdot \frac{1+\sqrt{\cone\cfour}}{1-\cfour} \cdot \frac{1+\sqrt{2}+\sqrt{\cfour/\cone}}{\cfour} \cdot (\sqrt{2}+\sqrt{\cfour/\cone}) \ \bigg)\sqrt{\cnH\cnS}
\end{align*}
\fi
\begin{align*}
&\bigg\vert\iprod{\begin{bmatrix}\Cov&0\\0&0\end{bmatrix}}{ \bigg((\eyeT-\AL)^{-2}\AL+(\eyeT-\AR\T)^{-2}\AR\T\bigg)\phiv_{\infty}}\bigg\vert\leq\UC\cdot\sigma^2 d\sqrt{\cnH\cnS}
\end{align*}
Where, $\UC$ is a universal constant.
\end{lemma}
\begin{proof}
\input{files/10.8.lowerOrderTermProof.tex}
\end{proof}
\begin{lemma}\label{lem:bound-variance}
\iffalse
\begin{align*}
&\iprod{\begin{bmatrix}
\Cov & \zero \\ \zero & \zero
\end{bmatrix}}{\E{\thetavb^{\textrm{variance}} \otimes \thetavb^{\text{variance}}}}\leq 5\frac{\sigma^2d}{n-t} + 6912\cdot\sigma^2d\cdot\frac{(\cnH\cnS)^{7/4}}{\cthree\cfour(\cone\cthree)^{3/2}}\exp^{-(n+1)\cdot\frac{\ctwo\cthree\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnS}}}\\ &+ 4\cdot\frac{\sigma^2 d}{(n-t)^2} \cdot\bigg(\ \frac{2}{\cfour}\cdot\bigg(1+\big(\frac{1+\sqrt{\cone\cfour}}{1-\cfour}\big)^2\bigg) + 3 \cdot \frac{1+\sqrt{\cone\cfour}}{1-\cfour} \cdot \frac{1+\sqrt{2}+\sqrt{\cfour/\cone}}{\cfour} \cdot (\sqrt{2}+\sqrt{\cfour/\cone}) \ \bigg)\sqrt{\cnH\cnS} \\ &+ 41472\frac{\sigma^2d}{n-t}(\cnH\cnS)^{11/4}\alpha^{(n-t-1)/2}\frac{1}{\cthree\cfour^2(\cone\cthree)^{3/2}} + 41472\cdot\frac{\sigma^2d}{(n-t)^2}\cdot\frac{1}{\cfour^2(\cone\cthree^2)^3}\cdot\exp\bigg({-(n+1)\frac{\cone\cthree^2}{\sqrt{\cnH\cnS}}}\bigg)\cdot(\cnH\cnS)^{7/2}\cnS
\end{align*}
\fi
\begin{align*}
&\iprod{\begin{bmatrix}
\Cov & \zero \\ \zero & \zero
\end{bmatrix}}{\E{\thetavb^{\textrm{variance}} \otimes \thetavb^{\text{variance}}}}\leq 5\frac{\sigma^2d}{n-t} +\UC\cdot\frac{\sigma^2 d}{(n-t)^2} \cdot\sqrt{\cnH\cnS} \\ &+ \UC\cdot\frac{\sigma^2d}{n-t}(\cnH\cnS)^{11/4}\exp\bigg(-\frac{(n-t-1)\ctwo\sqrt{2\cone-\cone^2}}{4\sqrt{\cnH\cnS}}\bigg) \\&+ \UC\cdot\frac{\sigma^2d}{(n-t)^2}\cdot\exp\bigg({-(n+1)\frac{\cone\cthree^2}{\sqrt{\cnH\cnS}}}\bigg)\cdot(\cnH\cnS)^{7/2}\cnS+\UC\cdot\sigma^2d\cdot(\cnH\cnS)^{7/4}\exp\bigg({-(n+1)\cdot\frac{\ctwo\cthree\sqrt{2\cone-\cone^2}}{\sqrt{\cnH\cnS}}}\bigg)
\end{align*}
where, $\UC$ is a universal constant.
\end{lemma}
\begin{proof}
\input{files/10.9.varianceActBound.tex}
\end{proof}
\section{Proof of Theorem~\ref{thm:main}}\label{sec:proofMainTheorem}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
The proof of the theorem follows through various lemmas that have been proven in the appendix:
\begin{itemize}
\item Section~\ref{sec:tailAverageIterateCovariance} provides the bias-variance decomposition and provides an exact tensor expression governing the covariance of the bias error (through lemma~\ref{lem:average-covar-bias})and the variance error (lemma~\ref{lem:average-covar-var}).
\item Section~\ref{sec:biasContraction} provides a scalar bound of the bias error through lemma~\ref{lem:bound-bias}. The technical contribution of this section (which introduces a new potential function) is in lemma~\ref{lem:main-bias}.
\item Section~\ref{sec:varianceContraction} provides a scalar bound of the variance error through lemma~\ref{lem:bound-variance}. The key technical contribution of this section is in the introduction of a stochastic process viewpoint of the proposed accelerated stochastic gradient method through lemmas~\ref{lem:main-variance},~\ref{lem:var-main-1}. These lemmas provide a tight characterization of the stationary distribution of the covariance of the iterates of the accelerated method. Lemma~\ref{lem:var1N2bound} is necessary to show the sharp burn-in (up to log factors), beyond which the leading order term of the error is up to constants the statistically optimal error rate $\mathcal{O}(\sigma^2 d/n)$.
\end{itemize}
Combining the results of these lemmas, we obtain the following guarantee of algorithm~\ref{algo:TAASGD}:
\iffalse
\begin{align*}
\E{P(\xtilde)}-P(\xs) &\leq 10^7\cdot\frac{(\cnH\cnS)^{9/4}d}{(n-t)^2\delta}\cdot\exp\bigg(-\frac{t+1}{9\sqrt{\cnH\cnS}}\bigg)\norm{\thetat[0]}^2 +10^6\cdot\frac{(\cnH\cnS)^{5/4}d}{\delta}\exp\left(\frac{-n }{9\sqrt{\cnH\cnS}}\right) \cdot \norm{\thetat[0]}^2 + \\&5\frac{\sigma^2d}{n-t}+ 1360\cdot\frac{\sigma^2 d}{(n-t)^2} \sqrt{\cnH\cnS} + 10^6\cdot\sigma^2d\cdot(\cnH\cnS)^{7/4}\cdot\exp\bigg(\frac{-(n+1)}{9\sqrt{\cnH\cnS}}\bigg) \\ &+ 10^8\cdot\frac{\sigma^2d}{n-t}(\cnH\cnS)^{11/4}\alpha^{(n-t-1)/2} +10^{10}\cdot\frac{\sigma^2d}{(n-t)^2}\cdot\exp\bigg({-\frac{(n+1)}{9\sqrt{\cnH\cnS}}}\bigg)\cdot(\cnH\cnS)^{7/2}\cnS
\end{align*}
\fi
\begin{align*}
\E{P(\bar{\x}_{t,n})}-P(\xs) &\leq \UC\cdot\frac{(\cnH\cnS)^{9/4}d\cnH}{(n-t)^2}\cdot\exp\bigg(-\frac{t+1}{9\sqrt{\cnH\cnS}}\bigg)\cdot\big(P(\x_0)-P(\xs)\big) \\&+\UC\cdot(\cnH\cnS)^{5/4}d\cnH\cdot\exp\left(\frac{-n }{9\sqrt{\cnH\cnS}}\right) \cdot \big(P(\x_0)-P(\xs)\big) + 5\frac{\sigma^2d}{n-t}\\&+ \UC\cdot\frac{\sigma^2 d}{(n-t)^2} \sqrt{\cnH\cnS} + \UC\cdot\sigma^2d\cdot(\cnH\cnS)^{7/4}\cdot\exp\bigg(\frac{-(n+1)}{9\sqrt{\cnH\cnS}}\bigg) \\ &+ \UC\cdot\frac{\sigma^2d}{n-t}(\cnH\cnS)^{11/4}\exp\bigg(-\frac{(n-t-1)}{30\sqrt{\cnH\cnS}}\bigg) \\&+\UC\cdot\frac{\sigma^2d}{(n-t)^2}\cdot\exp\bigg({-\frac{(n+1)}{9\sqrt{\cnH\cnS}}}\bigg)\cdot(\cnH\cnS)^{7/2}\cnS
\end{align*}
Where, $\UC$ is a universal constant.
\end{proof}
\input{files/08.simulations.tex}
\iffalse
\section{Proof of Theorem~\ref{thm:par}}\label{sec:thm-par}
\begin{algorithm}[t]
\caption{ cde}
\begin{algorithmic}[1]
\INPUT Samples $(\ai[1],\bi[1]),\cdots,(\ai[n],\bi[n])$, Initial point $\xt[0]$, Mini-batch size $m$, Non-averaging phase $t$, Parameters $\alpha, \beta, \gamma, \delta$
\STATE $\vt[0] = \xt[0]$
\FOR{$j = 1, \cdots n/m$}
\STATE $\yt[j-1] \leftarrow \alpha \xt[j] + (1-\alpha) \vt[j]$
\STATE $\zt[j-1] \leftarrow \beta \yt[j] + (1-\beta) \vt[j]$
\STATE $\xt[j] \leftarrow \yt[j-1] - \delta \frac{1}{m} \sum_{k=(j-1)*m+1}^{j*m} \widehat{\nabla}_k P(\x)$
\STATE $\vt[j] \leftarrow \zt[j-1] - \gamma \frac{1}{m} \sum_{k=(j-1)*m+1}^{j*m} \widehat{\nabla}_k P(\x)$
\ENDFOR
\OUTPUT $\frac{1}{n-t}\sum_{j=t+1}^{n} \xt[j]$
\end{algorithmic}
\end{algorithm}
\begin{proof}[Proof of Theorem~\ref{thm:par}]
The proof follows easily by combining Theorem~\ref{thm:main} with the ideas of~\cite{JainKKNS16}. The analysis can be broken down into three parts.
\textbf{Initial phase}: In this phase, Algorithm~\ref{algo:PASGD} runs Algorithm~\ref{algo:MBTAASGD} with a mini-batch size of $m = \max\left(\frac{\infbound}{\twonorm{\Cov}}, \cnS\right)$. For this mini-batch size, recall that we have condition number $\cnH_m \leq \frac{m-1}{m}\kappa(\Cov) + \frac{1}{m} \cnH \leq 2 \kappa(\Cov)$, statistical condition number $\cnS_m \leq \frac{m-1}{m} + \frac{1}{m} \cnS \leq 2$ and noise level $\sigma_m^2 = \sigma^2 / m$. Applying Theorem~\ref{thm:main} for this setting tells us that after every invocation of Algorithm~\ref{algo:MBTAASGD}, the initial error decays by a factor of $2$. So the error of the iterate $\xt[j]$ at the end of the initial phase is at most $L(\xt[j]) \leq 8 \sigma^2 d/m$. If $n < 2qm \log \frac{E_0 m}{\sigma^2}$, note that the iterate after initial phase already satisfies the theorem's guarantee.
\textbf{Middle phase}: In this phase, again the condition number and statistical condition numbers are bounded by $2\kappa(\Cov)$ and $2$ respectively. We show that in every iteration $j$, after the function call to Algorithm~\ref{algo:MBTAASGD}, the following invariant is maintained:
\begin{align}
L(\xt[j]) \leq \frac{8 \sigma^2 d}{m_0*2^{j-1}},\label{eqn:geomdecay}
\end{align}
where $m_0$ is the initial mini-batch size.
The base case for $j=1$ holds by the guarantee from the initial phase. Suppose now that the above condition holds for some $j$ and we wish to establish it for $j+1$. From Theorem~\ref{thm:main}, we see that:
\begin{align*}
L(\xt[j+1]) &\leq \exp\left(\frac{-\sqrt{7}q}{4\sqrt{\kappa(\Cov)}}\right) L(\xt[j]) + \frac{4 \sigma^2 d}{m_0 2^{j}},
\end{align*}
where we used the fact that the variance with a mini-batch size of $m_0*2^j$ in $j+1^{\textrm{th}}$ iteration is $\frac{\sigma^2}{m_0 2^j}$ Substituting the value of number of iterations $q$ and the hypothesis on $L(\xt[j])$ finish the induction step.
\textbf{Final phase}: In the final phase, we again directly apply Theorem~\ref{thm:main} to conclude that the loss of output $\x$ can be bounded by
\begin{align*}
L(\x) \leq \exp\left(\frac{-\sqrt{7}q}{4\sqrt{\kappa(\Cov)}}\right) L(\xt[j]) + \frac{4 \sigma^2 d}{n/2} \leq \frac{9 \sigma^2 d}{n} \leq \exp\left(-n/qm\right) L(\xt[0]) + \frac{9 \sigma^2 d}{n},
\end{align*}
where we used~\eqref{eqn:geomdecay} in the second last step.
This proves the theorem.
\end{proof}
\fi
| 183,104
|
How to Prevent Burnout
Not many people get through life without experiencing burnout at some point. Whether it’s from working more than one job, taking care of a sick loved one, or juggling family life while going back to school, burnout is real, and it negatively impacts your life and health.
Common Signs of Burnout
If you think you may be experiencing burnout but are unsure, here are some of the most common signs:
- Physical and mental exhaustion
- Feeling overwhelmed
- A need to isolate
- Fantasies of escaping
- Irritability
- Frequent illnesses such as colds and flues
5 Ways to Prevent Burnout
Exercise
You know exercise is necessary for your physical health, but it is also fantastic for your mental and emotional health as well. Physical activity helps our bodies secrete feel-good hormones, which give our mood a boost. So be sure to commit to exercising at least 3-4 times a week.
Get Enough Rest
It’s essential to get enough restorative sleep each night. Sleep not only helps our bodies build and repair new tissue, but it helps us be able to feel calm and focus.
If you have trouble getting enough ZZZZZs each night, skip caffeinated beverages past 2 pm, ban smartphones and other electronics from the bedroom, and establish a relaxing nighttime ritual like meditation, reading, or taking a bath.
Validate Your Feelings
“Keep calm and carry on.” That’s a fun saying for a throw pillow, but it’s not always the best advice. Sometimes it’s important to admit that you are struggling and that you need a break.
Remember to Play
Just because you’re an adult, that doesn’t mean you don’t need some downtime to just have fun. Whether you want to play a sport, enjoy a hobby, or go to the theater, be sure to make time each week to enjoy yourself and your life.
Ask for Help
During stressful times, it’s important to reach out to others for help. Sometimes all we need is a friendly ear to listen to what’s on our minds and hearts. Let your friends and family know you could use a little support.
If your stress levels don’t seem to go down, you may want to consider working with a counselor who can help you navigate your feelings and offer coping strategies to deal with the issues you have going on.
If you’d like to speak with someone, please reach out to me.
SOURCES:
| 277,008
|
\begin{document}
\begin{abstract}
Here we investigate the property of effectivity for adjoint divisors.
Among others, we prove the following results:
A projective variety $X$ with at most canonical singularities is uniruled if and only if
for each very ample Cartier divisor $H$ on $X$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.
Let $X$ be a projective $4$-fold, $L$ an ample divisor and $t$ an integer with $t \ge 3$.
If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$.
\end{abstract}
\maketitle
\section{Introduction}
Let $X$ be a normal projective variety over the complex field $\mathbb{C}$; let $K_X$ be its canonical divisor.
We assume that $X$ has at most canonical singularities.
In the paper we fix a suitable Cartier divisor $H$ on $X$ and we discuss when the effectivity or non-effectivity of some
adjoint divisors $aK_X + bH$ determines the geometry of $X$.
\smallskip
In the first part we consider the notion of \emph{Termination of Adjunction}. This turns out to be rather delicate,
since in the literature there are different meanings for such a property. The following are four possibilities, where $m_0$ and $m$ are natural numbers.
\begin{itemize}
\item[(A)] For every (for some) big Cartier divisor $H$ there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline {Eff(X)}$
(i.e. it is not pseudo-effective) for $m \geq m_0$.
\item[(B)] For every big Cartier divisor $H$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.
\item[(C)] For every very ample Cartier divisor $H$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.
\item[(D)] For some (for every) big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.
\end{itemize}
It is clear that (A) $\Longrightarrow$ (B) $\Longrightarrow$ (C) $\Longrightarrow$ (D).
\smallskip
We prove that these four definitions are equivalent and moreover that {\sl Adjunction Terminates
in the above sense if and only if $X$ is uniruled} (see Theorem \ref{termination},
Corollary \ref{iff1} and Corollary \ref{iff2}).
The results follow by some characterizations of pseudo-effective Cartier divisor (see Theorem \ref{pseudo-effective}),
which are direct consequences of a fundamental result of Siu (\cite{S}).
The connection with uniruledness follows in turn from the fact that a projective variety $X$ with canonical singularities is uniruled if and only if $K_X$ is not pseudo-effective (see \cite{BDPP}, Corollary~0.3, or \cite{BCHM}, Corollary~1.3.3).
\smallskip
A characterization of rationally connected manifolds along the same lines has been given in \cite{CDP}.
\smallskip
The examples described in \cite{Ko}, Theorem 39, show that, for varieties with singularities worst then canonical, uniruledness is not connected to Termination of Adjunction.
\medskip
We consider also the following more general definition.
\begin{itemize}
\item[(C')] Let $H$ be an effective Cartier divisor on $X$.
We say that {\sl Adjunction Terminates in the classical sense} for $H$ if
there exists an integer $m_0 \ge 1$ such that
$$H^0(X, H+mK_X)=0$$
for every integer $m \ge m_0$.
\end{itemize}
We conjecture that such a definition is actually equivalent to the previous ones; a partial result
in this direction is provided by Proposition \ref{veryclassical}.
In dimension two, Castelnuovo and Enriques indeed proved that Condition (C') implies
that $X$ is uniruled (see \cite{CE} and also \cite{Mu}).
\bigskip
In the second part of the paper we assume that $X$ is a projective variety of dimension $n$
with at most terminal $\mathbb{Q}$-factorial singularities. We take a nef and and big Cartier divisor
$L$ on $X$ and we call $(X,L)$ a quasi polarized pair.
\smallskip
The following is a straightforward consequence of Theorem D in \cite{BCHM}, see Remark \ref{cascini} at the beginning of Section \ref{pabundance}.
\begin{Proposition}\label{first}
Let $(X,L)$ be a quasi polarized pair and $t >0$.
If $K_X+tL\in \overline {Eff(X)}$, then there exists $N \in \mathbb{N}$
such that $H^0(X, N(K_X+tL)) \ne 0$.
\end{Proposition}
Note that for $t=0$ the statement of the Proposition would amount to Abundance Conjecture, together with MMP.
\smallskip
The next Conjecture is an effective version of the above Proposition.
\begin{Conjecture}\label{second}
Let $(X,L)$ be a quasi polarized pair and $t > 0$.
If $K_X+tL \in \overline {Eff(X)}$, then $H^0(X, K_X+tL) \ne 0$.
\end{Conjecture}
The case $t=1$ is a version of the so-called Ambro-Ionescu-Kawamata conjecture,
which is true for $n \leq 3$ (see Theorem 1.5 in \cite{H}), while for $t=n-1$
we recover a conjecture by Beltrametti and Sommese (see \cite{BS}, Conjecture~7.2.7).
Note that if Conjecture \ref{second} holds for $t=1$ then it holds also for every $t > 0$.
\smallskip
In the paper we consider the following conjecture.
\begin{Conjecture}\label{third}
Let $(X,L)$ be a quasi polarized pair and $s> 0$.
Then $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le s$
if and only if $K_X+sL$ is not pseudo-effective.
\end{Conjecture}
Since $L$ is big, in particular pseudo-effective, then the \emph{if} part is obvious. Note that Conjecture \ref{third} for $s=1$ implies Conjecture \ref{second}.
We prove that {\bf Conjecture \ref{third} is true for $s = n$} (see Proposition \ref{as}); we actually show that this case happens if and only if
the pair $(X,L)$ is birationally equivalent (via a $0$-reduction, see the definition in the next section) to the pair $(\mathbb{P}^n, \mathcal{O}(1))$.
{\bf For $s=n-1$ the conjecture was essentially proved by H\"oring}, see \cite{H}, Theorem~1.2. We prove a slightly more explicit version of
his result (see Proposition \ref{bs}), namely, we show that this case happens if and only if the pair $(X,L)$ is birationally equivalent
to a finite list of pairs.
\smallskip
Finally, we focus on the case $n = 4$ (see Theorem \ref{fourfold} and Proposition \ref{fourfold2}) and we generalize previous work by Fukuma
(\cite{F2}, Theorem 3.1).
\section{Notation and preliminaries} \label{notation}
Let $X$ be a normal complex projective variety of dimension $n$.
We adopt \cite{KollarMori} and \cite{L1} as the standard references for our set-up.
In particular, we denote by $Div(X)$ the group of all Cartier divisors on $X$
and by $Num(X)$ the subgroup of numerically trivial divisors.
The quotient group $N^1(X) = Div(X)/Num(X)$ is the Neron-Severi group of $X$.
In the vector space $N^1(X)_{\R} := N^1(X) \otimes \R$, whose dimension is $\rho(X) := rk N^1(X)$, we consider some convex cones.
\begin{itemize}
\item[(a)] $Amp(X) \subset N^1(X)_{\R}$ the convex cone of all {\it ample} $\R$-divisor classes; it is an open convex cone.
\item[(b)] $Big(X) \subset N^1(X)_{\R}$ the convex cone of all {\it big} $\R$-divisor classes; it is an open convex cone.
\item[(e)] $Eff(X) \subset N^1(X)_{\R}$ the convex cone spanned by the classes of all effective $\R$-divisors.
\item[(n)] $Nef(X) = \overline {Amp(X)} \subset N^1(X)_{\R}$ the closed convex cone of all {\it nef} $\R$-divisor classes.
\item[(p)] $\overline {Eff(X)} = \overline {Big(X)} \subset N^1(X)_{\R}$ the closed convex cone of all {\it pseudo-effective} $\R$-divisor classes.
\end{itemize}
The above definitions actually lean on some fundamental results like the openess of the ample and big cones, the facts that $int\{\overline {Eff(X)}\} = {Big(X)}$ and
$Nef(X) = \overline {Amp(X)} $; for more details see \cite{L1}.
Note that $Amp(X) \subset Nef(X) \cap Big(X)$ and that there are no inclusions between $Nef(X)$ and $Big(X)$.
Note also that if $ \pi : X' \ra X$ is a birational morphism and $D$ is a Cartier divisor on $X$ then $D$ is big (resp. pseudo-effective) if and only if $\pi^* D$ is big (resp. pseudo-effective).
\medskip
We consider projective varieties with singularities of special type, as in the Minimal Model Program. For reader convenience we recall their definition
(see \cite{KollarMori}, Definition 2.11 and Definition 2.12).
\begin{Definition}\label{sing}
Let $X$ be a normal projective variety. We say that $X$ has {\sl canonical }(respectively {\sl terminal}) singularities if
\begin{itemize}
\item[i)] $K_X$ is $\Q$-Cartier, and
\item[ii)] $\nu_*\cO _{\tilde X}(m K_{\tilde X})= \cO _{X}(m K_{X})$ for one (or for any) resolution of the singularities $\nu: \tilde X \to X$
\end{itemize}
(respectively
\begin{itemize}
\item[ii)] $\nu_*\cO _{\tilde X}(m K_{\tilde X} - E)= \cO _{X}(m K_{X})$ for one (or for any) resolution of the singularities $\nu: \tilde X \to X$, where $E \subset \tilde X$ is the reduced exceptional divisor).
\end{itemize}
\end{Definition}
\smallskip
In the category of projective varieties with canonical singularities the pseudo-effectivity of the canonical bundle is a birational invariant, as noticed by Mori in \cite{M}, (11.4.1). He actually conjectured the following beautiful result
(\cite{M}, (11.4.2) and (11.5)), which was proved in \cite{BDPP}, Corollary~0.3 and in \cite{BCHM}, Corollary~1.3.3.
\begin{Theorem} \label{canpseudo}
Let $X$ be a projective variety with at most canonical singularities. Then $X$ is uniruled if and only if $K_X$ is not pseudoeffective.
\end{Theorem}
\smallskip
As for the invariance of the global sections of adjoint bundles (or of pluri-canonical bundles if $L$ is trivial) we have the following.
\begin{Lemma} \label{canonical}
Let $\pi : Y \ra X$ be a birational morphism between projective varieties with at most canonical singularities,
let $L$ be a Cartier divisor on $X$ and let $a, b \in \N$. Then
$$H^0(X, aK_X + b L) = H^0(Y, aK_Y + b \pi^*(L)).$$
\end{Lemma}
\proof Since $Y$ and $X$ have canonical singularities we have $\pi_* a K_Y = a K_X$.
This is straightforward from the definition of canonical singularities and by taking
a resolution of $Y$, $\nu: Y' \ra Y$, and $\pi \circ \nu: Y' \ra X$ as a resolution of $X$.
Since $L$ is Cartier, by projection formula it follows
$$\pi_* (a K_Y + b \pi^*(L))= \pi_* (a K_Y + \pi^*(b L)) = \pi_*(a K_Y) + b L = a K_X + b L;$$
by taking global sections we obtain our statement.
\qed
\section{Termination of Adjunction}
Much of this section is based on the following Lemma, which was proved in the analytic setting by Siu (see \cite{S}, Proposition 1).
For reader convenience we provide an algebraic proof relying on \cite{L2} (see also \cite{N}, Chapter V, Corollary 1.4).
\begin{Lemma}\label{siu}
Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be a very ample divisor on $X$.
If $G := (n+1)H+K_X$, then for every pseudo-effective divisor $F$ on $X$ we have $H^0(X, F+G) \ne 0$.
\end{Lemma}
\proof Since $F$ is pseudo-effective we have that $F+H$ is big, hence there exists a positive integer $m >0$
such that $m(F+H) \sim A+E$ with $A$ ample and $E$ effective (see for instance \cite{L1}, Corollary 2.2.7).
Let $D := \frac{1}{m}E$ and $L := F+H$, so that $L-D = \frac{1}{m}A$ is big and nef; apply
\cite{L2}, Proposition~9.4.23, to get $H^0(X, K_X+L+kH + \mathcal{I}(D)) \ne 0$.
Since the multiplier ideal $\mathcal{I}(D)$ is an ideal of $\mathcal{O}_X$,
it follows that $H^0(X, K_X+L+kH) \ne 0$ for every $k \ge n$, i.e. $H^0(X, K_X+F+(k+1)H) \ne 0$ as soon as $k+1 \ge n+1$.
\qed
\medskip
The following characterization of pseudo-effective divisors is probably well-known to the specialists;
however, we did not find it explicitly in the literature.
\begin{Theorem}\label{pseudo-effective}
Let $X$ be a smooth projective variety and let $F$ be a divisor on $X$.
The following statements, where $m$ and $N$ denote natural numbers, are equivalent:
\begin{itemize}
\item[i)] $F\in \overline {Eff(X)}$ (i.e it is pseudo-effective).
\item[ii)] There is a big divisor $G$ such that
$H^0(X, N(mF+G)) \ne 0$ for every $m > 0$ and for some $N >0$.
\item[iii)] There is a big divisor $G$ such that $H^0(X, mF+G) \ne 0$ for all $m > 0$.
\item[iv)] There is a very ample divisor $G$ such that $H^0(X, mF+G) \ne 0$ for all $m > 0$.
\item[v)] For every big divisor $H$ we have $H^0(X, mF+kH) \ne 0$ for all $m > 0$ and all $k \geq k_0(H)$.
\end{itemize}
\end{Theorem}
\proof
First of all note that the implications v) $\Longrightarrow$ iv), iv) $\Longrightarrow$ iii) and iii) $\Longrightarrow$ ii)
are obvious. Moreover ii) $\Longrightarrow$ i) follows from $F \equiv \lim_{m \to + \infty} \frac{mF+G}{m}$.
The difficult part is to prove i) $\Longrightarrow$ v); for this
we use Lemma \ref{siu} together with Kodaira's Lemma (see for instance \cite{L1}, Proposition 2.2.6).
Namely, let $G$ be the divisor of Lemma \ref{siu}; then $H^0(X, G) \ne 0$ (just take $F = \cO_X$).
If $H$ is a big divisor on $X$, then by Kodaira's Lemma $H^0(X, kH -G) \ne 0$ for every $k \geq k_0(H)$.
Hence
\begin{eqnarray*}
\dim H^0(X, mF+kH) &=& \dim H^0(X, mF+k_0H - G + G + (k-k_0)H)\geq \\
&\geq& \dim H^0(X, mF + (k-k_0)H +G) > 0,
\end{eqnarray*}
where the last inequality follows from Lemma \ref{siu} by taking as a pseudo-effective divisor $mF + (k-k_0)H$.
\qed
\begin{Remark}
Note that i) $\Longrightarrow$ iii) is just Lemma \ref{siu}, while
i) $\Longrightarrow$ ii) follows easily from $int\{\overline {Eff(X)}\} = {Big(X)}$; this last fact was first noticed by Mori in \cite{M}, (11.3) on p. 318.
Indeed, let $G \in Big(X)$ and $F \in \overline {Eff(X)}$; then the set $[G, F) := \{G +m F : m\in \R^+\}$ is contained in $int\{\overline {Eff(X)}\} =Big(X)$.
\end{Remark}
\bigskip
The next Theorem proves the equivalence of the different definitions of {\sl Termination of Adjunction} stated in the Introduction.
\begin{Theorem}\label{termination}
Let $X$ be a projective variety with at most canonical singularities.
The following statements, where $m$ and $m_0$ denote natural numbers, are equivalent:
(i) $X$ is uniruled (i.e. $K_X$ is not pseudo-effective).
(ii) For every big Cartier divisor $H$ there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline {Eff(X)}$ for $m \geq m_0$.
(iii) For every big Cartier divisor $H$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.
(iv) For every very ample Cartier divisor $H$ we have $H^0(X, m_0K_X+H)=0$ for some $m_0=m_0(H)>0$.
(v) For some big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.
\end{Theorem}
\proof (i) $\Longrightarrow$ (ii) is implied by the properties of the cone described in Section \ref{notation}; indeed, it follows
by contradiction from $K_X \equiv \lim_{m \to + \infty} \frac{mK_X+H}{m}$.
(ii) $\Longrightarrow$ (iii), (iii) $\Longrightarrow$ (iv) and (iv) $\Longrightarrow$ (v) are straightforward.
(v) $\Longrightarrow$ (i) requires a resolution of the singularities $\nu: \tilde X \to X$. Assume by contradiction that $X$ is not uniruled.
Therefore also $\tilde X$ is not uniruled and $K_{\tilde X}$ is pseudo-effective. If $H$ is any big Cartier divisor on $X$, then
${\tilde H}= \nu^*(H)$ is big and by \cite{L1}, Corollary 2.2.7, we have $l{\tilde H}=A+N$ with $A$ ample and $N$ effective for some $l>0$.
It follows that $hl{\tilde H}=hA+hN$ with $hA$ very ample for some $h>0$.
Hence, by Lemma \ref{canonical}, for every $m_0 >0$ we have
$\dim H^0(X, m_0K_X+(n+1)hlH) = \dim H^0(\tilde X, m_0K_{\tilde X}+(n+1)hl{\tilde H})= \dim H^0(\tilde X, (m_0-1)K_{\tilde X}+(K_{\tilde X}+(n+1)hA)+(n+1)hN) \ge
\dim H^0(\tilde X, (m_0-1)K_{\tilde X}+(K_{\tilde X}+(n+1)hA))$.
Lemma \ref{siu} says that this last term is positive, thus contradicting our assumption.
\qed
\begin{Remark}\label{history}
Note that Mori in \cite{M}, (11.4) on p. 318, suggests that in principle (i) could have been
stronger then (iv): \emph{We say that $X$ is $\kappa$-uniruled if $K_X$ is not
pseudo-effective. We note that $\kappa$-uniruledness is slightly stronger than saying that adjunction terminates, i.e.
$H^0(X, mK_X+H)=0$ for each very ample divisor $H$ and some $m=m(H)>0$}.
\end{Remark}
The following two corollaries show that the two formulations, respectively {\sl for some} and {\sl for every}, of (A) and (D) in the Introduction are equivalent.
\begin{Corollary}\label{iff1}
Let $X$ be a projective variety with at most canonical singularities.
The following statements, where $m$ and $m_0$ denote natural numbers, are equivalent:
(i) For every big Cartier divisor $H$ there exists $m_0=m_0(H)>0$ such that $mK_X+H \notin \overline {Eff(X)}$ for $m \geq m_0$.
(ii) For some big Cartier divisor $H_0$ there exists $m_0=m_0(H_0)>0$ such that $mK_X+H_0 \notin \overline {Eff(X)}$ for $m \geq m_0$.
\end{Corollary}
\proof It is obvious that (i) implies (ii). Conversely, if (ii) holds then $K_X$ is not pseudoeffective, hence $X$ is uniruled.
It follows from Theorem \ref{termination} that (i) holds.
\qed
\begin{Corollary}\label{iff2}
Let $X$ be a projective variety with at most canonical singularities.
The following statements, where $m$ and $m_0$ denote natural numbers, are equivalent:
(i) For some big Cartier divisor $H_0$ we have $H^0(X, m_0K_X+kH_0)=0$ for every $k>0$ and some $m_0=m_0(k)>0$.
(ii) For every big Cartier divisor $H$ we have $H^0(X, m_0K_X+kH)=0$ for every $k>0$ and some $m_0=m_0(k,H)>0$.
\end{Corollary}
\proof It is obvious that (ii) implies (i). Conversely, if (i) holds then by Theorem \ref{termination} $X$ is uniruled,
i.e. $K_X$ is not pseudoeffective. Assume by contradiction that there exist a big divisor $H$ and some $k_0>0$ such that
$H^0(X, mK_X+k_0H)\ne 0$ for every $m >0$. Then $K_X = \lim_{m \to + \infty} \frac{mK_X+k_0H}{m}$ is pseudo-effective, a contradiction.
\qed
\medskip
As pointed out by the referee, since every divisor is a difference of very ample ones, (C) is actually equivalent to
the following stronger condition.
\begin{itemize}
\item[(C*)] For every Cartier divisor $D$ we have $H^0(X, m_0K_X+D)=0$ for some $m_0=m_0(D)>0$.
\end{itemize}
\bigskip
The following is a more general definition of {\sl Termination of Adjunction}.
\begin{Definition} \label{classicaltermination} (Condition (C'))
Let $X$ be a normal projective variety; let $H$ be an effective Cartier divisor on $X$.
We say that {\sl Adjunction Terminates in the classical sense} for $H$ if
there exists an integer $m_0 \ge 1$ such that
$$H^0(X, H+mK_X)=0$$
for every integer $m \ge m_0$.
\end{Definition}
We conjecture that such a definition is actually equivalent to the previous ones. The following partial result in this direction is straightforward.
\begin{Proposition}\label{veryclassical}
Let $X$ be a projective variety with canonical singularities.
Let $H$ be any effective divisor and assume that Adjunction Terminates in the classical sense for $H$. Then $X$ has negative Kodaira dimension.
\end{Proposition}
\proof Recall that the Kodaira dimension of a singular variety is defined to be the Kodaira dimension of any smooth model (see for instance
\cite{L1}, Example 2.1.5). Assume by contradiction that $X$ has non-negative Kodaira dimension, i.e. $H^0(\tilde{X}, n_0 K_{\tilde{X}}) \ne 0$
for some integer $n_0 \ge 1$, where $\nu: \tilde{X} \to X$ is any resolution of the singularities. Since $X$ has canonical singularities,
from Lemma \ref{canonical} it follows that $H^0(X, n_0 K_X) = H^0(\tilde{X}, n_0 K_{\tilde{X}}) \ne 0$. Hence $H^0(X, H+ n n_0 K_X) \ne 0$
for every integer $n \ge 1$, contradicting the assumption that $H^0(X, H+mK_X)=0$ for $m >> 0$.
\qed
\smallskip
Together with the standard conjecture that negative Kodaira dimension implies uniruledness (see for instance \cite{M}, (11.5) on p. 319, and \cite{BDPP}, Conjecture 0.1),
from Proposition \ref{veryclassical} it would follow that Termination of Adjunction in the classical sense implies uniruledness.
In dimension two such an implication holds unconditionally, as it was proved by Castelnuovo and Enriques in \cite{CE} (for a modern proof we refer to \cite{Mu}).
\bigskip
We conclude this section with a characterization of uniruled varieties which
may suggest a different way to consider (effective) termination of adjunction. It follows as a straightforward consequence
of Lemma \ref{siu} and the main result in \cite{BDPP}.
\begin{Proposition}
Let $X$ be a smooth projective variety of dimension $n$ and let $H$ be a very ample divisor on $X$.
If $H^0(X, mK_X+(n+1)H)=0$ for some natural number $m \ge 1$, then $X$ is uniruled.
\end{Proposition}
\proof
Assume by contradiction that $X$ is not uniruled, so that $K_X$ is pseudo-effective by \cite{BDPP}. Lemma \ref{siu} with $F = (m-1)K_X$
gives the sought-for contradiction.
\qed
\smallskip
Theorem 3.1 in \cite{Di} gives a statement similar to the last proposition; there the variety is singular and $H$ is just nef and big. However
$m >1$ and $H$ has to be multiplied by a higher number, for instance $n^2$.
\section{Quasi polarized pairs}
A {\it quasi polarized pair} is a pair $(X,L)$ where $X$ is a projective variety
with at most $\mathbb{Q}$-factorial terminal singularities and
$L$ is a nef and big Cartier divisor on $X$.
If $L$ is ample we call the pair $(X,L)$ a {\it polarized pair}.
\smallskip
In \cite{A}, Section 4, following T. Fujita's ideas as revisited by A. H\"oring in \cite{H} and using
the MMP developed in \cite{BCHM}, we described a MMP with scaling related to divisors of type $K_X + rL$,
for $r$ a positive rational number.
In particular we introduced the {\bf $0$-reduction} of a quasi polarized pair $(X,L)$
(see \cite{A}, Definition 4.4) as quasi polarized pair $(X',L')$ birational to $(X, L)$
obtained from $(X,L)$ via a Minimal Model Program with scaling:
\centerline{$(X,L) \sim (X, \Delta) :=(X_0, \Delta_0)
\ra ----\ra (X_s, \Delta_s) \sim (X',L'),$}
which contracts or flips all extremal rays $\R^+[C]$ on $X$ such that $L\cdot C =0$.
At every step of the MMP given above, we have a quasi polarized variety $(X_i, L_i)$ with at most terminal $\Q$-factorial singularities.
If $\pi _i:(X_{i}, \Delta_i) \ra (X_{i+1}, \Delta_{i+1})$ is birational then $L_i = \pi_i^*(L_{i+1})$,
while if $\pi _i:(X_i, \Delta_i) \ra (X_{i+1}, \Delta_{i+1})$ is a flip then $L_{i}$ and $ \pi_i^*(L_{i+1})$
are isomorphic in codimension one.
\begin{Remark}
\label{reduction}
By using Lemma \ref{canonical} and Hartogs theorem we deduce
$$H^0(X, aK_{X} +bL) = H^0(X', aK_{X'} +bL')$$ for $a, b\in\N.$
\end{Remark}
\smallskip
The following has been proved in \cite{A}, Theorem 5.1 and in \cite{H1}, Proposition 1.3.
\begin{Theorem}\label{adjunction}
Let $(X,L)$ be a quasi polarized pair.
Then $K_X + t L$ is pseudo-effective for all $t \geq n$ unless the $0$-reduction $(X',L')$ is $(\proj^n, \cO (1))$.
Actually, $K_X + (n-1)L$ is pseudo-effective unless $(X',L')$ is one of the following pairs:
\begin{itemize}
\item $(\proj ^n, \cO(1))$,
\item $(Q, \cO(1)_{|Q})$, where $Q\subset \proj ^{n+1}$ is a quadric,
\item $C_n(\proj^2, \cO(2))$, a generalized cone over $(\proj^2, \cO(2))$,
\item $X$ has the structure of a $\proj^{n-1}$-bundle over a smooth curve $C$ and $L$ restricted to any fiber is $\cO(1)$.
\end{itemize}
Moreover, except in the above cases, $K_{X'} + (n-1)L'$ is nef.
\end{Theorem}
The {\bf first-reduction} of a quasi polarized pair $(X,L)$ (see \cite{A}, Definition 5.5) is a quasi polarized pair $(X'',L'')$ birational to $(X, L)$ obtained from
a $0$-reduction $(X',L')$ via a morphism $\rho: X' \ra X''$ consisting of a series of divisorial contractions to smooth points,
which are weighted blow-ups of weights $(1,1,b, \ldots, b)$ with $b \geq 1$ (see \cite{AT}, Theorem 1.1).
\begin{Remark}
\label{reduction1}
According to \cite{A}, Proposition 5.4, we have
$$H^0(K_{X} +tL) = H^0(K_{X''} +tL'')$$ for any $0\leq t \leq n-2$.
\end{Remark}
\smallskip
The following has been proved in \cite{A}, Theorem 5.7.
\begin{Theorem} \label{adjunction1}
Let $(X,L)$
be a quasi polarized pair.
$K_{X} + (n-2)L$ is not pseudo-effective if and only if any first-reduction $(X'', L'')$ is either
one of the pairs listed in the statement of Theorem \ref{adjunction} or one of the following pairs:
\begin{itemize}
\item a del Pezzo variety, that is $-K_{X''} \sim_\mathbb{Q}(n-1)L$ with $L$ ample,
\item $(\proj ^4, \cO(2))$,
\item $(\proj ^3, \cO(3))$,
\item $(Q, \cO(2)_{|Q})$, where $Q\subset \proj ^{4}$ is a quadric,
\item $X$ has the structure of a quadric fibration over a smooth curve $C$ and $L$ restricted to any fiber is $\cO(1)_{|Q}$,
\item $X$ has the structure of a $\proj^{n-2}$-bundle over a normal surface $S$ and $L$ restricted to any fiber is $\cO(1)$,
\item $n=3$, $X$ is fibered over a smooth curve $Z$ with general fiber $\proj^{2}$ and $L$ restricted to it
is $\cO(2)$.
\end{itemize}
If $K_{X} + (n-2)L$ is pseudo-effective then on any first-reduction $(X'', L'')$ the divisor
$K_{X''} + (n-2)L''$ is nef.
\end{Theorem}
\bigskip
The following definition was given by H\"oring (see (\cite{H}, Definition~1.2).
\begin{Definition}\label{scrolldef}
A quasi polarized pair $(X,L)$ is a (generalized) scroll if $X$ is smooth and there is
a fibration $X \ra Y$ onto a projective manifold $Y$ such
that the general fiber $F$ admits a birational morphism
$\tau : F \ra \mathbb{P}^m$ and that $\mathcal{O}_F(L) = \tau^*\mathcal{O}_{\mathbb{P}^m}(1)$.
A quasi polarized pair $(X,L)$ is birationally a scroll if there is a birational morphism $\nu: X' \ra X$ such that $(X', \nu^* L)$ is a (generalized) scroll.
\end{Definition}
The next is Theorem~1.4 in \cite{H}.
\begin{Theorem}
\label{hoering0}
Let $(X,L)$ be a quasi polarized pair. If $(X,L)$ is not birationally a scroll then $\Omega_X \otimes L$ is generically nef.
\end{Theorem}
\bigskip
A key step in the proofs of Theorem \ref{bs} and of Theorem \ref{fourfold} is the following lemma
due to H\"oring (see \cite{H}, p. 741, Step~2 in the proof of Theorem 1.2).
\begin{Lemma}
\label{hoering}
Let $(X,L)$ be a quasi polarized pair.
Assume that $K_X+(n-2)L$ is pseudo-effective and that $K_X+(n-1)L$ is nef and big.
Then
$$L^{n-2} [(2(K_X^2 +c_2 (X))+6nL K_X +(n+1)(3n-2)L^2] > 0.$$
\end{Lemma}
\bigskip
We consider now a quasi polarized pair $(X,L)$ and we assume moreover that $X$ is smooth. We borrow from Y. Fukuma the following set-up for the computation of the Hilbert polynomial of $K_X+tL$.
Let
\begin{eqnarray*}
F_{0}(t)&:=&\dim H^0(X, K_X +tL),\\
F_{i}(t)&:=&F_{i-1}(t+1) - F_{i -1}(t) \hbox{ for every integer } i \hbox{ with } 1\leq i\leq n.
\end{eqnarray*}
\medskip
The following statement can be easily checked by reverse induction on $b \le a$.
\begin{Lemma}\label{induction}
Fix an integer $a \ge 1$.
If $F_{0}(t)=0$ for every integer $t$ with $1 \le t \le a$,
then $F_{a-b}(c)=0$ for all integers $b,c$ with $1 \le c \le b \le a$.
\end{Lemma}
\medskip
If one defines
$$A_i(X,L) := F_{n-i}(1)$$
then it follows easily that
\begin{align}
\label{fukform}
\dim H^0(X, K_X+tL) = \sum_{j=0}^n \binom{t-1}{n-j} A_j(X,L).
\end{align}
Moreover, by taking $a = n-i+1$ and $b=c=1$ in Lemma \ref{induction}, we obtain
the following implication.
\begin{Corollary}\label{A2}
If $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le n-i+1$, then $A_i(X,L)=0$.
\end{Corollary}
On the other hand, by Kawamata-Viehweg vanishing theorem and Serre duality,
we have $\dim H^0(X, K_X+tL) = \chi(X,-tL)$; therefore from the Riemann-Roch theorem we obtain
the following explicit computations (for further details, see \cite{F0}, (2.2),
and \cite{F1}, Proposition~3.2).
\begin{Lemma}\label{fukuma}
Let $(X,L)$ be a polarized manifold of dimension $n$ and let $g(X,L)$ denote the sectional genus of $(X,L)$.
Then we have
\begin{eqnarray*}
A_0(X,L) &=& L^n \\
A_1(X,L) &=& g(X,L)+L^n-1 \\
24\cdot A_2(X,L) &=&L^{n-2} [(2(K_X^2+c_2(X))+6nL K_X +(n+1)(3n-2)L^2] \\
48\cdot A_3(X,L) &=& (n-2)(n^2-1)L^n+ n(3n-5)K_X L^{n-1} + \\
& & + 2(n-1) K_X^2 L^{n-2} + 2c_2(X)(K_X+(n-1)L) L^{n-3}.
\end{eqnarray*}
\end{Lemma}
\section{Polarized Abundance}
\label{pabundance}
The aim of this section is to argue around the Conjectures stated in the introduction.
We start showing that Proposition \ref{first} is a direct consequence of (the more general) Theorem D in \cite{BCHM}.
\begin{Remark} \label{cascini}
Let $(X,L)$ be a quasi-polarized variety
and let $t$ be a positive rational number.
Then there exists an effective $\Q$-divisor $\Delta^t$ on $X$ such that
$\Delta^t \sim_{\Q} t L $ \ and
$(X, \Delta^t)$ is Kawamata log terminal.
This is well-known to the specialists, a proof can be found in \cite{A}.
If $K_X+tL\in \overline {Eff(X)}$, then $K_X+\Delta^t \in \overline {Eff(X)}$ and
by \cite{BCHM}, Theorem D, there exists an $\R$-divisor $D \geq 0$
such that $K_X+\Delta^t \sim_{\R} D$. That is, there exists $N \in \mathbb{N}$
such that $H^0(X, N(K_X+tL))>0$.
\end{Remark}
\medskip
We consider Conjecture \ref{third};
for $s=n$ we recover the following easy fact.
\begin{Proposition} \label{as}
Let $(X,L)$ be a quasi polarized pair of dimension $n$.
We have $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le n$
if and only if $K_X+ nL$ is not pseudo-effective. Moreover this is the case if and only if
the $0$-reduction $(X',L')$ of the pair $(X,L)$ is $(\mathbb{P}^n, \mathcal{O}(1))$.
\end{Proposition}
\proof By Remark \ref{reduction} we have $H^0(X, K_{X} +tL) = H^0(X', K_{X'} +tL')$
for any $t\geq 0$. Hence if $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le n$
then from Corollary \ref{A2} it follows that $A_1(X',L')=g(X',L')+L'^n-1=0$. Since we
have $g(X',L')=0$ and $L'^n=1$ if and only if $(X',L') = (\mathbb{P}^n, \mathcal{O}(1))$,
the claim follows from \cite{A}, Theorem~5.1~(2).
\qed
\medskip
Next, for $s=n-1$, the following is a slightly more explicit version of \cite{H}, Theorem~1.2;
the proof is essentially the one of \cite{H}.
\begin{Theorem}\label{bs}
Let $(X,L)$ be a quasi polarized pair of dimension $n$.
We have $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le n-1$
if and only if $K_X+(n-1)L$ is not pseudo-effective.
That is, by Theorem \ref{adjunction}, if and only if
the $0$-reduction $(X',L')$ of the pair $(X,L)$ is one of the following:
(i) $(\proj ^n, \cO(1))$,
(ii) $(Q, \cO(1)_{|Q})$, where $Q\subset \proj ^{n+1}$ is a quadric,
(iii) $C_n(\proj^2, \cO(2))$, a generalized cone over $(\proj^2, \cO(2))$,
(iv) $X$ has the structure of a $\proj^{n-1}$-bundle over a smooth curve $C$
and $L$ restricted to any fiber $F$ is $\cO(1)$.
\end{Theorem}
\proof
Let $(X',L')$ be the $0$-reduction of the pair $(X,L)$ and let
$(\tilde X',\tilde L')$ be its desingularization
(namely, $\nu : \tilde X' \ra X'$ and $\tilde L'= \nu ^*(L'))$.
By Remark \ref{reduction} and Lemma \ref{canonical} we have
$$H^0(X, K_{X} +tL) = H^0(X', K_{X'} +tL')= H^0(\tilde X', K_{\tilde X'} +t\tilde L')$$
for any $t\geq 0$.
\medskip
The \emph{if} part is obvious. In order to prove the \emph{only if} part, assume that
$H^0(X, K_{X} +tL) = H^0(X', K_{X'}$ $+tL')= H^0(\tilde X', K_{\tilde X'} +t\tilde L') =0$
for every integer $t$ with $1 \le t \le n-1$. Corollary \ref{A2} implies that
\begin{equation}\label{vanish}
A_2({\tilde X'}, \tilde L')=0.
\end{equation}
Assume by contradiction that $(X',L')$ is not one of the pairs
in (i), (ii), (iii), (iv); then, by Theorem \ref{adjunction},
$K_{X'}+(n-1)L'$ is nef.
The required contradiction is provided by \cite{H}, Theorem 1.2.
\qed
\smallskip
The next step $s=n-2$ should work as follows.
\begin{Conjecture}
Let $(X,L)$ be a quasi polarized manifold of dimension $n$.
We have $H^0(X, K_X+tL)=0$ for every integer $t$ with $1 \le t \le n-2$
if and only if $K_X+(n-2)L$ is not pseudo-effective, that is if and only if
the first-reduction $(X'',L'')$ is one of the pairs $(X,L)$ listed in Theorems \ref{adjunction}
and \ref{adjunction1}.
\end{Conjecture}
Once again, the \emph{if} part is obvious.
Conversely, from Corollary \ref{A2} it follows that $A_3(X,L)=0$, but the proof of the \emph{only if} part
seems to be elusive.
\bigskip
From now on, we focus on the case $n=4$; here formula (\ref{fukform}) reads simply as:
\begin{eqnarray}\label{fukform2}
H^0(X, K_X+tL) =
\binom{t-1}{4} A_0(X,L) + \binom{t-1}{3} A_1(X,L) + \\
+\binom{t-1}{2} A_2(X,L) +\binom{t-1}{1} A_3(X,L) + \binom{t-1}{0} A_4(X,L) \nonumber
\end{eqnarray}
where
\begin{eqnarray*}
A_1(X,L) &=& g(X,L) + L^4 - 1, \\
A_2(X,L) &=& \dim H^0(X, K_X+3L)-2\dim H^0(X, K_X+2L)+ \\
& & + \dim H^0(X, K_X+L),\\
A_3(X,L) &=& \dim H^0(X, K_X+2L)-\dim H^0(X, K_X+L), \\
A_4(X,L) &=& \dim H^0(X, K_X+L).
\end{eqnarray*}
We prove the following generalization of \cite{F2}, Theorem 3.1.
\begin{Theorem}\label{fourfold}
Let $(X,L)$ be a polarized manifold of dimension $4$ and let $t$ be an integer with $t \ge 3$.
If $K_X+tL$ is pseudo-effective, then $H^0(X, K_X+tL) \ne 0$. In particular,
\begin{itemize}
\item $H^0(X, K_X+tL) \ne 0$ for $t\geq 5$
\item $H^0(X, K_X+4L)=0$ if and only if $(X,L)$ is $ (\mathbb{P}^4, \mathcal{O}(1))$
\item $H^0(X, K_X+3L)=0$ if and only if $(X,L)$ is either $(Q, \cO(1)_{|Q})$, where
$Q\subset \proj ^{5}$ is a quadric, or $X$ has the structure of a $\proj^{3}$-bundle
over a smooth curve $C$ and $L$ restricted to any fiber is $\cO(1)$.
\end{itemize}
\end{Theorem}
\proof
Since $L$ is ample $(X,L)$ is a 0-reduction, in particular by Theorem \ref{adjunction}
we can assume that $K_X+tL$ is nef for $t \geq 4$.
We can also assume that $K_X+3L$ is nef. Indeed, if not then $(X,L)$ is one of the exceptions listed
in the statement of Theorem \ref{adjunction}.
If $(X,L)$ is $(\mathbb{P}^4, \mathcal{O}(1))$ or $(Q, \mathcal{O}(1))$,
where $Q \subset \mathbb{P}^5$ is a quadric hypersurface, then Theorem \ref{fourfold} is obvious.
The case of a generalized cone over $(\mathbb{P}^2, \mathcal{O}(2))$ does not occur
since $X$ is smooth, while the case of a $\mathbb{P}^3$-bundle over a smooth curve
will be considered in Proposition~\ref{scroll}.
\smallskip
Now, assume that $\Omega_X \otimes L$ is generically nef. By using the formulas in Lemma~\ref{fukuma}
and Miyaoka inequality as stated in \cite{H}, Corollary~2.11, with $D:=4L$, we compute:
\begin{eqnarray*}
A_2(X,L) &\ge& \frac{1}{24}\left( 2(K_X+3L)^2 L^2 + 6 (K_X+3L) L^3 + 2L^4 \right) \\
A_3(X,L) &\ge& - \frac{1}{24} (K_X+3L) L^3.
\end{eqnarray*}
Hence from (\ref{fukform2}) and the nefness of $K_X+3L$ it follows that
$$
\dim H^0(X, K_X+tL) \ge (t-1) A_3(X,L) + \frac{(t-1)(t-2)}{2} A_2(X,L) > 0
$$
for every $t \ge 3$.
\smallskip
Finally, assume that $\Omega_X \otimes L$ is not generically nef. By Theorem~\ref{hoering0}
and Lemma \ref{canonical} we may assume that $X$ is a (generalized) scroll and the
claim is a consequence of the following proposition.
\qed
\begin{Proposition}\label{scroll}
Let $(X,L)$ be a generalized scroll of dimension $4$ and let $t$ be an integer such that $t \ge 3$.
If $K_X+tL$ is nef, then $H^0(X, K_X+tL) \ne 0$.
\end{Proposition}
\proof
Let $X \ra Y$ be the scroll fibration and let $F$ be the generic fiber with
a birational morphism $\tau : F \ra \mathbb{P}^m$ as in Definition \ref{scrolldef}.
\smallskip
If $X = \mathbb{P}^4$ the claim is obvious; therefore we can assume that $m \leq 3$ and that
$A_1(X,L) = g(X,L) + L^4 - 1 > 0 $ (since we have $g(X,L)=0$ and $L^4=1$ if and only if
$(X,L) = (\mathbb{P}^4, \mathcal{O}(1))$).
We also have that $A_0(X,L) = L^4 \geq 1$ and $A_4(X,L) = \dim H^0(X, K_X+L) \geq 0$.
\smallskip
If $m=3$, then $K_X+sL \vert_F = \tau^*\mathcal{O}_{\mathbb{P}^3}(-4+s)$, hence $H^0(X,K_X+sL)=0$
for $s \le 3$.
Thus we have $A_2(X,L)=A_3(X,L)=0$ and from (\ref{fukform2}) it follows that for $t \ge 4$
we have
$$
\dim H^0(X, K_X+tL) \ge A_1(X,L) > 0
$$
\smallskip
If $m = 2$, then $K_X+sL \vert_F = \tau^*\mathcal{O}_{\mathbb{P}^2}(-3+s)$, hence $H^0(X,K_X+sL)=0$
for $s \le 2$. In particular, we have $A_3(X,L)=0$ and $A_2(X,L)=\dim H^0(X, K_X+3L)$.
For $t=3$, i.e. if we assume $K_X+3L$ is nef, by Theorem~1.2 in \cite{H}
we must have $H^0(X,K_X+3L) \ne 0$ since $H^0(X,K_X+sL)=0$
for $s \le 2$.
For $t \ge 4$ we deduce from (\ref{fukform2}) that
$$
\dim H^0(X, K_X+tL) \ge A_1(X,L) > 0.
$$
\smallskip
If $m=1$, then $K_X+sL \vert_F = \tau^*\mathcal{O}_{\mathbb{P}^1}(-2+s)$, hence $H^0(X,K_X+L)=0$.
In particular, we have $A_3(X,L) \ge 0$.
If $H^0(X,K_X+2L)=0$, then $A_2(X,L)=\dim H^0(X, K_X+3L)$ and we
conclude exactly as in the previous case $m = 2$.
If $H^0(X,K_X+2L) \ne 0$, then
$K_X+2L$ is pseudo-effective and $K_X+3L$ is pseudo-effective and big.
Passing to the $0$-reduction we may assume that $K_X+3L$ is nef and big. Therefore Lemma~\ref{hoering} applies and by Lemma~\ref{fukuma} we get $A_2(X,L)>0$.
Hence from
(\ref{fukform2}) it follows that for $t \ge 3$ we have
$$
\dim H^0(X, K_X+tL) \ge A_2(X,L) > 0.
$$
\qed
\bigskip
The statement of Theorem \ref{fourfold} should hold also for $t=2$, but we have only the following partial result.
\begin{Proposition} \label{fourfold2}
Let $(X,L)$ be a polarized manifold of dimension $4$.
If $K_X+2L$ is pseudo-effective, then $H^0(X, K_X+2L) \ne 0$
unless $\Omega_X<\frac{1}{2}L>$ is not generically nef.
\end{Proposition}
\proof
By Theorem \ref{adjunction1} and Remark \ref{reduction1} we may assume that $K_X+2L$ is nef.
Assume that $\Omega_X<\frac{1}{2}L>$ is generically nef. By using the formula for $A_3(X,L)$ in Lemma~\ref{fukuma}
and Miyaoka inequality, as stated in \cite{H}, Corollary~2.11, with $D:=2L$, we compute:
$$
A_3(X,L) \ge \frac{1}{16} (K_X+2L)^2L^2 + \frac{1}{12}(K_X+2L)L^3 + \frac{1}{48} L^4.
$$
Hence from (\ref{fukform2}) it follows that
$$
\dim H^0(X, K_X+tL) \ge A_3(X,L) > 0.
$$
\qed
| 194,184
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GM will do a mid-cycle refresh currently in the following year, one of their cars that will have redesign is Chevy Equinox. For the 2020 Chevy Equinox edition, we will see an entire combination of adjustments, beginning with some deal with the designing. Besides uniqueness in regards to the outside layout, we will likewise see a good variety of modifications on the within.
Chevy Equinox New Exterior Layout For 2020 Version
These mid-cycle rejuvenates are mostly regarding aesthetical renovations as well as the 2020 Chevy Equinox should not be an exemption. This ought to be a traditional renovation, so rely on great deals of alterations to the outside layout. In regards to base style features, however, the following year will not bring more crucial adjustments. This tiny crossover will remain to ride on an acquainted D2 system, sharing technicians with the GMC Surface nearly totally. Beyond, we depend on a great deal of work with sheet steel.
The present version complies with the brand name’s newest layout language. Nonetheless, it looks a little traditional, so Chevy can spruce up points for a little bit. Naturally, the emphasis will get on the front end. In this field, we expect to see a great deal of service the grille, bumper and also perhaps on fronts lights also. Backside possibly will not endure that a lot, however a wonderful variety of refined adjustments promises. The remainder of the car must continue to be the exact same, though we rely on brand-new wheels layouts and also a couple of much more color choices in the deal.
Interior Design of 2020 Chevy Equinox
For The Inside, we wish that Chevy designers recognize that there is a great deal of job to be finished with this upgrade. The existing version includes a cabin that teems with low-cost products, particularly difficult plastics. In regards to indoor design and also an improvement, it most definitely can not take on course leaders. For that reason, we are rather certain that the control panel will include great deals of alterations, specifically in regards to brand-new products. Certainly, relying on even more common attributes also.
Engine Options of 2020 Chevy Equinox Engines
For the engine, the 2020 Chevy Equinox will proceed with a 1.5-liter turbo inline-four, which supplies around 170 horsepower and also 200 pound-feet of torque. Optional 2.0-liter turbo-four supplies around 252 hp as well as 260 lb-ft of torque, that makes this crossover among one of the most effective designs in the course. If you are extra for something cost-effective, an excellent aspect of 2020 Chevy Equinox will be its diesel choice. Certainly, we are discussing a 1.6-liter turbodiesel that provides around 138 horsepower and also warranties 28 mpg in the city and also 39 mpg on the freeway.
See Also: The Upcoming 2020 Kia Sorento Review
Launching & Price Estimate For 2020 Chevy Equinox
A significant upgrade typically suggests an earlier arrival. For that reason, we anticipate seeing the 2020 Chevy Equinox at some point in the 3rd quarter of the following year. We are quite certain that Chevy will not make larger adjustments in regards to the rate. Base designs must be offered at around 24.000, up to 36.000 in US Dollar.
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Antioxidant Slow Progression Not Restore VisionJuly 4th, 2009
Antioxidant Slow Progression Not Restore Vision
Remember that the antioxidant treatment will not restore vision that has already been lost, but it can prevent more vision loss in the years to come. If you are someone who already has intermediate macular degeneration and is at risk for developing an advanced case, you’ll find that this is something that you need to consider. Your doctor should recommend the treatment for macular degeneration that best fits your needs. Eye vitamins are good. Make a regular appointment with your eye doctor and makes sure that you know how the disease is progressing. If you can tell that an early case of AMD is progressing into an intermediate stage, it is time to stop and really consider what this treatment can do for you.
keyword 5
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Classy Lady: Liz Ansley Photoshoot
What prompted the shoot?
I am often approached by models via Facebook because they see my work with other models. In this case, this model Nyana, contacted me. I noticed from her images on social media that she has modeled for a dress designer that I have worked with before. I like to work with people I know and have worked with because they understand how I work and how my work is different than most other photographers so I suggested we all work together. I contacted the designer, Cobilee West, and we set a date that worked for all of us.
Did something in particular inspire you?
I know that my portfolio is lacking in diverse ethnicities and that is something I have wanted to rectify.
How do you usually choose your subjects?
I belong to a group of photographers, models, MUAH’s and designers here in Minnesota. When I step outside my circle of friends and family for models, I look there.
Do you study trends, or do any research into the needs of clients?
I definitely study trends. I do this by searching new releases on all the amazon sites worldwide, visiting bookstores here in America, and reading articles on trends. I also study the Arcangel collection looking for gaps. In this case, I have noticed there are a lot of YA fantasy novels featuring woman in big flowing dresses but most of the models used in these shoots are caucasian. I do see other ethnicities on these covers but do not see this option in the Arcangel collection.
How do you choose the location, or any objects you’ll use?
I rent a studio in downtown Minneapolis during the winter months because it is just so cold here in Winter. If I do not use that studio then I rent out local venues that allow shoots for a fee. I also try to get outside for shoots in the snow when weather permits. In the summer I shoot EVERYWHERE but am particularly fond of using the many lakes we have here in Minnesota as neutral backdrops. As far as props, I scour thrift store and garage sales looking for potential props. I then have them handy in case they would be useful to telling a story in an image.
Do you usually work with professional models, or family and friends?
I do use professional models but if they are not interested in working for trade this gets expensive so whenever possible I employ friends and family that are willing to work for ice cream or wine (plus free images).
How important are costumes, and where do you usually source them?
Costumes are an important part of my shoots. I source them in MANY different ways. Whenever possible I try to work with models that own appropriate clothing for the shoot. I try to find costumes shops that are reasonably priced (not easy) or beg and borrow from friends or family. I once helped clean and organize High School costume space in exchange for use of some costumes. I also have a huge amount of costumes that I have found at thrift stores over the years. These costumes and props take up a large area of my basement (much to my husbands dismay).
What is your must-have gear for the type of shoot in question?
For this shoot, I just wanted a white background that would allow me to cut her out and place her into my images that I shoot around the world. I used extra continuous lights in addition to window light just to brighten her up.
Do you work with any assistants?
I sometimes have assistants but rarely. In this case, my assistant was the dress designer. She dressed the model and made alterations to the clothing on the spot. She also steamed any last minute wrinkles out of the the dresses once they were on the model. She shot the behind the scenes images for me as well.
What are your strategies for getting the best out of your models?
I try to make sure the shoot is fun. I’ve been told my shoots were much more fun than the models expected. I am pretty easy going and like to work collaboratively with my models so they feel we are a team.
How does your approach change for shooting landscapes, objects, and people?
Shooting landscapes for me is always done alone. I really enjoy being by myself shooting early in the morning or at sunset typically. It is more of a meditative process vs shooting models which requires me to be upbeat and probably more social than I am comfortable with. Objects are my least favorite, for whatever reason, still-life photography gives me a headache.
Do you plan and control your shoots tightly? How much do you improvise around opportunities and ideas that arise during the shoot?
I don’t plan my shoots tightly at all. I peruse book covers on the internet before I go to a shoot to have inspiration in my head but typically when I plan too much it’s a waste of time. I do something completely different once I start shooting. I LOVE to be inspired in the moment. This is the thrill I get from this job. Being creative in the moment based on light, backgrounds, props or furniture available etc gives me a thrill. I actually like to shoot in brand new locations, whenever possible, without visiting the place first because the newness is so inspiring otherwise I get bored too easily.
How important is photoshop or other photo editor apps to your work?
Very important. I use photoshop and add on programs with all of my images.
How much time do you usually spend editing your images?
It depends, composites can take up to 2 hours each depending on how elaborate they are. When I travel around the world I shoot potential backgrounds or just elements to create a scene later. For example, in this shoot I knew the goal was to cut the model out and put her into one of my other images or to simply make the background white to make life easier for a designer who might want to put her into another photographers background. If I am not changing the background, an image can be processed in 5 minutes.
Do you design your own photo actions?
I do design my own actions and recipes within add-on programs but I also am a junkie and buy many different actions, filters, and textures from other sources. They are all like toys to me and I love getting new toys. I don’t use textures very often anymore but if I do use them with still- life it is for vintage or grudge effect.
| 21,262
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Hexavalent Chromium in Cement Manufacturing: Literature Review
- by Linda Hills and Vagn C. Johansen(PCA R&D Serial No. 2983) Ferrous (II) Sulfate Monohydrate - Free flowing dry powder and……
产品详细
Ferrous (II) Sulfate Monohydrate - Free flowing dry powder and granular for Cement Industry
INTRODUCTION
The purpose of this review is to summarize information related to hexavalent chromium, Cr (VI), in the portland cement industry. The catalyst for initiating this project was the content restrictions and labeling/marketing of cements based on the level of Cr (VI) in cement in Europe, and the potential for a similar situation in North America in the near future. European Directive 2003/53/EC was implemented January 2005 and is binding in the UK and other EU member states (European Parliament 2003). As outlined in reference BCA (2006), the directive: 1) prohibits the placing on the market or use of cement or cement preparations which contain, when hydrated, more than 2 ppm (0.0002%) of soluble Cr (VI); 2) requires that where cement or cement preparations have a soluble Cr (VI) content of 2 ppm or less, when hydrated, due to the presence of a reducing agent, their packaging should be marked with information on the period of time for which the reducing agent remains effective (i.e. packing date, suggested storage conditions, and suggested storage period); and 3) permits the placing on the market and use of cement or cement preparation not meeting the two requirements above only when it is for use in totally automated and fully enclosed processes where there is no possibility of contact with the skin. In regards to possible similar restrictions in North America, at the time of this report, the Occupational Safety and Health Administration (OSHA) has exempted portland cement from its standard for occupational exposure to hexavalent chromium (OSHA 2006).
This report includes information on: 1) potential sources of chromium to the manufacturing process, 2) the process of chromium oxidation in the cement kiln, including the dependant variables, and 3) the use of additives to reduce the level of hexavalent chromium formation in hydrated cement.
CHROMIUM IN CEMENT
The “chromium” content of cement generally refers to compounds containing chromium. An important consideration is the oxidation state of chromium in these compounds. The most often discussed forms in the cement industry are Cr (III) and Cr (VI). Cr (III) because it is the major form of chromium in cement, and Cr (VI) because it has received the most attention regarding health issues. Chromium has also been detected in the form of Cr (IV) and Cr (V) (Mishulovich 1995) although during cement hydration, these forms disproportionate to Cr (III) and Cr (VI) (Johansen 1972). A description of the trivalent and hexavalent states of chromium in clinker and examples of the compounds in each are provided below:
• Trivalent chromium, also Cr (III), Cr3+. Compounds include chromic oxide, chromic sulfate, chromic chloride, and chromic potassium sulfate (APCA 1998). Compounds with Cr (III) are stable, and therefore the form found in quarried materials, and most prevalent in clinker and cement. Since these compounds are the most stable, having low solubility and low reactivity, their impact on the environment and living systems is low.
• Hexavalent chromium, also Cr (IV), Cr6+. Compounds include chromium trioxide, chromic acid, sodium chromate, sodium dichromate, potassium dichromate, ammonium dichromate, zinc chromate, calcium chromate, lead chromate, barium chromate, and strontium chromate (APCA 1998). Compounds of hexavalent chromium are strong oxidizers and unstable (Mishulovich 1995). It’s solubility in water is related to reported health risks, as described further below.
Chromium in some form is present in portland cement in generally trace amounts. The form of this chromium is important to reported health risks. The form of particular interest is Cr (VI) due to it’s solubility in water. For example, when dissolved, Cr (VI) can penetrate unprotected skin and is transformed into Cr (III), which combines with epidermal proteins to form the allergen that causes sensitivity to certain people (Chandelle 2003). This allergic problem only occurs in certain individuals who are particularly sensitive; once sensitization is induced, this condition, allergic contact dermatitis (ACD), may be triggered by very small amounts of subsequent exposure to chromate ions. This sensitivity can exacerbate the severity of chemical burns brought on by the high pH of hydrating cement.
The amount of Cr (VI) in clinker and cement can originate from: 1) oxidation of total chromium from the raw materials or fuel entering the system based on conditions of the clinker burning process, 2) magnesia-chrome kiln refractory brick, if used, 3) wear metal from total chromium and soluble hexavalent chromium found in clinker and in hydraulic cements may originate from a variety of sources, as exemplified in this section.
Raw Materials
All quarried raw materials for cement manufacture contain very small or trace quantities of total chromium, which is a common element in the earth's crust. The increasing use of many by-product raw materials such as metallurgical slag, spent catalyst fines, flue gas desulfurization gypsum, lime sludge, etc., may contribute additional amounts, however little published data was found on many of these by-product materials. Total chromium from the primary raw materials varies with the type and origin; typical values are given in Table 1. Most quarried raw materials contain no water soluble chromium as Cr (VI), and chromium is usually in oxidation state Cr (III) (ATILH 2003). Cr (VI) levels in fly ashes and electro filter dust are reported in the range of about 0.5 ppm and 0.3 ppm respectively (ATILH 2003).
Table 1. Reported Chromium Content of Raw Materials
*Bhatty references 1992 Environmental Toxicology Institute report
NA: not available (no values provided)
Fuels
Many fuel types are used in the cement industry, and the chromium content will vary accordingly. Overall, considering the fuel consumption is 10-15% of the kiln feed, the contribution to the total chromium content of the clinker is minor. Table 2 provides some typical values for fuels. Coal contributes minimal total chromium, while supplemental fuels may contribute more. For example, with usage of tire-derived fuel, steel belts would theoretically contribute more total chromium.
Table 2. Reported Chromium Content of Fuels
*Bhatty references 1992 Environmental Toxicology Institute report
NA: not available (no values provided)
Refractory Brick
While low-chromium brick is currently more common in use today, refractory brick containing high levels of chrome have been used in cement kilns. Use of this refractory could contribute to a surge in chromium levels to the clinker during the first use of kiln after re-bricking
(Klemm 1994). Klemm (2000) refers that these bricks could also contribute Cr (VI) during its service life in the rotary kiln, chrome-bonded refractory brick was exposed to the clinker coating and reactive alkalies circulating in the hot kiln gas stream. The chemical reactions that take place can result in a significant amount of trivalent chromium being oxidized to
hexavalent chromium on the surface and within the bulk of the refractory, and the formation of alkali chromates and calcium chromate. Potassium chromate and sodium chromate are highly soluble in water, whereas calcium chromate has only a limited solubility.
Grinding Media
If chromium alloys are used in grinding media and crushers, they may contribute metallic chromium. Klemm (1994) reports that in clinker ground with chrome alloy balls containing
17-28% chromium, the hexavalent chromium content of the cement may increase to over twice that in the original clinker. However, the reduction in use of such materials over recent years makes this a less likely source of chromium. Regarding conversion to Cr (VI) during finish grinding, possible favorable conditions are discussed in the following section of this report.
Additions
Additions of gypsum, pozzolans, ground granulated blast furnace slag, mineral components, cement kiln dust, and set regulators may be potential sources of chromium. ATILH (2003) reports total chromium content in gypsum to be 3.3-33 ppm.
FORMATION OF HEXAVALENT CHROMIUM.
The source of chromium input in the kiln feed is primarily as Cr (III). The conditions in the kiln include high amount of CaO, free lime, and alkalis due to the internal circulation of volatiles. Such conditions are favorable for oxidation of chromium to Cr (VI), the amount of which will depend on the oxygen pressure in the kiln atmosphere.
The process is similar to the production of sodium chromate by which chromite ore [(Mg,FeII)(Cr,Al,FeIII)2O4] is mixed with sodium carbonate and free lime and roasted in a rotary kiln with excess air at around 1200 °C. The sodium chromate is water soluble and washed out of the product. In the cement burning process, depending on the partial pressure of oxygen and availability of potassium and sodium chromates of these will form, and due to some chemical similarity between sulfate and chromate, the chromate will follow alkali sulfates in clinker (Fregert 1974). Mishulovich (1995) also indicates the importance of alkali concentration, since Cr (VI) in clinker is principally in the form of chromates.
Numerous studies emphasize the importance of oxidizing conditions for conversion to Cr (VI), including Reifenstein and Paetzold (from Bhatty 1993), Boikova (from Bhatty 1993), and Fregert (1974). Mishulovich (1995) showed in laboratory studies the relationship between degrees of oxidation to oxygen content and concluded that limiting oxygen in the burning zone would decrease formation of Cr (VI) in the clinker. Lizarraga (2003) observed that insufficiently calcined clinker had low amounts of Cr (VI), supporting the view that the oxidation of chromium and formation of Cr (VI) takes place in the burning zone. In this study, plant tests were performed to investigate the possibility to decrease the amount of Cr (VI) in clinker by having reducing conditions in the kiln. This was obtained by adding fuel oil to the cooler, pet coke to various positions in the preheater, and to kiln feed at various rates. Determination of Cr (VI) in the clinker indicated some effect in terms of decreasing Cr (VI) to 0 mg/kg clinker from around 5 mg/kg. However it was concluded that the operational complications involved were much greater than addition of ferrous sulfate to the cement.
Other than as alkali chromate, the chromium is distributed in solid solution in the clinker minerals as a function of burning conditions. In a laboratory study combing C3S with Cr2O3 heated in air, Johansen (from Bhatty, 1993) concluded that Cr (III) oxidizes to Cr (V) above 700 °C and then is reduced to Cr (IV) above 1400 °C, resulting in the presence of Cr (IV) and Cr(V) in solid solution.
Table 3 shows the distribution of chromium in the individual clinker minerals according to Hornain (1971). Table 4 shows the relative distribution of chromium in laboratory clinker burned at different temperatures and oxidation conditions (the fact that belite holds a smaller fraction of the total amount of chromium reflects the low amount of belite in this clinker).
Table 3. Distribution of Chromium in Clinker Minerals (Hornain 1971)
Table 4. Distribution of Chromium Between Clinker Phases (ATILH 2003)
Formation in Finish Mill
Conversion to Cr (VI) in the cement may also take place in the finish mill. Wear metal from chrome alloy grinding media may provide metallic chromium. The finish mill provides thermodynamically favorable conditions for oxidation of metallic Cr to Cr (VI), including high air sweep, moisture from gypsum dehydration, cooling water injection, and grinding aids,
along with the high pH conditions characteristic of portland cement (Klemm 2000)
However, report APCA (1998) states that chromium from finish grinding will not oxidize to Cr (VI). “Chromium from finish grinding remains either as metal or gradually oxidizes to divalent chromium, trivalent chromium, or perhaps Cr (IV), but not hexavalent chromium.”
Chromium Levels in Clinker and Cement
The chromium content in clinker samples from a Belgian study are shown in Table 5. In regards to Cr (VI) levels in clinker, a Spanish investigation reports Cr (VI) content of clinker between 8 and 20% of the total chromium content (Lizarraga 2003). The content of total chromium and Cr (VI) of cements are shown in Tables 6 and 7. Table 7 is a compilation of data from different countries and shows large variation from location to location.
Table 5. Chromium Content of Clinker (ATILH 2003)
Table 6. Chromium Content of 94 Cement Samples (PCA 1992)
Table 7. Chromium Content of Cement (ATILH 2003)
NA: not available (no values provided)
REDUCING AGENTS
The use of materials to reduce the level of Cr (VI) formation is especially prevalent in the European cement industry due to the 2003 European Directive which prohibits sale of cement containing more than 2 ppm of soluble Cr (VI) when hydrated. Cement companies under this directive are adding reducing agents to comply with this directive. The description of materials used for this purpose, reported effectiveness, limitations, and other items of note are provided below. Examples of capital investment for storage and metering systems for reducing agents, and estimated cost of these agents for the European industry are provided in Cement International (2004).
Ferrous Sulfate
• Natural heptahydrate (FeSO4*7H2O) is found as an alteration product of iron sulfides as the mineral melanterite, or can be an industrial by-product. It is soluble in water, its aqueous solutions are oxidized slowly by air when cold and rapidly when hot, and the oxidation rate is increased under alkaline conditions (Klemm 1994). Should not be overheated to avoid partial oxidization and dehydration, leading to reduced solubility.
• Various ferrous sulfate types differ in active-substance content, particle size, and chemical and physical properties; the product selected for use is determined on metering location and temperature, and storage conditions (Kuehl 2006).
• Monohydrate form (FeSO4*H2O) has been demonstrated to be a successful reducing agent by Valverde, Lobato, Fernandez, Marijuan, Perez-Mohedano, and Talero (2005)
• Addition rate is usually 0.5% by mass and is generally added as powder which may require addition and blending equipment.
• The dosing process can influence the effectiveness of the reducer. The addition of the reducer in the cement mill involves thermal, mechanical and chemical stress, which can accelerate the chemical reaction of the reducers and decrease its effectiveness (Kehrmann and Bremers 2006). These authors conclude heptahydrates are particularly effective if added to cement in format of granulates versus ground in the finish mill, whereas monohydrate, which contain less crystallized water are less susceptible to high temperature and can be ground with clinker.
• May effect cement quality. Excess sulfate may result in lower concrete strength, expansion, and possible internal sulfate attack. At high dosages, there can be concerns of increased water demand, long setting time, and possible concrete discoloration or mottling (Sumner, Porteneuve, Jardine, and Macklin 2006).
• Regarding reduction results, 0.35% ferrous sulfate reduced 20 ppm of Cr (VI) in cement to less than 0.01 ppm in aqueous slurry, however, high temperature and humidity in simulated grinding minimized this reduction capacity (Fregert, Gruvberger and Sandahl from Klemm 1994).
Modifications:
• Bhatty (1993) reports several studies in which the ferrous sulfate was dissolved as 20% solution and added as admixture in concrete/mortar.
• A free-flowing powder was reported by Rasmussen (from Klemm 1994) by mixing with fly ash, gypsum, or other absorbing powder and drying at 20-60ºC.
• The use of “ferrogypsum” is described by mixing the ferrous sulfate with “green salt” (waste product from titanium dioxide manufacture) and gypsum (Norelius from Klemm 1994).
• A Belgium patent application includes a method for fluidization of moist ferrosulfate heptahydrate by adding flyash or fumed silica as a desiccant (Degre, Duron, and Vecoven, 2005).
• A patent by Kehrmann and Paulus (2004) involves cement, iron (II) sulfate, and an acidifying agent for reducing the pH. The acidifying agent provides an acidic environment in the cement, thereby influencing the reactivity of iron (II)-sulfate and increasing the storage life.
Stannous Sulfate
• Manufactured product which can withstand relatively high temperatures without degradation, enabling addition to finish mill.
• Storage stability is longer than ferrous sulfate (Bonder 2005).
• More effective at low dosages compared to ferrous sulfate (Sumner, Porteneuve,
Jardine, and Macklin 2006). These authors also discuss a patent pending liquid additive to increase storage stability.
• Can be available in form of suspension, which may not require expensive metering/addition equipment.
Manganese Sulfate
Larsen (from Klemm 1994) discusses that a cement composition containing manganous sulfate is effective in reducing the content of water-soluble chromate. Mangenese compounds are much more oxidation stable than iron compounds in dry cement at high temperatures and have a high chromate-reduction efficiency. Klemm (1994) reports a study in which clinker with
19.7 ppm Cr (VI) interground in laboratory ball mill with 5% gypsum and 0.75% manganous sulfate resulted in water-soluble chromate after leaching of 0.0 ppm.
Stannous Chloride
The use of tin salts is described by Magistri and Padovani (2005), who describe higher reduction efficiency over iron salts, superior stability and duration of reduction with time, and absence of surface discoloration. The disadvantage of use in the cement industry, as stated by these authors, is the high cost and reduced availability.
A patent using liquid additive with stannous chloride is provided by Jardine, Cornman, Chun, and Gupta (2005). Included in the patent description is the use of an antioxidant and/or oxygen scavenger, which is believed to extend the shelf-life and effectiveness of the stannous chloride reducing agent. A similar methodology is described by Magistri and Padovani (2005), in which a triple emulsion is employed involving the aqueous reducing agent solution in an emulsion surrounded by a layer of organic solvent, which is dispersed in a second aqueous solution. The organic solvent functions as a barrier which impedes contact between the reducing agent and atmospheric oxygen, preventing loss in performance with time.
Zinc (II) Salts
A patent by Alter and Rudert (2004) involves zinc (II) or stannous salts mixed with a fine hydraulic binder or finely ground (10000-18000 cm2/g) blast furnace slag to provide a “physiologically-effective industrial protective means for preventing the harmful effects of tetravalent chromium compounds in cements.”
Others Agents
• A method described by Schremmer, Oelschlaeger and Boege (from Klemm 1994) to achieve a low-chromate cement involved calcining the clinker under oxidizing atmosphere followed by granulating or selecting sizes less than 10 mm, heating to
550ºC with waste coal dust to produce reducing atmosphere and cooling to 300ºC.
• A Japanese patent said to be “effective in diminishing hexavalent chromium” includes a cement admixture comprised of an air-cooled blast furnace slag powder which consists mainly of a melilite and has a CO2 absorption of 2% or higher and an ignition loss of
5% or lower. In the powder, the content of sulfur not in the form of sulfuric acid is
0.5% or higher and/or the concentration of non-sulfuric-acid-form sulfur which dissolves away is 100 mg/L or higher (Morioka, Nakashima, Higuchi, Takahashi, Yamamoto, Sakai, Daimon 2003).
• A European patent involves the addition of ammonium, alkaline metal or earth alkaline metal disulphides and/or polysulphides (Cabria 2004).
• Sodium thiosulfate, sodium metabisulfite, and ascorbic acid was found unsatisfactory due to incomplete chemical reduction of Cr (VI); zinc and aluminum powder required large amounts to be effective and handling difficulties were encountered; and sodium dithionate was effective at low concentrations but deteriorated rapidly with storage (Fregert and Gruvberger from Klemm 1994).
Possible Feed Points
Figure 1. Locations of possible feed points for FeSO4 at the cement plant.
Storage and Shelf Life
With at least some of the materials described above, there is reduced effectiveness with exposure (time/temperature/humidity). The European directive requires that delivery documents and cement bags be marked with information on the period of time for which the reducing agents remain effective; BCA member companies have initially declared shelf life as
61 days (BCA 2006). Brookbanks (2005) outlines the recommended storage for packed
cement as stored in unopened bags clear of the ground in cool dry conditions and protected from excessive draught, whereas recommended storage for bulk cement is to be stored in silos that are waterproof, clean, and protected from contamination, dry with stock rotated in chronological order with dispatch dates marked on delivery documents.
One documentation of storage conditions was reported by Lizarraga (2003). In this study, different levels of FeSO4 was added to cement stored in sacks. The amount of Cr (VI) was followed up to 88 days in individual sacks stored at ambient and up to 180 days for sacks stored in plastic cover. Tables 8 and 9 show the results demonstrating the dependence of storage conditions. Important differences in analytical results were observed between samples from the outer and the inner part of a sack, so homogenization of samples before analysis was needed. A later study by this author investigated an accelerated test to determine period of effectiveness for reducing agents (Lizarraga 2006).
A study reported by Valverde, Lobato, Fernandez, Marijuan, Perez-Mohedano, and Talero (2005) did not show a decline in the reducing power of either ferrous sulfate heptahydrate or monohydrate after storage in a dry environment for three months.
Table 8. Amount of Cr (VI) as Function of FeSO4 Dosage and Time for Cement Stored at
Ambient Conditions in Individual Sacks (Lizarraga 2003)
Table 9. Amount of Cr (VI) as Function of FeSO4 Dosage and Time for Cement Stored Under
Plastic Cover in Individual Sacks (Lizarraga 2003).
CONCLUSION
Chromium in the cement can originate from a variety of sources, including raw materials, fuel, refractory, grinding media, and additions. With regard to health and safety aspects, the water- soluble compounds of chromium in cement are most relevant, specifically compounds of the form Cr (VI). The manufacturing process, specifically kiln conditions, can influence how much Cr (VI) will form. Oxidizing atmosphere will play the largest role, with more oxygen in the burning zone leading to increased Cr (VI) formation. Alkali concentration is also of
importance, since Cr (VI) in clinker is primarily in the form of chromates. The finish mill may play a role, as the thermodynamically favorable conditions for oxidation to Cr (VI) exists, including high air sweep, moisture from gypsum dehydration, cooling water injection, and grinding aids, along with the high pH of the cement.
The most widely used material used to reduce the level of soluble Cr (VI) formation in hydrated cement is ferrous sulfate; other materials include stannous sulfate, manganese sulfate, and stannous chloride. Some of these materials have limitations such as limited stability, limited supply, and possible influence on cement performance. In all cases, some form of dosing and mixing equipment is required.
ACKNOWLEDGEMENTS
The information reported in this paper (SN2983) was conducted by CTLGroup with the sponsorship of the Portland Cement Association (PCA Project Index No. M05-04). The contents of this report reflect the views of the authors, who are responsible for the facts and accuracy of the data presented. The contents do not necessarily reflect the views of the Portland Cement Association.
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Bhatty, Javed, Chromium in Portland Cement: Literature Review, SN1971, Portland Cement
Association, Skokie, Illinois, USA, 1993, 10 pages.
BCA (British Cement Association), Cement Industry Update: Special Issue on Chromium (VI) Directive,, accessed from website 2006.
Bonder, Wolfgang, “Chromate VI Reduction”, World Cement, November 2005, pages 19-110. Brookbanks, Peter, The New Chromium (VI) Directive, Institute of Concrete Technology,
Briefing Note BN 04/05, 2005, 3 pages.
Cabria, Flavio, Process for Preparing Cement with a Low Hexavalent Chromium Content, European Patent Office, Patent Application No. 04425201.3, 2005.
Cement International, “Chromate Reduction – A New Challenge for the Cement and Concrete
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Chandelle, Jean-Marie, Chromium VI in Cement, Global Cement and Lime Magazine, Surry, U.K, July-August 2003, pages 12-13.
11
Degre, Geoffrey ; Duron, Jacques ; and Vecoven, Jacques, Improving the Flow of Moist Ferrous Sulfate Heptahydrate, e.g. Useful for Reducing Hexavalent Chromium in Cement, Comprises Adding Flyash or Fumed Silica, European Patent Office, Patent Application No.
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Fregert (1974) Private communication.
Fregert, S. and Gruvberger, B., “Correlation Between Alkali Sulphate and Water-Soluble
Chromate in Cement”, Acta Dermatovener (Stockholm) Vol. 53, No. 3, 1973, pages 225-228.
Fregert, S. and Gruvberger, B., “Factors Decreasing the Content of Water-Soluble Chromate in
Cement”, Acta Dermatovener (Stockholm), Vol. 53, No. 4, 1973, pages 267-278.
Hornain, H., “The Distribution of Transition Elements and Their Influence on Some Properties of Clinker and Cement”, Revue des Materiaux de Construction, Lafayette, Paris, Trav. Publ. No. 671-672, 1971, pages 203-218.
Hjorth, L., “The Occurrence and Prevention of Cement Eczema”, World Cement Research and
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Jardine, Leslie, A.; Cornman, Charles, R.; Chun, Byong-Wa; and Gupta, Vijay; Liquid Additive for Intergrinding Cement, World Intellectual Property Organization, Patent Publication No.
2005/076858/WO-A2, 2005.
Johansen, Vagn, “Solid Solution of Chromium in Ca3SiO5”, Cement and Concrete Research, Elmsford, New York, 1972, Volume 2, page 33.
Kehrmann, Alexander and Andreas, Paulus, Chromate Reduced Hydraulic Binder, European
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Lizarraga, Serafin, “Study of the Behaviour of Ferrous Sulphate Heptahydrate as a Hexavalent
Chromium Reducing Agent”, Cemento Hormingon, No.887, April 2006, pages 4-8.
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Review, October, 2005, pages 49-56.
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©Portland Cement Association 2007
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1 Chor 2 Mastikhor: Description
1 Chor 2 Mastikhor (2017) Full Movie Watch Online Free TodayPk, Latest Indian Movies Download Free HD mkv 720p, Movierulz Tamilrockers
1 Chor 2 Mastikhor is an upcoming Hindi movie scheduled to be released on 10 November, 2017. The movie is directed by Prabhakar Sharan and will feature Prabhakar Sharan, Scott Steiner and Nancy Dobles as lead characters.
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TITLE: Does the Fourier transform preserve the separation property?
QUESTION [4 upvotes]: The space of Schwartz functions on the plane is denoted by $\mathcal{S}$.
The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.
The Fourier transform $\mathcal{F}$ on $\mathcal{S}$ give a bijective correspondence between the $m_1$-subalgebras of $\mathcal{S}$ and $m_2$- subalgebras of $\mathcal{S}$.
We say that a subset $A$ of $\mathcal{S}$ separates compact subsets of $\mathbb{R}^2$ if for every disjoint compact set $K_1,K_2, \ldots,K_n$, there exist a function $f\in A$ such that $f(K_i) \cap f(K_j)$ is null, for $i\neq j$.
Is it true to say that the Fourier transform gives a bijective correspondence between separating $m_1$-subalgebras of $\mathcal{S}$ and its $m_2$ separating subalgebras?
The question is motivated by the following post:
Are these function spaces appropriate to be considered as the domain of certain differential operator?
REPLY [1 votes]: I can't give a precise or complete answer, but based on vague, spontaneous thoughts I suggest considering the subalgebra of the Schwartz space generated by all bump functions of the form
$$
f^c_x(y):=e^{-\frac{c}{\Vert x-y\Vert^2}},\quad x,y\in \mathbb{R}^2,\; c>0.
$$
Clearly, this algebra is separating.
In these notes it is shown that the Fourier transform $\hat f^1_x$ of $f^1_x$ fulfills
$$
\hat f^1_x(\xi)\sim 2 \mathrm{Re}\,\bigg(\sqrt{\frac{-i\pi}{\sqrt{2i}\xi^{3/2}}}e^{i\xi-1/2-\sqrt{2i\xi}}\bigg)\qquad \text{as }\xi \to +\infty.
$$
Deducing from this the asymptotic behaviour of the Fourier transforms of the functions $f^c_x$ for $c\neq 1$, one should be able to conclude that the Fourier transforms $\hat f^c_x(\xi)$ oscillate very quickly as $\xi$ becomes large. This should imply that also their convolutions oscillate very quickly as $\xi$ becomes large, hence the algebra formed by the $\hat f^c_x(\xi)$ under convolution is not separating anymore because when one chooses two compact sets that are large enough and far enough away from the origin, then there won't be any two functions in the algebra that don't oscillate at least once inside each of the two compact sets.
Of course, when looking more closely at things I might be totally wrong and these considerations might be misleading, but maybe they can at least start some discussions.
| 67,545
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TITLE: Is Lindeberg's condition sufficient to apply the central limit theorem to dependent random variables?
QUESTION [2 upvotes]: I have a triangular array $(X_{i,n})$ that is not independent across $i$ for each $n$. But I can show that $\operatorname{Var}\left(n^{-1/2}\sum_{i = 1}^n X_{i,n}\right)$ is bounded and converges to $\Sigma$ as $n$ grows to infinity. I think that this is sufficient to establish Lindeberg's condition. Is it true?
For simplicity, I assume that $E(X_{i,n}) = 0$ for all $i$ and $n$.
Which central limit theorem can I apply on $n^{-1/2}\sum_{i = 1}^n X_{i,n}$? That is, I want to show that $\displaystyle n^{-1/2}\sum_{i = 1}^n X_{i,n} \to N(0, \Sigma)$.
REPLY [1 votes]: No miracle can happen without any further assumption on the dependence across the rows $(X_{i,n})_{1\leqslant i\leqslant n}$. For example, let $X_{i,n}=Y_{i,n}-Y_{i-1,n}$, where $Y_{0,n}=0$, $\mathbb E[Y_{i,n}]=0$ and $\operatorname{Var}\left(n^{-1/2}Y_{n,n}\right)=1$. The central limit theorem would mean in this context that $n^{-1/2}Y_{n,n}$ in distribution to a standard normal random variable, which does not need to be the case, as we only required $Y_{n,n}$ to be centered and have variance $n$.
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Certain Orthodox Christians around the world, however, celebrate Christmas on Jan. 7, which is according to the Julian calendar — pre-dating the Gregorian calendar.
At St. Vladimir Ukrainian Orthodox Cathedral on State Road in Parma, the celebration of Christmas began on Christmas Eve Jan. 6 with a dinner prior to the Christmas Eve vigil service.
The Rev. John Nakonachy, pastor of St. Vladimir’s, said that the dinner on Friday evening was a chance for many of the elderly parishioners to enjoy the traditional Ukrainian Christmas dinner.
As with Christmas services in other Christian churches, this is a time for reflection and healing, when families gather to give thanks for everything they have.
St. Vladimir’s Ukrainian Orthodox Cathedral is in the heart of Parma’s Ukrainian Village on State Road, and the church is the “mother church” to other Ukrainian Orthodox churches further out in the suburbs of northeast Ohio.
Many of the parishioners live outside of Parma but return every Sunday to attend liturgy there.
The Ukrainian Museum and Archives is at 1202 Kenilworth Avenue, Cleveland. Visit the museum’s website, umacleveland.org.
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\title{ {\bf Commutator Theory and Abelian Algebras}}
\date{August 2013}\author{P.Ouwehand\\Department of Mathematical Sciences\\Stellenbosch
University\\\url{peter\_ouwehand@sun.ac.za}}\maketitle
\abstract{This is the first draft of a set of lecture notes developed for one-half of a seminar on two approaches to the notion of ``Abelian", namely those of universal algebra, and of category theory. The half pertaining to the universal-algebraic perspective is covered here. All material has been adapted from the sources \cite{Freese_McKenzie}, \cite{Gumm} and \cite{McKenzie_Snow}.
}
\newpage
\tableofcontents
\newpage
\section{``Abelian" in Groups}
\fancyhead[RE]{``Abelian" in Groups}
\subsection{Congruences and Normal Subgroups}
Let $G$ be a group. Let $N(G)$ be the lattice of normal subgroups of $G$, and let $\Con(G)$ the congruence lattice of $G$. It is well--known that there is a one-to-one correspondence between the normal subgroups of $G$ and the congruence relations on $G$:
\begin{proposition} Let $G$ be a group.
\begin{enumerate}[(a)]\item For each $H\trianglelefteq G$, define $\Theta_H\subseteq G\times G$ by
\[(a,b)\in\Theta_H\quad\text{iff}\quad a^{-1}b\in H\]
Then $\Theta_H$ is a congruence relation on $G$.
\item For each $\theta\in \Con(G)$, define $N_\theta\subseteq G$ by
\[N_\theta:=1/\theta\qquad\text{i.e.}\quad a\in N_\theta \text{ iff } a\theta 1\]
Then $N_\theta$ is a normal subgroup of $G$.
\item The maps $\Theta: N(G)\to \Con(G):H\mapsto\Theta_H$ and $N:\Con(G)\to N(G):\theta\mapsto N_\theta$ are isotone and inverses of each other, and hence are lattice isomorphisms.
\end{enumerate}
\end{proposition}
\bproof Straightforward exercise.
\eproof
The lattice of normal subgroups $N(G)$ has the following operations: If $H,K\in N(G)$, then
\[H\land K:= H\cap K\qquad H\lor K= HK:=\{hk:h\in H,k\in K\}\]
(Indeed, if $h_1k_1, h_2k_2\in HK$, then $(h_1k_1)^{-1}h_2k_2 = (k_1^{-1}(h_1^{-1}h_2)k_1)\cdot (k_1^{-1}k_2)$. Now $k_1^{-1}(h_1^{-1}h_2)k_1\in H$ because $H$ is a normal subgroup of $G$, and $ k_1^{-1}k_2\in K$. Hence $(h_1k_1)^{-1}h_2k_2\in HK$, proving that $HK$ is a subgroup of $G$. Furthermore, $g^{-1}hkg = (g^{-1}hg)(g^{-1}kg)$ shows that $HK\trianglelefteq G$.)
It is also well--known that the subsets of groups satisfy the following {\em modular law}
\begin{proposition} \begin{enumerate}[(a)]\item If $H,K\subseteq G$, $L\leq G$ and $H\subseteq L$, then \[HK\cap L=H(K\cap L)\]
\item The lattice $N(G)$ is a {\em modular lattice}, i.e. satisfies the modular law:
If $H,K,L\in N(G)$ and $H\leq L$, then
\[(H\lor K)\land L = H\lor (K\land L)\]
\end{enumerate}
\end{proposition}
\bproof (a) It is easy to see that $HK\cap L\supseteq H(K\cap L)$.
Conversely, suppose that $h\in H, k\in K$ are such that $l:=hk\in L$ (i.e. $hk\in HK\cap L$). We must show that $hk\in H(K\cap L)$. Now since $H\subseteq L$, we have $h\in L$. Since $L$ is a subgroup of $G$, we have $h^{-1}l=k\in L$. Thus $k\in K\cap L$, so $hk\in H(K\cap L)$, as required.
(b) follows immediately from (a).
\eproof
\begin{corollary} Groups are {\em congruence modular}, i.e. the congruence lattices of groups are modular lattices: If $\theta,\varphi,\psi\in\Con(G)$ and $\theta\subseteq \psi$, then
\[(\theta\lor\varphi)\land \psi = \theta\lor(\varphi\land\psi)\]
\end{corollary}
\subsection{Commutators of Groups}
For $h,g\in G$, define the group action of $G$ on itself by $h^g:=g^{-1}hg$. Observe that $H\trianglelefteq G$ if and only if $h^g\in H$ whenever $g\in G$ and $h\in H$.
Recall that the commutator operation on groups is defined by
\[[g,h]:=g^{-1}h^{-1}gh\]
If $H,K\leq G$, we define
\[[H,K]:=\text{subgroup generated by }\{[h,k]:h\in H, k\in K\}\]
It is easy to show that the commutator satisfies the following identities: \[[h,k]^{-1}=[k,h]\qquad [hk,l]=[h,l]^k[k,l]\]
The fact that \[[h,k]^g = [h^g,k^g]\]
shows that $[H,K]\trianglelefteq G$ when $H,K\trianglelefteq G$.
The commutator can therefore be regarded as an additional binary operation on the lattice $N(G)$.
\begin{proposition}For $H,K,L\trianglelefteq G$, we have:\begin{enumerate}[(a)]\item $[H,K]\subseteq H\cap K$
\item $[H,K]=[K,H]$.
\item $[HK,L]=[H,L][K,L]$, i.e. $[H\lor K,L] = [H,L]\lor [K,L]$
\end{enumerate}
\end{proposition}
\bproof (a) If $h\in H,k\in K$, then $h^{-1}k^{-1}h \in K$ (by normality of $K$), and hence $[h,k]\in K$. In the same way we see that $[h,k]\in H$, and hence that $[h,k]\in H\cap K$.
\noindent (b) $[h,k]=[k,h]^{-1}\in [K,H]$.
\noindent (c) Clearly $[HK,L]\supseteq[H,L], [K,L]$ and hence $[HK,L]\supseteq [H,L][K,L]$.
Conversely, if $h\in H, k\in K$ and $l\in L$, then $[hk,l]=[h,l]^k[k,l]$. Now $[h,l]^k\in [H,L]$, since $[H,L]\trianglelefteq G$. It follows that $[hk,l]\in [H,L][K,L]$ as required.
\eproof
\subsection{Abelian Groups}
For any group $G$, we can define a subset $\Delta(G)\subseteq G\times G$ by\[\Delta(G):=\{(g,g):g\in G\}\]
Clearly $\Delta(G)$ is a subgroup of $G\times G$.
\begin{theorem}\label{thm_Abelian_group} Let $G$ be a group.
The following are equivalent:
\begin{enumerate}[(a)]\item $G$ is Abelian.
\item $[G,G]=\{1\}$ is the least element of $N(G)$
\item $\Delta(G)$ is a normal subgroup of $G\times G$.
\item $\Delta(G)$ is a coset of a congruence on $G\times G$.
\end{enumerate}
\end{theorem}
\bproof
(a) $\Leftrightarrow$ (b): Obvious, since $ab=ba$ iff $[a,b]=1$.
(a) $\Leftrightarrow$ (c): If $G$ is Abelian, then so is $G\times G$, and hence any subgroup of $G\times G$ is a normal subgroup. Conversely, if $\Delta(G)\trianglelefteq G\times G$, then $(a^{-1}ga, b^{-1}gb) = (g,g)^{(a,b)}\in \Delta(G)$ for any $a,b,g\in G$. In particular,
\[(ab,ab)^{(a,1)}=(a^{-1}aba, ab)=(ba,ab)\in\Delta(G),\qquad\text{so}\quad ab=ba\]
(c) $\Leftrightarrow$ (d): Any normal subgroup $N$ is a coset of a congruence, namely $N=1/\Theta_N$. Conversely, any coset of a congruence which contains the group identity element is a normal subgroup. In particular $\Delta(G)$ is a coset of a congruence if and only if $\Delta(G)\trianglelefteq G\times G$.
\eproof
The lattice $M_3$ is the lattice
\[
\begin{tikzpicture}
\draw[fill] (0,0) circle[radius = 0.1];
\draw[fill] (0,-1) circle[radius = 0.1];
\draw[fill] (0,1) circle[radius = 0.1];
\draw[fill] (-1,0) circle[radius = 0.1];
\draw[fill] (1,0) circle[radius = 0.1];
\draw (0,1)--(0,0)--(0,-1)--(1,0)--(0,1)--(-1,0)--(0,-1);
\end{tikzpicture}\]
A $(0,1)$--lattice homomorphism is a lattice homomorphism $f:L_1\to L_2$ between two lattices having top element (denoted $1$) and bottom element (denoted $0$), such that $f(0_{L_1})=0_{L_2}$, and $f(1_{L_1}) = 1_{L_2}$.
\begin{theorem}\label{thm_Abelian_group_M_3}\begin{enumerate}[(a)]\item
Suppose that $M_3$ is a $(0,1)$--sublattice of $N(G)$. Then $G$ is Abelian.
\item $G$ is Abelian if and only if there is a $(0,1)$--homomorphism $M_3\to N(G\times G)$.
\end{enumerate}
\end{theorem}
\bproof
(a) Suppose that $H,K,L\in N(G)$ have
\[H\cap K=H\cap L=K\cap L=\{1\}=0_{N(G)}\qquad HK=HL=KL = G=1_{N(G)}\]
Then $[H,K]\leq H\cap K$ implies $[H,K]=\{1\}$. Similarly $[H,L]=[K,L]=\{1\}$.
Now\[[G,G]=[HK,HL]=[H,H]\,[H,L]\,[K,H]\,[K,L] =[H,H]\leq H\]
Similarly, $[G,G]\leq K$ and $[G,G]\leq L$, and hence $[G,G]=\{1\}$.
(b) Observe that $M_3$ is a simple lattice. If a $(0,1)$--homomorphism $f:M_3\to N(G\times G)$ is not an embedding, then $0=1$ in $N(G\times G)$, and hence $G$ is trivial, so certainly Abelian. Else, $M_3$ is a $(0,1)$--sublattice of $N(G\times G)$. Then $G\times G$ is Abelian, by (a), and hence $G$ is Abelian.
Conversely, if $G$ is Abelian (and non--trivial), then consider the following subgroups of $G\times G$:
\[G_0:=\{(1,g):g\in G\}\qquad G_1:=\{(g,1):g\in G\}\qquad \Delta(G):=\{(g,g):g\in G\}\]
Since $G$ is Abelian, these are normal subgroups of $G\times G$.
Now it is easy to see that
\[G_0\cap G_1= G_0\cap \Delta(G)=G_1\cap\Delta(G) = \{(1,1)\}=0_{N(G\times G)}\]
Furthermore, if $g_1,g_2\in G$, then $(g_1,g_2)=(1,g_2)(g_1,1)\in G_0G_1$, so that $G_0\lor G_1=G\times G=1_{N(G\times G)}$. Also, $(g_1,g_2) = (1, g_1^{-1}g_2)(g_1,g_1)\in G_0\Delta(G)$, so that $G_0\lor\Delta_(G) = G\times G=1_{N(G\times G)}$. Similarly $G_1\lor \Delta(G) = 1_{N(G\times G)}$
Hence \[G_0\lor G_1 = G_0\lor \Delta(G) = G_1\lor\Delta(G) = 1_{N(G\times G)}\] as required.
\eproof
\section{``Abelian" in Rings, \dots Leaning Towards General Algebra}
\fancyhead[RE]{``Abelian" in Rings}
\subsection{Congruences and Ideals}
Let $R$ be a ring. Let $\mathcal I(R)$ be the lattice of (twosided) ideals of $R$, and let $\Con(R)$ the congruence lattice of $R$. It is well--known that there is a one-to-one correspondence between the ideals of $R$ and the congruence relations on $R$:
\begin{proposition} Let $R$ be a ring.
\begin{enumerate}[(a)]\item For each $I\in\mathcal I$, define $\Theta_I\subseteq R\times R$ by
\[(a,b)\in\Theta_I\quad\text{iff}\quad a-b\in I\]
Then $\Theta_I$ is a congruence relation on $R$.
\item For each $\theta\in \Con(R)$, define $I_\theta\subseteq R$ by
\[I_\theta:=0/\theta\qquad\text{i.e.}\quad a\in I_\theta \text{ iff } a\theta 0\]
Then $I_\theta$ is an ideal of $R$.
\item The maps $\Theta: \mathcal I(R)\to \Con(R):I\mapsto\Theta_I$ and $I:\Con(R)\to N(R):\theta\mapsto I_\theta$ are isotone and inverses of each other, and hence are lattice isomorphisms.
\end{enumerate}
\end{proposition}
\bproof Straightforward exercise.
\eproof
As is easily verified, the lattice of ideals of $\mathcal I(R)$ has the following operations:
\[I\land J:=I\cap J\qquad I\lor J:=I+J:=\{i+j:i\in I, j\in J\}\]
We saw in the previous section that the lattice of normal subgroups is a modular lattice. One can show similarly that the lattice $\mathcal I(R)$ of ideals is a modular lattice as well. We'll take a slightly different tack\dots
\subsection{Congruence Lattices of General Algebras}
Recall that a congruence relation on an algebra $A$ is an equivalence relation $\theta$ which is compatible with the fundamental operations (and thus with any polynomial operation) on $A$. If $\theta,\varphi\in\Con(A)$, then clearly
\[\theta\circ\theta=\theta,\qquad \theta,\varphi\subseteq \theta\circ\varphi\]
Clearly $\theta\circ\varphi$ is a binay relation on $A$ which is reflexive and symmetric. Moreover, $\theta\circ\varphi$ is a {\em subalgebra} of $A\times A$, i.e. it is compatible with the fundamental operations on $A$: If $t$ is an $n$--ary operation and $x_i(\theta\circ\varphi)y_i$ for $i=1,\dots, n$, then there are $z_i$ such that $x_i\,\theta\, z_i\,\varphi \, y_i$.
It follows that $t(\mathbf x)\,\theta\, t(\mathbf z)\,\varphi \, t(\mathbf y)$, so that $t(\mathbf x)(\theta\circ\varphi)t(\mathbf y)$. Hence $\theta\circ\varphi$ is compatible with the fundamental operations on $A$. One reason $\theta\circ\varphi$ need not be a congruence is that it nmay not be transitive. This defect is easily fixed: For $n\in\mathbb N$, d $(\theta\circ\varphi)^{n}$ inductively by
\[\aligned (\theta\circ\varphi)^{0}&:= \Delta_A=\{(a,a):a\in A\}\\
(\theta\circ\varphi)^{n+1}&:=(\theta\circ\varphi)^n\circ\theta\circ\varphi\endaligned\]Just as for $\theta\circ\varphi$, it is straightforward to show that each $(\theta\circ\varphi)^n$ is compatible with the fundamental operations (i.e. is a subalgebra of $A\times A$). Moreover \[(\theta\circ\varphi)^{0}\subseteq(\theta\circ\varphi)^{1}\subseteq (\theta\circ\varphi)^{2}\subseteq\dots\subseteq (\theta\circ\varphi)^{n}\subseteq \dots\] It follows easily that
$\bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$ compatible with the fundamental operations of $A$ (being the union of an increasing chain of subalgebras of $A\times A$). It is obvious also that $\bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$ is reflexive and transitive. Finally, $\bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$ is symmetric: For if $ (a,b)\in \bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$, then $a\,(\theta\circ\varphi)^m\,b$ for some $m\in\mathbb N$. Then
$a\,\varphi \,a\,(\theta\circ\varphi)^m\,b\,\theta\, b$, i.e. $a\,(\varphi\circ \theta)^{m+1}\,b$. It follows that $b\,(\theta\circ\varphi)^{m+1}\, a$, i.e. that $(b,a)\in \bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$ as well.
We may therefore conclude that $\bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$ is a congruence relation on $A$.
Now clearly $\theta,\varphi\subseteq \bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$. Furthermore, any equivalence relation which contains $\theta$ and $\varphi$ must contain $\bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n$. We have shown:
\begin{proposition} The lattice $\Con(A)$ has the following operatoions:
\[\theta\land\varphi=\theta\cap\varphi\qquad \theta\lor\varphi= \bigcup_{n\in\mathbb N} (\theta\circ\varphi)^n\]\endbox
\end{proposition}
An algebra $A$ is said to be {\em congruence permutable} if $\theta\circ\varphi = \varphi\circ\theta$ for any $\theta,\varphi\in\Con(A)$.
In that case, for $n\geq 1$, we have $\theta\circ\varphi\circ\theta = \theta\circ\theta\circ\varphi=\theta\circ\varphi$. Similar reasoning show that $(\theta\circ\varphi)^n = \theta\circ\varphi$ for any $n\geq 1$. Hence if $A$ is congruence permutable, then in $\Con(A)$, the join operation is composition:
\[\theta\lor\varphi = \theta\circ\varphi\qquad\text{in }\Con(A)\]
\begin{proposition} If $A$ is congruence permutable, then $A$ is congruence modular (i.e. $\Con(A)$ is a modular lattice).
\end{proposition}
\bproof Suppose that $\theta,\varphi,\psi\in\Con(A)$ and that $\theta\leq\psi$.
It is always the case that $(\theta\lor\varphi)\land\psi\geq\theta\lor(\varphi\land\psi)$. To prove the $\leq$--direction, suppose that $(a,b)\in (\theta\lor\varphi)\land\psi$. Then $(a,b)\in (\theta\circ\varphi)\cap\psi$. It follows that $a\,\psi\, b$ and that $a\,\theta\,c\,\varphi\,b$ for some $c\in A$. Since $\theta\subseteq\psi$, we also have $a\,\psi\,c$, and thus $a,b,c$ belong to the same $\psi$--coset. In particular $c\,\varphi\, b$ and $c\,\psi\, b$ imply that $c\,(\varphi\cap\psi)\,b$. Hence $a\,\theta\,c\,(\varphi\cap\psi)\,b$, so that $(a,b)\in \theta\circ(\varphi\cap\psi)$ and hence $(a,b)\in \theta\lor(\varphi\land\psi)$, as required.
\eproof
\begin{proposition}Suppose that $A$ is an algebra with a ternary polynomial $p(x,y,z)$ such that
\[A\vDash p(x,x,y)=y\land p(x,y,y)=x\]
Then $A$ is congruence permutable.
\end{proposition}
\bproof
Suppose that $(a,b)\in \theta\circ\varphi$. Then there is $c\in A$ such that $a\,\theta\,c\,\varphi\,b$.
Keeping the $x$-- and $z$--places fixed in the term polynomial $p(x,y,z)$, it follows that
\[p(a,a,b)\,\theta \,p(a,c,b)\,\varphi\, p(a,b,b)\qquad\text{i.e.}\qquad b\,\theta\,p(a,c,b)\,\varphi\, a\]
We conclude that $(b,a)\in \theta\circ\varphi$, i.e. that $(a,b)\in\varphi\circ\theta$. It follows that $\theta\circ\varphi\subseteq\varphi\circ\theta$.
\begin{center}
\begin{tikzcd}
a\arrow[dash]{r}{\theta} &c\dar[dash]{\varphi}\\
&b
\end{tikzcd} \qquad$\Longrightarrow$\qquad\begin{tikzcd}
{a=p(a,b,b)}\arrow[dash]{r}{\theta}\arrow[dash]{d}{\varphi} &p(c,b,b)=c\dar[dash]{\varphi}\\
p(a,c,b)\arrow[dash]{r}{\theta} &p(c,c,b)=b
\end{tikzcd}\end{center}
By symmetry, also $\varphi\circ\theta\subseteq\theta\circ \varphi$, and hence $\theta\circ\varphi = \varphi\circ \theta$.
\eproof
Using the term $p(x,y,z) := xy^{-1}z$ for groups, and $p(x,y,z):=x-y+z$ for rings, we obtain the following:
\begin{corollary} Groups and rings are congruence permutable, hence congruence modular.
\end{corollary}
In particular, the ideal lattice $\mathcal I(R)$ of a ring $R$ is a modular lattice (as it is isomorphic to the modular lattice $\Con(R)$).
\subsection{Commutators of Rings}
Let $I,J$ be subrings of a ring $R$. Define $[I,J]:= \text{subring generated by }\{ij, ji:i\in I, j\in J\}=\text{subring generated by } IJ+JI$.
Observe that $r(ij) = (ri)j\in IJ$ and $(ij)r = i(jr)\in IJ$ for any $r\in R, i\in I$ and $j\in J$. It follows that if $I,J$ are ideals, then so is $[I,J]$.
Analogous to the situation for groups, but much easier to prove because the underlying additive group of $R$ is commutative, we have
\begin{proposition} For ideals $I,J,K\in\mathcal I(R)$, we have:
\begin{enumerate}[(a)]\item $[I,J]\subseteq I\cap J$.
\item $[I,J]=[J,I]$
\item $[I+J,K]=[I,K]+[J,K]$, i.e. $[I\lor J,K]=[I,K]\lor[J,K]$.
\end{enumerate}
\end{proposition}\bproof (a) If $i\in I, j\in J$, then $ij, ji\in I\cap J$.
(b) Obvious from symmetry of $I,J$ in definition of $[I,J]$.
(c) It is clear that $[I+J,K]\supseteq [I,K], [J,K]$ and hence that $[I+J,K]\supseteq [I,K]+[J,K]$. Conversely the types of generators $(i+j)k, k(i+j)$ of $[I+J,K]$ clearly belong to $IK+JK$ and $KI+KJ$ respectively. Hence each generator of $[I+J,K]$ belongs to $[I,K]+[J,K]$
\subsection{Abelian Rings}
\begin{theorem}\label{thm_Abelian_ring} For a ring $R$, the following are equivalent:
\begin{enumerate}[(a)]\item $R$ is a {\em zeroring}: $rs=0$ for all $r,s\in R$.
\item $[R,R]=\{0\} $ is the least element of $\mathcal I(R)$.
\item $\Delta(R):=\{(r,r):r\in R\}$ is an ideal of $R\times R$.
\item $\Delta(R)$ is a coset of a congruence on $R$.
\item $M_3$ is a $(0,1)$--sublattice of $\mathcal I(R)$.
\end{enumerate}
\end{theorem}
\bproof
(a) $\Leftrightarrow$ (b): Clearly if $R$ is a zeroring, then $[R,R]=\{0\}$. Conversely, $RR\subseteq [R,R]$, so if $[R,R]=\{0\}$, then $R$ is a zeroring.
(a) $\Leftrightarrow$ (c): If $R$ is a zeroring, then so is $R\times R$. It is clear that any subring of a zeroring is an ideal, and hence $\Delta(R)$ is an ideal. Conversely, if $\Delta(R)$ is an ideal, then $(a,b)\cdot (r,r)\in\Delta(R)$ for any $a,b,r\in R$. In particular, $(s,0)\cdot(r,r) = (sr,0)\in \Delta(R)$, so $sr = 0$ for any $r,s\in R$.
(c) $\Leftrightarrow$ (d): follows from the isomorphism between $\mathcal I(R)$ and $\Con(R)$.
(a) $\Rightarrow$ (e): Define subrings $R_0, R_1$ of $R\times R$ by\[R_0:=\{(0,r):r\in R\}\qquad R_1:=\{(r,0):r\in R\}\]
Since $R$ is a zeroring, every subring of $R$ is an ideal.
Clearly\[R_0\cap R_1=R_0\cap\Delta(R)=R_1\cap\Delta(R)=\{(0,0)\}=0_{\mathcal I(R\times R)}\]
Also $(r,s)=(0,s)+(r,0)\in R_0+R_1$, so $R_0\lor R_1=R_0+R_1= R\times R=1_{\mathcal I(R\times R)}$.
Next, $(r,s) = (r-s,0)+(s,s)\in R_0+\Delta(R)$, so that $R_0\lor\Delta(R)=R_0+\Delta(R)=R\times R=1_{\mathcal I(R\times R)}$. Similarly, $R_1\lor\Delta(R)=1_{\mathcal I(R\times R)}$.
It follows that \[R_0\land R_1=R_0\land\Delta(R)=R_1\land\Delta(R)=0_{\mathcal I(R\times R)}\qquad R_0\lor R_1=R_0\lor\Delta(R)=R_1\lor\Delta(R)=1_{\mathcal I(R\times R)}\]
(e) $\Rightarrow$ (b): Suppose that $I,J,K$ are ideals in $R\times R$ so that $I\land J=I\land K = J\land K=0$ and $I\lor J=I\lor K=J\lor K=1$.
Then $[R,R] = [I\lor J,I\lor K]=[I,I]\lor[I,K]\lor[J,I]\lor[J,K]$
Now $[I,K]\subseteq I\cap K = 0$, and similarly $[J,I], [J,K]=0$, so $[R,R]=[I,I]\subseteq I$. Similarly $[R,R]\subseteq J$ and $[R,R]\subseteq K$. Hence $[R,R]=0$.
\eproof
It thus makes sense to call a ring {\em Abelian} precisely when it is a zeroring. In particular, an Abelian ring is not the same as a commutative ring.
\begin{remarks} \rm\begin{enumerate}[(a)]\item The proof (e) $\Rightarrow$ (b) of Theorem \ref{thm_Abelian_ring} shows that if $M_3$ is a $(0,1)$--sublattice of the ideal lattice $\mathcal I(R)$ of some ring $R$, then $R$ is a zeroring.
\item
Observe that any Abelian group can be turned into a zeroring simply by defining the multiplication to be trivial. Conversely, any ring has an underlying additive group, which is Abelian. Hence, in some sense, Abelian rings are really no different from Abelian groups.
\item Furthermore, a group $G$ is Abelian if and only if every subgroup of $G\times G$ is normal. Similarly, a ring $R$ is Abelian if and only if every subring of $R\times R$ is an ideal: For in that case $R_0, R_1,\Delta(R)$ are ideals, and the argument (a) $\Rightarrow$ (e) of Theorem \ref{thm_Abelian_ring} goes through.
\end{enumerate}\endbox
\end{remarks}
\section{The Term Condition, the Commutator, and the Center}
\fancyhead[RE]{The Term Condition, the Commutator, and the Center}
\subsection{The Term Condition in {\bf Groups}}\label{subsec_TC_Grp}
Recall that in group theory, the commutator is a binary operation on the lattice of normal subgroups $N(G)$ of a group $G$. It has the following properties:
\begin{enumerate}[(1)]\item $[M,N]\subseteq M\cap N$
\item
$[M,N]=[N,M]$
\item $[M,\bigvee_{i\in I}N_i]=\bigvee_{i\in I}[M,N_i]$
\end{enumerate}
We now begin to investigate commutativity and the commutator. First observe that
\[[M,N]=\{1\}\qquad\Longleftrightarrow\qquad \text{elements of $M$ commute with elements of $N$}\]
i.e. iff $mn = nm$ whenever $m\in M, n\in N$.
\begin{proposition} Suppose that $M,N,K\trianglelefteq G$, and that $K\subseteq M\cap N$.\begin{enumerate}[(a)]\item We have
\[[M/K, N/K]=[M,N]K/K\qquad\text{i.e.}\qquad [M/K,N/K]=([M, N]\lor K)/K\quad\text{in }N(G/K)\]
\item Elements of $M/K$ commute with elements of $N/K$ (in $G/K$) if and only if $[M,N]\subseteq K$.
\end{enumerate}
\end{proposition}
\bproof (a)
Let $H\trianglelefteq G$ be such that $[M/K,N/K]=H/K$ (i.e. $H:=\{h\in G: hK\in[M/K,N/K]\}$). It is clear that $[M,N], K\subseteq H$, and thus that $[M,N]K\subseteq H$.
Conversely, if $h\in H$, then $hK\in[M/K,N/K]:=\Sg(\{mK,nK]:m\in M, n\in N\})$, so
\[hK=\prod_{i=1}^n[m_iK,n_iK]^{\pm 1}=(\prod_{i=1}^n[m_i,n_i]^{\pm 1})K\qquad\text{ some $m_i\in M, n_i\in N$}\] Hence $h\in (\prod_{i=1}^n[m_i,n_i]^{\pm 1})K\subseteq [M,N]K$ for all $h\in H$.
(b) Elements of $M/K$ commute with those of $N/K$ iff $[M/K,N/K]=\{1\}=\{K/K\}$, iff $[M,N]K = K$, iff $[M,N]\subseteq K$.
\eproof
\begin{corollary}Let $G$ be a group, and let $M,N\trianglelefteq G$. Then $[M,N]$ is the smallest normal subgroup $K\trianglelefteq G$ such that in $G/K$ every element of $M/K$ commutes with every element of $N/K$.
\endbox\end{corollary}
Now suppose that every element of $M$ commutes with every element of $N$, i.e. that $[M,N]=\{1\}$.
Let $t(x_1,\dots, x_m,y_1,\dots, y_n)$ be an $(m+n)$--ary term, and let $\mathbf m^1,\mathbf m^2\in M^m$ and $\mathbf n^1,\mathbf n^2\in N^n$. We then have the following situation (where $\Theta_M,\Theta_N$ are the congruences induced by $M,N\trianglelefteq G$):
\begin{center}\begin{tikzcd}
t(\mathbf m^1,\mathbf n^1)\arrow[dash]{r}{\Theta_N}\arrow[dash]{d}{\Theta_M} & t(\mathbf m^1, \mathbf n^2)\dar[dash]{\Theta_M}\\
t(\mathbf m^2,\mathbf n^1)\arrow[dash]{r}{\Theta_N} & t(\mathbf m^2, \mathbf n^2)
\end{tikzcd}
\end{center}
Each term $t(\mathbf m,\mathbf n)$ is a product of powers of $m_i$'s and $n_j$'s.
Now since elements of $M$ commute with those of $N$, it is easy to see that there exists an $m$--ary term $r(x_1,\dots, x_m)$ and an $n$-ary term $s(y_1,\dots, y_n)$ such that
$t(\mathbf m,\mathbf n)=r(\mathbf m)s(\mathbf n)$ for any $\mathbf m\in M^m,\mathbf n\in N^n$.
It follows that
\begin{center}
\begin{tabular}{>{$}r<{$} >{$}r<{$} >{$}c<{$} >{$}l<{$}}\phantom{aaaa} & t(\mathbf m^1,\mathbf n^1)&=& t(\mathbf m^1,\mathbf n^2)\\
\Rightarrow &r(\mathbf m^1)s(\mathbf n^1)&=& r(\mathbf m^1)s(\mathbf n^2)\\
\Rightarrow &s(\mathbf n^1)&=& s(\mathbf n^2)\\
\Rightarrow& r(\mathbf m^2)s(\mathbf n^1)&=& r(\mathbf m^2)s(\mathbf n^2)\\
\Rightarrow &t(\mathbf m^2,\mathbf n^1)&=& t(\mathbf m^2,\mathbf n^2)\end{tabular}
\end{center}
i.e that
\[t(\mathbf m^1,\mathbf n^1)= t(\mathbf m^1,\mathbf n^2)\quad\Longrightarrow \quad t(\mathbf m^2,\mathbf n^1)= t(\mathbf m^2,\mathbf n^2)\]
We thus have the following, (where $\Delta_G$ is the trivial congruence, i.e. the equality relation on $G$):
\begin{center}\begin{tikzcd}
t(\mathbf m^1,\mathbf n^1)\arrow[dash, bend left]{r}{\Delta_G}\arrow[dash]{r}{\Theta_N}\arrow[dash]{d}{\Theta_M} & t(\mathbf m^1, \mathbf n^2)\dar[dash]{\Theta_N}\\
t(\mathbf m^2,\mathbf n^1)\arrow[dash, bend left, dashed]{r}{\Delta_G}\arrow[dash]{r}{\Theta_N} & t(\mathbf m^2, \mathbf n^2)
\end{tikzcd}
\end{center}
Suppose now that we relativize this to the group $G/K$. If elements of $M/K$ commute with elements of $N/K$, i.e. if $[M,N]\subseteq K$, then similar reasoning yields the following: For any $(m+n)$--ary term $t$ and any $\mathbf m^1,\mathbf m^2\in M^m$ and $\mathbf n^1,\mathbf n^2\in N^n$, we have
\begin{center}\begin{tikzcd}
t(\mathbf m^1,\mathbf n^1)\arrow[dash, bend left]{r}{\Theta_K}\arrow[dash]{r}{\Theta_N}\arrow[dash]{d}{\Theta_M} & t(\mathbf m^1, \mathbf n^2)\dar[dash]{\Theta_N}\\
t(\mathbf m^2,\mathbf n^1)\arrow[dash, bend left, dashed]{r}{\Theta_K}\arrow[dash]{r}{\Theta_N} & t(\mathbf m^2, \mathbf n^2)
\end{tikzcd}
\end{center}
i.e. \[t(\mathbf m^1,\mathbf n^1)\;\Theta_K\;t(\mathbf m^1,\mathbf n^2)\quad\Longrightarrow \quad t(\mathbf m^2,\mathbf n^1)\;\Theta_K\; t(\mathbf m^2,\mathbf n^2)\]
If the above situation holds, we say that $\Theta_M$ centralized $\Theta_N$, modulo $\Theta_K$, and denote it by $C(\Theta_M,\Theta_N;\Theta_K)$.
\subsection{The Term Condition and the Commutator in General Algebras}\label{subsection_TC_commutator} We attempt to imitate the constructions of the previous section in general algebras:
Let $A$ be an algebra, and let $\alpha,\beta,\delta\in \Con(A)$.
The algebra $A^4$ can be regarded as the collection of all $2\times 2$--matrices\[(a,b,c,d)\text{ corresponds to }\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\qquad a,b,c,d\in A\]
\begin{definition}\rm For $\alpha, \beta\in\Con(A)$, let $M(\alpha,\beta)$ be the subalgebra of $A^4$ generated by all matrices
\[\left(\begin{matrix}a&a\\a'&a'\end{matrix}\right),\qquad \left(\begin{matrix}b&b'\\b&b'\end{matrix}\right)
\qquad\text{where } a\,\alpha \,a' \text{ and }b\,\beta \,b'\]\endbox
\end{definition}
It is easy to see that:
\begin{proposition} $M(\alpha,\beta)$ consists of all $2\times 2$--matrices of the form
\[\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\]
where, for any $n,m\in\mathbb N$, $t(\cdot)$ is an $(m+n)$--ary term, and $\mathbf a^1,\mathbf a^2\in A^m$, $\mathbf b^1,\mathbf b^2\in A^n$ are such that
\[\mathbf a^1_i\,\alpha\,\mathbf a^2_i\quad\text{for all }i\leq m\qquad \mathbf b^1_j\,\beta\,\mathbf b^2_j\quad\text{for all }j\leq n\]
In particular, we have
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center} for all elements of $M(\alpha,\beta)$.
\endbox
\end{proposition}
\begin{definition}\rm Suppose that $A$ is an algebra, and that $\alpha,\beta,\delta\in \Con(A)$. We say that $\alpha$ {\em centralizes} $\beta$ modulo $\delta$ (and denote this by $C(\alpha,\beta;\delta)$) if and only if whenever $\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\in M(\alpha,\beta)$, we have
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash, bend left]{r}{\delta}\arrow[dash]{r}{\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash, bend left, dashed]{r}{\delta}\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
i.e. \[t(\mathbf a^1,\mathbf b^1)\;\delta\;t(\mathbf a^1,\mathbf b^2)\quad\Longrightarrow \quad t(\mathbf a^2,\mathbf b^1)\;\delta\; t(\mathbf a^2,\mathbf a^2)\]
In that case, we also say that $A$ satisfies the {\em $\alpha,\beta$--term condition modulo $\delta$}.
\endbox
\end{definition}
Of course, by symmetry, we will have $t(\mathbf a^1,\mathbf b^1)\;\delta\;t(\mathbf a^1,\mathbf b^2)\Longleftrightarrow t(\mathbf a^2,\mathbf b^1)\;\delta\; t(\mathbf a^2,\mathbf a^2)$, i.e. the $\Longrightarrow$ can be replaced by a $\Longleftrightarrow$.
Here are some easy consequences of the definition:
\begin{proposition} \label{propn_properties_term_commutator} Suppose that $\alpha,\beta,,\delta,\alpha_i,\delta_j\in\Con(A)$.
\begin{enumerate}[(a)]\item If $C(\alpha,\beta;\delta_j)$ holds for all $j\in J$, then $C(\alpha,\beta;\bigwedge_{j\in J}\delta_j)$.
\item If $C(\alpha_i,\beta;\delta)$ holds for all $i\in I$, then $C(\bigvee_{i\in I}\alpha_i,\beta;\delta)$.
\item $C(\alpha,\beta;\alpha\land\beta)$.
\end{enumerate}
\end{proposition}
\bproof
(a) If $\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\in M(\alpha,\beta)$ is such that
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{\bigwedge_{j\in J}\delta_j}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
then for all $j\in J$, we have\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{\delta_j}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left,dashed]{r}{\delta_j} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
because $C(\alpha,\beta;\delta_j)$, i.e. $t(\mathbf a^2,\mathbf b^1)\,\delta_j\, t(\mathbf a^2, \mathbf b^2)$ for all $j\in J$. Hence also $t(\mathbf a^2,\mathbf b^1)\,\bigwedge_{j\in J}\delta_j\, t(\mathbf a^2, \mathbf b^2)$, as required.
(b) Suppose next that $C(\alpha_i,\beta;\delta)$ for all $i\in I$, and that
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{\delta}\arrow[swap, dash]{d}{\bigvee_{i\in I}\alpha_i} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\bigvee_{i\in I}\alpha_i}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
where $a^1_k\,\bigvee_{i\in I}\alpha_i\, a^2_k$ for $k=1,\dots, m$ and $b^1_k\,\beta\,b^2_k$ for $k=1,\dots,n$.
Then there exist $i_1,\dots,i_l\in I$ such that $a^1_k\,(\alpha_{i_1}\circ\alpha_{i_2}\circ\dots\circ\alpha_{i_l})\, a^2_k$ for each $k=1,\dots, m$, and hence there exist $u^1_k,\dots, u^{l-1}_k$ such that
\[a^1_k\,\alpha_{i_1}\,u^1_k\,\alpha_{i_2}\,u^2_k\,\alpha_{i_3} \dots \alpha_{i_{l-1}}\, u^{l-1}_k\,\alpha_{i_l}\,a^2_k\]
Hence if $\mathbf u^j:=(u^j_1,\dots, u^j_m)$ for $j=1,\dots, l-1$, we see that
\begin{center}\begin{tikzcd}t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{\delta}\arrow[swap, dash]{d}{\alpha_{i_1}} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha_{i_1}}\\
t(\mathbf u^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left,dashed]{r}{\delta}\arrow[swap, dash]{d}{\alpha_{i_2}} & t(\mathbf u^1, \mathbf b^2)\dar[dash]{\alpha_{i_2}}\\
t(\mathbf u^2,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left,dashed]{r}{\delta}\arrow[swap, dash]{d}{\alpha_{i_3}} & t(\mathbf u^2, \mathbf b^2)\dar[dash]{\alpha_{i_3}}\\
\vdots&\vdots\\
t(\mathbf u^{l-1},\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left,dashed]{r}{\delta}\arrow[swap, dash]{d}{\alpha_{i_{l-1}}} & t(\mathbf u^{l-1}, \mathbf b^2)\dar[dash]{\alpha_{i_{l-1}}}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left,dashed]{r}{\delta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}\end{center}
where $t(\mathbf u^{j},\mathbf b^1)\,\delta\,t(\mathbf u^j,\mathbf b^2)$ follows (inductively) from the fact that $t(\mathbf u^{j-1},\mathbf b^1)\,\delta\,t(\mathbf u^{j-1},\mathbf b^2)$ and that $C(\alpha_{i_j},\beta;\delta)$.
(c) If we have\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{\alpha\cap\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
then the path $t(\mathbf a^2,\mathbf b^1)\,\alpha\,t(\mathbf a^1,\mathbf b^1)\,(\alpha\cap\beta)\, t(\mathbf a^1,\mathbf b^2)\,\alpha\, t(\mathbf a^2,\mathbf b^2)$ shows that $t(\mathbf a^2,\mathbf b^1)\,\alpha\, t(\mathbf a^2,\mathbf b^2)$. Since also $t(\mathbf a^2,\mathbf b^1)\,\beta\, t(\mathbf a^2,\mathbf b^2)$, we see that $t(\mathbf a^2,\mathbf b^1)\,(\alpha\cap\beta)\, t(\mathbf a^2,\mathbf b^2)$, as required.
\eproof
Since $C(\alpha,\beta;\delta_j)$ for all $j\in J$ implies that $C(\alpha,\beta,\bigcap_{j\in J}\delta_j)$, it follows that there is a {\em smallest} congruence $\delta$ such that $C(\alpha,\beta;\delta)$. This congruence is called the {\em commutator} of $\alpha$ and $\beta$, and denoted by $[\alpha,\beta]$:
\begin{definition}\rm
\begin{enumerate}[(a)]\item If $\alpha,\beta\in\Con(A)$, then {\em commutator} of $\alpha$ and $\beta$ is the congruence
\[[\alpha,\beta]:=\bigwedge\{\delta\in\Con(A): C(\alpha,\beta;\delta)\}\]
\item An algebra $A$ is said to be {\em Abelian} if and only if \[[1_A,1_A]=0_A\] (where $1_A,0_A$ are respectively, the top and bottom elements of $\Con(A)$).
\end{enumerate}
\endbox
\end{definition}
\begin{proposition} \begin{enumerate}[(a)]\item The commutator is monotone in both variables, i.e. if $\alpha\subseteq\alpha',\beta\subseteq\beta'$, then $[\alpha,\beta]\subseteq[\alpha',\beta']$\item $[\alpha,\beta]\subseteq \alpha\cap\beta$
\end{enumerate}\end{proposition}
\bproof (a) This follows easily from the fact that $M(\alpha,\beta)\subseteq M(\alpha',\beta')$: If the $\alpha',\beta'$--term condition holds (modulo $\delta$), then the $\alpha,\beta$--term condition holds, i.e. $C(\alpha',\beta';\delta)$ implies $C(\alpha,\beta;\delta)$. In particular $C(\alpha,\beta;[\alpha',\beta'])$, so $[\alpha,\beta]\subseteq[\alpha',\beta']$.
(b) We have shown $C(\alpha,\beta;\alpha\cap\beta)$.
\eproof
In particular, if $A$ is Abelian, then $[\alpha,\beta]\subseteq [1_A,1_A]=0_A$, i.e. $[\alpha,\beta]=0_A$ for any $\alpha,\beta\in \Con(A)$.
\subsection{Some Examples}
\begin{example}\rm Let us verify that the definition of commutator is the same as that for groups, i.e. that \[[\Theta_M,\Theta_N]=\Theta_{[M,N]}\] when $M,N\trianglelefteq G$. In particular, Abelian groups are exactly those groups $G$ satisfying $[1_G,1_G]=0_G$.
At the end of section \ref{subsec_TC_Grp}, we showed that
if $[M,N]\subseteq K\subseteq M\cap N$, then for any $(m+n)$--ary term $t$ and any $\mathbf m^1,\mathbf m^2\in M^m$ and $\mathbf n^1,\mathbf n^2\in N^n$, we have
\begin{center}\begin{tikzcd}
t(\mathbf m^1,\mathbf n^1)\arrow[dash, bend left]{r}{\Theta_K}\arrow[dash]{r}{\Theta_N}\arrow[dash]{d}{\Theta_M} & t(\mathbf m^1, \mathbf n^2)\dar[dash]{\Theta_N}\\
t(\mathbf m^2,\mathbf n^1)\arrow[dash, bend left, dashed]{r}{\Theta_K}\arrow[dash]{r}{\Theta_N} & t(\mathbf m^2, \mathbf n^2)
\end{tikzcd}
\end{center}
i.e. \[t(\mathbf m^1,\mathbf n^1)\;\Theta_K\;t(\mathbf m^1,\mathbf n^2)\quad\Longrightarrow \quad t(\mathbf m^2,\mathbf n^1)\;\Theta_K\; t(\mathbf m^2,\mathbf n^2)\]
i.e. that $C(\Theta_M,\Theta_N;\Theta_K)$ whenever $[M,N]\subseteq K\subseteq M\cap N$.
In particular, we have $C(\Theta_M,\Theta_N;\Theta_{[M,N]})$, and thus $[\Theta_M,\Theta_N]\subseteq \Theta_{[M,N]}$.
For the converse, suppose that the normal subgroup $K$ corresponds to the congruence $[\Theta_M,\Theta_N]$, i.e. that $K:=\{g\in G: g\,[\Theta_M,\Theta_N]\,1\}$, so that $[\Theta_M,\Theta_N]=\Theta_K$. We must show that $[\Theta_M,\Theta_N]\supseteq \Theta_{[M,N]}$, i.e. that $K\supseteq [M,N]$.
Consider the term $t(x,y):= [x,y]=x^{-1}y^{-1}xy$, and let $m\in M,n\in N$. Then $m\,\Theta_M\, 1$ and $n\, \Theta_N\,1$.
As $[m,1]=1=[1,n]$, and as $(1,1)$ belongs to any congruence, we have \begin{center}\begin{tikzcd}
1= [m,1]\arrow[dash]{r}{\Theta_N}\arrow[dash, bend left]{r}{\Theta_K}\arrow[dash]{d}{\Theta_M} & {[1,1]=1}\dar[dash]{\Theta_M}\\
{[m,n]}\arrow[dash]{r}{\Theta_N} & {[1,n]=1}
\end{tikzcd}
\end{center}
Since $C(\Theta_M,\Theta_N;\Theta_K)$ (by definition of $K$), we see that $[m,n]\,\Theta_K\,1$, i.e. that $[m,n]\in K$. It follows that $[M,N]\subseteq K$ as required.
\endbox
\end{example}
\begin{example}\rm {\bf Abelian Sets:}\newline A {\em set} $A$ can be regarded as an algebra with an empty set of fundamental operations, so that the clone of term operations on $A$ consists only of the projection mappings: $t(x_1,\dots, x_n)=x_i$ for some $i=1,\dots, n$.
It is then obvious that if $t(\mathbf a^1,\mathbf b^1)= t(\mathbf a^1,\mathbf b^2)$, then also $t(\mathbf a^2,\mathbf b^1)= t(\mathbf a^2,\mathbf b^2)$ Hence if \begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash]{r}{1_A}\arrow[dash, bend left]{r}{0_A}\arrow[dash]{d}{1_A} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{1_A}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{1_A} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
then $t(\mathbf a^2,\mathbf b^1)\,0_A\, t(\mathbf a^2,\mathbf b^2)$, so that $C(1_A,1_A;0_A)$. It follows that $[1_A,1_A]=0_A$, i.e. that every set is Abelian.
The same argument also applies, for example, to the variety of semigroups satisfying the equation $xy=x$. In that case, too, the clone of term operations consists of projections only, and every such semigroup is Abelian.
\endbox
\end{example}
\begin{example}\rm {\bf Abelian Lattices:}
Consider the term operation $t(x,y,z)=x\land y\land z$ in the language of lattice theory.
In any lattice $A$, we have, for any $a,b\in A$, that\begin{center}\begin{tikzcd}
a\land a\land b\arrow[dash]{r}{1_A}\arrow[dash, bend left]{r}{0_A}\arrow[dash]{d}{1_A} & a\land b\land b\dar[dash]{1_A}\\
b\land a\land b\arrow[dash]{r}{1_A} & b\land b\land b
\end{tikzcd}\end{center}
Hence if $[1_A,1_A]=0_A$, i.e. if $C(1_A,1_A;0_A)$, it must follow that $(b\land a\land b)\,0_A\,(b\land b\land b)$, i.e. that $a\land b = b$. Thus the ony Abelian lattices are the trivial lattices.
As the above argument only used the $\land$--operation, the same is true for semilattices: Only the trivial semilattices are Abelian.
\endbox
\end{example}
\begin{example}\rm {\bf The Commutator in Lattice Theory:}\newline
A term $m(x,y,z)$ is called a {\em majority term} if it satisfies the following equations:
\[t(x,x,y)\approx t(x,y,x) \approx t(y,x,x)\approx x\]
In lattice theory, the term $m(x,y,z):=(x\land y)\lor (x\land z)\lor(y\land z)$ is clearly a majority term.
Now suppose that $A$ is a lattice, that $a,b\in A$, $\alpha,\beta\in \Con(A)$, and that $a\,(\alpha\cap\beta)\, b$.
Then
\begin{center}\begin{tikzcd}
m(a,a,a)\arrow[dash]{r}{\beta}\arrow[dash, bend left]{r}{[\alpha,\beta]}\arrow[dash]{d}{\alpha} & m(a,a,b)\dar[dash]{\alpha}\\
m(b,a, a)\arrow[dash]{r}{\beta} &m(b,a,b)
\end{tikzcd}\end{center}
Since $C(\alpha,\beta;[\alpha,\beta])$, we conclude that $m(b,a,a)\,[\alpha,\beta]\, m(b,a,b)$, i.e. that $a\,[\alpha,\beta]\,b$. It follows that $\alpha\cap\beta\subseteq [\alpha,\beta]$, and thus that \[[\alpha, \beta]=\alpha\cap \beta\] for any algebra that has a majority term.
In fact, it can be shown that any variety that has a majority term is congruence distributive. Moreover, it can be shown that the commutator operation in any congruence distributive variety is intersection.
\endbox
\end{example}
\subsection{The Center}
\begin{definition}\rm Let $A$ be an algebra, and let $\alpha\in\Con(A)$.
\begin{enumerate}[(a)]\item $\alpha$ is said to be an {\em Abelian congruence} if $[\alpha,\alpha]=0_A$.
\item The algebra $A$ is said to be {\em Abelian} if $[1_A,1_A]=0_A$.
\end{enumerate}\endbox
\end{definition}
Observe that an algebra $A$ is Abelian if and only if every $\alpha\in\Con(A)$ is an Abelian congruence. This follows from the monotonicity of the commutator: $[\alpha,\alpha]\leq[1_A,1_A]$.
Let $\{\alpha_i:i\in I\}$ be the set of all $\alpha\in\Con(A)$ with the property that $[\alpha,1_A]=0_A$. Then $C(\alpha_i,1_A;0_A)$ for all $i\in I$. By Proposition \ref{propn_properties_term_commutator}, we see that $C(\bigvee_{i\in I}\alpha_i;1_A;0_A)$, and thus that $[\bigvee_{i\in I}\alpha_i,1_A]=0_A$. It follows that $\bigvee_{i\in I}\alpha_i$ is the {\em largest} congruence $\alpha$ such that $[\alpha,1_A]=0_A$. We give this congruence a name:
\begin{definition}\rm Let $A$ be an algebra. The {\em center} of $A$ is the largest congruence $\alpha\in\Con(A)$ with the property that $[\alpha,1_A]=0_A$. It is denoted by $\zeta_A$.
\endbox
\end{definition}
Clearly, an algebra $A$ is Abelian if and only if $\zeta_A=1_A$. An algebra is said to be {\em centerless} if $\zeta_A=0_A$.
\begin{remarks}\rm Recall that if $G$ is a group, then the group--theoretic center is the normal subgroup $Z(G)$ defined by
\[Z(G):=\{z\in G:\forall g\in G\;([z,g]=1)\}\]
Clearly, therefore, $[Z(G),G]=\{1\}$ is trivial. Moreover, if $K\trianglelefteq G$ has $[K,G]=\{1\}$, then $K\leq Z(G)$. Thus $Z(G)$ is indeed the largest normal subgroup $K$ of $G$ with the property that $[K,G]=\{1\}$.
\endbox
\end{remarks}
Now since $C(\zeta_A, 1_A;0_A)$ we have
\begin{center}\begin{tikzcd}
t(x,\mathbf a)\rar[dash, bend left]{0_A}\rar[dash]{1_A}\dar[dash,swap]{\zeta_A}&t(x,\mathbf b)\dar[dash]{\zeta_A}\\
t(y,\mathbf a)\rar[dash,bend left, dashed]{0_A}\rar[dash]{1_A}&t(y,\mathbf b)
\end{tikzcd}\qquad\qquad whenever $x\;\zeta_A\;y$
\end{center} i.e. we have that for any $(n+1)$--ary term $t$ and any $x,y\in A$, $\mathbf a,\mathbf b\in A^n$
\begin{center} If $x\;\zeta_A\;y$\qquad then \qquad $t(x,\mathbf a) = t(x,\mathbf b) \longrightarrow t(y,\mathbf a)=t(y,\mathbf b)$
\end{center} This property characterizes the center:
\begin{proposition}\label{propn_char_center} Let $A$ be an algebra.
Then $(x,y)\in\zeta_A$ if and only if for any $n\in\mathbb N$, any $(n+1)$--ary term $t$ and any $\mathbf a,\mathbf b\in A^n$
\[t(x,\mathbf a) = t(x,\mathbf b) \longleftrightarrow t(y,\mathbf a)=t(y,\mathbf b)\]
\end{proposition}
\bproof Let $\Gamma$ be the set of all ordered pairs $(x,y)$ which satisfy the condition that for any term $t$ and any $\mathbf a,\mathbf b\in A^n$ we have
\[t(x,\mathbf a) = t(x,\mathbf b) \longleftrightarrow t(y,\mathbf a)=t(y,\mathbf b)\]
We must prove that $\Gamma=\zeta_A$. Now we have already seen that $\zeta_A\subseteq \Gamma$, and hence there remain two thing to prove
\begin{enumerate}[(i)]\item that $\Gamma$ is a congruence, and
\item that $[\Gamma, 1_A]=0_A$.
\end{enumerate}
To prove that $\Gamma$ is a congruence, it suffices to show that $\Gamma$ is an equivalence relation which is compatible with all the unary polynomials, by Proposition \ref{propn_congruence_unary_pol}. That $\Gamma$ is an equivalence relation is straightforward. So let $p(x)$ be a unary polynomial. We must show that \[ (x,y)\in \Gamma\qquad\Longrightarrow\qquad (p(x),p(y))\in\Gamma\] i.e. that
\[t(p(x),\mathbf a)=t(p(x),\mathbf b)\longleftrightarrow t(p(y),\mathbf a)=t(p(y),\mathbf b)\] for any $(n+1)$--ary term $t$ and $\mathbf a,\mathbf b\in A^n$. But as $p(x)$ is a unary polynomial, it is of the form $p(x) = s(x,\mathbf c)$ for some $(m+1)$--ary term $s$ and $\mathbf c\in A^m$.
Thus
\[\aligned t(p(x),\mathbf a)&=t(p(x),\mathbf b)\\
\Longleftrightarrow\qquad t(s(x,\mathbf c),\mathbf a)&= t(s(x,\mathbf c),\mathbf b)\\
\Longleftrightarrow\qquad t(s(y,\mathbf c),\mathbf a)&= t(s(y,\mathbf c),\mathbf b)\\&\qquad\text{because $t(s(x,\mathbf x),\mathbf y)$ is a term and $(x,y)\in\Gamma$}\\
\Longleftrightarrow\qquad\quad t(p(y),\mathbf a)&=t(p(y),\mathbf b)
\endaligned\] Hence $\Gamma$ is compatible with the unary polynomials, and is thus a congruence.
It remains to show that $[\Gamma,1_A]=0_A$, i.e. that $C(\Gamma,1_A;0_A)$. Suppose therefore that $t(x_1,\dots, x_m,y_1,\dots, y_n)$ is an $(m+n)$--ary term, and that $\mathbf x,\mathbf y\in A^m$, and $\mathbf a,\mathbf b\in A^n$, where $x_i\;\Gamma\; y_i$ for all $i\leq m$ (and, of course, automatically $a_j\;1_A\;b_j$ for all $j\leq n$). We must show that
\begin{center}\begin{tikzcd}
t(\mathbf x,\mathbf a)\arrow[dash]{r}{1_A}\rar[equal,bend left]{0_A}\arrow[dash,swap]{d}{\Gamma} & t(\mathbf x, \mathbf b)\dar[dash]{\Gamma}\\
t(\mathbf y,\mathbf a)\arrow[dash]{r}{1_A} & t(\mathbf y, \mathbf b)
\end{tikzcd}\qquad $\Longrightarrow\qquad t(\mathbf y,\mathbf a) = t(\mathbf y,\mathbf b)$ \end{center}
So suppose that $t(\mathbf x,\mathbf a) = t(\mathbf x,\mathbf b)$.
Then
\[ \aligned \text{We have}\qquad t(\fbox{$x_1$},x_2,\dots x_m,\mathbf a) &= t(\fbox{$x_1$},x_2,\dots, x_m,\mathbf b)\\\text{hence}\qquad
t(\fbox{$y_1$},x_2,\dots, x_m,\mathbf a) &= t(\fbox{$y_1$},x_2,\dots, x_m,\mathbf b)\endaligned\]because
$x_1\;\Gamma\;y_1$.
\[\aligned\text{Now}\qquad t(y_1,\fbox{$x_2$},x_3,\dots x_m,\mathbf a) &= t(y_1\fbox{$x_2$},x_3,\dots, x_m,\mathbf b)\\\text{and thus}\qquad t(y_1,\fbox{$y_2$},x_3,\dots, x_m,\mathbf a) &= t(y_1,\fbox{$y_2$},x_3,\dots, x_m,\mathbf b)\endaligned\]because
$x_2\;\Gamma\;y_2$. Proceeding inductively in this way, we obtain
\[t(y_1,\dots, y_m,\mathbf a) = t(y_1,\dots, y_m,\mathbf b)\] as required.
\eproof
\section{Terms and Congruence Identities: Mal'tsev Conditions}
\fancyhead[RE]{Terms and Congruence Identities}
\subsection{Permutability: Mal'tsev Terms}
Recall that an algebra $A$ is said to be {\em congruence permutable} if and only if $\theta\circ\varphi =\varphi\circ\theta$ holds for any $\theta,\varphi\in\Con(A)$. In that case $\theta\lor\varphi=\theta\circ\varphi$ in $\Con(A)$.
A variety is said to be congruence permutable if each of its member algebras is congruence permutable.
\begin{theorem} A variety $\mathcal V$ is congruence permutable if and only if there is a ternary term $p(x,y,z)$ (called a {\em Mal'tsev term}) such that $\mathcal V$ satisfies the following identities:
\[p(x,x,z)\approx z\qquad\qquad p(x,z,z)\approx x\]
\end{theorem}
\bproof We have already seen that any algebra which has a ternary polynomial satisfying the Mal'tsev identities is congruence permutable, to whit:
\begin{center}\begin{tikzcd} a\rar[dash]{\theta}&c\dar[dash]{\varphi}\\&b\end{tikzcd}\qquad\qquad$\Longrightarrow$\qquad\qquad
\begin{tikzcd} a=p(a,b,b)\rar[dash]{\theta}\dar[dash,swap]{\varphi}&p(c,b,b)=c\dar[dash]{\varphi}\\p(a,c,b)\rar[dash]{\theta}&p(c,c,b)=b\end{tikzcd}
\end{center}
Conversely, suppose that $\mathcal V$ is congruence permutable, and let $F=F_{\mathcal V}(x,y,z)$ be the $\mathcal V$--free algebra on three generators.
Let $f,g:F\to F$ be the unique homomorphisms satisfying
\[\left\{\aligned f(x)&=f(y)=x\\f(z)&=z\endaligned\right.\qquad\qquad \left\{\aligned g(x)&=x\\
g(y)&=g(z)=z\endaligned\right.\] and define $\theta:=\ker f,\varphi:=\ker g$.
Then certainly $(x,z)\in \theta\circ\varphi$, since $x\;\theta\;y\;\varphi\; z$. By permutability, there is $w\in F$ such that $x\;\varphi\;w\;\theta\;z$. Now since $w$ is an element of the free algebra $F$, it is of the form $w=p^F(x,y,z)$ for some term $p(x,y,z)$.
Since $x\;\varphi\; p^F(x,y,z)$, we must have $g(x)= g(p^F(x,y,z))=p^F(g(x),g(y),g(z))$, and hence $x=p(x,z,z)$.
Similarly, since $p^F(x,y,z)\;\theta\;z$, we have $f(z)=f(p^F(x,y,z))=p^F(f(x),f(y),f(z))$, so that $z= p^F(x,x,z)$.
Hence the Malt'sev identities hold in $F$. Since $F$ is the $\mathcal V$--algebra on 3 generators, they hold in $\mathcal V$.
\eproof
From the proof of the preceding theorem, we obtain:
\begin{corollary} A variety is congruence permutable if and only if its free algebra on 3 generators is congruence permutable.\endbox
\end{corollary}
\subsection{Distributivity: J\'onsson Terms}
An algebra $A$ is said to be {\em congruence distributive} if and only if $\Con(A)$ is a distributive lattice, i.e. if and only if \[\theta\land(\varphi\lor\psi)=(\theta\land\varphi)\lor(\theta\land\psi)\qquad\text{for all }\quad \theta,\varphi,\psi\in\Con(A)\]
(The other distributive identity $\theta\lor(\varphi\land\psi)=(\theta\lor\varphi)\land(\theta\lor\psi)$ follows automatically: Any lattice satisfying one of the distributive identities can easily be shown to satisfy also the other.)
A variety of algebras is said to be congruence distributive if and only if each of its member algebras is congruence distributive.
\begin{theorem} A variety $\mathcal V$ of algebras is congruence distributive if and only if for some $n\in\mathbb N$ there is a sequence $d_0, d_1,\dots, d_n$ of ternary terms (called {\em J\'onsson terms}), such that the following identities hold in $\mathcal V$:
\begin{enumerate}[\rm ({J}1)]\item $d_0(x,y,z)\approx x$
\item $d_{i}(x,y,x)\approx x$ for all $i\leq n$.
\item $d_i(x,x,z)\approx d_{i+1}(x,x,z)$ if $i<n$ is even.
\item $d_{i}(x,z,z)\approx d_{i+1}(x,z,z)$ if $i<n$ is odd.
\item $d_n(x,y,z)\approx z$.
\end{enumerate}
\end{theorem}
\bproof
$(\Longrightarrow)$: Suppose that $\mathcal V$ has J\'onsson terms $d_0,\dots, d_n$. Let $A\in\mathcal V$ and let $\theta,\varphi,\psi\in\Con(A)$. To prove that $\Con(A)$ is distributive, it suffices to show that \[\theta\land(\varphi\lor\psi)\leq (\theta\land\varphi)\lor(\theta\land\psi)\] since the opposite inequality $\geq$ always holds.
Suppose, therefore that $(a,c)\in \theta\land(\varphi\lor\psi)$. Then $(a,c)\in\theta$, and $(a,c)\in (\varphi\lor\psi)=\bigcup_{i=1}^\infty,(\varphi\lor\psi)^n$, so there is a chain $a=x_0,x_1,x_2,\dots, x_m=c$ such that
\[a=x_0\;\varphi\;x_1\;\psi\;x_2\;\varphi\;x_3\;\psi\;x_4\;\dots\;x_m=c\]
Now as $a=d_i(a,x_k,a)$ for all $i\leq n$, and since $a\;\theta\;c$, we see that
\[d_i(a,x_k,c)\;\theta\;d_i(a,x_k,a)=a=d_{i}(a,x_{k+1},a)\;\theta\; d_{i}(a, x_{k+1},c)\]
i.e. that $d_{i}(a,x_k,c)\;\theta\;d_i(a,x_{k+1},c)$ for all $i\leq n$ and $k<m$. Thus
\begin{center}\begin{tikzcd} a=d_0(a,a,c)\dar[equals]\\
d_1(a,a,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_1(a,x_1,c)\rar[dash]{\psi}\rar[dash, bend left]{\theta}&d_1(a,x_2,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_1(a,x_3,c)\rar[dash,dashed]&d_1(a,c,c)\dar[equals]\\
d_2(a,a,c)\dar[equals]\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_2(a,x_1,c)\rar[dash]{\psi}\rar[dash, bend left]{\theta}&d_2(a,x_2,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_2(a,x_3,c)\rar[dash,dashed]&d_2(a,c,c)\\
d_3(a,a,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_3(a,x_1,c)\rar[dash]{\psi}\rar[dash, bend left]{\theta}&d_3(a,x_2,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_3(a,x_3,c)\rar[dash,dashed]&d_3(a,c,c)\dar[equals]\\
d_4(a,a,c)\dar[dash,dashed]\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_4(a,x_1,c)\rar[dash]{\psi}\rar[dash, bend left]{\theta}&d_4(a,x_2,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_4(a,x_3,c)\rar[dash,dashed]&d_4(a,c,c)\\{}\\
d_n(a,a,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_n(a,x_1,c)\rar[dash]{\psi}\rar[dash, bend left]{\theta}&d_n(a,x_2,c)\rar[dash]{\varphi}\rar[dash, bend left]{\theta}&d_n(a,x_3,c)\rar[dash,dashed]&d_n(a,c,c)
\end{tikzcd}
\end{center}where each element of the bottom row is equal to $c$. It follows that $(a,c)\in(\theta\land\varphi)\lor(\theta\land\psi)$, as required.
\vskip0.3cm\noindent$(\Longleftarrow)$: Conversely, suppose that $\mathcal V$ is congruence distributive. Let $F:=F_{\mathcal V}(x,y,z)$ be the $\mathcal V$--free algebra on 3 generators. Define $f,g,h:F\to F$ to be the unique homomorphisms such that
\[\left\{\aligned f(x)&=f(y)=x\\f(z)&=z\endaligned\right.\qquad
\left\{\aligned g(x)&=x\\g(y)&=g(z)=z\endaligned\right.\qquad
\left\{\aligned h(x)&=h(z)=x\\
h(y)&=y\endaligned\right. \] and let $\varphi:=\ker f,\psi:=\ker g$ and $\theta:=\ker h$.
Then $(x,z)\in \theta$. Also $x\;\varphi\;y\;\psi\;z$, so $(x,z)\in(\varphi\lor\psi)$. It follows that $(x,z)\in\theta\land(\varphi\lor\psi)$ so that $(x,z)\in(\theta\land\varphi)\lor(\theta\land\psi)$ by congruence distributivity.
Since $F$ is generated by $x,y,z$, there are, for some $n\in\mathbb N$, ternary terms $d_0,d_1,\dots,d_n$
such that
\[x=d_0^F(x,y,z)\;(\theta\land\varphi)\;d_1^F(x,y,z)\;(\theta\land\psi)\;d_2^F(x,y,z)\;(\theta\land\varphi)\;d_3^F(x,y,z)\;(\theta\land\psi)\;\dots d_n^F(x,y,z)=z\]
We now verify that $d_0,\dots, d_n$ satisfy the J\'onsson identities (J1)-(J5) in $F$:
\begin{enumerate}[\rm({J}1)]\item $d_0^F(x,y,z)=x$ by definition of $d_0$.
\item We have \[x=d_0^F(x,y,z)\;\theta\;d_1^F(x,y,z)\;\theta\;d_2^F(x,y,z)\;\theta\;d_3^F(x,y,z)\;\theta\;\dots d_n^F(x,y,z)=z\] and thus
\[h(x)=h(d_0^F(x,y,z)) =h(d_1^F(x,y,z))=h(d_2^F(x,y,z)) =h(d_3^F(x,y,z)=\dots h(d_n^F(x,y,z))\]
Now as $h(x)=x$ and $h(d_i^F(x,y,z)) = d_i^F(h(x),h(y),h(z)) = d_i(x,y,x)$, we obtain $x=d_i(x,y,x)$ for all $i\leq n$.
\item We have $d_i^F(x,y,z)\;\varphi\;d_{i+1}^F(x,y,z)$ when $i<n$ is even. It follows that $f(d_i^F(x,y,z))=f(d_{i+1}^F(x,y,z))$, and thus that $d_i^F(x,x,z)=d_{i+1}^F(x,x,z)$ when $i<n$ is even.
\item We have $d_i^F(x,y,z)\;\psi\;d_{i+1}^F(x,y,z)$ when $i<n$ is odd. It follows that $g(d_i^F(x,y,z))=g(d_{i+1}^F(x,y,z))$, and thus that $d_i^F(x,z,z)=d_{i+1}^F(x,z,z)$ when $i<n$ is odd.
\item $d_n^F(x,y,z)=z$ by definition of $d_n$.
\end{enumerate}
Hence the J\'onsson identities hold in $F$. Since $F$ is $\mathcal V$--free on 3 generators, they hold in $\mathcal V$.
\eproof
\subsection{Modularity: Day Terms}
Let $\mathcal V$ be a variety of algebras. Consider the following statements:
\begin{enumerate}[I.]\item {\bf CM}: $\mathcal V$ is {\em congruence modular}, i.e. for all algebras $A\in\mathcal V$ and all $\alpha,\beta,\gamma\in \Con(A)$,
\[\alpha\geq\gamma\qquad\Longrightarrow\qquad \alpha\land(\beta\lor\gamma) = (\alpha\land\beta)\lor\gamma\]
\item {\bf SL}: $\mathcal V$ satisfies the {\em Shifting Lemma}: If $\alpha,\beta,\gamma\in\Con(A)$ such that $\alpha\land\beta\leq\gamma$, then
\begin{center}
\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}&c\end{tikzcd}\qquad implies\qquad\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}&c\end{tikzcd}\end{center}
\item {\bf D}: $\mathcal V$ has {\em Day terms}: There are terms $m_0(x,y,z,u), m_1(x,y,z,u),\dots, m_n(x,y,z,u)$ such that $\mathcal V$ satisfies the following identities:
\begin{enumerate}[\rm({D}1)]
\item $m_0(x,y,z,u)\approx x$.
\item $m_i(x,y,y,x)\approx x$\qquad for all $0\leq i\leq n$.
\item $m_i(x,x,z,z)\approx m_{i+1}(x,x,z,z)$ \qquad if $0\leq i<n$ is even.
\item $m_i(x,y,y,u)\approx m_{i+1}(x,y,y,u)$ \qquad if $0\leq i<n$ is odd.
\item $m_n(x,y,z,u)\approx u$.
\end{enumerate}
\item {\bf SP}: $\mathcal V$ satisfies the {\em Shifting Principle}: If $\alpha,\gamma\in\Con(A)$, and $\beta\subseteq A^2$ is a compatible reflexive binary relation\footnote{i.e. $\beta$ is a subalgebra of $A^2$ which contains the diagonal $\{(a,a):a\in A\}$.} such that $\alpha\cap\beta\subseteq\gamma$, then
\begin{center}
\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}&c\end{tikzcd}\qquad implies\qquad\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}&c\end{tikzcd}\end{center}
\end{enumerate}
We shall prove
\begin{theorem} \label{thm_char_modularity} The following are equivalent:
\begin{enumerate}[(i)]\item {\bf CM}
\item {\bf SL}
\item {\bf D}
\item {\bf SP}
\end{enumerate}
\endbox
\end{theorem}
Our method will be to show, in a series of lemmas, that\[
\text{\bf CM}\quad\Longrightarrow\quad \text{\bf SL}\quad\Longrightarrow\quad \text{\bf D}\quad\Longrightarrow\quad
\text{\bf SP}\quad\Longrightarrow\quad \text{\bf CM}\]
\begin{lemma} $\text{\bf CM}\quad\Longrightarrow\quad \text{\bf SL}$
\end{lemma}
\bproof Suppose that $A\in\mathcal V$, that $\alpha,\beta,\gamma\in\Con(A)$ and that $\gamma\geq\alpha\land\beta$ are such that \begin{center}
\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}&c\end{tikzcd}\end{center}
We must show that $(a,c)\in\gamma$. Now $(a,c)\in \alpha\land(\beta\circ(\alpha\land\gamma)\circ\beta)\subseteq \alpha\land(\beta\lor(\alpha\land\gamma))$. and hence by modularity we have $(a,c)\in(\alpha\land\beta)\lor(\alpha\land\gamma)\subseteq\gamma$.
\eproof
\begin{lemma} $\text{\bf SL}\quad\Longrightarrow\quad \text{\bf D}$
\end{lemma}
\bproof
Let $F=F_{\mathcal V}(x,y,z,u)$ be the $\mathcal V$--free algebra on four generators.
Define $f,g,h:F\to F$ to be the unique homomorphisms such that
\[\left\{\aligned &f(x)=f(u)=x\\
&f(y)=f(z)=y\endaligned\right. \qquad \left\{\aligned &g(x)=g(y)=x\\
&g(z)=g(u)=z\endaligned\right. \qquad \left\{\aligned &h(x)=x\\
&h(y)=h(z)=y\\
&h(u)=u\endaligned\right. \]
Also define $\alpha,\beta,\gamma\in\Con(F)$ by
\[\alpha :=\ker f=\Cg(x,u)\lor\Cg(y,z)\qquad \beta=\ker g= \Cg(x,y)\lor \Cg(u,z)\qquad \gamma:=\ker h = \Cg(y,z)\]
Clearly $\gamma\leq\alpha$.
Now \begin{center}
\begin{tikzcd}y\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&z\dar[dash]{\beta}\\
x\rar[dash]{\alpha}&u\end{tikzcd}\end{center}
Define $\bar{\gamma}:=(\alpha\land\beta)\lor\gamma$. Then $\alpha\land\beta\leq\bar{\gamma}$, and also \begin{center}
\begin{tikzcd}y\rar[dash]{\alpha}\rar[dash, bend left]{\bar{\gamma}}\dar[dash]{\beta}&z\dar[dash]{\beta}\\
x\rar[dash]{\alpha}&u\end{tikzcd}\end{center} so, by the Shifting Lemma, we may deduce that $(x,u)\in \bar{\gamma}$, i.e. that $(x,u)\in \gamma\lor(\alpha\land\beta)$.
It follows that there are terms $m_0(x,y,z,u), m_1(x,y,z,u),\dots, m_n(x,y,z,u)$ such that in the algebra $F$ we have
\[x=m_0^F(x,y,z,u)\;(\alpha\land\beta)\; m_1^F(x,y,z,u)\; \gamma\; m_2^F(x,y,z,u)\;(\alpha\land\beta)\; m_3^F(x,y,z,u)\;\gamma\dots m_n^F(x,y,z,u)=u\]
In particular $m_0^F(x,y,z,u)=x$ and $m_n^F(x,y,z,u)= u$, showing that (D1) and (D5) of the Day term identities are satisfied.
Now since $\gamma\leq\alpha$, we see that
\[x=m_0^F(x,y,z,u)\;\alpha\; m_1^F(x,y,z,u)\; \alpha\; m_2^F(x,y,z,u)\;\alpha\; m_3^F(x,y,z,u)\;\alpha\dots m_n^F(x,y,z,u)=u\] and thus, because $\alpha:=\ker f$, that
\[f(x)=m_0^F(f(x),f(y),f(z),f(u))=m_1^F(f(x),f(y),f(z),f(u)) =\dots = m_n^F(f(x),f(y),f(z),f(u))\]
It follows that
\[x=m_0^F(x,y,y,x) = m_1^F(x,y,y,x)=\dots=m_n^F(x,y,y,x)\] which shows that (D2) of the Day term identities is satisfied.
Finally, we have
\[m_0^F(x,y,z,u)\;\beta\; m_1^F(x,y,z,u)\; \gamma\; m_2^F(x,y,z,u)\;\beta\; m_3^F(x,y,z,u)\;\gamma\dots m_n^F(x,y,z,u)\]
Thus \[\left\{\aligned m_i^F(x,y,z,u)\;\beta\;m_{i+1}^F(x,y,z,u)\quad&\text{if } i<n\text{ is even}\\
m_i^F(x,y,z,u)\;\gamma\; m_{i+1}^F(x,y,z,u)\quad&\text{if } i<n\text{ is odd}\endaligned\right.\]
Since $\beta:=\ker g$, and $\gamma:=\ker h$, we obtain
\[\left\{\aligned m_i^F(g(x),g(y),g(z),g(u)) =m_{i+1}^F(g(x),g(y),g(z),g(u))\quad&\text{if } i<n\text{ is even}\\
m_i^F(h(x),h(y),h(z),h(u)) = m_{i+1}^F(h(x),h(y),h(z),h(u))\quad&\text{if } i<n\text{ is odd}\endaligned\right.\]
and thus \[\left\{\aligned m_i^F(x,x,z,z) =m_{i+1}^F(x,x,z,z)\quad&\text{if } i<n\text{ is even}\\
m_i^F(x,y,y,u) = m_{i+1}^F(x,y,y,u)\quad&\text{if } i<n\text{ is odd}\endaligned\right.\]
Thus the Day term identities (D1)-(D5) hold in $F$. Since $F$ is the $\mathcal V$--free algebra on 4 generators, they hold in $\mathcal V$.
\eproof
Next, we want to show that {\bf D} $\Longrightarrow$ {\bf SP}. To that end, we first prove the following lemma:
\begin{lemma}\label{lemma_a_gamma_c} Suppose $\mathcal V$ has Day terms $m_0,\dots, m_n$ satisfying identities (D1)-(D5). Let $A\in\mathcal V$, $\gamma\in\Con(A)$, with $a,b,c,d\in A$ such that $b\;\gamma\;d$. Then
\[a\;\gamma\; c\qquad \Longleftrightarrow\qquad m_i(a,a,c,c,)\;\gamma\;m_i(a,b,d,c)\quad\text{ for all }i\leq n\]
\end{lemma}
\bproof $(\Longrightarrow)$: If $b\;\gamma\; d$ and $a\;\gamma\;c$, then certainly for all $i\leq n$ we have
\[m_{i}(a,a,c,c)\;\gamma\;m_i(a,a,a,a) = a = m_i(a,b,b,a)\;\gamma\; m_i(a,b,d,c)\]
$(\Longrightarrow$): If $b\;\gamma\; d$ and $m_i(a,a,c,c)\;\gamma\;m_i(a,b,d,c)$ for all $i\leq n$, then
\begin{center}
\begin{tikzcd}
a=m_0(a,a,c,c)\dar[equals]\\
m_1(a,a,c,c)\rar{\gamma}&m_1(a,b,d,c)\rar{\gamma}&m_1(a,b,b,c)\dar[equals]\\
m_2(a,a,c,c)\dar[equals]\rar[leftarrow]{\gamma}&m_2(a,b,d,c)\rar[leftarrow]{\gamma}&m_2(a,b,b,c)\\
m_3(a,a,c,c)\rar{\gamma}&m_3(a,b,d,c)\rar{\gamma}&m_3(a,b,b,c)\dar[equals]\\
{}\dar[dash,dashed]\rar[dash,dashed]&{}\rar[dash,dashed]\dar[dash,dashed]&{}\dar[dash,dashed]\\
m_n(a,a,c,c)\rar[equals]&c\rar[equals]&m_n(a,b,b,c)
\end{tikzcd}
\end{center}
\eproof
\begin{lemma} {\bf D} $\Longrightarrow$ {\bf SP}.
\end{lemma}
\bproof
Supose $\mathcal V$ has Day terms $m_0,\dots, m_n$, and that $A\in\mathcal V$. Suppose further that $\alpha,\gamma\in\Con(A)$ and that $\beta$ is a compatible reflexive binary relation on $A$ such that $\gamma\supseteq\alpha\cap\beta$. Further suppose that
\begin{center}\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{\gamma}\dar[dash]{\beta}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}&c\end{tikzcd} \end{center} We must show that $a\;\gamma\; c$, and by Lemma \ref{lemma_a_gamma_c}, it suffices to show that $m_i(a,a,c,c)\;\gamma\; m_i(a,b,d,c)$ for all $i\leq n$.
Since $\beta$ is compatible and reflexive, we have $m_i(a,a,c,c)\;\beta\; m_i(a,b,d,c)$ for all $i\leq n$.
Furthermore,
\[m_i(a,a,c,c,)\;\alpha\;m_i(a,a,a,a) = a = m_i(a,b,b,a)\;\alpha\; m_i(a,b,d,c)\] and thus $m_i(a,a,c,c)\;\alpha\; m_i(a,b,d,c)$.
It follows that $m_i(a,a,c,c)\;(\alpha\cap\beta)\; m_i(a,b,d,c)$ for all $i\leq n$. Since $\gamma\supseteq \alpha\cap\beta$, the result follows.
\eproof
To complete the proof of Theorem \ref{thm_char_modularity}, it remains to show that:
\begin{lemma} {\bf SP} $\Longrightarrow$ {\bf CM}.
\end{lemma}
\bproof
Suppose that $A\in\mathcal V$ and that $\alpha,\beta,\gamma\in\Con(A)$ are such that $\alpha\geq\gamma$. We must show, assuming the Shifting Principle holds, that $\alpha \land(\beta\lor\gamma)\leq (\alpha\land\beta)\lor\gamma$. Define binary relations $\beta_k$ (for $k\in\mathbb N$) inductively by:
\[\beta_0=\beta\qquad \beta_{k+1}=\beta_k\circ \gamma\circ\beta_k\]Observe that each $\beta_k$ is a reflexive compatible binary relation on $A$. Moreover, $\beta\lor\gamma=\bigcup_{k=1}^\infty \beta_k$, and hence $\alpha\land(\beta\lor\gamma) =\bigcup_{k=1}^\infty (\alpha\cap \beta_k)$.
It therefore suffices to show that $\alpha\cap\beta_k\subseteq (\alpha\land\beta)\lor\gamma$ for all $k\in\mathbb N$. We accomplish this by induction and the Shifting Principle. For the base case, note that indeed $\alpha\cap\beta_0=\alpha\cap\beta\subseteq (\alpha\cap \beta)\lor\gamma$. Assume now that $\alpha\cap\beta_k\subseteq(\alpha\land\beta)\lor\gamma)$, and that $(a,c)\in\alpha\cap\beta_{k+1}$. We wil show that $(a,c) \in (\alpha\land\beta)\lor\gamma$. Now since $\beta_{k+1}=\beta_k\circ\gamma\circ\beta_k$, there are $b,d\in A$ such that
\begin{center}
\begin{tikzcd}b\rar[dash, bend left]{\gamma}\dar[dash]{\beta_k}&d\dar[dash]{\beta_k}\\
a\rar[dash]{\alpha}&c\end{tikzcd}\end{center}
Since $\gamma\leq\alpha$, and $\gamma\leq(\alpha\land\beta)\lor\gamma$, we see that
\begin{center}
\begin{tikzcd}b\rar[dash]{\alpha}\rar[dash, bend left]{(\alpha\land\beta)\lor\gamma}\dar[dash]{\beta_k}&d\dar[dash]{\beta_k}\\
a\rar[dash]{\alpha}&c\end{tikzcd}\end{center}
Finally, since $\alpha\cap\beta_k\subseteq (\alpha\land\beta)\lor\gamma$ (by induction hypothesis) and $\beta_k$ is reflexive and compatible, we obtain from the Shifting Principle that $(a,c)\in (\alpha\cap\beta)\lor\gamma$, as required.
\eproof
\begin{remarks}\rm \begin{enumerate}[(a)]\item Suppose that $\mathcal V$ has a Mal'tsev term $p(x,y,z)$, so that $\mathcal V$ is congruence permutable.
Then \[m_0(x,y,z,u):=x\qquad m_1(x,y,z,u) := p(y,z,u)\qquad m_2(x,y,z,u):=u\] are Day terms.
\item Conversely, if a variety $\mathcal V$ has Day terms $m_0, m_1, m_2$, i.e. $n=2$, then $p(x,y,z):=m_1(x,x,y,z)$ is a Mal'tsev term:
\[p(x,y,y)=m_1(x,x,y,y)=m_0(x,x,y,y) = x\qquad p(x,x,y) = m_1(x,x,x,y)=m_2(x,x,x,y) = y\]
\item Suppose that $\mathcal V$ has J\'onsson terms.
Then $\mathcal V$ is congruence distributive, and hence congruence modular. In particular, $\mathcal V$ should have Day terms. How these can be defined from the J\'onsson terms will become clear in the proof of Lemma \ref{lemma_Gumm_to_Day}.
\end{enumerate}
\endbox
\end{remarks}
\subsection{Modular $\equiv$ (Permutable $\circ$ Distributive): Gumm Terms}
A variety $\mathcal V$ is said to have {\em Gumm terms} if there are ternary terms $p, q_1,\dots, q_n$ such that $\mathcal V$ satisfies the following identities:
\begin{enumerate}[\rm({G}1)]\item $p(x,z,z)\approx x$.
\item $p(x,x,z)\approx q_1(x,x,z)$
\item $q_i(x,y,x)\approx x$ for all $i\leq n$.
\item $q_i(x,x,z)\approx q_{i+1}(x,x,z)$ if $i<n$ is even.
\item $q_i(x,z,z)=q_{i+1}(x,z,z)$ if $i<n$ is odd.
\item $q_n(x,y,z)=z$
\end{enumerate}
We shall prove the following:
\begin{theorem} For any variety $\mathcal V$, the following are equivalent:
\begin{enumerate}[(a)]\item $\mathcal V$ is congruence modular.
\item For any $A\in\mathcal V$ and any $\alpha,\beta,\gamma\in\Con(A)$, we have \[(\alpha\circ \beta)\cap\gamma\subseteq (\beta\circ\alpha)\circ[(\alpha\land\gamma)\lor(\beta\land\gamma)]\]
\item $\mathcal V$ has Gumm terms.
\end{enumerate}\end{theorem}
For ease of exposition, we shall break up the proof into a number of lemmas which show that
\[\text{(a)}\quad\Longrightarrow\quad\text{(b)}\quad\Longrightarrow\quad\text{(c)}\quad\Longrightarrow\quad\text{(a)}\]
Observe that (G1) and (G2) of the identities are reminiscent of the Mal'tsev identities for congruence permutability, whereas (G3)-(G5) are the same as the J\'onsson identities for congruence distributivilty. Hence Gumm's dictum:\begin{center} Modularity = Permutability composed with Distributivity\end{center}
\begin{lemma} If $\mathcal V$ is congruence modular,
then for any $A\in\mathcal V$ and any $\alpha,\beta,\gamma\in\Con(A)$, we have \[(\alpha\circ \beta)\cap\gamma\subseteq(\beta\circ\alpha)\circ[(\alpha\land\gamma)\lor(\beta\land\gamma)]\]\end{lemma}
\bproof
Suppose that $\mathcal V$ is congruence modular, so that $\mathcal V$ has Day terms $m_0,m_1,\dots, m_n$. Let $A\in\mathcal V$ and $\alpha,\beta,\gamma\in\Con(A)$. Define $\theta:=(\alpha\land\gamma)\lor(\beta\land\gamma)$. We must show that
\[(\alpha\circ\beta)\cap\gamma\subseteq\beta\circ\alpha\circ\theta\]
Let $(x,z)\in (\alpha\circ\beta)\cap\gamma$. Then there is $y\in A$ such that $x\;\alpha\; y\;\beta\;z$.
Define sequences $u_0,u_1,\dots, u_n$ and $v_0, v_1,\dots, v_n$ of elements of $A$ inductively as follows:
\[u_0 = x\qquad\qquad u_{i}=\left\{\aligned m_i(u_{i-1},x,z,u_{i-1})\quad&\text{if $0<i\leq n$ is even}\\
m_i(u_{i-1},z,x, u_{i-1})\quad&\text{if $0<i\leq n$ is odd}\endaligned\right.\]
\[v_0 = x\qquad\qquad v_{i}=\left\{\aligned m_i(v_{i-1},y,z,v_{i-1})\quad&\text{if $0<i\leq n$ is even}\\
m_i(v_{i-1},z,y, v_{i-1})\quad&\text{if $0<i\leq n$ is odd}\endaligned\right.\]
We will show that
\[x\;\beta\;v_n\;\alpha\;u_n\;\theta\; z\qquad\text{so that}\quad (x,z)\in \beta\circ\alpha\circ\theta\]as required.
\vskip0.3cm\noindent{\bf Claim I:} $x\;\beta\;v_n$.\newline To see this, note that
\begin{center}\begin{tikzcd}
x=v_0 \rar[equals]& m_1(v_0,y,y,v_0)\arrow[dash]{dl}{\beta}\\m_1(v_0,z,y,v_0)= v_1\rar[equals]&m_2(v_1,y,y,v_1)\arrow[dash]{dl}{\beta}\\m_2(v_1,y,z,v_1)=v_2\rar[equals]&m_2(v_2,y,y,v_2)\arrow[dash]{dl}{\beta}\\m_2(v_2,z,y,v_2)=v_3\rar[dash,dashed]&\arrow[dash,dashed]{dl}\\v_n\end{tikzcd}\end{center}
\vskip0.3cm\noindent{\bf Claim II:} $v_n\;\alpha\;u_n$.\newline We prove by induction that $u_i\;\alpha\;v_i$ for all $i\leq n$. This is obvious if $i=0$. Assuming now that $u_{i-1}\;\alpha\;v_{i-1}$, we see that both
\[m_{i}(u_{i-1}, x,z,u_{i-1})\;\alpha\;m_i(v_{i-1},y,z,v_{i-1})\qquad\text{and}\qquad m_{i}(u_{i-1}, z,x,u_{i-1})\;\alpha\;m_i(v_{i-1},z,y,v_{i-1})\] so that at any rate also $u_i\;\alpha\;v_i$.
\vskip0.3cm\noindent{\bf Claim III:} $u_i\;\gamma\;x\;\gamma z$ for all $0\leq i\leq n$.\newline
This follows by induction: $(x,z)\in(\alpha\circ\beta)\cap\gamma$, so $x=u_0\gamma z$.
Assuming now that $u_{i-1}\;\gamma\;x\;\gamma\; z$. we see that both
\[m_i(u_{i-1}, x,z, u_{i-1})\;\gamma\; m_{i-1}(x,x,x,x)=x\qquad\text{and}\qquad m_i(u_{i-1}, z,x, u_{i-1})\;\gamma\; m_{i-1}(x,x,x,x)=x\] and hence $u_i\;\gamma\; x$ also.
\vskip0.3cm\noindent{\bf Claim IV:} $u_n\;\theta\;z$\newline
Note that $x=m_0(x,x,z,z)=m_1(x,x,z,z)$ and hence certainly $m_1(u_0,x,x,u_0)\;\theta\;m_1(x,x,z,z)$. Using the facts that $u_i\;\gamma\;x\;\gamma\; z$ and the Day term identities, we see that we have:
\begin{center}\begin{tikzcd}
x=m_1(u_0,x,x,u_0)\rar[dash, bend left]{\theta}\rar[dash]{\gamma}\dar[dash,swap]{\alpha}&m_1(x,x,z,z)\dar[dash]{\alpha}\\m_1(u_0,y,x,u_0)\dar[dash,swap]{\beta}\rar[dash]{\gamma}&m_1(x,y,z,z)\dar[dash]{\beta}\\
u_1=m_1(u_0,z,x,u_0)\dar[equals]\rar[dash]{\gamma}&m_1(x,z,z,z)\dar[equals]\\u_1=m_2(u_1,z,z,u_i)\dar[dash,swap]{\beta}\rar[dash]{\gamma}&m_2(x,z,z,z)\dar[dash]{\beta}\\
m_2(u_1,y,z,u_1)\dar[dash,swap]{\alpha}\rar[dash]{\gamma}&m_2(x,x,z,z)\dar[dash]{\alpha}\\u_2=m_2(u_1,x,z,u_1)\dar[equals]\rar[dash]{\gamma}&m_2(x,x,z,z)\dar[equals]\\
u_2=m_3(u_2,x,x,u_2)\dar[dash, swap]{\alpha}\rar[dash]{\gamma}&m_3(x,x,z,z)\dar[dash]{\alpha}\\{}\dar[dash,dashed]&{}\dar[dash,dashed]\\
u_n\rar[dash]{\gamma}&m_n(x,x,z,z)\text{ or }m_n(x,z,z,z) =z
\end{tikzcd}
\end{center}
Since $\theta\geq\alpha\land\gamma$ and $\theta\geq\beta\land\gamma$, we can use the Shifting Lemma to hop down each of the above rectangles, and conclude that $u_n\;\theta\; z$.\eproof
\begin{lemma}
If for any $A\in\mathcal V$ and any $\alpha,\beta,\gamma\in\Con(A)$, we have \[(\alpha\circ \beta)\cap\gamma\subseteq (\beta\circ\alpha)\circ[(\alpha\land\gamma)\lor(\beta\land\gamma)]\]
then $\mathcal V$ has Gumm terms.
\end{lemma}
\bproof Let $F:=F_{\mathcal V}(x,y,z)$ be the $\mathcal V$--free algebra on 3 generators. Define $f,g,h:F\to F$ to be the unique homomorphisms such that
\[\left\{\aligned f(x)&=f(y)=x\\f(z)&=z\endaligned\right.\qquad \left\{\aligned g(x)&=x\\g(y)&=g(z)=z\endaligned\right.\qquad \left\{\aligned h(x)&=h(z)=x\\h(y)&=y\endaligned\right.\]
and let $\alpha:=\ker f,\beta:=\ker g$ and $\gamma:=\ker h$.
Since $(x,z)\in(\alpha\circ\beta)\cap\gamma$, it follows that $(x,z)\in(\beta\circ\alpha)\circ[(\alpha\land\gamma)\lor(\beta\land\gamma)]$.
Observing that $ (\alpha\land\gamma)\lor(\beta\land\gamma)=\bigcup_{n=1}^\infty[ (\alpha\land\gamma)\circ(\beta\land\gamma)]^n$, we see that there must be terms
$ p(x,y,z), q_1(x,y,z), q_2(x,y,z),\dots, q_n(x,y,z)=z$ such that
\[x\;\beta\;p^F(x,y,z)\alpha \;q_1^F(x,y,z)\;(\alpha\land\gamma)\;q_2^F(x,y,z)\;(\beta\land\gamma)\; q_3^F(x,y,z)\;(\alpha\land\gamma)\;q_4^F(x,y,z)\dots \;q_n^F(x,y,z)=z\]
We now check that these terms satisfy the Gumm identities.
\begin{enumerate}[\rm ({G}1)]\item Since $x\;\beta\;p^F(x,y,z)$, we have $x=p^F(x,z,z)$.
\item From $p^F(x,y,z)\alpha \;q_1^F(x,y,z)$ we deduce that $p^F(x,x,z)= q_1^F(x,x,z)$.
\item
\[q_1^F(x,y,z)\;\gamma\;q_2^F(x,y,z)\;\gamma\; q_3^F(x,y,z)\;\gamma\;q_4^F(x,y,z)\dots \;\gamma\;q_n^F(x,y,z)=z\]
and hence\[ q_1^F(x,y,x)=q_2^F(x,y,x)= q_3^F(x,y,x)=q_4^F(x,y,z)\dots=q_n^F(x,y,x)=x\]
\item We have $q_i^F(x,y,z)\;\beta\; q_{i+1}^F(x,y,z)$ if $i<n$ is even, and hence $q_i^F(x,z,z)=q_{i+1}^F(x,z,z)$ for even $i<n$.
\item We have $q_i^F(x,y,z)\;\alpha\; q_{i+1}^F(x,y,z)$ if $i<n$ is odd, and hence $q_i^F(x,x,z)\;\alpha\; q_{i+1}^F(x,x,z)$ for odd $i<n$.
\item $q_n^F(x,y,z)=z$ by definition of $q_n$.
\end{enumerate}
Hence in $F$ the terms $p,q_1,\dots, q_n$ satisfy the Gumm identities. Since $F$ is free on 3 generators in $\mathcal V$, the Gumm identities hold in $\mathcal V$.
\eproof
\begin{lemma} \label{lemma_Gumm_to_Day} If $\mathcal V$ has Gumm terms, then $\mathcal V$ is congruence modular.
\end{lemma}
\bproof It suffices to show that $\mathcal V$ has Day terms. Given Gumm terms $p,q_1,\dots, q_n$, define \[\aligned
m_0(x,y,z,u)&=m_1(x,y,z,u)=x\\
m_2(x,y,z,u)&=p(x,y,z)\\
m_3(x,y,z,u)&=q_1(x,y,u)\\
m_4(x,y,z,u)&=q_1(x,z,u)\\
&\vdots\\
m_{4i+1}(x,y,z,u)&=q_{2i}(x,z,u)\\
m_{4i+2}(x,y,z,u)&=q_{2i}(x,y,u)\\
m_{4i+3}(x,y,z,u)&=q_{2i+1}(x,y,u)\\
m_{4i+4}(x,y,z,u)&=q_{2i+1}(x,z,u)\\
&\vdots
\endaligned\]
We now verify the Day identities (D1)-(D5) for the above defined $m_0, m_1, m_2,\dots$.
\begin{enumerate}[\rm ({D}1)]
\item Clearly $m_0(x,y,z,u)\approx x$.
\item For each $i$, we have have $m_{i}(x,y,y,x)=q_j(x,y,x)$ for some $j$. Since $q_j(x,y,x)\approx x$, we see that $m_i(x,y,y,x)\approx x$.
\item For $i=2$, we have $m_2(x,x,z,z)\approx p(x,z,z)\approx x$. If $i<n$ is even and $i>2$, then
either $i=4k+2$ or else $i=4k+4$. Now
\[m_{4k+2}(x,x,z,z)= q_{2k}(x,x,z)=q_{2k+1}(x,x,z)=m_{4k+3}(x,x,z,z)\]
Similarly,
\[m_{4k+4}(x,x,z,z)=q_{2k+1}(x,z,z)=q_{2k+1}(x,z,z)= m_{4k+5}(x,x,z,z)\]
\item If $i=1$, then $m_1(x,y,y,z)= x= p(x,y,y)=m_2(x,y,y,z)$. If $i<n$ is odd and $i>1$, then either $i=4k+1$ or else $i=4k+3$. Now
\[m_{4k+1}(x,y,y,u)= q_{2k}(x,y,u)= m_{4k+2}(x,y,y,u)\]
and
\[m_{4k+3}(x,y,y,u)=q_{2k+1}(x,y,u)=m_{4k+4}(x,y,y,u)\]
\item Finally, if $m_l$ is the last definable Day term, then $m_l(x,y,z,u)$ is either $q_n(x,y,u)$ or $q_n(x,z,u)$, where $q_n$ is the last Gumm term.
In either case we have $m_l(x,y,z,u)=u$.
\end{enumerate}
\eproof
\section{The Modular Commutator}
\fancyhead[RE]{The Modular Commutator}
\subsection{The Modular Commutator via the Term Condition}
Recall that in Section \ref{subsection_TC_commutator} we defined the commutator operation on the congruence lattices of general algebras, as follows: For $\alpha, \beta,\delta\in\Con(A)$, we defined $M(\alpha,\beta)$ to be the subalgebra of $A^4$ of all matrices
\[\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\]
where, for any $n,m\in\mathbb N$, $t(\cdot)$ is an $(m+n)$--ary term, and $\mathbf a^1,\mathbf a^2\in A^m$, $\mathbf b^1,\mathbf b^2\in A^n$ are such that
\[ a^1_i\,\alpha\,a^2_i\quad\text{for all }i\leq m\qquad b^1_j\,\beta\, b^2_j\quad\text{for all }j\leq n\]
We then said that $\alpha$ {\em centralizes} $\beta$ modulo $\delta$ (and denoted this by $C(\alpha,\beta;\delta)$) if and only if whenever $\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\in M(\alpha,\beta)$, we have
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash, bend left]{r}{\delta}\arrow[dash]{r}{\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash, bend left, dashed]{r}{\delta}\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
We observed that \begin{itemize}\item If $C(\alpha,\beta;\delta_j)$ holds for all $j\in J$, then $C(\alpha,\beta;\bigwedge_{j\in J}\delta_j)$.
\item If $C(\alpha_i,\beta;\delta)$ holds for all $i\in I$, then $C(\bigvee_{i\in I}\alpha_i,\beta;\delta)$.
\item $C(\alpha,\beta;\alpha\land\beta)$.
\end{itemize} and thus were able to define the commutator $[\alpha,\beta]$ to be the smallest $\delta$ such that $C(\alpha,\beta;\delta)$.
We verified that this definition of commutator is the same as that for groups (when we identify congruences with normal subgroups). Nevertheless, the only properties of the commutator that we were able to prove for general algebras were the following:
\begin{enumerate}[1.]\item The commutator is monotone in both variables, i.e. if $\alpha\subseteq\alpha',\beta\subseteq\beta'$, then $[\alpha,\beta]\leq[\alpha',\beta']$\item $[\alpha,\beta]\leq \alpha\land\beta$
\end{enumerate}
The group commutator has three additional properties:
\begin{enumerate}[1.]\setcounter{enumi}{2}\item $[M,N]=[N,M]$, i.e. $[\alpha,\beta]=[\beta,\alpha]$
\item $[M,\bigvee_{i\in I}M_i]=\bigvee_{i\in I}[M,N_i]$, i.e. $[\alpha,\bigvee_{i\in I}\beta_i]=\bigvee_{i\in I}[\alpha,\beta_i]$
\item $[M/K,N/K]=[M,N]K/K$ for $K\subseteq M\cap N$, i.e. $[\alpha/\pi,\beta/\pi]=[\alpha,\beta]\lor \pi/\pi$ for $\pi\leq\alpha\land\beta$.
\end{enumerate}
We shall show in this section that modularity suffices to obtain the conditions 3. and 4. Condition 5. will be obtained in Section \ref{section_commutator_HSP}.
Recall that if $\mathcal V$ is a congruence modular variety, the there are Day terms $m_0(x,y,z,u), m_1(x,y,z,u)$\dots, $m_n(x,y,z,u)$ satisfying the equations (D1)-(D5). Further, in Lemma \ref{lemma_a_gamma_c} we saw that if $A\in\mathcal V$, $\gamma\in\Con(A)$, and $a,b,c,d\in A$ are such that $b\;\gamma\;d$, then
\[a\;\gamma\; c\qquad \Longleftrightarrow\qquad m_i(a,a,c,c,)\;\gamma\;m_i(a,b,d,c)\quad\text{ for all }i\leq n\]
We use this result as motivation for the following definition:
\begin{definition}\rm Suppose that $A$ is an algebra in a variety with Day terms $m_0,\dots, m_n$, and that $\alpha,\beta\in\Con(A)$. Let $X(\alpha,\beta)$ be the set of all pairs
\[\Big(m_i(x,x,u,u),\; m_i(x,y,z,u)\Big)\qquad\text{where } \left(\begin{matrix}x&y\\u&z\end{matrix}\right)\in M(\alpha,\beta)\text{ and } 0\leq i\leq n\]
\endbox
\end{definition}
\begin{lemma}\label{lemma_m_alpha_beta} If $\left(\begin{matrix}x&y\\u&z\end{matrix}\right)\in M(\alpha,\beta)$, then $\left(\begin{matrix} m_i(x,u,u,x)&m_i(x,z,z,x)\\m_i(x,x,u,u)&m_i(x,y,z,u)\end{matrix}\right)\in M(\alpha,\beta)$ for $0\leq i\leq n$.\end{lemma}
\bproof If $\left(\begin{matrix}x&y\\u&z\end{matrix}\right)\in M(\alpha,\beta)$, then there is an $(m+n)$--ary term $t$ and $\mathbf a^1,\mathbf a^2\in A^m$, $\mathbf b^1,\mathbf b^2\in A^n$ such that $a^1_i\;\alpha\;a^2_i$ and $b^1_j\;\beta\;b^2_j$ for $i\leq m, j\leq n$ such that
\[\left(\begin{matrix}x&y\\u&z\end{matrix}\right)=\left(\begin{matrix}t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\t(\mathbf a^2,\mathbf b^1)& t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)\]
Define a term $T$ by
\[T(\mathbf x^1,\mathbf x^2,\mathbf x^3,\mathbf x^4, \mathbf y^1,\mathbf y^2,\mathbf y^3,\mathbf y^4):= m_i(t(\mathbf x^1,\mathbf y^1),t(\mathbf x^2,\mathbf y^2),t(\mathbf x^3,\mathbf y^3),t(\mathbf x^4,\mathbf y^4),)\]
Then
\[\aligned \left(\begin{matrix} m_i(x,u,u,x)&m_i(x,z,z,x)\\m_i(x,x,u,u)&m_i(x,y,z,u)\end{matrix}\right)&=\left(\begin{matrix}T(\mathbf a^1,\mathbf a^2,\mathbf a^2,\mathbf a^1, \mathbf b^1,\mathbf b^1,\mathbf b^1,\mathbf b^1)&T(\mathbf a^1,\mathbf a^2,\mathbf a^2,\mathbf a^1, \mathbf b^1,\mathbf b^2,\mathbf b^2,\mathbf b^1)\\T(\mathbf a^1,\mathbf a^1,\mathbf a^2,\mathbf a^2, \mathbf b^1,\mathbf b^1,\mathbf b^1,\mathbf b^1)&T(\mathbf a^1,\mathbf a^1,\mathbf a^2,\mathbf a^2, \mathbf b^1,\mathbf b^2,\mathbf b^2,\mathbf b^1)\end{matrix}\right)\\&=\left(\begin{matrix}T(\mathbf A^1,\mathbf B^1)&T(\mathbf A^1,\mathbf B^2)\\T(\mathbf A^2,\mathbf B^1)& T(\mathbf A^2,\mathbf B^2)\end{matrix}\right)\endaligned\]
where $\mathbf A^1:=(\mathbf a^1,\mathbf a^2,\mathbf a^2,\mathbf a^1)$, $\mathbf A^2:=(\mathbf a^1,\mathbf a^1,\mathbf a^2,\mathbf a^2)$, $\mathbf B^1:=(\mathbf b^1,\mathbf b^1,\mathbf b^1,\mathbf b^1)$, and $\mathbf B^2:=(\mathbf b^1,\mathbf b^2,\mathbf b^2,\mathbf b^1)$.
\eproof
\begin{proposition}\label{propn_C_X} Suppose that $\mathcal V$ is a congruence modular variety, that $A\in\mathcal V$, and that $\alpha,\beta,\delta\in\Con(A)$. The following are equivalent:
\begin{enumerate}[(a)]\item $C(\alpha,\beta;\delta)$
\item $X(\alpha,\beta)\subseteq\delta$
\item $C(\beta,\alpha;\delta)$
\item $X(\beta,\alpha)\subseteq\delta$
\end{enumerate}
\end{proposition}
\bproof We prove that (a) $\Longrightarrow$ (b) $\Longrightarrow$ (c). Interchanging $\alpha$ and $\beta$ then immediately yields
(c) $\Longrightarrow$ (d) $\Longrightarrow$ (a), proving the theorem.
\vskip0.3cm\noindent (a) $\Longrightarrow$ (b): Let $m_0,\dots, m_n$ be Day terms for $\mathcal V$. We must prove that if $\left(\begin{matrix}x&y\\u&z\end{matrix}\right)\in M(\alpha,\beta)$, then $\Big(m_i(x,x,u,u),\; m_i(x,y,z,u)\Big)\in\delta$.
Now by Lemma \ref{lemma_m_alpha_beta}, we have $\left(\begin{matrix} m_i(x,u,u,x)&m_i(x,z,z,x)\\m_i(x,x,u,u)&m_i(x,y,z,u)\end{matrix}\right)\in M(\alpha,\beta)$. Thus
\begin{center}\begin{tikzcd}
m_i(x,u,u,x)\dar[dash,swap]{\alpha}\rar[dash]{\beta}&m_i(x,z,z,x)\dar[dash]{\alpha}\\
m_i(x,x,u,u)\rar[dash]{\beta}&m_i(x,y,z,u)
\end{tikzcd}\end{center}
Furthermore, by the Day term equation (D2), we have $m_i(x,u,u,x)=x=m_i(x,z,z,x)$, and thus
\begin{center}\begin{tikzcd}
m_i(x,u,u,x)\rar[dash, bend left]{\delta}\dar[dash,swap]{\alpha}\rar[dash]{\beta}&m_i(x,z,z,x)\dar[dash]{\alpha}\\
m_i(x,x,u,u)\rar[dash]{\beta}&m_i(x,y,z,u)
\end{tikzcd}\end{center}
The fact that $C(\alpha,\beta;\delta)$ now allows us to conclude that also $m_i(x,x,u,u)\;\delta\;m_i(x,y,z,u)$.
\vskip0.3cm\noindent (b) $\Longrightarrow$ (c): Suppose now that $X(\alpha,\beta)\subseteq\delta$, and that $\left(\begin{matrix}b&d\\a&c\end{matrix}\right)\in M(\beta,\alpha)$ are such that
\begin{center}\begin{tikzcd}
b\dar[dash,swap]{\beta}\rar[dash,bend left]{\delta}\rar[dash]{\alpha}&d\dar[dash]{\beta}\\
a\rar[dash]{\alpha}&c
\end{tikzcd}\end{center} We must show that $a\;\delta\;c$. Now since $b\;\delta\;d$, it suffices (by Lemma \ref{lemma_a_gamma_c}) to show that $m_i(a,a,c,c)\;\delta\; m_i(a,b,d,c)$ for all $i\leq n$. Now clearly $\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in M(\alpha,\beta)$, and hence $\Big(m_i(a,a,c,c), m_i(a,b,d,c)\Big)\in X(\alpha,\beta)\subseteq\delta$, as required.
\eproof
\begin{corollary}\label{corr_C_X} \begin{enumerate}[(a)]\item $[\alpha,\beta]= \Cg_A(X(\alpha,\beta))=\Cg_A(X(\beta,\alpha))$.
\item $C(\alpha,\beta;\delta)$ if and only if $[\alpha,\beta]\leq\delta$.
\item $C(\alpha,\beta;\delta)$ if and only if $C(\beta,\alpha;\delta)$.
\end{enumerate}
\end{corollary}
\bproof (a) Let $\gamma:= \Cg(X(\alpha,\beta)$. Then $X(\alpha,\beta)\subseteq\gamma$, and hence $C(\alpha,\beta;\gamma)$. Furthermore, if $C(\alpha,\beta;\delta)$, then $X(\alpha,\beta)\subseteq\delta$, so $\gamma\subseteq\delta$. Hence $\gamma$ is the smallest congruence such that $\alpha$ centralizes $\beta$ modulo $\gamma$, i.e. $\gamma=[\alpha,\beta]$. In the same way it can be shown that $[\alpha,\beta] =\Cg_A(X(\beta,\alpha))$.
(b) Clearly if $C(\alpha,\beta\;\delta)$, then $\delta\geq[\alpha,\beta]$, by definition of the commutator. Conversely, suppose that $[\alpha,\beta]\leq\delta$. We have $X(\alpha,\beta)\subseteq [\alpha,\beta]$ (by (a)), and hence $X(\alpha,\beta)\subseteq\;\delta$. It follows that $C(\alpha,\beta;\delta)$.
(c) is obvious from the equivalence of Proposition \ref{propn_C_X} (b) and (d).
\eproof
We can now easily prove two additional properties of the commutator in congruence modular varieties:
\begin{theorem} Suppose that $\mathcal V$ is a congruence modular variety, that $A\in\mathcal V$, and that $\alpha,\beta,\alpha_i\in\Con(A)$ for $i\in I$. Then:
\begin{enumerate}[(a)]\item $[\alpha,\beta]=[\beta,\alpha]$
\item $[\bigvee_{i\in I}\alpha_i,\beta]=\bigvee_{i\in I}[\alpha_i,\beta]$
\end{enumerate}
\end{theorem}
\bproof
(a) is obvious from Corollary \ref{corr_C_X}(a).
\vskip 0.3cm\noindent (b) Let $\delta:=\bigvee_{i\in I}[\alpha_i,\beta_i]$. By monotonicity, it is clear that $\delta\subseteq[\bigvee_{i\in I}\alpha_i,\beta]$. Now by definition of the commutator, we see that
$C(\alpha_i,\beta;[\alpha_i,\beta])$ holds for all $i\in I$. Since $[\alpha_i,\beta]\leq\delta$, it follows by Corollary \ref{corr_C_X}(b) that $C(\alpha_i,\beta;\delta)$ for all $i\in I$, and thus that $C(\bigvee_{i\in I}\alpha_i,\beta;\delta)$. By definition of commutator, we have $[\bigvee_{i\in I}\alpha_i,\beta]\subseteq \delta$, and hence $[\bigvee_{i\in I}\alpha_i,\beta]=\delta= \bigvee_{i\in I}[\alpha_i,\beta_i]$.
\eproof
\subsection{ Congruences on Congruences}\label{subsection_congruences_on_congruences} Let $A$ be an algebra in a congruence modular variety $\mathcal V$, and let $\alpha,\beta\in\Con(A)$. We can think of $\alpha$ as a subalgebra $A_\alpha$ of the product algebra $A\times A$, where we think of th elements of $\alpha$ as {\em column vectors}, to whit:
\[A_\alpha:=\left\{\left[\begin{matrix}x\\y\end{matrix}\right]: x\;\alpha\;y\right\}\]
Similarly, we think of $\beta$ as a subalgebra $A^\beta$ of $A\times A$, consisting of {\em row vectors}:
\[A^\beta:=\{[x\quad y]: x\;\beta\;y\}\]
Naturally, we now define
\[A_\alpha^\beta:=\left\{\left[\begin{matrix}x&y\\u&z\end{matrix}\right]: x\;\beta\; y, u\;\beta\;z, x\;\alpha\; u, y\;\alpha z\right\}\] to be the matrix whose rows belong to $A^\beta$ and whose columns to $A_\alpha$:
\begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]{\beta}&y\dar[dash]{\alpha}\\
u\rar[dash]{\beta}&z\end{tikzcd}\end{center}
Observe that $A^\beta_\alpha$ may be regarded as a both subalgebra of $A_\alpha\times A_\alpha$ and a subalgebra of $A^\beta\times A^\beta$. Hence all the algebras $A_\alpha, A^\beta, A^\beta_\alpha$ belong to any variety that has $A$ as member. In particular, if $\mathcal V$ is congruence modular and $A\in\mathcal V$, then $A_\alpha$ has a modular congruence lattice, so the Shifting Lemma applies to $\Con(A_\alpha)$, etc. This will be important in the sequel.
We now proceed to define the following congruences:
\begin{definition}\rm $\Delta_\alpha(\beta)$ is the congruence on $A_\alpha$ generated by all pairs $\left(\left[\begin{matrix}b\\b\end{matrix}\right], \left[\begin{matrix}b'\\b'\end{matrix}\right]\right)$, where $(b,b')\in\beta$, i.e. \[\Delta_\alpha(\beta):=\Cg_{A_\alpha}\left(\left[\begin{matrix}b&b'\\b&b'\end{matrix}\right]: b\;\beta\; b'\right)\] Similarly,
\[\Delta^\beta(\alpha):= \Cg_{A^\beta}\left(\left[\begin{matrix}a&a'\\a&a'\end{matrix}\right]: a\;\alpha\; a'\right)\] is the congruence on $A^\beta$ generated by all pairs $\Big([a\quad a], [a'\quad a']\Big)$, where $(a,a')\in\alpha$.\endbox
\end{definition}
\begin{lemma}\label{lemma_Delta_tr_cl} \begin{enumerate}[(a)]\item $\Delta_\alpha(\beta)$ is the transitive closure of $M(\alpha,\beta)$ (where each matrix is regarded as a pair of column vectors).
\item $\Delta_\alpha(\beta)\subseteq A_\alpha^\beta$
\item $\left[\begin{matrix} x&y\\u&v\end{matrix}\right]\in\Delta_\alpha(\beta)$ if and only if $\left[\begin{matrix} u&v\\x&y\end{matrix}\right]\in\Delta_\alpha(\beta)$
\end{enumerate}
\end{lemma}
\bproof
(a) Recall that $M(\alpha,\beta)$ is the subalgebra of $A^4$ generated by all matrices of the form
\[\left[\begin{matrix} a&a\\a'&a'\end{matrix}\right]\qquad\text{and }\qquad \left[\begin{matrix} b&b'\\b&b'\end{matrix}\right]
\qquad\text{where } a\;\alpha\; a'\text{ and } b\;\beta\;b'\]
Now $\Delta_\alpha(\beta)$ is a congruence on $A_\alpha$, and $\left[\begin{matrix} a\\a'\end{matrix}\right]\in A_\alpha$ when $a\;\alpha\;a'$. Hence certainly $\left[\begin{matrix}a\\a'\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix}a\\a'\end{matrix}\right]$ (by reflexivity of congruence relations), and thus $\left[\begin{matrix} a&a\\a'&a'\end{matrix}\right]\in \Delta_\alpha(\beta)$. Moreover, if $b\;\beta\;b'$, then $\left[\begin{matrix} b&b'\\b&b'\end{matrix}\right]\in \Delta_\alpha(\beta)$ by definition. It follows that $M(\alpha,\beta)\subseteq \Delta_\alpha(\beta)$.
In particular, $M(\alpha,\beta)$ is a binary relation on $A_\alpha$. It is easy to see that $M(\alpha,\beta)$ is a reflexive, symmetric and compatible relation on $A_\alpha$, where $\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1)&t(\mathbf a^1,\mathbf b^2)\\
t(\mathbf a^2,\mathbf b^1)&t(\mathbf a^2,\mathbf b^2)\end{matrix}\right)$ is regarded as a pair of column vectors $\left(\left[\begin{matrix}t(\mathbf a^1,\mathbf b^1)\\t(\mathbf a^2,\mathbf b^1)\end{matrix}\right], \left[\begin{matrix}t(\mathbf a^1,\mathbf b^2)\\t(\mathbf a^2,\mathbf b^2)\end{matrix}\right]\right)$. Hence the transitive closure of $M(\alpha,\beta)$ is a congruence relation $\Theta$ on $A_\alpha$. We must show that $\Theta=\Delta_\alpha(\beta)$.
Certainly $\Theta\subseteq\Delta_\alpha(\beta)$, because $M(\alpha,\beta)\subseteq \Delta_\alpha(\beta)$. Conversely, each
generator $\left[\begin{matrix}b&b'\\b&b'\end{matrix}\right]$ of $\Delta_\alpha(\beta)$ (where $b\;\beta\;b'$) is clearly a member of $M(\alpha,\beta)$ and hence of $\Theta$. Hence also $\Delta_\alpha(\beta)\subseteq\Theta$.
\vskip0.3cm \noindent(b) It is easy to see that $A_\alpha^\beta$ is transitive, in the sense that
\[\left[\begin{matrix} x&y\\u&z\end{matrix}\right], \left[\begin{matrix} y&v\\z&w\end{matrix}\right]\in A_\alpha(\beta)\quad\Longrightarrow\quad \left[\begin{matrix} x&v\\u&w\end{matrix}\right]\in A_\alpha(\beta)\]
\begin{center}\begin{tikzcd}x\dar[dash]{\alpha}\rar[dash]{\beta}&y\dar[dash]{\alpha}\rar[dash]{\beta}&v\dar[dash]{\alpha}\\
u\rar[dash]{\beta}&z\rar[dash]{\beta}&w\end{tikzcd}\end{center}
Moreover, clearly $M(\alpha,\beta)\subseteq A_\alpha^\beta$. Since $\Delta_\alpha(\beta)$ is the transitive closure of $M(\alpha,\beta)$, the result follows.
\vskip0.3cm \noindent(c) Observe that $\left[\begin{matrix} x&y\\u&v\end{matrix}\right]\in M(\alpha,\beta)$ if and only if $\left[\begin{matrix} u&v\\x&y\end{matrix}\right]\in M(\alpha,\beta)$. Since $\Delta_\alpha(\beta)$ is the transitive closure of $M(\alpha,\beta)$, the result follows.
\eproof
Now observe that each coset of $\Delta_\alpha(\beta)$ is a set of ordered pairs from $A$, i.e. each coset of $\Delta_\alpha(\beta)$ is a binary relation on $A$. The connection between $\Delta_\alpha(\beta)$ and $[\alpha,\beta]$ is as follows:
\begin{proposition} $[\alpha,\beta]$ is the smallest congruence on $A$ which is a union of $\Delta_\alpha(\beta)$--classes.\end{proposition}
\bproof
By definition, $[\alpha,\beta]$ is the smallest congruence for which $C(\alpha,\beta;\delta)$. Since $C(\alpha,\beta;\delta)\Longleftrightarrow C(\beta,\alpha;\delta)$, we see that $[\alpha,\beta]$ is the smallest $\delta$ for which it is the case that if one row or column of a matrix in $M(\alpha,\beta)$ belongs to $\delta$, then the other row or column belongs to $\delta$ also.
\begin{center}\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash, bend left]{r}{[\alpha,\beta]}\arrow[dash]{r}{\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash, bend left, dashed]{r}{[\alpha,\beta]}\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}\qquad\text{and}\qquad
\begin{tikzcd}
t(\mathbf a^1,\mathbf b^1)\arrow[dash, bend right, swap]{d}{[\alpha,\beta]}\arrow[dash]{r}{\beta}\arrow[dash]{d}{\alpha} & t(\mathbf a^1, \mathbf b^2)\dar[dash]{\alpha}\arrow[dash, bend right, dashed,swap]{d}{[\alpha,\beta]}\\
t(\mathbf a^2,\mathbf b^1)\arrow[dash]{r}{\beta} & t(\mathbf a^2, \mathbf b^2)
\end{tikzcd}
\end{center}
Now suppose that $\left[\begin{matrix} x\\u\end{matrix}\right]\in [\alpha,\beta]$, and that $\left[\begin{matrix} x\\u\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y\\z\end{matrix}\right]$. Since $\Delta_\alpha(\beta)$ is the transitive closure of $M(\alpha,\beta)$, there is a sequence of matrices in $M(\alpha,\beta)$
\[\left[\begin{matrix} x&v_1\\u&w_1\end{matrix}\right], \left[\begin{matrix} v_1&v_2\\w_1&w_2\end{matrix}\right], \left[\begin{matrix} v_2&v_3\\w_2&w_3\end{matrix}\right],\dots, \left[\begin{matrix} v_n&y\\w_n&z\end{matrix}\right]\]
This can be though of as a sequence of column vectors from $\left[\begin{matrix} x\\u\end{matrix}\right]$ to $\left[\begin{matrix} y\\z\end{matrix}\right]$ where consecutive pairs of columns form a matrix in $M(\alpha,\beta)$.
\begin{center}\begin{tikzcd}
x\rar[dash]{\beta}\dar[dash]{\alpha}\dar[dash,bend right, swap]{[\alpha,\beta]}&v_1\dar[dash]{\alpha}\rar[dash]{\beta}&v_2\dar[dash]{\alpha}\rar[dash]{\beta}&v_3\dar[dash]{\alpha}\rar[dash, dotted]&v_n\dar[dash]{\alpha}\rar[dash]{\beta}&y\dar[dash]{\alpha}\\
u\rar[dash]{\beta}&w_1\rar[dash]{\beta}&w_2\rar[dash]{\beta}&w_3\rar[dash,dotted]&w_n\rar[dash]{\beta}&z
\end{tikzcd}
\end{center}
It follows that $\left[\begin{matrix} y\\z\end{matrix}\right]\in [\alpha,\beta]$ also. We have thus shown that
\[\left[\begin{matrix} x\\u\end{matrix}\right]\in [\alpha,\beta]\quad\text{and}\quad \left[\begin{matrix} x\\u\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y\\z\end{matrix}\right]\quad\text{implies}\quad \left[\begin{matrix} y\\z\end{matrix}\right]\in [\alpha,\beta]\] from which it is clear that $[\alpha,\beta]$ is a union of cosets of $\Delta_\alpha(\beta)$.
Now suppose that $\delta\in\Con(A)$ is a union of cosets of $\Delta_\alpha(\beta)$. Then it must be the case that \[\left[\begin{matrix} x\\u\end{matrix}\right]\in \delta \quad\text{and}\quad \left[\begin{matrix} x\\u\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y\\z\end{matrix}\right]\quad\text{implies}\quad \left[\begin{matrix} y\\z\end{matrix}\right]\in \delta\] In particular, since $M(\alpha,\beta)\subseteq\Delta_\alpha(\beta)$, we have
$\left[\begin{matrix} t(\mathbf a^1,\mathbf b^1)\\t(\mathbf a^2,\mathbf b^1)\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} t(\mathbf a^1,\mathbf b^2)\\t(\mathbf a^2,\mathbf b^1)\end{matrix}\right]$. Thus if $\left[\begin{matrix} t(\mathbf a^1,\mathbf b^1)\\t(\mathbf a^2,\mathbf b^1)\end{matrix}\right]\in\delta$, then also $\left[\begin{matrix} t(\mathbf a^1,\mathbf b^2)\\t(\mathbf a^2,\mathbf b^1)\end{matrix}\right]\in\delta$. But that means $C(\beta,\alpha;\delta)$, and hence $\delta\geq [\beta,\alpha]=[\alpha,\beta]$.
\eproof
\begin{theorem}\label{thm_commutator_Delta} For $x,y\in A$, the following are equivalent:
\begin{enumerate}[(a)]\item $(x,y)\in[\alpha,\beta]$
\item $\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)$
\item There exists $a\in A$ such that $\left[\begin{matrix} x&a\\y&a\end{matrix}\right]\in\Delta_\alpha(\beta)$
\item There exists $b\in A$ such that $\left[\begin{matrix} x&y\\b&b\end{matrix}\right]\in\Delta_\alpha(\beta)$
\end{enumerate} \end{theorem}
\bproof We first prove that (b), (c) and (d) are equivalent. It should be clear that (b) $\Longrightarrow$ (c) and that (b) $\Longrightarrow$ (d).
\vskip0.3cm\noindent (c) $\Longrightarrow$ (b): Suppose that $\left[\begin{matrix} x&a\\y&a\end{matrix}\right]\in\Delta_\alpha(\beta)$. Then $y\;\beta\;a$, and hence $\left[\begin{matrix} a&y\\a&y\end{matrix}\right]\in\Delta_\alpha(\beta)$. By transitivity, we see that $\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)$.
\vskip0.3cm\noindent (d) $\Longrightarrow$ (b): Suppose that $\left[\begin{matrix} x&y\\b&b\end{matrix}\right]\in\Delta_\alpha(\beta)$. Since the columns are in $A_\alpha$, we must have $x\;\alpha\;b\;\alpha\;y$, and hence $\left[\begin{matrix}x\\b\end{matrix}\right], \left[\begin{matrix}y\\b\end{matrix}\right], \left[\begin{matrix}x\\y\end{matrix}\right], \left[\begin{matrix}y\\y\end{matrix}\right]\in A_\alpha$.
Let $\eta_0,\eta_1$ be the kernels of the projections $\pi_0,\pi_1$, where \[A_\alpha\buildrel{\pi_0}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto x\qquad\qquad A_\alpha\buildrel{\pi_1}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto y\]Then in $A_\alpha$ we have
\begin{center}\begin{tikzcd}
\left[\begin{matrix}x\\b\end{matrix}\right]\dar[dash]{\eta_0}\rar[dash]{\eta_1}\rar[dash, bend left]{\Delta_\alpha(\beta)}&\left[\begin{matrix}y\\b\end{matrix}\right]\dar[dash]{\eta_0}\\
\left[\begin{matrix}x\\y\end{matrix}\right]\rar[dash]{\eta_1}&\left[\begin{matrix}y\\y\end{matrix}\right]
\end{tikzcd}\end{center} By the Shifting Lemma (since $\eta_0\land\eta_1=0\leq\Delta_\alpha(\beta)$), we see that $\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)$.
\vskip0.3cm\noindent We now know that (b), (c) and (d) are equivalent. Next, we show that (b) $\Longleftrightarrow$ (a)
\vskip0.3cm\noindent (b) $\Longrightarrow$ (a): Suppose that $\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)$, i.e. that $\left[\begin{matrix} x\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y\\y\end{matrix}\right]$. Since $\left[\begin{matrix} y\\y\end{matrix}\right]\in[\alpha,\beta]$ (by reflexivity), and since $[\alpha,\beta]$ is a union of cosets of $\Delta_\alpha(\beta)$, we conclude that $\left[\begin{matrix} x\\y\end{matrix}\right]\in [\alpha,\beta]$ also.
\vskip0.3cm\noindent (a) $\Longrightarrow$ (b): Define a binary relation $\Theta$ on $A$ by
\[x\;\Theta\;y\qquad\Longleftrightarrow\qquad\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)\] We will show that $\Theta$ is a congruence relation.
By the equivalence of (b), (c), (d) we see that
\[x\;\Theta\;y \qquad\Longleftrightarrow\qquad\exists a\in A\left(\left[\begin{matrix} x&a\\y&a\end{matrix}\right]\in\Delta_\alpha(\beta)\right)\qquad\Longleftrightarrow\qquad\exists b\in A\;\;\left(\left[\begin{matrix} x&y\\b&b\end{matrix}\right]\in\Delta_\alpha(\beta)\right)\]\begin{itemize}\item $\Theta$ is reflexive, because $\left[\begin{matrix} x&x\\x&x\end{matrix}\right]\in\Delta_\alpha(\beta)$.\item $\Theta$ is symmetric, because $\left[\begin{matrix} x&y\\b&b\end{matrix}\right]\in\Delta_\alpha(\beta)$ implies $\left[\begin{matrix} y&x\\b&b\end{matrix}\right]\in\Delta_\alpha(\beta)$ (by symmetry of $\Delta_\alpha(\beta)$).
\item
$\Theta$ is transitive: If $x\;\Theta\; y$ and $y\;\Theta\;z$, then \[\left[\begin{matrix} x\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} z\\z\end{matrix}\right]\qquad\text{i.e.}\qquad \left[\begin{matrix} x&z\\y&z\end{matrix}\right]\in\Delta_\alpha(\beta)\] by transitivity of $\Delta_\alpha(\beta)$, from which $x\;\Theta\;z$.
\item $\Theta$ is compatible: If $f$ is an $n$--ary operation and $x_i\;\Theta\;y_i$ for $i=1,\dots,n$, then
$\left[\begin{matrix} x_i\\y_i\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} y_i\\y_i\end{matrix}\right]$ for each $i=1,\dots,n$, and hence $\left[\begin{matrix} f(x_1,\dots, x_n)\\f(y_1,\dots,y_n)\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} f(y_1,\dots,y_n)\\f(y_1,\dots,y_n)\end{matrix}\right]$ (because $\Delta_\alpha(\beta)$ is compatible). It follows that $f(x_1,\dots,x_n)\;\Theta\; f(y_1,\dots,y_n)$
\end{itemize}
it follows that $\Theta$ is indeed a congruence relation on $A$. Now observe that if $x\;\Theta\;y$ and $\left[\begin{matrix} x\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} v\\w\end{matrix}\right]$, then
\[\left[\begin{matrix} v\\w\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix} x\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix} y\\y\end{matrix}\right]\] so that $\left[\begin{matrix} v&y\\w&y\end{matrix}\right]\in\Delta_\alpha(\beta)$, from which we deduce that $v\;\Theta\;w$. Thus we have shown that
\[\text{if}\quad\left[\begin{matrix} x\\y\end{matrix}\right]\in \Theta\quad\text{and}\quad \left[\begin{matrix} x\\y\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix} v\\w\end{matrix}\right]\quad\text{then}\quad \left[\begin{matrix} v\\w\end{matrix}\right]\in \Theta\] from which it follows that $\Theta$ is a congruence on $A$ which is a union of cosets of $\Delta_\alpha(\beta)$. Since $[\alpha,\beta]$ is the smallest such, we obtain that $[\alpha,\beta]\leq\Theta$.
Hence $(x,y)\in[\alpha,\beta]$ implies $(x,y)\in\Theta$, and this in turn implies that $\left[\begin{matrix} x&y\\y&y\end{matrix}\right]\in\Delta_\alpha(\beta)$, proving that (a) implies (b).
\eproof
\begin{remarks}\rm
Gumm defined his commutator via (b) of Theorem \ref{thm_commutator_Delta}, whereas Hagemann and Hermann used (d).
\endbox
\end{remarks}
Here is a Lemma that we will shortly need:
\begin{lemma}\label{lemma_eta_Delta} Let $A$ be an algebra in a congruence modular variety, and let $\alpha,\beta\in\Con(A)$. Let
\[A_\alpha\buildrel{\pi_0}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto x\qquad\text{and}\qquad A_\alpha\buildrel{\pi_1}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto y\]
be the projection mappings, and let $\eta_0=\ker\pi_0,\eta_1=\ker\pi_1$.
Define $\beta_0,\beta_1, [\alpha,\beta]_0, [\alpha,\beta]_1\in\Con(A_\alpha)$ by
\[\beta_0:=\pi_0^{-1}(\beta)\qquad\beta_1:=\pi_1^{-1}(\beta)\qquad [\alpha,\beta]_0:=\pi_0^{-1}([\alpha,\beta])\qquad \qquad [\alpha,\beta]_1:=\pi_1^{-1}([\alpha,\beta])\]
Then
\begin{enumerate}[(a)]\item $\eta_1\land\Delta_\alpha(\beta)\subseteq [\alpha,\beta]_0$
\item $\eta_0\land\Delta_\alpha(\beta)\subseteq [\alpha,\beta]_1$
\item $\Delta_\alpha(\beta)\lor\eta_0=\beta_0$
\item $\Delta_\alpha(\beta)\lor\eta_1=\beta_1$
\end{enumerate}
\end{lemma}
\bproof Set $\Delta:=\Delta_\alpha(\beta)$.\newline
(a)
If $\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in \eta_1\cap\Delta$ (where the matrix is regarded as a pair of column vectors), then $u=v$, and hence
\begin{center}\begin{tikzcd}
\left[\begin{matrix}x\\u\end{matrix}\right]\dar[dash]{\eta_1}\dar[dash,bend right,swap]{\Delta}\rar[dash]{\eta_0}&\left[\begin{matrix}x\\y\end{matrix}\right]\dar[dash]{\eta_1}\\
\left[\begin{matrix}y\\u\end{matrix}\right]\rar[dash]{\eta_0}&\left[\begin{matrix}y\\y\end{matrix}\right]
\end{tikzcd}\end{center}
so that we may conclude that $\left[\begin{matrix}x&y\\y&y\end{matrix}\right]\in\Delta$, by the Shifting Lemma.
It follows from Theorem \ref{thm_commutator_Delta} that $(x,y)\in [\alpha,\beta]$, so that $\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in[\alpha,\beta]_0$.
\vskip0.3cm\noindent (b) We have \[\aligned &\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in \eta_0\cap\Delta\\
\Longrightarrow\quad &\left[\begin{matrix}y&x\\v&u\end{matrix}\right]\in \eta_1\cap\Delta\\
\Longrightarrow\quad &\left[\begin{matrix}y&x\\v&u\end{matrix}\right]\in [\alpha,\beta]_0\\
\Longrightarrow\quad &\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in [\alpha,\beta]_1\endaligned\]
\vskip0.3cm\noindent (c) Clearly $\eta_0, \Delta\subseteq\beta_0$, so $\eta_0\lor\Delta \subseteq\beta_0$. Now if $\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in\beta_0$, then $x\;\alpha\;u$, $y\;\alpha\; v$ and $x\;\beta\; u$ (because $\beta_0$ is a congruence on $A_\alpha$, whose elements are column vectors belonging to $\alpha$). We thus have
\[\left[\begin{matrix}x\\u\end{matrix}\right]\;\eta_0\; \left[\begin{matrix}x\\x\end{matrix}\right]\;\Delta\; \left[\begin{matrix}y\\y\end{matrix}\right]\;\eta_0 \left[\begin{matrix}y\\v\end{matrix}\right]\] so that $\left[\begin{matrix}x&y\\u&v\end{matrix}\right]\in \eta_0\lor\Delta$. Thus also $\beta_0\subseteq \eta_0\lor\Delta$.
\vskip0.3cm\noindent (d) can be proved similarly.
\eproof
\subsection{The Modular Commutator under {\bf H}, {\bf S} and {\bf P}}\label{section_commutator_HSP}
In this section we investigate how the commutator behaves under the formation of homomorphic images, subalgebras and products. We begin by showing that the 5$^{th}$ property of the group commutator holds in general (in congruence modular varieties) i.e. that \[ [M/K,N/K]=[M,N]K/K \text{ for } K\subseteq M\cap N,\quad\text{ i.e. }\quad [\alpha/\pi,\beta/\pi]=([\alpha,\beta]\lor \pi)/\pi\text{ for } \pi\leq\alpha\land\beta\]
Let $f:A\twoheadrightarrow B$ be a surjective epimorphism. For each $\theta\in \Con(A)$, let $f(\theta)$ be the congruence on $B$ generated by the set $\{(f(x),f(y)): (x,y)\in\theta)\}$. Recall from Proposition \ref{propn_generated_con_forward} the following facts:\begin{itemize}\item
If $\ker f\subseteq\theta$, then \[(x,y)\in\theta\qquad\Longleftrightarrow\qquad (f(x),f(y))\in f(\theta)\]
\item If $\pi = \ker f$, then $f(\theta\lor\pi) = f(\theta)$.
\item For $\Theta\in\Con(B)$, $f(f^{-1}(\Theta))=\Theta$
\end{itemize}
\begin{theorem}\label{thm_commutator_H} Let $f:A\twoheadrightarrow B$ be a surjective homomorphism in a congruence modular variety, and let $\alpha,\beta\in \Con(A)$. Then
\begin{enumerate}[(a)]\item $[f(\alpha), f(\beta)] = f([\alpha,\beta])$
\item If $\pi:=\ker f$, then in $\Con(A/\pi)\cong \Con(B)$, we have \[[(\alpha\lor\pi)/\pi,(\beta\lor\pi)/\pi]=([\alpha,\beta]\lor\pi)/\pi\]
\item $[\alpha,\beta]\lor\pi = f^{-1}([f(\alpha),f(\beta)])$
\end{enumerate}
\end{theorem}
\bproof (a) Let $\pi:=\ker f$. First observe that $f(\alpha\lor\pi) = f(\alpha)$ and $f(\beta\lor\pi) = f(\beta)$, by Proposition \ref{propn_generated_con_forward}(d). Furthermore, \[[\alpha\lor\pi,\beta\lor\pi] \lor\pi = [\alpha,\beta]\lor[\alpha,\pi]\lor[\pi,\beta]\lor[\pi,\pi]\lor\pi=[\alpha,\beta]\lor\pi\] so that $f([\alpha,\beta]) = f([\alpha,\beta]\lor\pi) = f([\alpha\lor\pi,\beta\lor\pi]\lor\pi) = f([\alpha\lor\pi,\beta\lor\pi])$. Hence we may assume that $\alpha,\beta\geq \pi$ (by replacing $\alpha$ with $\alpha\lor\pi$ and $\beta$ with $\beta\lor \pi$ if necessary). It therefore suffices to show that \[[f(\alpha), f(\beta)] = f([\alpha,\beta])\qquad \alpha,\beta\geq\pi\] Recall, by Proposition \ref{propn_generated_con_forward}(c), that since $\alpha\geq \pi:=\ker f$, we have $(x,y)\in \alpha$ if and only if $(f(x),f(y))\in f(\alpha)$; the same holds for $\beta$. Further recall that $[\alpha,\beta]=\Cg_A(X(\alpha,\beta))$. Hence
\[\left(\begin{matrix} t(\mathbf a^1,\mathbf b^1) & t(\mathbf a^1,\mathbf b^2) \\ t(\mathbf a^2,\mathbf b^1) & t(\mathbf a^2,\mathbf b^2) \end{matrix}\right)\in M(\alpha,\beta)\quad\Longleftrightarrow\quad \left(\begin{matrix} t(f(\mathbf a^1),f(\mathbf b^1)) & t(f(\mathbf a^1),f(\mathbf b^2)) \\ t(f(\mathbf a^2),f(\mathbf b^1)) & t(f(\mathbf a^2),f(\mathbf b^2)) \end{matrix}\right)\in M(f(\alpha), f(\beta))\] \newline
It then follows that $(m_i(x,x,u,u), m_i(x,y,z,u))\in X(\alpha,\beta)$ if and only if \newline $\Big(m_i(f(x),f(x),f(u),f(u)), m_i(f(x),f(y),f(z),f(u))\Big)\in X(f(\alpha),f(\beta))$, i.e. that
\[(x,y)\in X(\alpha,\beta) \quad\Longleftrightarrow\quad (f(x),f(y))\in X(f(\alpha),f(\beta))\]
Thus by Proposition \ref{propn_generated_con_forward}(d), $f(\Cg_A(X(\alpha,\beta)) = \Cg_B\Big(X(f(\alpha),f(\beta))\Big)$, i.e. $f([\alpha,\beta]) = [f(\alpha),f(\beta)]$.
\vskip0.3cm\noindent (b)
We have $f(\theta) = (\theta\lor\pi)/\pi$, and hence \[[\tfrac{\alpha\lor\pi}{\pi},\tfrac{\beta\lor\pi}{\pi}] = [f(\alpha),f(\beta)] = f([\alpha,\beta]) = \tfrac{[\alpha,\beta]\lor\pi}{\pi}\]
\vskip0.3cm\noindent (c) We have \[\aligned &(x,y)\in f^{-1}([f(\alpha),f(\beta)])\\\Longleftrightarrow\quad &(f(x),f(y)) \in [f(\alpha),f(\beta)]\\\Longleftrightarrow\quad& (f(x),f(y))\in f([\alpha,\beta])\\\Longleftrightarrow\quad& (f(x),f(y))\in f([\alpha,\beta]\lor\pi)\\\Longleftrightarrow\quad &(x,y)\in[\alpha,\beta]\lor\pi\endaligned\]
\eproof
Hagemann and Hermann obtained the following nice characterization of the commutator:
\begin{theorem}\label{thm_HH_characterization} Suppose that $\mathcal V$ is a congruence modular variety. Then the commutator is the greatest binary operation defined on the congruence lattices of members of $\mathcal V$ such that for any surjective homomorphism
$f:A\twoheadrightarrow B$ in $\mathcal V$ and any $\alpha,\beta\in\Con(A)$:
\begin{enumerate}[(i)]\item $C(\alpha,\beta)\leq\alpha\land\beta$.
\item $f(C(\alpha,\beta))=C(f(\alpha),f(\beta))$
\end{enumerate}
\end{theorem}
\bproof Certainly, $[\alpha,\beta]$ has both those properties (i) and (ii), since $[\alpha,\beta]=[\beta,\alpha]$ and $f([\alpha,\beta]) = [f(\alpha),f(\beta)]$.
Now let $C$ be any binary operation defined on the congruence lattices of all algebras in a variety $\mathcal V$ satisfying (i) and (ii).
We will show that $C(\alpha,\beta)\subseteq [\alpha,\beta]$.
To that end, we invoke Lemma \ref{lemma_eta_Delta}. As in that Lemma, let $A\in\mathcal V$, and let $\alpha,\beta\in\Con(A)$. Further let
\[A_\alpha\buildrel{\pi_0}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto x\qquad\text{and}\qquad A_\alpha\buildrel{\pi_1}\over\longrightarrow A:\left[\begin{matrix}x\\y\end{matrix}\right]\mapsto y\]
be the projection mappings, and let $\eta_0=\ker\pi_0,\eta_1=\ker\pi_1$.
Define $\alpha_0,\beta_0, [\alpha,\beta]_0 \in\Con(A_\alpha)$ by
\[\alpha_0:=\pi_0^{-1}(\alpha)\qquad\beta_0:=\pi_0^{-1}(\beta)\qquad [\alpha,\beta]_0:=\pi_0^{-1}([\alpha,\beta])\] and write $\Delta:=\Delta_\alpha(\beta)$. Observe that
$\left[\begin{matrix} x&y\\u&v\end{matrix}\right]\in\alpha_0$ if and only if $x,y,u,v$ belong to the same $\alpha$--class, if and only if \[\left[\begin{matrix} x\\u\end{matrix}\right]\;\eta_0\;\left[\begin{matrix} x\\v\end{matrix}\right]\;\eta_1\; \left[\begin{matrix} y\\v\end{matrix}\right]\] so that $\alpha_0=\eta_0\lor\eta_1$. By Lemma \ref{lemma_eta_Delta} we have also that $\beta_0=\eta_0\lor\Delta$. Hence \[\alpha=\pi_0(\eta_0\lor\eta_1)\qquad\qquad \beta=\pi_0(\eta_0\lor\Delta)\]
By the assumptions on $C$ and the Lemma, we have \[C(\eta_1,\Delta)\subseteq \eta_1\cap\Delta\subseteq[\alpha,\beta]_0\]
Thus
\[\aligned C(\alpha,\beta)&= C(\pi_0(\eta_0\lor\eta_1), \pi_0(\eta_0\lor\Delta))\\
&=C(\pi_0(\eta_1),\pi_0(\Delta))\qquad\text{by Proposition \ref{propn_quotient_con}, since $\eta_0=\ker\pi_0$}\\
&=\pi_0(C(\eta_1,\Delta))\\&\subseteq\pi_0([\alpha,\beta]_0)\\
&=[\alpha,\beta]
\endaligned\]\eproof
Next, we consider the behaviour of the commutator when restricted to subalgebras:
\begin{theorem}\label{thm_commutator_S} Let $A$ be an algebra in a congruence modular variety, and let $B$ be a subalgebra of $A$.
If $\alpha,\beta\in\Con(A)$, then the restrictions of $\alpha,\beta$ to $B$ satisfy
\[[\alpha\!\!\restriction_B,\beta\!\!\restriction_B]\leq [\alpha,\beta]\!\!\restriction_B\]
\end{theorem}
\bproof
It is easy to verify that if $C(\alpha,\beta;\delta)$ in $\Con(A)$, then $C(\alpha\!\!\restriction_B,\beta\!\!\restriction_B;\delta\!\!\restriction_B)$ in $\Con(B)$.
It follows that $C(\alpha\!\!\restriction_B,\beta\!\!\restriction_B;[\alpha,\beta]\!\!\restriction_B)$. Thus by definition of $[\alpha\!\!\restriction_B,\beta\!\!\restriction_B]$, we have that $[\alpha,\beta]\!\!\restriction_B\geq [\alpha\!\!\restriction_B,\beta\!\!\restriction_B]$.
\eproof
Finally, we study the behaviour of the commutator under products: Suppose that $A_i$ are algebras, for $i\in I$, and that $\theta_i\in\Con(A_i)$. We can then define two congruences on $A:=\prod_{i\in I}A_i$ as follows:
\[\aligned \prod_{i\in I}\theta_i&:=\left\{(\mathbf a,\mathbf b)\in A^2: a_i\;\theta_i\; b_i\text{ for all }i\in I\right\}\\
\bigoplus_{i\in I}\theta_i&:=\left\{(\mathbf a,\mathbf b)\in A^2 : a_i\;\theta_i\; b_i\text{ for all }i\in I\text{ and $a_i\not=b_i$ for at most finitely many $i\in I$}\right\}
\endaligned\]
\begin{proposition} Let $A_i\: (i\in I)$ be algebras, and let $A:=\prod_{i\in I}A_i$ The map \[h:\prod_{i\in I}\Con(A_i)\to \Con(A):(\theta_i)_{i\in I}\mapsto\bigoplus_{i\in I}\theta_i\]
is a lattice embedding.
\end{proposition}
\bproof It is easy to see that $h$ is a one--to--one mapping.
Suppose that $\theta_i,\psi_i\in\Con(A_i)$ for $i\in I$. Clearly $\bigoplus_I (\theta_i\lor\psi_i)\geq \bigoplus_I\theta_i\lor\bigoplus_I\psi_i$. Conversely, if $(\mathbf a,\mathbf b)\in \bigoplus_I (\theta_i\lor\psi_i)$, then there are at most finitely many $i\in I$ such that $a_i\not= b_i$. Let $\{i_1,\dots, i_n\}$ be the set of those $i$'s. It is easy to see that we can find $m\in \mathbb N$ so that \[(a_i, b_i)\in \underbrace{\theta_i\circ\psi_i\circ\theta_i\circ\dots}_{m\text{ ``factors}"}\qquad\text{for all }i\in \{i_1,\dots,i_n\}\text{ and thus for all }i\in I\]
Consequently, there are $c_{i,1},\dots, c_{i,m-1}\in A_i$ such that
\[\underbrace{a_i\;\theta_i\;c_{i,1}\;\psi_i\;c_{i,2}\;\theta_i\;\dots b_i}_{m\text{ ``factors"}}\qquad\text{for all }i\in I\] where $c_{i,j}=a_i=b_i$ for all $i\not\in\{i_1,\dots, i_n\}$ and all $j<m$. Hence \[\underbrace{\mathbf a\;\bigoplus_I\theta_i\;\mathbf c_1\;\bigoplus_I\psi_i\;\mathbf c_2\;\bigoplus_I\theta_i\dots \mathbf b}_{m\text{ ``factors"}}\qquad \text{where }\mathbf c_j:=(c_{i,j})_{i\in I}\] proving that $(\mathbf a,\mathbf b)\in \bigoplus_I\theta_i\lor\bigoplus_I\psi_i$. Hence $\bigoplus_I (\theta_i\lor\psi_i)= \bigoplus_I\theta_i\lor\bigoplus_I\psi_i$.
Similarly (and rather more easily) it can be shown that $\bigoplus_I (\theta_i\land\psi_i)= \bigoplus_I\theta_i\land\bigoplus_I\psi_i$, so that the map $h$ is a lattice embedding.
\eproof
\begin{proposition}\label{propn_embedding_preserves_commutator}
Let $A_i\: (i\in I)$ be algebras in a congruence modular variety, and let $A:=\prod_{i\in I}A_i$ The map lattice embedding \[h:\prod_{i\in I}\Con(A_i)\to \Con(A):(\theta_i)_{i\in I}\mapsto\bigoplus_{i\in I}\theta_i\] preserves the commutator operation:
\[\bigoplus_{i\in I}[\theta_i,\psi_i] = \left [\bigoplus_{i\in I}\theta_i, \;\bigoplus_{i\in I}\psi_i\right]\]
\end{proposition}
\bproof Let $\pi_i:A\twoheadrightarrow A_i$ be the $i^{th}$ projection, and put $\eta_i:=\ker\pi_i$. Define $\lambda:=\bigoplus_{i\in I}1_{A_i}$, so that $(\mathbf a,\mathbf b)\in \lambda$ if and only if $a_i=b_i$ for all but finitely many $i\in I$. Also define $\eta_i':=\bigwedge\limits_{j\not=i}\eta_j$, and for $\theta_i\in\Con(A_i)$, let $\overleftarrow{\theta_i}:=\pi_i^{-1}(\theta_i)\in\Con(A)$.
Observe that by modularity we have
\[\overleftarrow{\theta_i}=\overleftarrow{\theta_i}\land(\eta_i'\lor\eta_i) =( \overleftarrow{\theta_i}\land\eta_i')\lor\eta_i\tag{$\dagger$}\]
Also,
\[\bigoplus_{i\in I}\theta_i = \lambda\land \bigwedge_{i\in I}\overleftarrow{\theta_i}\]
Furthermore $(\mathbf a,\mathbf b)\in \bigvee_{i\in I}(\overleftarrow{\theta_i}\land\eta_i')$ if and only if there exist $\mathbf c_1,\dots,\mathbf c_n$ and $i_1,\dots, i_n\in I$ such that $\mathbf a\; (\overleftarrow{\theta_{i_1}}\land\eta_{i_1}')\;\mathbf c_1\; (\overleftarrow{\theta_{i_2}}\land\eta_{i_2}')\;\mathbf c_2\; (\overleftarrow{\theta_{i_3}}\land\eta_{i_3}')\;\dots\;\mathbf b$, which is the case if and only if
\[a_{i_1}\;\theta_{i_1}\;c_{1,i_1}\qquad\text{and }\qquad a_{k}=c_{1,k}\text{ for }k\not=i_{1}\]
\[c_{1,i_2}\;\theta_{i_2}\;c_{2,i_2}\qquad\text{and }\qquad c_{1,k}=c_{2,k}\text{ for }k\not=i_{2}\tag{$\star$}\]
\[\vdots\]
\[c_{n,i_n}\;\theta_{i_n}\;b_{i_n}\qquad\text{and }\qquad c_{n,k}=b_k\text{ for }k\not=i_{n}\]
Now clearly if $(\star)$ holds then $\mathbf a\;\bigoplus_{i\in I}\theta_i \;\mathbf c_1\bigoplus_{i\in I}\theta_i \;\mathbf c_2 \bigoplus_{i\in I}\theta_i \;\dots\;\mathbf b$, so $ \bigvee_{i\in I}(\overleftarrow{\theta_i}\land\eta_i')\subseteq\bigoplus_{i\in I}\theta_i$. Conversely, if $(\mathbf a,\mathbf b) \in\bigoplus_{i\in I}\theta_{i}$, then there are $i_1,\dots, i_n$ such that $a_{i_j}\;\theta_{i_j}\;b_{i_j}$, and such that $a_i=b_i$ for all other $i\in I$. If one defines $\mathbf c_1,\dots,\mathbf c_n$ by
\[c_{1,i_1}=b_{i_1}\qquad c_{1,k}=a_k\text{ for }k\not=i_1\]
\[c_{2,i_2}=b_{i_2}\qquad c_{2,k}= c_{1,k}\text{ for }k\not=i_2\]
\[\vdots\]
\[c_{n,i_2}=b_{i_n}\qquad c_{n,k}= c_{n-1,k}\text{ for }k\not=i_2\]
then $(\star)$ holds for $\mathbf a,\mathbf c_1,\mathbf c_2,\dots,\mathbf b$, and hence $(\mathbf a,\mathbf b)\in \bigvee_{i\in I}(\overleftarrow{\theta_i}\land\eta_i')$. We deduce that
\[ \bigvee_{i\in I}(\overleftarrow{\theta_i}\land\eta_i')=\bigoplus_{i\in I}\theta_i = \lambda\land \bigwedge_{i\in I}\overleftarrow{\theta_i}\tag{$\ddagger$}\]
Now let $\alpha:= \left [\bigoplus_{i\in I}\theta_i, \;\bigoplus_{i\in I}\psi_i\right]$ and $\beta:= \bigoplus_{i\in I}[\theta_i,\psi_i] $. Our aim is to prove that
$\alpha=\beta$.
Observe that
\[\overleftarrow{[\theta_i,\psi_i]}=\pi_i^{-1}([\pi_i(\overleftarrow{\theta_i}),\pi_i(\overleftarrow{\psi_i})]) = [\overleftarrow{\theta_i},\overleftarrow{\psi_i}]\lor\eta_i\tag{$\star\star$}\] by Theorem \ref{thm_commutator_H}.
Hence\[\beta= \bigoplus_{i\in I}[\theta_i,\psi_i] = \lambda\land\bigwedge_{i\in I} \overleftarrow{[\theta_i,\psi_i]} = \lambda\land\bigwedge_{i\in I}\left([\overleftarrow{\theta_i},\overleftarrow{\psi_i}]\lor\eta_i\right)\] Moreover, by the monotone properties of the commutator, we have
\[\alpha=\left[\lambda\land\bigwedge_{i\in I}\overleftarrow{\theta_i}, \lambda\land\bigwedge_{i\in I}\overleftarrow{\psi_i}\right]\leq\lambda\land\bigwedge_{i\in I}[ \overleftarrow{\theta_i}, \overleftarrow{\psi_i}]\]
It is now clear that $\alpha\leq\beta$.
To prove the reverse inequality, note that by $(\ddagger$) we have
\[\beta= \bigoplus_{i\in I}[\theta_i,\psi_i] =\bigvee_{i\in I}\left(\overleftarrow{[\theta_i,\psi_i]}\land\eta_i'\right) =\bigvee_{i\in I}\left(([\overleftarrow{\theta_i},\overleftarrow{\psi_i}]\lor\eta_i)\land\eta_i'\right) \]
But by ($\dagger$), $\overleftarrow{\theta_i}=( \overleftarrow{\theta_i}\land\eta_i')\lor\eta_i$, and hence
\[[\overleftarrow{\theta_i},\overleftarrow{\psi_i}]\lor\eta_i = [(\overleftarrow{\theta_i}\land\eta_i')\lor\eta_i ,(\overleftarrow{\psi_i}\land\eta_i')\lor\eta_i]\lor\eta_i =[\overleftarrow{\theta_i}\land\eta_i',\overleftarrow{\psi_i}\land\eta_i']\lor\eta_i\]
Hence by modularity,
\[ \aligned ([\overleftarrow{\theta_i},\overleftarrow{\psi_i}]\lor\eta_i)\land\eta_i' &= ([\overleftarrow{\theta_i}\land\eta_i',\overleftarrow{\psi_i}\land\eta_i']\lor\eta_i)\land\eta_i' \\&=[\overleftarrow{\theta_i}\land\eta_i',\overleftarrow{\psi_i}\land\eta_i']\lor( \eta_i\land\eta_i' )\\&= [\overleftarrow{\theta_i}\land\eta_i',\overleftarrow{\psi_i}\land\eta_i']\endaligned \]
so that
\[\beta= \bigvee_{i\in I} [\overleftarrow{\theta_i}\land\eta_i',\overleftarrow{\psi_i}\land\eta_i']\]
On the other hand by $(\ddagger)$,
\[\alpha= \left [\bigoplus_{i\in I}\theta_i, \;\bigoplus_{j\in I}\psi_j\right] = \left[ \bigvee_{i\in I}(\overleftarrow{\theta_i}\land\eta_i'), \bigvee_{j\in I}(\overleftarrow{\psi_j}\land\eta_j')\right] =\bigvee_{i,j\in I}[\overleftarrow{\theta_i}\land\eta_i', \overleftarrow{\psi_j}\land\eta_j'] \] from which it is clear that $\beta\leq\alpha$.
\eproof
\begin{theorem}\label{thm_commutator_P} Let $\mathcal V$ be a congruence modular variety. \begin{enumerate}[(a)]\item Suppose that $A,B\in\mathcal V$, that $\alpha_0,\alpha_1\in \Con(A)$ and that $\beta_0,\beta_1\in\Con(B)$. Then
\[[\alpha_0\times\beta_0,\alpha_1\times\beta_1]=[\alpha_0,\alpha_1]\times[\beta_0,\beta_1]\]
\item Suppose that $A_i\in\mathcal V$ and that $\theta_i,\psi_i\in\Con(A_i)$ (for $i\in I$).
Then
\[\left[\prod_{i\in I}\theta_i ,\prod_{i\in I}\psi_i\right]\leq \prod_{i\in I}[\theta_i,\psi_i]\]\end{enumerate}
\end{theorem}
\bproof (a) follows directly from Proposition \ref{propn_embedding_preserves_commutator}, since $\alpha_0\times\alpha_1 = \alpha_0\oplus\alpha_1$.\vskip0.3cm\noindent (b) We use the notation of the proof of Proposition \ref{propn_embedding_preserves_commutator}. Then $\prod_{i\in I}\theta_i = \bigwedge_{i\in I}\overleftarrow{\theta_i}$.
Hence\[\left[\prod_{i\in I}\theta_i ,\prod_{i\in I}\psi_i\right] =\left[\bigwedge_{i\in I}\overleftarrow{\theta_i}, \bigwedge_{i\in I}\overleftarrow{\psi_i}\right] \leq \bigwedge_{i\in I}[\overleftarrow{\theta_i}, \overleftarrow{\psi_i}]\leq \bigwedge_{i\in I}([\overleftarrow{\theta_i}, \overleftarrow{\psi_i}] \lor\eta_i)\]
Now by ($\star\star$) of the proof of Proposition \ref{propn_embedding_preserves_commutator}, we have $[\overleftarrow{\theta_i}, \overleftarrow{\psi_i}] \lor\eta_i =\overleftarrow{[\theta_i,\psi_i]}$. hence
\[\left[\prod_{i\in I}\theta_i ,\prod_{i\in I}\psi_i\right] \leq \bigwedge_{i\in I}\overleftarrow{[\theta_i,\psi_i]} =\prod_{i\in I}[\theta_i,\psi_i]\]
\eproof
\section{The Distributive Commutator}
We have already seen that the commutator reduces to intersection in the variety of lattices, or indeed, in any variety that has a ternary majority term.
The possession of such a term implies congruence distributivity. The following theorem shows that $[\alpha,\beta]=\alpha\cap\beta$ in any congruence distributive variety.
\begin{theorem} Let $\mathcal V$ be a variety (not assumed a priori to be congruence modular). Then $\mathcal V$ is congruence distributive if and only if for any $A\in\mathcal V$ and any $\alpha,\beta,\gamma\in\Con(A)$:
\begin{enumerate}[(i)]\item
$[\alpha,\beta] = \alpha\cap\beta$
\item $[\alpha\lor\beta,\gamma] = [\alpha,\gamma]\lor[\beta,\gamma]$
\end{enumerate}\end{theorem}
\bproof First suppose that $\mathcal V$ satisfies (i) and (ii). Then
\[(\alpha\lor\beta)\land\gamma = [\alpha\lor\beta,\gamma] = [\alpha,\gamma]\lor[\beta,\gamma] =(\alpha\land\gamma)\lor(\beta\land\gamma)\]
so that $\mathcal V$ is congruence distributive.
Conversely, suppose that $\mathcal V$ is congruence distributive. Then it is congruence modular. We now show that (i) holds by using the Hagemann--Hermann characterization of the commutator (in Theorem \ref{thm_HH_characterization}) as the largest binary operation $C$ defined on all the congruence lattices of algebras in $\mathcal V$ satisfying
\[C(\alpha,\beta)\leq\alpha\land\beta\qquad f([\alpha,\beta]) = [f(\alpha), f(\beta)]\] We prove that $C(\alpha,\beta):=\alpha\land\beta$ satisfies both these conditions. Clearly, we need only show that $f(\alpha\cap \beta) = f(\alpha)\cap f(\beta)$. Suppose, therefore, that $f:A\twoheadrightarrow B$ is a surjective homomorphism in $\mathcal V$, and that $\alpha,\beta\in\Con(A)$.
Let $\pi:=\ker f$.
Note that by Proposition \ref {propn_generated_con_backward}, $f^{-1}(f(\alpha))=\alpha\lor\pi$.
Thus
\[\aligned &(f(x),f(y))\in f(\alpha)\cap f(\beta)\\
\Longleftrightarrow\quad & (x,y) \in f^{-1}(f (\alpha)\cap(\beta))\\
\Longleftrightarrow\quad & (x,y) \in f^{-1}(f (\alpha))\cap f^{-1}(f(\beta))\\
\Longleftrightarrow\quad & (x,y) \in (\alpha\lor\pi)\cap(\beta\lor\pi)\\
\Longleftrightarrow\quad &(x,y)\in (\alpha\cap\beta)\lor\pi\qquad\qquad\qquad\qquad\qquad\quad \text{by congruence distributivity}\\
\Longleftrightarrow\quad & (f(x),f(y))\in f((\alpha\cap\beta)\lor\pi) =f(\alpha\cap\beta)\qquad \text{by Proposition \ref {propn_generated_con_forward}}
\endaligned\]
Hence $f(\alpha)\cap f(\beta) = f(\alpha\cap\beta)$, as required.
Now congruence distributivity and (i) are easily seen to imply (ii):
\[[\alpha\lor\beta,\gamma] = (\alpha\lor\beta)\land\gamma = (\alpha\land\gamma)\lor(\beta\land\gamma) = [\alpha,\gamma]\lor[\beta,\gamma]\]
\eproof
\begin{remarks}\rm
\begin{enumerate}[(a)]\item The above argument shows that if we {\em define} the commutator to be the largest binary operation satisfying the two conditions of Theorem \ref{thm_HH_characterization}, then the commutator reduces to intersection in any congruence distributive variety.
\item One can also prove that the commutator reduces to intersection in a congruence distributive variety without recourse to
Theorem \ref{thm_HH_characterization}. If a variety $\mathcal V$ is congruence distributive, then it has J\'onsson terms $d_0,\dots, d_n$ satisfying the J\'onsson term equations (J1)-(J5). Now let $(x,y)\in\alpha\cap\beta$. We will show by induction that \[(d_i(x,y,x), d_i(x,y,y))\in[\alpha,\beta]\tag{$\star$}\] for all $i=0,\dots, n$. $(\star)$ clearly holds for $i=0$, since $d_0(x,y,x)=x=d_0(x,y,y)$ by (J1). Suppose now that $(\star)$ holds for $i$. To prove that it also holds for $i+1$ we consider two cases:
\vskip0.3cm\noindent \underline{$i$ is even:} Then \[\left(\begin{matrix}d_i(x,y,x)&d_i(x,y,y)\\d_i(x,x,x)&d_i(x,x,y)\end{matrix}\right)\in M(\alpha,\beta)\] Since the elements of the top row are $[\alpha,\beta]$ related (by hypothesis), and since $C(\alpha,\beta;[\alpha,\beta])$, we can deduce that $(d_i(x,x,x), d_i(x,x,y))\in [\alpha,\beta]$ also. But $d_i(x,x,y) = d_{i+1}(x,x,y)$ by (J3), since $i$ is even. Since $d_i(x,y,x)=x$ (by (J2)), we have $d_{i+1}(x,x,x)=d_i(x,x,x)\;[\alpha,\beta]\; d_i(x,x,y)=d_{i+1}(x,x,y)$, i.e. $d_{i+1}(x,x,x)\;[\alpha,\beta]\;d_{i+1}(x,x,y)$. Observe now that \[\left(\begin{matrix}d_{i+1}(x,x,x)&d_{i+1}(x,x,y)\\d_{i+1}(x,y,x)&d_{i+1}(x,y,y)\end{matrix}\right)\in M(\alpha,\beta)\] and use centrality once again to deduce that $d_{i+1}(x,y,x)\;[\alpha,\beta]\;d_{i+1}(x,y,y)$.
\vskip0.3cm\noindent\underline{$i$ is odd:} In that case $d_{i}(x,y,y) =d_{i+1}(x,y,y)$ by (J4). Hence \[d_{i+1}(x,y,x) = d_i(x,y,x)\;[\alpha,\beta]\;d_i(x,y,y) = d_{i+1}(x,y,y)\] immediately yields $d_{i+1}(x,y,x)\;[\alpha,\beta]\;d_{i+1}(x,y,y)$.
\vskip0.3cm\noindent This completes the induction. In particular $d_n(x,y,x)[\alpha,\beta]\;d_n(x,y,y)$, so $x\;[\alpha,\beta]\; y$ by (J2) and (J5). Since $(x,y)$ is an arbitrary member of $\alpha\cap\beta$, we obtain $\alpha\cap\beta\subseteq[\alpha,\beta]$. The reverse inclusion is always valid. \end{enumerate}
\endbox
\end{remarks}
\section{The Gumm Difference Term}
\fancyhead[RE]{The Gumm Difference Term}
\subsection{A Congruence Modular Variety has a Gumm Difference Term}
Suppose that $\mathcal V$ is a congruence modular variety, with Day terms $m_0,\dots, m_n$ satisfying the Day term equations (D1)-(D5).
Define ternary terms $p_0,\dots, p_n$ inductively as follows:
\[\aligned p_0(x,y,z)&=z\\
p_{i+1}(x,y,z)&=m_{i+1}(p_i(x,y,z),x,y,p_i(x,y,z))\qquad\text{if $i<n$ is even}\\
p_{i+1}(x,y,z)&=m_{i+1}(p_i(x,y,z),y,x,p_i(x,y,z))\qquad\text{if $i<n$ is odd}
\endaligned\]
\begin{proposition}\label{propn_difference_term}\begin{enumerate}[(a)]\item
$\mathcal V$ satisfies the identity $p_i(x,x,y)\approx y$ for all $i=0,\dots, n$.
\item If $A\in\mathcal V$ and $\alpha\in\Con(A)$, then \[x\;\alpha\;y \qquad\Longrightarrow\qquad p_n(x,y,y)\;[\alpha,\alpha]\;x\]
\end{enumerate}
\end{proposition}
\bproof (a) By definition, $p_0(x,x,y)= y$. Supppose now that $p_i(x,x,y)=y$ for some $i<n$. Then
\[p_{i+1}(x,x,y) =m_{i+1}(p_i(x,x,y),x,x,p_i(x,x,y))= m_{i+1}(y,x,x,y) = y\qquad\text{ whether $i$ even or odd,}\]
by (D2), so the result follows by induction.
\vskip0.3cm\noindent (b) Let $A\in\mathcal V$ and let $\alpha\in\Con(A)$. We shall show, again by induction, that
\[\aligned p_i(x,y,y)\;[\alpha,\alpha]\; m_i(y,y,x,x)\quad&\text{if $i$ is even}\\
p_i(x,y,y)\;[\alpha,\alpha]\; m_i(y,y,y,x)\quad&\text{if $i$ is odd}\endaligned\]
It will then follow that $p_n(x,y,y)\;[\alpha,\alpha]\;x$ by (D5).
Since $p_0(x,y,y)=y=m_0(x,x,y,y)$, the case $i=0$ is trivial. To proceed, we consider two cases:
\vskip0.3cm\noindent\underline{Case 1}: $i<n$ is even, and $p_i(x,y,y)\;[\alpha,\alpha]\; m_i(y,y,x,x)$.\newline We must show that $p_{i+1}(x,y,y)\;[\alpha,\alpha]\;m_{i+1}(y,y,y,x)$. Now we have
\[\aligned p_{i+1}(x,y,y)&=m_{i+1}(p_i(x,y,y),x,y,p_i(x,y,y))\\\;[\alpha,&\alpha]\; m_{i+1}(m_i(y,y,x,x),x,y,m_i(y,y,x,x))\endaligned\tag{$\star$}\]
Now since $x\;\alpha\;y$, we observe that
\begin{center}\begin{tikzcd} m_{i+1}(m_i(y,y,x,x), x, \fbox{$x$}, m_i(y,y,x,x))\dar[dash,swap]{\alpha}\rar[dash]{\alpha}&m_{i+1}(m_i(y,y,y,y), y, \fbox{$x$}, m_i(x,x,x,x))\dar[dash]{\alpha}\\m_{i+1}(m_i(y,y,x,x), x, \fbox{$y$}, m_i(y,y,x,x))\rar[dash]{\alpha}&m_{i+1}(m_i(y,y,y,y), y, \fbox{$y$}, m_i(x,x,x,x))
\end{tikzcd}\end{center}
Furthermore, \[\aligned m_{i+1}(m_i(y,y,x,x), x, \fbox{$x$}, m_i(y,y,x,x)) &= m_i(y,y,x,x)\quad\qquad\qquad\qquad\qquad\text{by (D2)}\\
&=m_{i+1}(y,y,x,x)\qquad\qquad\qquad\qquad\text{ by (D3)}\\
&= m_{i+1}(m_i(y,y,y,y),y,\fbox{$x$},m_i(x,x,x,x))\endaligned\] and thus the two elements in the top row of the matrix in the diagram above are equal, and thus $[\alpha,\alpha]$--related. Since $C(\alpha,\alpha;[\alpha,\alpha])$, we may deduce that the elements of the bottom row are $[\alpha,\alpha]$--related as well, i.e. \[m_{i+1}(m_i(y,y,x,x), x, y, m_i(y,y,x,x))\;[\alpha,\alpha]\;m_{i+1}(m_i(y,y,y,y), y,y, m_i(x,x,x,x))\] and thus \[m_{i+1}(m_i(y,y,x,x), x, y, m_i(y,y,x,x))\;[\alpha,\alpha]\; m_{i+1}(y,y,y,x)\tag{$\dagger$}\]
But we saw in $(\star)$ that
\[ p_{i+1}(x,y,y)\;[\alpha,\alpha]\; m_{i+1}(m_i(y,y,x,x),x,y,m_i(y,y,x,x))\] Combining this with $(\dagger)$, it follows that
\[ p_{i+1}(x,y,y)\;[\alpha,\alpha]\; m_{i+1}(y,y,y,x)\] which is what we had to prove.
\vskip0.3cm\noindent\underline{Case 2}: $i<n$ is odd, and $p_i(x,y,y)\;[\alpha,\alpha]\; m_i(y,y,y,x)$.\newline We must show that $p_{i+1}(x,y,y)\;[\alpha,\alpha]\;m_{i+1}(y,y,x,x)$. Now we have
\[\aligned p_{i+1}(x,y,y)&=m_{i+1}(p_i(x,y,y),x,y,p_i(x,y,y))\\\;[\alpha,&\alpha]\; m_{i+1}(m_i(y,y,y,x),x,y,m_i(y,y,y,x))\endaligned\tag{$\star\star$}\]
Now since $x\;\alpha\;y$, we observe that
\begin{center}\begin{tikzcd} m_{i+1}(m_i(y,y,y,x), y, \fbox{$y$}, m_i(y,y,y,x))\dar[dash,swap]{\alpha}\rar[dash]{\alpha}&m_{i+1}(m_i(y,y,y,y), y, \fbox{$y$}, m_i(x,x,x,x))\dar[dash]{\alpha}\\m_{i+1}(m_i(y,y,y,x), y, \fbox{$x$}, m_i(y,y,y,x))\rar[dash]{\alpha}&m_{i+1}(m_i(y,y,y,y), y, \fbox{$x$}, m_i(x,x,x,x))
\end{tikzcd}\end{center}
Furthermore, \[\aligned m_{i+1}(m_i(y,y,y,x), y, \fbox{$y$}, m_i(y,y,y,x)) &= m_i(y,y,y,x)\quad\qquad\qquad\qquad\qquad\text{by (D2)}\\
&=m_{i+1}(y,y,y,x)\qquad\qquad\qquad\qquad\text{ by (D4)}\\
&= m_{i+1}(m_i(y,y,y,y),y,\fbox{$y$},m_i(x,x,x,x))\endaligned\] and thus the two elements in the top row of the matrix in the diagram above are equal, and thus $[\alpha,\alpha]$--related. Since $C(\alpha,\alpha;[\alpha,\alpha])$, we may deduce that the elements of the bottom row are $[\alpha,\alpha]$--related as well, i.e. \[m_{i+1}(m_i(y,y,y,x), y, x, m_i(y,y,y,x))\;[\alpha,\alpha]\;m_{i+1}(m_i(y,y,y,y), y,x, m_i(x,x,x,x))\] and thus \[m_{i+1}(m_i(y,y,y,x), y, x, m_i(y,y,y,x))\;[\alpha,\alpha]\; m_{i+1}(y,y,x,x)\tag{$\ddagger$}\]
But we saw in $(\star\star)$ that
\[ p_{i+1}(x,y,y)\;[\alpha,\alpha]\; m_{i+1}(m_i(y,y,y,x),y,x,m_i(y,y,y,x))\] Combining this with $(\ddagger)$, it follows that
\[ p_{i+1}(x,y,y)\;[\alpha,\alpha]\; m_{i+1}(y,y,x,x)\] which is what we had to prove.
\eproof
Observe that the term $d(x,y,z):=p_n(x,y,z)$ constructed above satisfies the following two conditions:
\begin{enumerate}[(i)]\item $d(x,x,y)\approx y$ in $\mathcal V$.
\item For all $A\in\mathcal V$ and $\alpha\in\Con(A)$, \[x\;\alpha\;y\qquad\Longrightarrow\qquad d(x,y,y)\;[\alpha,\alpha]\; x\]
\end{enumerate}
We call any term which satisfies the above two conditions in a variety $\mathcal V$ a {\em Gumm difference term} for $\mathcal V$.
Thus:
\begin{corollary}\label{corollary_modular_implies_Gumm_difference} Every congruence modular variety $\mathcal V$ has a Gumm difference term, i.e. a ternary term $d(x,y,z)$ satisfying: \begin{enumerate}[(i)]\item $d(x,x,y)\approx y$ in $\mathcal V$.
\item For all $A\in\mathcal V$ and $\alpha\in\Con(A)$, \[x\;\alpha\;y\qquad\Longrightarrow\qquad d(x,y,y)\;[\alpha,\alpha]\; x\]
\end{enumerate}
\end{corollary}
Observe that if $p(x,y,z)$ is a Mal'tsev term for a congruence permutable variety, then it is clearly a Gumm difference term.
\begin{theorem} \label{thm_difference_term_equiv_shift}
\begin{enumerate}[(a)]\item Suppose that $\mathcal V$ is congruence modular, and that $d(x,y,z)$ is a Gumm difference term for $\mathcal V$. Then for all $A\in \mathcal V$, if $\alpha,\beta,\gamma\in \Con(A)$ are such that $\alpha\land\beta\leq \gamma$, then
\begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\qquad implies \begin{tikzcd}
{}&x\dlar[dash,swap,dashed]{\gamma}\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
d(u,v,y)\rar[dash,dashed]{\beta}&y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\end{center}
\item Conversely, if there is a ternary term $d(x,y,z)$ such that for all $A\in \mathcal V$, if $\alpha,\beta,\gamma\in \Con(A)$ are such that $\alpha\land\beta\leq \gamma$, then
\begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\qquad implies \begin{tikzcd}
{}&x\dlar[dash,swap,dashed]{\gamma}\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
d(u,v,y)\rar[dash,dashed]{\beta}&y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\end{center}
then $\mathcal V$ is congruence modular, and $d$ is a Gumm difference term for $\mathcal V$.
\end{enumerate}
\end{theorem}
\bproof
(a) $\Longrightarrow$ (b): Suppose that $d(x,y,z)$ is a Gumm difference term, and that we have \begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\end{center} for some $\alpha,\beta,\gamma\in \Con(A)$, where $A\in\mathcal V$ and $\alpha\land\beta\leq\gamma$. We will show that $d$ has the property stated in (b), i.e. that $d(u,v,y)\;\beta\;y$ and that $d(u,v,y)\;\gamma\;x$
Now clearly $d(u,v,y)\;\beta\;d(y,y,y)=y$, so that $d(u,v,y)\;\beta\; y$.
To prove that $d(u,v,y)\;\gamma\;x$ is more complicated:
\begin{enumerate}[(1)]\item First observe that we have $(x,y)\in\beta\circ\gamma\circ\beta$, so that $x\;(\beta\lor\gamma)\;y$. Hence $x\;(\alpha\land(\beta\lor \gamma))\;y$.
\item By (ii) it follows that $d(x,y,y)\;[\alpha\land(\beta\lor\gamma), \alpha\land(\beta\lor\gamma)]\; x$.
\item $[\alpha\land(\beta\lor\gamma), \alpha\land(\beta\lor\gamma)]\leq [\alpha,\beta\lor\gamma]=[\alpha,\beta]\lor[\alpha,\gamma]\leq(\alpha\land\beta)\lor (\alpha\land\gamma) = \alpha\land\gamma$, as $\alpha\land\beta\leq\gamma$.\\Thus $d(x,y,y)\;(\alpha\land\gamma)\;x$.
\item Also $d(x,y,y)\;\beta\; d(z,v,y)$, so $d(z,v,y)\;(\beta\lor(\alpha\land\gamma))\; x$. \item Moreover, $d(z,v,y)\;\alpha\;d(z,z,x)$ and $d(z,z,x)=x$ by (i). \\Hence using (4) we obtain that $d(z,v,y)\;(\alpha\land(\beta\lor(\alpha\land\gamma)))\;x$.
\item By modularity, $\alpha\land(\beta\lor(\alpha\land\gamma))=(\alpha\land\beta)\lor(\alpha\land\gamma) = \alpha\land\gamma$.\\We conclude from (5) that
$d(z,v,y)\;(\alpha\land\gamma)\; x$.
\item Finally, we clearly have $d(u,v,y)\;\gamma\; d(z,v,y)$, so by (6) we see that $ d(u,v,y)\;\gamma\;x$, as required.
\end{enumerate}
\vskip0.3cm\noindent (b) $\Longrightarrow$ (a): Conversely, suppose now that we have a term $d(x,y,z)$ satisfying the implication in (b). We must show that $\mathcal V$ is congruence modular, and that $d$ is a Gumm difference term, i.e. that
\begin{enumerate}[(i)]\item $d(x,x,y)\approx y$ in $\mathcal V$.
\item For all $A\in\mathcal V$ and $\alpha\in\Con(A)$, \[x\;\alpha\;y\qquad\Longrightarrow\qquad d(x,y,y)\;[\alpha,\alpha]\; x\]
\end{enumerate}
Now by assumption,
\begin{center}\begin{tikzcd}
y\dar[dash,swap]{0}\rar[dash]&{1}\rar[dash]& x\dar[dash]{0}\dlar[dash,swap]{0}\\
y\rar[dash]{1}&x\rar[dash]{1}&x
\end{tikzcd}\qquad implies \begin{tikzcd}
{}&y\dlar[dash,dashed,swap]{0}\dar[dash,swap]{0}\rar[dash]&{1}\rar[dash]& x\dar[dash]{0}\dlar[dash,swap]{0}\\
d(x,x,y)\rar[dash,dashed]{1}&y\rar[dash]{1}&x\rar[dash]{1}&x
\end{tikzcd}\end{center} so that $d(x,x,y)=y$, proving (i).
To prove that $\mathcal V$ is congruence modular, it suffices to show that the Shifting Lemma holds in $\mathcal V$, by Theorem \ref{thm_char_modularity}. Now if $\alpha\land\beta\leq\gamma$, then
\begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]{\beta}& z\dar[dash,swap]{\alpha}\dar[dash, bend left]{\gamma}\\
y\rar[dash]{\beta}&u
\end{tikzcd}\qquad implies \qquad \begin{tikzcd}x\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
y\rar[dash]{\beta}&u\rar[dash]{\beta}&u
\end{tikzcd}\end{center} so that we may conclude
\begin{center}\begin{tikzcd}
{}&x\dlar[dash,swap,dashed]{\gamma}\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
d(u,u,y)\rar[dash,dashed]{\beta}&y\rar[dash]{\beta}&u\rar[dash]{\beta}&u
\end{tikzcd}\end{center}
Since $d(u,u,y) = y$ by (i), we see that $x\;\gamma\;y$,\begin{center}\begin{tikzcd}
x\dar[dash,bend left]{\gamma} \dar[dash, swap]{\alpha}\rar[dash]{\beta}& z\dar[dash,swap]{\alpha}\dar[dash, bend left]{\gamma}\\
y\rar[dash]{\beta}&u
\end{tikzcd}\end{center}
and hence the Shifting Lemma holds in $\mathcal V$. It follows that $\mathcal V$ is congruence modular.
To prove that (ii) holds, we will use Theorem \ref{thm_commutator_Delta}. Suppose therefore that $A\in\mathcal V$ and that $\alpha\in\Con(A)$. We want to show that \[x\;\alpha\;y\qquad\text{implies}\qquad d(x,y,y)\;[\alpha,\alpha]\;x\] By Theorem \ref{thm_commutator_Delta}, it suffices to show that in the algebra $A_\alpha$, we have \[\left[\begin{matrix} x&y\\d(x,y,y)&y\end{matrix}\right] \in \Delta_\alpha(\alpha)\]
Here $\Delta_\alpha(\alpha)$ is the congruence on $A_\alpha$ generated by all matrices (regarded as pairs of column vectors) of the form $\left[\begin{matrix}a&a'\\a&a'\end{matrix}\right]$, where $a\;\alpha\; a'$. We apply (b) to $A_\alpha$ as follows: Let $\eta_0,\eta_1$ be the kernels of the projections $A_\alpha\twoheadrightarrow A$. Then we see that
\begin{center}\begin{tikzcd}
\left[\begin{matrix}y\\y\end{matrix}\right]\dar[dash,swap]{\eta_1}\rar[dash]&{\eta_0}\rar[dash]& \left[\begin{matrix}y\\y\end{matrix}\right]\dar[dash]{\eta_1}\dlar[dash,swap]{\Delta_\alpha(\alpha)}\\
\left[\begin{matrix}x\\y\end{matrix}\right]\rar[dash]{\eta_0}&\left[\begin{matrix}x\\x\end{matrix}\right]\rar[dash]{\eta_0}&\left[\begin{matrix}x\\y\end{matrix}\right]
\end{tikzcd}\end{center} so that we may conclude that
\[\left[\begin{matrix}d(x,x,x)\\d(x,y,y)\end{matrix}\right]\;\Delta_\alpha(\alpha)\; \left[\begin{matrix}y\\y\end{matrix}\right]\qquad\text{i.e.}\qquad \left[\begin{matrix} x&y\\d(x,y,y)&y\end{matrix}\right] \in \Delta_\alpha(\alpha)\] as required.
\eproof
Here is a nice application of the above result:
\begin{corollary} Let $\mathcal V$ be a congruence modular variety. \begin{enumerate}[(a)]\item Suppose that $A\in\mathcal V$. Let $\alpha,\beta,\gamma\in\Con(A)$ be such that $\alpha\land\beta\leq\gamma\leq\alpha\lor\beta$. If $\alpha$ permutes with $\beta$, then $\gamma$ permutes with $\alpha,\beta$.
\item Let $A_i\in\mathcal V$ for $i\in I$, and let $A:=\prod_iA_i$ be the direct product. If $\eta_i=\ker\pi_i$ is the kernel of the $i^{th}$ projection, then $\eta_i$ permutes with every $\alpha\in\Con(A)$.
\end{enumerate}
\end{corollary}
\bproof (a) We show that $\gamma$ permutes with $\beta$. Suppose that $x\;(\alpha\circ\gamma)\;u$. Then there is $z$ such that $x\;\beta\;z\;\gamma\;u$. Now $\gamma\leq\alpha\lor\beta=\alpha\circ\beta=\beta\circ\alpha$, so
\begin{itemize}\item $z\;(\alpha\circ\beta)\;u$, i.e. there is $v$ such that $z\;\alpha\;v\;\beta\;u$.\end{itemize} Furthermore, since $x\;(\beta\circ\gamma)\;u$, and $\beta\circ\gamma\subseteq \beta\circ\alpha\circ\beta=\alpha\circ\beta$, we see that
\begin{itemize}\item $x\;(\alpha\circ\beta)\;u$, i.e. there is $y$ such that $x\;\alpha\;y\;\beta\; u$
\end{itemize}
We thus have
\begin{center}\begin{tikzcd}
x\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\qquad so that by Theorem \ref{thm_difference_term_equiv_shift}\begin{tikzcd}
{}&x\dlar[dash,swap,dashed]{\gamma}\dar[dash,swap]{\alpha}\rar[dash]&{\beta}\rar[dash]& z\dar[dash]{\alpha}\dlar[dash,swap]{\gamma}\\
d(u,v,y)\rar[dash,dashed]{\beta}&y\rar[dash]{\beta}&u\rar[dash]{\beta}&v
\end{tikzcd}\end{center} where $d$ is a Gumm difference term for $\mathcal V$, i.e.
$x\;\gamma\;d(u,v,y)\;\beta\; u$. Hence if $x\;(\alpha\circ\gamma)\;u$, then also $x\;(\gamma\circ\alpha)\;u$.
\vskip0.3cm\noindent(b) For $i\in I$, let $A_i':=\prod\limits_{j\in I, j\not=i}A_j$, so that $A=A_i\times A_i'$. Let $\eta_i'$ be the kernel of the natural homomorphism $A\twoheadrightarrow A_i'$. Then $\eta_i\circ\eta_i'=1_A$, so that $\eta_i,\eta_i'$ permute and $\eta_i\lor\eta_i'=1_A$. Furthermore $\eta_i\land \eta_i'= 0_A$. Now if $\alpha\in\Con(A)$, then $\eta_i\land\eta_i'\leq\alpha\leq \eta_i\lor\eta_i'$. By (a), $\alpha$ and $\eta_i$ permute.
\eproof
\subsection{The Gumm Difference Term and the Commutator}
We tackle one more important fact about Gumm difference terms:
\begin{definition}\label{def_commuting_operations}\rm
Suppose that $f:A^n\to A$, and that $g:A^m\to A$. We say that $f$ and $g$ {\em commute} on the matrix
\[\left(\begin{matrix} x^1_1&x^1_2&\dots &x^1_n\\
x^2_1&x^2_2&\dots& x^2_n\\
\vdots&\vdots&\dots&\vdots\\
x^m_1&x^m_2&\dots&x^m_n\end{matrix}\right) \qquad x^i_j\in A\] if and only if
\[g\left(\begin{matrix}f(x^1_1,x^1_2,\dots, x^1_n)\\f(x^2_1,x^2_2,\dots, x^2_n)\\\vdots\\f(x^m_1,x^m_1,\dots, x^m_n)\end{matrix}\right) = f\left(g\left(\begin{matrix}x^1_1\\x^2_1\\\vdots\\x^m_1\end{matrix}\right), g\left(\begin{matrix}x^1_2\\x^2_2\\\vdots\\x^m_2\end{matrix}\right),\hdots, g\left(\begin{matrix}x^1_n\\x^2_n\\\vdots\\x^m_n\end{matrix}\right)\right)\] i.e. if and only if we get the same result whether \begin{itemize} \item we apply $f$ to every row of the matrix, and $g$ to the resulting column vector, or
\item we apply $g$ to every column of the matrix, and $f$ to the resulting row vector.\end{itemize}\vskip0.3cm\noindent
We say that $f$ and $g$ commute if and only if they commute on every $m\times n$--matrix of members of $A$.\endbox
\end{definition}
\begin{theorem}\label{thm_difference_term_commutes}
Let $A$ be an algebra in a congruence modular variety $\mathcal V$ with Gumm difference term $d(x,y,z)$. Let $\alpha,\beta\in\Con(A)$ be such that $\alpha\geq\beta$. The following are equivalent:
\begin{enumerate}[(a)]\item $[\alpha,\beta]=0_A$.
\item \begin{enumerate}[(i)]\item
For any term $t(x_1,\dots, x_n)$, $d$ and $t$ commute on any $n\times 3$--matrix of the form
\[\left(\begin{matrix}x_1&y_1&z_1\\x_2&y_2&z_2\\\vdots&\vdots&\vdots\\x_n&y_n&z_n\end{matrix}\right)\qquad\text{such that }\quad x_i\;\beta\;y_i\;\alpha\;z_i\quad (i\leq n)\]
\item $y\;\beta\; z\quad\Longrightarrow\quad d(y,z,z)=y=d(z,z,y)$
\end{enumerate}
\end{enumerate}
\end{theorem}
\bproof (a) $\Longrightarrow$ (b):
Suppose that $[\alpha,\beta]=0_A$. Let $\eta_0,\eta_1$ be the kernels of the projections $A_\alpha\twoheadrightarrow A$. If we have $x\;\beta\;y\;\alpha\;z$, then
\begin{center}\begin{tikzcd} \left[\begin{matrix}z\\y\end{matrix}\right] \rar[dash]\dar[dash,swap]{\eta_0}&\eta_1\rar[dash] &\left[\begin{matrix}y\\y\end{matrix}\right] \dar[dash]{\eta_0}\dlar[dash,swap]{\Delta_\alpha(\beta)}\\
\left[\begin{matrix}z\\x\end{matrix}\right] \rar[dash]{\eta_1}&\left[\begin{matrix}x\\x\end{matrix}\right] \rar[dash]{\eta_1}&\left[\begin{matrix}y\\x\end{matrix}\right]
\end{tikzcd}\end{center}
Hence by Theorem \ref{thm_difference_term_equiv_shift}, we obtain \[\left[\begin{matrix}d(x,y,z)\\d(x,x,x)\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix}z\\y\end{matrix}\right]\qquad \text{i.e.}\qquad \left[\begin{matrix}d(x,y,z)\\x\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix}z\\y\end{matrix}\right]\tag{$\star$}\]
Since $(\star)$ hold for any $x,y,z$ such that $x\;\beta\;y\;\alpha\;z$, we obtain
\[\left[\begin{matrix} d(t(\mathbf x),t(\mathbf y), t(\mathbf z))\\t(\mathbf x)\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix}t(\mathbf z)\\t(\mathbf y)\end{matrix}\right]\]
whenever $\mathbf x,\mathbf y,\mathbf z\in A^n$ are such that $x_i\;\beta\;y_i\;\alpha\;z_i$ for $i\leq n$.
We also havethat \[\left[\begin{matrix}d(x_i,y_i,z_i)\\x_i\end{matrix}\right]\;\Delta_\alpha(\beta)\; \left[\begin{matrix}z_i\\y_i\end{matrix}\right]\qquad\text{for each} i\leq n\] and hence applying the term $t$, we obtain\[\left[\begin{matrix}t\Big(d(x_1,y_1,z_1),\dots,d(x_n,y_n,z_n)\Big)\\t(\mathbf x)\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix}t(\mathbf z)\\t(\mathbf y))\end{matrix}\right] \] It follows that \[\left[\begin{matrix} d(t(\mathbf x),t(\mathbf y), t(\mathbf z))\\t(\mathbf x)\end{matrix}\right]\;\Delta_\alpha(\beta)\;\left[\begin{matrix}t\Big(d(x_1,y_1,z_1),\dots,d(x_n,y_n,z_n)\Big)\\t(\mathbf x)\end{matrix}\right]\] and hence that $d(t(\mathbf x),t(\mathbf y), t(\mathbf z))\; [\alpha,\beta]\;t\Big(d(x_1,y_1,z_1),\dots,d(x_n,y_n,z_n)\Big)$, by Theorem \ref{thm_commutator_Delta}. Since $[\alpha,\beta]=0_A$, we obtain that $d(t(\mathbf x),t(\mathbf y), t(\mathbf z))=t\Big(d(x_1,y_1,z_1),\dots,d(x_n,y_n,z_n)\Big)$, i.e. that $t, d$ commute on the matrix $[\mathbf x\;\;\mathbf y\;\;\mathbf z]$. This prove that (b)(i) holds.
To prove (b)(ii), we note that since $d$ is a Gumm difference term, we have that $d(z,z,y)=y$ for all $y,z$ and that
\[y\;\beta\; z\qquad\Longrightarrow\qquad d(y,z,z)[\beta,\beta]\;y\] Now, observe since $\beta\leq\alpha$, we have $[\beta,\beta]=0$.
Hence $d(z,z,y)=y=d(y,z,z)$ whenever $y\;\beta\; z$.
\vskip0.3cm\noindent (b) $\Longrightarrow$ (a): Suppose that (i) and (ii) of (b) hold, and that $\beta\leq\alpha$ in $\Con(A)$. To prove that $[\alpha,\beta]=0_A$ it suffices to prove that
\[\left[\begin{matrix}x\\y\end{matrix}\right] \;\Delta_\beta(\alpha)\; \left[\begin{matrix}x\\x\end{matrix}\right]\qquad\Longrightarrow\qquad x=y\]
by Theorem \ref{thm_commutator_Delta} and the fact that $[\alpha,\beta]=[\beta,\alpha]$.
To accomplish that, we will show that
\[\left[\begin{matrix}x\\y\end{matrix}\right] \;\Delta_\beta(\alpha)\; \left[\begin{matrix}u\\v\end{matrix}\right]\qquad\Longleftrightarrow\qquad x\;\beta\;y\;\alpha\; u\quad\text{and}\quad d(y,x,u)=v\tag{$\dagger$}\]
Let $\Gamma$ be the binary relation on $A_\beta$ defined by $(\dagger)$, i.e. $(x,y)\;\Gamma\;(u,v)$ if and only if $x\;\beta\;y\;\alpha\;u$ and $d(y,x,u) = v$. We must show that $\Gamma=\Delta_\beta(\alpha)$. To begin, we prove that $\Gamma$ is a congruence relation on $A_\beta$:
\vskip0.3cm\noindent\underline{$\Gamma$ is reflexive}: Because $x\;\beta\;x\;\alpha\;x$ and $d(x,x,x)=x$.
\vskip0.3cm\noindent\underline{$\Gamma$ is symmetric:} Suppose that $(x,y)\;\Gamma\;(u,v)$. We must show that $u\;\beta\;v\;\alpha x$ and that $d(v,u,x)=y$. But $(x,y),(u,v)\in A_\beta$, and $\beta\leq\alpha$. Moreover, $x\;\alpha\;u$ and $y\;\alpha\; v$ by Lemma \ref{lemma_Delta_tr_cl}. It follows that $u\;\beta\;v\;\alpha\; x$. By (b)(i), $d$ commutes with any term on a matrix of the appropriate form, and hence with itself. Moreover, $v=d(y,x,u)$, since $(x,y)\;\Gamma\;(u,v)$. Hence using (b)(ii)
\[\aligned d(v,u,x) = d\left(\begin{matrix} d(y,x,u)\\x\\x\end{matrix}\right)
&=d\left(\begin{matrix} d(y,x,u)\\d(x,x,u)\\d(x,x,x)\end{matrix} \right)\\
&=d\left(d\left(\begin{matrix}y\\x\\x\end{matrix}\right),d\left(\begin{matrix}x\\x\\x\end{matrix}\right),d\left(\begin{matrix}u\\u\\x\end{matrix}\right)\right) \\
&= d(y,x,x)\\&=y\endaligned\] since the matrix
\[\left(\begin{matrix} y&x&u\\x&x&u\\x&x&x\end{matrix}\right)\]
is of the appropriate form.
\vskip0.3cm\noindent\underline{$\Gamma$ is transitive}: Suppose that $(x,y)\;\Gamma\;(u,v)\;\Gamma\;(a,b)$. It is easy to see that $x\;\beta\;y \;\alpha\;a$.
From the matrix
\[\left(\begin{matrix} y&x&u\\x&x&u\\x&x&a\end{matrix}\right)\] and (b)(i),(ii)
we may conclude that $d(d(y,x,u),u,a) = d(y,x,a)$. But $d(d(y,x,u),u,a) = d(v,u,a) = b$. Hence $b = d(y,x,a)$, and thus $(x,y)\;\Gamma\;(a,b)$.
\vskip0.3cm\noindent\underline{$\Gamma$ is compatible}: Suppose that $f$ is an $n$--ary operation, and that $(x_i,y_i)\;\Gamma\;(u_i,v_i)$ for $i\leq n$. We must show that $(f(\mathbf x), f(\mathbf y))\;\Gamma\;f(\mathbf u),f(\mathbf v))$. Now since $x_i\;\beta\;y_i\;\alpha\;u_i$ for each $i\leq n$, it easily follows that $f(\mathbf x)\;\beta\;f(\mathbf y)\;\alpha\;f(\mathbf u)$. It remains to show that $f(\mathbf v) = d(f(\mathbf y), f(\mathbf x),f(\mathbf u))$. But if $x_i\;\beta\;y_i\;\alpha\;u_i$, then also $y_i\;\beta\;x_i\;\alpha\;u_i$, so by (b)(i), $d$ commutes with $f$ on the implied matrix below:
\[\aligned d(f(\mathbf y), f(\mathbf x), f(\mathbf u))&=d\left(f\left(\begin{matrix}y_1\\y_2\\\vdots\\y_n\end{matrix}\right), f\left(\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right), f\left(\begin{matrix}u_1\\u_2\\\vdots\\u_n\end{matrix}\right)\right) \\&=f\left(\begin{matrix}d(y_1,x_1,u_1)\\d(y_2,x_2,u_2)\\\vdots\\d(y_n,x_n,u_n)\end{matrix}\right) \\&= f\left(\begin{matrix}v_1\\v_2\\
\vdots\\v_n\end{matrix}\right)\\&=f(\mathbf v)\endaligned\]
It follows that $\Gamma$ is a congruence relation on $A_\beta$.
\vskip0.3cm\noindent\underline{$\Gamma=\Delta_\beta(\alpha)$}: By definition,
\[\Delta_\beta(\alpha):=\Cg\left(\left[\begin{matrix}x&u\\x&u\end{matrix}\right]:x\;\alpha \;u\right)\]
Since clearly $(x,x)\;\Gamma\;(u,u)$ if $x\;\alpha\; u$, we see that $\Delta_\beta(\alpha)\subseteq\Gamma$. Conversely, if $(x,y)\;\Gamma\;(u,v)$, then
\[\aligned \left[\begin{matrix}x\\y\end{matrix}\right] &=
d\left(\left[\begin{matrix}x\\y\end{matrix}\right], \left[\begin{matrix}x\\x\end{matrix}\right], \left[\begin{matrix}x\\x\end{matrix}\right]\right)\\
\Delta_\beta&(\alpha)\;d\left(\left[\begin{matrix}x\\y\end{matrix}\right], \left[\begin{matrix}x\\x\end{matrix}\right], \left[\begin{matrix}u\\u\end{matrix}\right]\right) \qquad\text{because}\quad \left[\begin{matrix}x\\x\end{matrix}\right]\;\Delta_\beta(\alpha)\;\left[\begin{matrix}u\\u\end{matrix}\right] \\
&= \left[\begin{matrix}u\\v\end{matrix}\right]
\endaligned\] since $d(y,x,u)=v$.
Hence $(x,y)\;\Gamma\;(u,v)$ implies $(x,y)\;\Delta_\beta(\alpha)\;(u,v)$, i.e. $\Delta_\beta(\alpha)\supseteq\Gamma$.
\vskip0.3cm\noindent\underline{$[\alpha,\beta]=0_A$}: Observe that
\[\aligned x\;[\alpha,\beta]\;y \qquad&\Longleftrightarrow\qquad \left[\begin{matrix}x\\x\end{matrix}\right] \;\Delta_\beta(\alpha)\; \left[\begin{matrix}x\\y\end{matrix}\right]\\
&\Longleftrightarrow\qquad
x\;\beta\;x\;\alpha\;x \quad\text{and}\quad d(x,x,x)=y\\
&\Longleftrightarrow\qquad x=y
\endaligned\]
\eproof
\section{Abelian Algebras}
\fancyhead[RE]{Abelian Algebras}
\subsection{``Abelian" under {\bf H}, {\bf S} and {\bf P} in Modular Varieties}
We state here an immediate corollary of Theorems \ref{thm_commutator_H}, \ref{thm_commutator_S} and \ref{thm_commutator_P}:
\begin{theorem}\label{thm_Abelian_HSP} Let $\mathcal V$ be a congruence modular variety. Homomorphic images, subalgebras and products of Abelian algebras are Abelian. Hence the class of Abelian algebras in $\mathcal V$ is a subvariety of $\mathcal V$.
\end{theorem}
\bproof Suppose that $A$ is Abelian and that $f:A\twoheadrightarrow B$ is a surjective homomorphism. Then by Theorem \ref{thm_commutator_H},
\[[1_B,1_B] = [f(1_A), f(1_A)] = f([1_A,1_A])= f(0_A) = 0_B\]
Next, suppose that $B$ is a subalgebra of an Abelian algebra $A$.
Then by Theorem \ref{thm_commutator_S},
\[[1_B,1_B] = [1_A\!\!\restriction_B, 1_A\!\!\restriction_B]\leq [1_A,1_A]\!\!\restriction_B = 0_A\!\!\restriction_B = 0_B\]
Finally, suppose that $A_i, (i\in I)$ are Abelian algebras, and that $A=\prod_{i\in I}A_i$. By Theorem \ref{thm_commutator_P}, we have
\[[1_A,1_A] = \left[\prod_{i\in I}1_{A_i}, \prod_{i\in I}1_{A_i}\right]\leq\prod_{i\in I}[1_{A_i},1_{A_i}] =\prod_{i\in I}0_{A_i}= 0_A\]
\eproof
As in the case of groups, we have the following result:
\begin{proposition} Let $A$ be an algebra in a congruence modular variety. Then $A/[1_A,1_A]$ is an Abelian algebra. Moreover, if $\theta\in\Con(A)$, then $A/\theta$ is Abelian if and only if $[1_A,1_A]\leq\theta$.
\end{proposition}
\bproof Suppose that $B:=A/[1_A,1_A]$, and that $f:A\twoheadrightarrow B$ is the induced surjective homomorphism. Then $\ker f =[1_A,1_A]$. Hence $[1_B,1_B]=[f(1_A),f(1_A)] = f([1_A,1_A])=0_B$ --- cf. Proposition \ref{propn_generated_con_forward}. Hence $B$ is Abelian.
Suppose now that $\theta\in\Con(A)$. If $\theta\geq[1_A,1_A]$, then $A/\theta$ is a homomorphic image of $A/[1_A,1_A]$, and hence is itself Abelian, by Theorem \ref{thm_Abelian_HSP}. Conversely, if $C:=A/\theta$ is Abelian, then $\theta/\theta=0_C=[1_C,1_C]=([1_A,1_A]\lor\theta)/\theta$, by Theorem \ref{thm_commutator_H}. Thus $[1_A,1_A]\lor\theta=\theta$, which yields $[1_A,1_A]\leq\theta$.
\eproof
Hence in a congruence modular variety $\mathcal V$, every homomorphism from an algebra $A$ onto an Abelian algebra $B$ will factor through $A/[1_A,1_A]$. It is therefore not hard to see that if $\mathcal V_{Ab}$ is the the subvariety of Abelian algebras in $\mathcal V$, then we have an adjunction $F\dashv U$, where
\[F:\mathcal V\to\mathcal V_{Ab}:A\mapsto A/[1_A,1_A]\] gives the {\em Abelianization} of an algebra $A$, and \[U:\mathcal V_{Ab}\to
\mathcal V:B\mapsto \] is the ``forgetful" functor (which forgets that the algebra $B$ is Abelian).
\subsection{Congruence Properties in Modular Varieties}
We develop analogues of Theorems \ref{thm_Abelian_group} and \ref{thm_Abelian_group_M_3}. Suppose that $A$ is an algebra. Recall the definitions of $A_\alpha$ and $\Delta_\alpha(\beta)$ in Section \ref{subsection_congruences_on_congruences}. We have $A_{1_A}=A\times A$, and $\Delta_{1_A}(1_A)$ is a congruence on $A_{1_A}=A\times A$, defined by:\[\Delta_{1_A}(1_A)=\Cg_{A\times A}\left(\left\{\left[\begin{matrix}a&b\\a&b\end{matrix}\right]:a,b\in A\right\}\right)=\Cg_{A\times A}(\Delta)\] where the matrices are regarded as pairs of column vectors.
Thus $\Delta_{1_A}(1_A)$ is the congruence on $A\times A$ generated by collapsing all elements of the diagonal. In particular, the diagonal $\Delta$ is contained in a $\Delta_{1_A}(1_A)$--coset, namely the coset of any diagonal element. The next theorem asserts that an algebra is Abelian if and only if that coset contains no non--diagonal elements:
\begin{theorem}\label{thm_Abelian} For an algebra $A$ in a congruence modular variety $\mathcal V$, the following are equivalent:\begin{enumerate}[(a)]\item $A\in\mathcal V$ is Abelian.
\item The diagonal $\Delta:=\left\{\left[\begin{matrix}a\\a\end{matrix}\right]:a\in A\right\}$ is a coset
of a congruence on $A\times A$.
\item The diagonal $\Delta:=\left\{\left[\begin{matrix}a\\a\end{matrix}\right]:a\in A\right\}$ is a coset
of $\Delta_{1_A}(1_A)$.
\end{enumerate}
\end{theorem}
\bproof (a) $\Longrightarrow$ (b): Suppose that $A$ is Abelian. If $\left[\begin{matrix}b\\c\end{matrix}\right]\;\Delta_{1_A}(1_A)\;\left[\begin{matrix}a\\a\end{matrix}\right]$, then $b\;[1_A,1_A]\;c$, by Theorem \ref{thm_commutator_Delta}, and hence $b=c$. It follows that every element of the $\Delta_{1_A}(1_A)$--coset of a diagonal element is itself a diagonal element. Thus $\Delta$ is the $\Delta_{1_A}(1_A)$--coset of any diagonal element.
\vskip0.3cm\noindent(b) $\Longrightarrow$ (c): Next, suppose that $\Delta$ is a coset of some congruence on $A\times A$. Since $\Delta_{1_A}(1_A)=\Cg_{A\times A}(\Delta)$, any congruence $\theta\in\Con(A\times A)$ with the property that $\Delta$ is contained in a $\theta$--coset must have $\theta\geq \Delta_{1_A}(1_A)$. Thus if $\Delta$ is a coset of some congruence, then $\Delta$ is a coset of $\Delta_{1_A}(1_A)$.
\vskip0.3cm\noindent(c) $\Longrightarrow$ (a): If $b,c\in A$, then
\[\aligned &b\;[1_A,1_A]\;c \\
\Longrightarrow\qquad &\left[\begin{matrix}b&a\\c&a\end{matrix}\right]\in \Delta_{1_A}(1_A)\qquad\text{for some $a\in A$, by Theorem \ref{thm_commutator_Delta}}\\
\Longrightarrow\qquad &\left[\begin{matrix}b\\c\end{matrix}\right]\in\Delta\\
\Longrightarrow\qquad &b=c\endaligned\]
Thus $[1_A,1_A]=0_A$, i.e. $A$ is Abelian.
\eproof
\begin{theorem}\label{thm_Abelian_M_3}
\begin{enumerate}[(a)]\item If $M_3$ is a $(0,1)$--sublattice of $\Con(A)$, then $A$ is Abelian.
\item For an algebra $A$ in a congruence modular variety $\mathcal V$, the following are equivalent:
\begin{enumerate}[(i)]\item $A$ is Abelian.
\item If $\eta_0,\eta_1$ are the kernels of the projections $\pi_0,\pi_1:A\times A\twoheadrightarrow A$, then $\eta_0,\eta_1$ have a common complement in $\Con(A)$ (and $\Delta_{1_A}(1_A)$ is such a common complement).
\item There is a $(0,1)$--homomorphism $M_3\to\Con(A\times A)$.
\item There is a $(0,1)$--homomorphism of $M_3$ to the congruence lattice of some subdirect product of two copies of $A$.
\end{enumerate}
\end{enumerate}
\end{theorem}
\bproof (a) We imitate the proof of Theorem \ref{thm_Abelian_group_M_3}(a): If $M_3$ is a $(0,1)$--sublattice of $\Con(A)$, thenthere are $\theta,\psi,\chi\in\Con(A)$ such that
\[\theta\land\psi = \theta\land\chi=\psi\land\chi =0_A\qquad\theta\lor\psi = \theta\lor\chi=\psi\lor\chi =1_A\]
Then, using the fact that $[\alpha,\beta]\leq\alpha\land\beta$ and the join distributivity of the commutator, we have \[[1_A,1_A]=[\theta\lor\psi,\theta\lor\chi] =[\theta,\theta]\lor[\theta,\chi]\lor[\psi,\theta]\lor[\psi,\chi]\leq \theta\]
Similarly, $[1_A,1_A]\leq\psi,\chi$ and hence $[1_A,1_A]\leq \theta\land\psi\land\chi = 0_A$.
\eproof
\vskip0.3cm\noindent (b) (i) $\Longrightarrow$ (ii): Observe that $\eta_0\land\eta_1=0_A$ and that $\eta_0\lor\eta_1=1_A$, consider now the congruence $\Delta_{1_A}(1_A)$ on $A\times A$. If $A$ is Abelian, then it follows by Theorem \ref{thm_Abelian} that the diagonal $\Delta$ is a coset of $\Delta_{1_A}$. Now clearly if $(x,y), (a,b)\in A\times A$ and $\left[\begin{matrix}x\\y\end{matrix}\right] \;(\eta_0\land\Delta_{1_A}(1_A))\;\left[\begin{matrix}a\\b\end{matrix}\right]$, then $x=a$, so $y\;[1_A,1_A]\;b$ by Theorem \ref{thm_commutator_Delta} and lemma \ref{lemma_Delta_tr_cl}. Since $A$ is Abelian, we have $y=b$, i.e. $\left[\begin{matrix}x\\y\end{matrix}\right]=\left[\begin{matrix}a\\b\end{matrix}\right]$. Hence $\eta_0\land \Delta_{1_A}(1_A) = 0_A$. Similarly, $\eta_1\land \Delta_{1_A}(1_A) = 0_A$. Next, if $(x,y), (a,b)\in A\times A$, then
\[\left[\begin{matrix}x\\y\end{matrix}\right]\;\eta_0\; \left[\begin{matrix}x\\x\end{matrix}\right] \;\Delta_{1_A}(1_A)\;\left[\begin{matrix}a\\a\end{matrix}\right]\;\eta_0\;\left[\begin{matrix}a\\b\end{matrix}\right] \]
It follows that $\left[\begin{matrix}x\\y\end{matrix}\right]\; (\eta_0\lor\Delta_{1_A}(1_A))\; \left[\begin{matrix}a\\b\end{matrix}\right]$, so that $\eta_0\lor\Delta_{1_A}(1_A)=1_A$. Similarly, $\eta_1\lor\Delta_{1_A}(1_A)=1_A$.
\vskip0.3cm\noindent (ii) $\Longrightarrow$ (iii) and \\ (iii) $\Longrightarrow$ (iv) are obvious.
\vskip0.3cm\noindent (b) (iv) $\Longrightarrow$ (i): Suppose that $f:B\hookrightarrow A\times A$ is a subdirect embedding, and that there is a $(0,1)$--homomorphism of $M_3$ into $\Con(B)$. Then $B$ is Abelian, by (a). Moreover, $A$ is a homomorphic image of $B$, because $B$ is a subdirect product of $A$ and $A$. Since homomorphic images of Abelian algebras are Abelian (by Theorem \ref{thm_Abelian_HSP}), $A$ is Abelian.
\eproof
\subsection{Affine Algebras}
We do not assume congruence modularity in this section.
Observe that any variety of modules over a ring $R$ is congruence permutable, as it has a Mal'tsev term $p(x,y,z) := x-y+z$. Thus any variety of modules is congruence modular. Moreover, if $A$ is a module over $R$, then each polynomial of $A$ is of the form
\[p(x_1,\dots,x_n) = r_1x_1+ r_2x_2+\dots +r_nx_n + a\qquad r_1,\dots, r_n\in R, a\in A\]
i.e. each polynomial in a module is an {\em affine} function of $x_1,\dots, x_n$.
From the above observations it follows easily that:
\begin{proposition} Every module is Abelian.
\end{proposition}
\bproof Let $A$ be a module over a ring $R$. Recall that by Proposition \ref{propn_char_center} the center $\zeta_A$ has the following characterization:
$(x,y)\in\zeta_A$ if and only if for any $(n+1)$--ary term $t$, and any $\mathbf a,\mathbf b\in A^n$:\[ t(x,\mathbf a)=t(x,\mathbf b)\qquad\longleftrightarrow \qquad t(y,\mathbf a)=t(y,\mathbf b) \] Now $t(x,\mathbf z) = rx+\sum_{i=1}^nr_iz_i$ for some $r,r_i\in R$.
Clearly, therefore, for any $x,y\in A$ and any $\mathbf a,\mathbf b\in A^n$, we have
\begin{center}
\begin{tabular}{ >{$}r<{$} >{$}r<{$} >{$}r<{$} >{$}l<{$}}\phantom{aaaaaaa}\qquad &t(x,\mathbf a)&=&t(x,\mathbf b)\\
\Longrightarrow \qquad &rx+\sum_{i=1}^nr_ia_i &=& rx+ \sum_{i=1}^nr_ib_i\\
\Longrightarrow \qquad &\sum_{i=1}^nr_ia_i &=& \sum_{i=1}^nr_ib_i\\
\Longrightarrow \qquad &\sum_{i=1}^nr_ia_i &=& ry+ \sum_{i=1}^nr_ib_i\\
\Longrightarrow\qquad &t(y,\mathbf a)&=&t(y,\mathbf b)
\end{tabular}\end{center}
Hence $(x,y)\in\zeta_A$ for any $x,y\in A$, i.e. $\zeta_A=1_A$. It follows that $A$ is Abelian.
\eproof
\begin{definition}\rm \begin{enumerate}[(a)]\item Two algebras are said to be {\em polynomially equivalent} if and only if they have the same underlying set, and the same polynomials.
\item An algebra is said to be {\em affine} if and only if it is polynomially equivalent to a module over a ring.\end{enumerate}
\endbox
\end{definition}
Clearly, two algebras are polynomially equivalent if and only if every fundamental operation of one of the algebras is a polynomial operation of the other. Because an equivalence relation is a congruence if and only if it is compatible with every polynomial operation, it follows that \begin{center}\em Polynomially equivalent algebras have the same congruences.\end{center} Polynomially equivalent algebras need not have the same subalgebras, however (when thinking about their underlying sets).
\begin{proposition} \begin{enumerate}[(a)]\item If $A,\hat{A}$ are polynomially equivalent, then
\[C(\alpha,\beta;\delta)\text{ holds in $A$}\qquad \Longleftrightarrow\qquad C(\alpha,\beta;\delta)\text{ holds in $\hat{A}$}\]
\item $[\alpha,\beta]$ in $\Con(A)$ equals $[\alpha,\beta]\in\Con(\hat{A})$
\end{enumerate}
\end{proposition}
\bproof (a) Suppose that $A,\hat{A}$ are polynomially equivalent, and that $\alpha,\beta,\delta\in \Con(A)=\Con(\hat{A})$
Suppose further that $\hat{C}(\alpha,\beta;\delta)$ holds in $\hat{A}$, and that $t$ is an $(m+n)$--ary term of $A$ such that
\begin{center}\begin{tikzcd} t(\mathbf x,\mathbf a)\rar[dash]{\beta}\rar[dash,bend left]{\delta}\dar[dash,swap]{\alpha}&t(\mathbf x,\mathbf b)\dar[dash]{\alpha}\\
t(\mathbf y,\mathbf a)\rar[dash]{\beta}&t(\mathbf y,\mathbf b)
\end{tikzcd} \qquad\text{ where $x_i\;\alpha\; y_i$ and $a_j\;\beta\;b_j$}
\end{center}
By polynomial equivalence of $A$ and $\hat{A}$, there is a polynomial of $\hat{A}$ which is equal to $t$ (as functions from $A^{n+1}$ to $A$). This means that there is a term $\hat{t}(x,\mathbf z,\mathbf u)$ of the algebra $\hat{A}$ and a tuple $\mathbf c$ in $A$ such that $\hat{t}(x,\mathbf z,\mathbf c) = t(x,\mathbf z)$ for any $x\in A$ and any $\mathbf z\in A^n$.
Thus
\begin{center}\begin{tikzcd} \hat{t}(\mathbf x,\mathbf a,\mathbf c)\rar[dash]{\beta}\rar[dash,bend left]{\delta}\dar[dash,swap]{\alpha}&\hat{t}(\mathbf x,\mathbf b,\mathbf c)\dar[dash]{\alpha}\\
\hat{t}(\mathbf y,\mathbf a,\mathbf c)\rar[dash]{\beta}&\hat{t}(\mathbf y,\mathbf b,\mathbf c)
\end{tikzcd}
\end{center} and hence $\hat{t}(y,\mathbf a,\mathbf c)\;\delta\;\hat{t}(y,\mathbf b,\mathbf c)$, by $\hat{C}(\alpha,\beta;\delta)$.
We conclude therefore that $t(y,\mathbf a) \;\delta\;t(y,\mathbf b)$, and thus that $C(\alpha,\beta;\delta)$
Similarly, $C(\alpha,\beta;\delta)$ implies $\hat{C}(\alpha,\beta;\delta)$ when $A,\hat{A}$ are polynomially equivalent.
\vskip0.3cm\noindent (b) In both $A$ and $\hat{A}$ we have
\[[\alpha,\beta]=\bigwedge\{\delta:C(\alpha,\beta;\delta)\}\]
\eproof
We thus have:
\begin{theorem} If $A$ and $\hat{A}$ are polynomially equivalent, then $\Con(A)=\Con(\hat{A})$ are identical lattices. Moreover, the commutator on $\Con(A)$ coincides with the commutator on $\Con(\hat{A})$. \endbox
\end{theorem}
Since modules are Abelian, we see immediately that:
\begin{corollary} \label{corollary_affine_Abelian} An affine algebra is Abelian.\endbox
\end{corollary}
\subsection{Difference Terms and Ternary Abelian Groups}
Observe that if $(A,+,-,0)$ is an Abelian group, then any term $t(x_1,\dots, x_n)$ is of the form
\[t(x_1,\dots, x_n) =\sum_{i=1}^n m_i x_i\qquad m_i\in\mathbb Z\] It is therefore easy to see that any two term operations on $A$ commute (cf. Definition \ref{def_commuting_operations}).
Now a general algebra $A$ may not have a distinguished element $0$. In an Abelian group, we can ``forget" about 0 if we take as basic operation the Mal'tsev term\[t(x,y,z):=x-y+z\] Such a term will commute with itself.
\begin{definition}
\rm A {\em ternary Abelian group} is an algebra $A$ with a single ternary fundamental operation $t(x,y,z)$ satisfying
\begin{enumerate} [(i)]\item $t$ is a Mal'tsev term, i.e.\qquad $t(x,x,y)\approx y\qquad t(x,y,y)\approx x$.
\item $t$ commutes with itself.
\end{enumerate}
\endbox
\end{definition}
Recall that any congruence modular variety has a Gumm difference term.
\begin{theorem} \label{thm_difference_term_Abelian} If $d(x,y,z)$ is a Gumm difference term for $\mathcal V$ and $A\in\mathcal V$ is an Abelian algebra, then
\begin{enumerate}[(i)]\item $d$ is a Mal'tsev term for $A$, i.e.
\[d(x,x,y)=y=d(y,x,x)\qquad\text{in }A\]
\item $d$ commutes with every polynomial operation on $A$.
\end{enumerate} Hence every Abelian algebra has permuting congruences. Moreover, if $d$ is a difference term and $A$ is Abelian, then $(A,d)$ is a ternary Abelian group.
\end{theorem}
\bproof
If $[1_A,1_A] = 0_A$, then it follows from Theorem \ref{thm_difference_term_commutes} that $d$ is Mal'tsev and commutes with every term operation. But $d(a,a,a)=a$, i.e. $d$ is idempotent. It follows easily that $d$ commutes with every polynomial operation: If $p(\mathbf x):=t(\mathbf x,\mathbf a)$ is an $n$--ary polynomial, where $t$ is an $(n+m)$--ary term, and $\mathbf a=(a_1,\dots,a_m)\in A^m$, then
\[\aligned &d(p(\mathbf x),p(\mathbf y),p(\mathbf z))=d\left(t\left(\begin{matrix}\mathbf x\\\mathbf a\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf a\end{matrix}\right), t\left(\begin{matrix}\mathbf z\\\mathbf a\end{matrix}\right)\right)\\
= \;\;&t\left(\begin{matrix}d(x_1,y_1,z_1)\\\vdots\\d(x_n,y_n,z_n)\\d(a_1,a_1,a_1)\\\vdots\\d(a_m,a_m,a_m)\end{matrix}\right)=t\left(\begin{matrix}d(x_1,y_1,z_1)\\\vdots\\d(x_n,y_n,z_n)\\a_1\\\vdots\\a_m\end{matrix}\right)\\
&=p\left(\begin{matrix}d(x_1,y_1,z_1)\\\vdots\\d(x_n,y_n,z_n)\end{matrix}\right)
\endaligned\]
\eproof
The conditions specified in the definition of a ternary Abelian group completely capture all the properties of an Abelian group, in the following sense:
\begin{theorem}\label{thm_ternary_Abelian_group} The following are equivalent:\begin{enumerate}[(a)]\item $(A,t)$ is a ternary Abelian group.
\item There is an Abelian group structure $(A,+,-,0)$ on the set $A$ such that $t(x,y,z)=x-y+z$.
\end{enumerate}\end{theorem}
\bproof (b) $\Longrightarrow$ (a) is straightforward.
\vskip 0.3cm\noindent (a) $\Longrightarrow$ (b): Let $(A,t)$ be a ternary Abelian group. Pick an arbitrary element of $A$ and call it $0$. Define a binary operation $+$ and a unary operation $-$ on $A$ by
\[x+y:=t(x,0,y) \qquad\qquad -x := t(0,x,0)\]
We will show that $(A,+,-,0)$ is an Abelian group.
Firstly, $+$ is associative:
\[\aligned
x+(y+z) &= t(x, 0, t(y,0,z))\\
&= t\left( t\left(\begin{matrix}x\\0\\0\end{matrix}\right),t\left(\begin{matrix}0\\0\\0\end{matrix}\right), t\left(\begin{matrix}y\\0\\z\end{matrix}\right)\right)\\&=t\left(\begin{matrix} t(x,0,y)\\t(0,0,0)\\t(0,0,z)\end{matrix}\right)\\
&=t(t(x,0,y),0,z)\\&=(x+y)+z
\endaligned\]
That $0$ is an additive identity follows from
\[x+0=t(x,0,0)=x\]
Next, to check that $x+(-x)=0$, one can check that $t$ commutes with itself on the following matrix:
\[\left(\begin{matrix}x&0&0\\0&0&x\\0&0&0\end{matrix}\right)\]
to obtain $x+(-x)=t(x,0,t(0,x,0))=t(x,x,0)=0$.
To check that $+$ is commutative, observe that
\[\aligned x+y &= t(x,0,y)\\
&= t\left( t\left(\begin{matrix}y\\y\\x\end{matrix}\right),t\left(\begin{matrix}0\\y\\y\end{matrix}\right), t\left(\begin{matrix}x
\\x\\y\end{matrix}\right)\right)\\&=t\left(\begin{matrix} t(y,0,x)\\t(y,y,x)\\t(x,y,y)\end{matrix}\right)\\
&=t\left(\begin{matrix} t(y,0,x)\\x\\x\end{matrix}\right)\\&=t(y,0,x)\\&=y+x
\endaligned\]
It follows that $(A,+,-,0)$ is an Abelian group.
Finally, observe that
\[\aligned x-y+z&= t(x-y,0,z)\\
&=
t\left( t\left(\begin{matrix}x\\0\\t(0,y,0)\end{matrix}\right),0,z\right)\\
&= t\left( t\left(\begin{matrix}x\\0\\t(0,y,0)\end{matrix}\right),t\left(\begin{matrix}y\\y\\0\end{matrix}\right), t\left(\begin{matrix}z
\\0\\0\end{matrix}\right)\right)\\
&=t\left(\begin{matrix} t(x,y,z)\\t(0,y,0)\\t(t(0,y,0),0,0)\end{matrix}\right)\\
&=t\left(\begin{matrix} t(x,y,z)\\t(0,y,0)\\t(0,y,0)\end{matrix}\right)\\
&=t(x,y,z)
\endaligned\] as required.
\eproof
\subsection{The Fundamental Theorem of Abelian Algebras}
\begin{lemma} \label{lemma_Abelian+Maltsev_implies_commute} If an Abelian algebra $A$ has a Mal'tsev polynomial $m(x,y,z)$, then $m$ commutes with every polynomial operation of $A$.
\end{lemma}
\bproof Let $p(\mathbf x)$ be an arbitrary $n$--ary polynomial of $A$. Then there exists an $(n+m)$--ary term $t$ and a $\mathbf c\in A^m$ such that $p(\mathbf x) = t(\mathbf x,\mathbf c)$.
Now if $\mathbf x,\mathbf y,\mathbf z\in A^n$, then \[\aligned
m\left(t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf z\\\mathbf c\end{matrix}\right)\right) = t(\mathbf z,\mathbf c) = m\left(t\left(\begin{matrix}\mathbf z\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right)\right)
\endaligned\]
and thus
\[\aligned
m\left(t\left(\begin{matrix}m(\fbox{$y_1$},y_1,y_1)\\m(\fbox{$y_2$},y_2,y_2)\\\vdots\\m(\fbox{$y_n$},y_n,y_n)\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf z\\\mathbf c\end{matrix}\right)\right) = m\left(t\left(\begin{matrix}m(\fbox{$y_1$},y_1,z_1)\\m(\fbox{$y_2$},y_2,z_2)\\\vdots\\m(\fbox{$y_n$},y_n,z_n)\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right)\right) \endaligned\]
Since $A$ is Abelian, we have $C(1_A,1_A;0_A)$ and hence we may relace the boxed $y_i$ by $x_i$ to conclude that
\[\aligned
m\left(t\left(\begin{matrix}m(\fbox{$x_1$},y_1,y_1)\\m(\fbox{$x_2$},y_2,y_2)\\\vdots\\m(\fbox{$x_n$},y_n,y_n)\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf z\\\mathbf c\end{matrix}\right)\right) = m\left(t\left(\begin{matrix}m(\fbox{$x_1$},y_1,z_1)\\m(\fbox{$x_2$},y_2,z_2)\\\vdots\\m(\fbox{$x_n$},y_n,z_n)\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right)\right) \endaligned\]
Since $m$ is a Malt'sev polynomial, we see that
\[\aligned
m\left(t\left(\begin{matrix}x_1\\x_2\\\vdots\\x_n\\\mathbf c\end{matrix}\right) , t\left(\begin{matrix}\mathbf y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}\mathbf z\\\mathbf c\end{matrix}\right)\right)&=t\left(\begin{matrix}m(x_1,y_1,z_1)\\m(x_2,y_2,z_2)\\\vdots\\m(x_n,y_n,z_n)\\\mathbf c\end{matrix}\right)
\endaligned\]
i.e. that \[m\left(p\left(\begin{matrix}x_1\\x_2\\\vdots\\x_n\end{matrix}\right),p\left(\begin{matrix}y_1\\y_2\\\vdots\\y_n\end{matrix}\right), p\left(\begin{matrix}z_1\\z_2\\\vdots\\z_n\end{matrix}\right)\right) = p\left(\begin{matrix}m(x_1,y_1,z_1)\\m(x_2,y_2,z_2)\\\vdots\\m(x_n,y_n,z_n)\end{matrix}\right)\]
Hence $m$ commutes with $p$.,
\eproof
\begin{lemma}\label{lemma_Maltsev_pol_=_term} Suppose that $m(x,y,z)$ is a Malt'sev polynomial on an algebra $A$ with the property that $m$ commutes with every polynomial operation on $A$. Then $m(x,y,z)$ is a (equal to) a term operation on $A$.
\end{lemma}
\bproof
Since $m(x,y,z)$ is a polynomial, there is an $(n+3)$--ary term $t$ and an $\mathbf c\in A^n$ such that $m(x,y,z) = t(x,y,z,\mathbf c)$.
Define $\mathbf y\in A^n$ vy $\mathbf y:=(y,y,\dots,y)$. We will show that
\[m(x,y,z) = t(x,t(y,y,y,\mathbf y),z,\mathbf y)\] from which it follows that $m$ is equivalent to a polynomial.
Define $\alpha:=t(0,0,0,\mathbf y)$. Observe that because $m$ will commute with any polynomial (and hence with itself) that
\[\aligned m(x,y,z)&=
m\left( m\left(\begin{matrix}x\\y\\z\end{matrix}\right), m\left(\begin{matrix}0\\0\\0\end{matrix}\right), m\left(\begin{matrix}\alpha\\\alpha\\0\end{matrix}\right)\right) \\
&= m\left(\begin{matrix}m(x,0,\alpha)\\m(y,0,\alpha)\\m(z,0,0)\end{matrix}\right)
\\&= m\left(\begin{matrix}m(x,\qquad\qquad\qquad0,\qquad\qquad\alpha)\\m\Big(m(y,0,\alpha), \quad m(y,0,\alpha),\quad m(y,0,\alpha)\Big)\\m\Big(z,\qquad\qquad m(y,0,\alpha),\qquad m(y,0,\alpha)\Big)\end{matrix}\right)\\&=
m\left(m\left(\begin{matrix}x\\m(y,0,\alpha)\\z\end{matrix}\right), m\left(\begin{matrix}0\\m(y,0,\alpha)\\m(y,0,\alpha)\end{matrix}\right), m\left(\begin{matrix}\alpha\\m(y,0,\alpha)\\m(y,0,\alpha)\end{matrix}\right)\right)\\
&= m\left(m\left(\begin{matrix}x\\m(y,0,\alpha)\\z\end{matrix}\right),0,\alpha\right)
\endaligned\]
Now using the fact that $m$ commutes with $t$, the expression $m(y,0,\alpha)$ cane be transformed as follows:
\[\aligned m(y,0,\alpha)&= m(m(y,y,y), m(0,0,0),\alpha)\\
&=m\left(t\left(\begin{matrix}y\\y\\y\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}0\\0\\0\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}0\\0\\0\\\mathbf y\end{matrix}\right)\right)\\
&= t\left(\begin{matrix}m(y,0,0)\\m(y,0,0)\\m(y,0,0)\\m(c_1,c_1,y)\\m(c_2,c_2,y)\\\vdots\\m(c_n,c_n,y)\end{matrix}\right)\\
&= t(y,y,y,\mathbf y)
\endaligned\]
It follows that
\[\aligned
m(x,y,z) &= m\left(m\left(\begin{matrix}x\\ t(y,y,y,\mathbf y)\\z\end{matrix}\right),0,\alpha\right)\\
&= m\left(t\left(\begin{matrix}x\\ t(y,y,y,\mathbf y)\\z\\\mathbf c\end{matrix}\right) , t\left(\begin{matrix}0\\0\\0\\\mathbf c\end{matrix}\right), t\left(\begin{matrix}0\\0\\0\\\mathbf y\end{matrix}\right) \right)\\
&=t\left(\begin{matrix} m(x,0,0)\\
m(t(y,y,y,\mathbf y),0,0)\\m(z,0,0)\\m(c_1,c_1,y)\\m(c_2,c_2,y)\\\vdots\\m(c_n,c_n,y)\end{matrix}\right)\\
&= t(x,t(y,y,y,\mathbf y),z,\mathbf y)
\endaligned\]
as required.
\eproof
\begin{lemma} \label{lemma_Maltsev_commute_Abelian} If an algebra $A$ has a Mal'tsev term which commutes with every term operation of $A$, then $A$ is an affine algebra.
\end{lemma}
\bproof
Suppose that $t(x,y,z)$ is a Mal'tsev term on $A$ which commutes with every term operation on $A$. Then $(A,t)$ is a ternary Abelian group. By Theorem \ref{thm_ternary_Abelian_group}, we can can take any element $0\in A$, and define operations $+,-$ so that $(A,+,-,0)$ is an Abelian group so that $t(x,y,z)=x-y+z$. We now show how to define a module whose base Abelian group is $\hat{A}:=(A,+,-,0)$, and such that the original algebra $A$ is polynomially equivalent to this module.
Firstly, we must define a ring $R$ so that $\hat{A}$ becomes a left $R$-module, which we will denote by $_RA$. Now if each $A$--polynomial is to be an $_R$--polynomial, then this is true in particular for the unary polynomials. We therefore define $R$ to be the set of all unary polynomials $r(x)$ on the original algebra $A$ with the property that $p(0)=0$. Observe that $R$ is non--empty, as the constant polynomial with value 0 is in $R$.
Since $t$ commutes with all the terms of $A$ and has $t(a,a,a)=a$, it follows as in the proof of Theorem \ref{thm_difference_term_commutes} that $t$ commutes with every polynomial on $A$. In particular, if $r\in R$, then since $r(0) = 0$ we have
\[r(a+b)= r(t(a,0,b))=t(r(a), 0,r(b)) =r(a)+r(b)\]
and
\[r(-a) = r(t(0,a,0)) = t(0,r(a),0) = -r(a)\]
Hence each $r\in R$ is an endomorphism of the Abelian group $\hat{A}$.
We now define operations of addition and multiplication on $R$ in the obvious way, to make it a subring of the usual ring of endomorphisms of $\hat{A}$:
\[\underbrace{(r+s)}_{\text{addition in $R$}}(x):=\underbrace{r(x)+s(x)}_{\text{addition in $\hat{A}$}}\qquad\qquad (rs)(x):=\underbrace{(r\circ s)}_{\text{composition}}(x)\]
To complete our definition of $_RA$, we have to define the action of elements of $R$ on elements of $A$:
\[ra:=r(a)\qquad r\in R, a\in A\]
It is easy to see that $_RA$ is a module. What we still have to show is that this module is polynomially equivalent to $A$, i.e. that $\text{Pol}(A)=\text{Pol}(_RA)$. Now each operation of $_RA$ is defined in terms of polynomials on $A$, so certainly $\text{Pol}(_RA)\subseteq\text{Pol}(A)$.
Next we show by onduction on $n$ that every $n$--ary polynomial of $A$ is a polynomial of $_RA$. First consider the case $1$: If $p(x)$ is a unary polynomial of $A$, then $r(x):=p(x)-p(0)$ has $r\in R$. Hence if we define $a:=p(0)$, then $p(x)=r(x)+p(0) = rx+a$ is a polynomial in $_RA$. Thus $A$, $_RA$ have the same unary polynomials.
For the induction step, assume that $A$, $_RA$ have the same $n$--ary polynomials, and let $p(x_1,\dots, x_{n+1})$ be an $(n+1)$--ary polynomial of $A$. Observe that
\[\aligned p\left(\begin{matrix}x_1\\x_2\\\vdots\\x_n\\x_{n+1}\end{matrix}\right) &= p\left(\begin{matrix} t(x_1,0,0)\\t(x_2,0,0)\\\vdots\\t(x_n,0,0)\\t(0,0,x_{n+1})\end{matrix}\right)\\
&= t\left( p\left(\begin{matrix}x_1\\x_2\\\vdots\\x_n\\0\end{matrix}\right), p\left(\begin{matrix}0\\0\\\vdots\\0\\0\end{matrix}\right), p\left(\begin{matrix}0\\0\\\vdots\\0\\x_{n+1}\end{matrix}\right)\right)\endaligned\]
Hence \[p(x_1,\dots, x_n,x_{n+1})=p_0(x_1,\dots, x_n)-a+r(x_{n+1})\]
where
\[p_0(x_1,\dots,x_n):= p(x_1,\dots, x_n,0)\qquad a:= p(0,\dots, 0)\qquad r(x_{n+1}):=p(0,\dots,0,x_{n+1})\]
Thus $p_0$ is an $n$--ary polynomial, $r$ is a unary polynomial, and $a$ is a constant. Each of these belongs to $\text{Pol}(_RA)$, by induction hypothesis. It follows that $p\in\text{Pol}(_RA)$ also.
\eproof
If we put together the preceding lemmas, we obtain the following:
\begin{theorem} The following are equivalent:
\begin{enumerate}[(a)]\item $A$ is an affine algebra.
\item $A$ is Abelian and has a Malt'sev polynomial.
\item $A$ has a Malt'sev polynomial which commutes with every polynomial operation on $A$.
\item $A$ has a Malt'sev term which commutes with every term operation on $A$.
\item $A$ has a Malt'sev term and is Abelian.
\end{enumerate}
\end{theorem}
\bproof
(a) $\Longrightarrow$ (b) is clear, as every affine algebra is Abelian with Mal'tsev term $t(x,y,z):=x-y+z$.
\vskip0.3cm\noindent (b) $\Longrightarrow$ (c) follows directly from Lemma \ref{lemma_Abelian+Maltsev_implies_commute}.
\vskip0.3cm\noindent (c) $\Longrightarrow$ (d) is Lemma \ref{lemma_Maltsev_pol_=_term}.
\vskip0.3cm\noindent (d) $\Longrightarrow$ (a) is Lemma \ref{lemma_Maltsev_commute_Abelian}.
\vskip0.3cm\noindent Hence (a), (b), (c) and (d) are equivalent. Furthermore, clearly (e) $\Longrightarrow$ (b) $\Longrightarrow$ (a) $\Longrightarrow$ (e), so all the above statements are equivalent.
\eproof
As an immediate corollary, we have:
\begin{theorem} {\rm(\bf Fundamental Theorem of Abelian Algebras)} In a congruence modular variety, an algebra is Abelian if and only if it is affine.\endbox
\end{theorem}
\subsection{Abelian Congruences}
Recall that a congruence $\theta\in\Con(A)$ is said to be Abelian if and only if $[\theta,\theta]=0_A$.
In this short section, we discuss two results:
\begin{itemize}\item An Abelian congruence permutes with every other congruence.
\item The cosets of an Abelian congruence can be endowed with a ternary group structure.
\end{itemize}
For $\alpha\in\Con(A)$, we define a sequence of iterated commutators $[\alpha,\alpha]^n$ inductively by
\[[\alpha,\alpha]^0:=\alpha\qquad [\alpha,\alpha]^{n+1}= \Big[[\alpha,\alpha]^n,\; [\alpha,\alpha]^n\Big]\]
In analogy with groups, we say that a congruence $\alpha$ is $n$--step {\em solvable} if $[\alpha,\alpha]^n=0_A$. Clearly, $\alpha$ is 1--step solvable if and only if $\alpha$ is Abelian.
Now recall that in a congruence modular variety, every Abelian algebra has a Mal'tsev term, and is therefore congruence permutable. We can do better than this, however:
\begin{proposition} Let $A$ be an algebra in a congruence modular variety. For any $\alpha,\beta\in\Con(A)$ and every $n\in\mathbb N$, we have \[\alpha\circ\beta\subseteq[\alpha,\alpha]^n\circ\beta\circ\alpha\tag{$\star$}\]
\end{proposition}
\bproof By induction on $n$. $(\star)$ clearly holds when $n=0$. Suppose now that $\alpha\circ\beta\subseteq[\alpha,\alpha]^n\circ\beta\circ\alpha$, and that $x\;(\alpha\circ\beta)\;z$. We must show that $x\; ([\alpha,\alpha]^{n+1}\circ\beta\circ\alpha)\;z$. Let $u,v$ be such that $x\;[\alpha,\alpha]^n\;u\;\beta\;v\;\alpha \;z$. If $d(x,y,z)$ is a Gumm difference term for $\mathcal V$ (which exists by Corollary \ref{corollary_modular_implies_Gumm_difference}), then
$d(x,u,u)\;[\alpha,\alpha]^{n+1}\;x$ (by definition of Gumm difference term). Thus
\[x\;[\alpha,\alpha]^{n+1}\;d(x,u,u)\;\beta\;d(x,u,v)\;\alpha\;d(u,u,z)=z\]
because $x\; [\alpha,\alpha]^n\;u$ implies $x\;\alpha\;u$. Hence $x\; ([\alpha,\alpha]^{n+1}\circ\beta\circ\alpha)\;z$, completing the induction.\eproof
Immediately, we have:
\begin{corollary} Let $A$ be an algebra in a congruence modular variety. Every solvable, and hence every Abelian, congruence permutes with every other congruence on $A$.
\endbox
\end{corollary}
Suppose now that $A$ is an algebra in a congruence modular variety $\mathcal V$. Let $\alpha\in\Con(A)$, and let $d$ be a Gumm difference term for $\mathcal V$.
\begin{proposition} Suppose that $A$ is an algebra in a congruence modular variety $\mathcal V$. Let $\alpha\in\Con(A)$, and let $d$ be a Gumm difference term for $\mathcal V$.Then $\alpha$ is an Abelian congruence if and only if
\begin{enumerate}[(a)] \item Each coset $a/\alpha$ equipped with $d$ becomes a ternary Abelian group $(a/\alpha,d)$.\item Whenever $t(x_1,\dots, x_n)$ is an $n$--ary term of $A$ and $a_1,\dots, a_n, a$ have $t(a_1,\dots, a_n)=a$, then
\[t:a_1/\alpha\times\dots\times a_n/\alpha\to a/\alpha\qquad\text{is a ternary group homomorphism}\]
\end{enumerate}
\end{proposition}
\bproof This is just a rephrasing of Theorem \ref{thm_difference_term_commutes} in the case that $\beta=\alpha$. If $a\in A$, then the coset $a/\alpha$ is closed under the ternary operation $d$, because $x,y,z\in a/\alpha$ implies $d(x,y,z)\;\alpha\;d(a,a,a)=a$. Now according to Theorem \ref{thm_difference_term_commutes}, the congruence $\alpha$ is Abelian if and only if \begin{enumerate}[(i)]\item
For any term $t(x_1,\dots, x_n)$, $d$ and $t$ commute on any $n\times 3$--matrix of the form
\[\left(\begin{matrix}x_1&y_1&z_1\\x_2&y_2&z_2\\\vdots&\vdots&\vdots\\x_n&y_n&z_n\end{matrix}\right)\qquad\text{such that }\quad x_i\;\alpha\;y_i\;\alpha\;z_i\quad (i\leq n)\]
\item $y\;\alpha\; z\quad\Longrightarrow\quad d(y,z,z)=y=d(z,z,y)$
\end{enumerate}
Thus if $\alpha$ is Abelian, then $d$ is a Mal'tsev term which commutes with itself on $a/\alpha$, making $(a/\alpha,d)$ a ternary Abelian group. Furthermore, if $t(a_1,\dots, a_n) = a$, and if $x_i,y_i,z_i\in a_i/\alpha$ for $i=1,\dots, n$, then
\[t(d(x_1,y_1,z_1),\dots,d(x_n,y_n,z_n)) = d(t(x_1,\dots, x_n), t(y_1,\dots, y_n), t(z_1,\dots, z_n))\] shows that $t$ preserves the operation $d$, i.e. is a ternary group homomorphism.
Conversely, if each $(a/\alpha,d)$ is a ternary Abelian group and satisfies (b), then $[\alpha,\alpha] = 0_A$ by Theorem \ref{thm_difference_term_commutes}.\eproof
\newpage
\appendix
\section{Congruence Relations}
\fancyhead[RE]{Congruence Relations}
\begin{proposition}\label{propn_congruence_unary_pol} \begin{enumerate}[(a)]\item If a transitive relation $R$ on an algebra $A$ is compatible with the unary polynomials on $A$, then it is compatible with the fundamental operations on $A$.
\item An equivalence relation relation $R$ on $A$ is a congruence relation if and only if it is compatible with the unary polynomials on $A$.
\end{enumerate}
\end{proposition}
\bproof (a) Suppose that $f$ is a fundamental $n$--ary operation on $A$. Let $(r_1,s_1),\dots, (r_n,s_n)\in R$, and define unary polynomials
$p_1,\dots, p_n$ by
\[p_i(x) := f(s_1,\dots, s_{i-1}, x, r_{i+1},\dots, r_n)\]
\[\aligned f(r_1,\dots, r_n)&=p_1(r_1)\\
p_1(r_1)\;&R\; p_1(s_1)\\
p_1(s_1)&=p_2(r_2)\\
p_2(r_2)\;&R\;p_2(s_2)\\
&\vdots\\
p_n(r_n)\;&R\;p_n(s_n)\\p_n(s_n)&=f(s_1,\dots, s_n)\endaligned\]
By transitivity, it follows that $f(r_1,\dots, r_n)\;R\;f(s_1,\dots, s_n)$ whenever $r_1\;R s_1,\dots, r_n\;R\;s_n$.
(b) follows immediately.
\eproof
Mal'tsev gave the following description of $\Cg_A(X)$:
\begin{proposition} Let $A$ be an algebra. Suppose that $X\subseteq A\times A$ is reflexive and symmetric. Then $(a,b)\in\Cg_A(X)$ if and only if there are unary polynomials $p_0,\dots,p_n$ and pairs $(x_0,y_0),\dots,(x_n,y_n)\in X$
such that \[\left.\aligned a&=p_0(x_0)\\
p_i(y_i)&=p_{i+1}(x_{i+1})\quad\text{for }0\leq i<n\qquad\\
p_{n}(y_n)&=b\endaligned\right\}\tag{$\dagger$}\]
\end{proposition}
The chain given $(\dagger)$ is called a {\em Mal'tsev chain that witnesses} $(a,b)\in\Cg_A(X)$, or a Mal'tsev chain from $a$ to $b$.
\bproof Suppose that $R$ is the relation defined by $(\dagger)$. Since clearly $p_{i}(x_i)\;\Cg_A(X)\; p_{i}(y_i)$, it is easy to see that $X\subseteq R\subseteq \Cg_A(X)$.
Thus to prove that $R=\Cg_A(X)$, it suffices to check that $R$ is a congruence.
$R$ is reflexive, because $X$ is reflexive and $X\subseteq R$. Furthermore, if $(a,b)\in R$, then there is a Mal'tsev chain from $a$ to $b$. The chain in reverse order witnesses that $(b,a)\in R$, and hence $R$ is symmetric. It is also easy to see that $R$ is transitive: If $(a,b), (b,c)\in R$, then we obtain a Mal'tsev chain from $a$ to $c$ by taking a Mal'tsev chain from $a$ to $b$ and adjoining a Mal'tsev chain from $b$ to $c$.
It therefore remains to show that $R$ is compatible with the fundamental operations of the algebra $A$. Since $R$ is transitive, it suffices to show that $R$ is compatible with the unary polynomials on $A$. Now suppose that $a\; R\; b$ and that $q$ is a unary polynomial. If $a\;R\;b$ is witnessed by a Mal'tsev chain
\[a = p_0(x_0)\;R\;p_0(y_0) = p_1(x_1)\;R\;p_1(y_1)=p_2(x_2)\;R\;p_2(y_2)\dots =p_n(x_n)\;R\;p_n(y_n)=b\]and if $q_i:= q\circ p_i$, then we have a Mal'tsev chain
\[q(a) = q_0(x_0)\;R\;q_0(y_0) = q_1(x_1)\;R\;q_1(y_1)=q_2(x_2)\;R\;q_2(y_2)\dots =q_n(x_n)\;R\;q_n(y_n)=b\] witnessing $q(a)\;R\;q(b)$.\eproof
\begin{proposition} \label{propn_generated_con_forward} Suppose that $f:A\twoheadrightarrow B$ is a surjective homomorphism, and that $\theta\in \Con(A)$. Let \[f(\theta) := \Cg_B(\{(f(a_1), f(a_2)):(a_1,a_2)\in \theta\})\] Also define \[\Phi_\theta:=\{(f(a_1), f(a_2)):(a_1,a_2)\in \theta\}\] so that $f(\theta)=\Cg_B(\Phi_\theta)$.
\begin{enumerate}[(a)]\item $\Phi_\theta:=\{(f(a_1), f(a_2)):(a_1,a_2)\in \theta\}$ is a symmetric, reflexive and compatible relation. Hence \[f(\theta) = \Phi_\theta\cup(\Phi_\theta\circ\Phi_\theta)\cup(\Phi_\theta\circ\Phi_\theta\circ\Phi_\theta)\cup\dots\]
\item $\Phi_\theta=f(\theta)$ if and only if $\theta\circ \ker f\circ\theta\subseteq \ker f\circ\theta\circ\ker f $.\item If $\ker f\subseteq\theta$, then \[f(\theta) =\Phi_\theta\] and \[(x,y)\in\theta\qquad \Longleftrightarrow\qquad (f(x),f(y))\in f(\theta)\]
\item $f(\theta)=f(\theta\lor\ker f)$.
\item If $X\subseteq A^2$ generates $\theta$, then $\{(f(x),f(y)):(x,y)\in X\}\cup \ker f$ generates $f(\theta)$.
\end{enumerate}
\end{proposition}
\bproof
(a) Symmetry and reflexivity are clear. Now suppose that $t$ is an $n$--ary operation, and that $(b^1_i,b^2_i)\in \Phi_\theta$ for $i=1,\dots, n$. Then there are $(a^1_i, a^2_i)\in\theta$ such that $(f(a^1_i),f(a^2_i))=(b^1_i, b^2_i)$. Now $(t(a^1_1,\dots, a^1_n), t(a^2_1,\dots a^2_n))\in\theta$, and hence \[(t(b^1_1,\dots b^1_n), t(b^2_1,\dots, b^2_n))= (f(t(a^1_1,\dots, a^1_n)), f(t(a^2_1,\dots a^2_n)))\in\Phi_\theta\]
Hence $\Phi_\theta$ is almost a congruence relation, but it need not be transitive. The relation $\Phi_\theta\cup(\Phi_\theta\circ\Phi_\theta)\cup(\Phi_\theta\circ\Phi_\theta\circ\Phi_\theta)\cup\dots$ is clearly transitive as well, and hence a congruence relation.
\vskip0.3cm\noindent (b) Observe from (a) that $f(\theta)=\Phi_\theta$ if and only if $\Phi_\theta$ is transitive. Now if $\Phi_\theta$ is transitive and $(x,z)\in\theta\circ\ker f\circ\theta$, then there are $y,y'\in A$ such that $x\;\theta \;y\ker f\;y'\;\theta\;z$. Hence $f(x)\;\Phi_\theta\;f(y)$, $f(y)=f(y')$ and $f(y')\;\Phi_\theta\; f(z)$. By transitivity, we have $f(x)\;\Phi_\theta\;f(z)$, and hence there are $x', z' \in A$ such that $f(x)=f(x')$, $x'\;\theta\; z'$ and $f(z')= f(z)$, i.e. $x\;\ker f\;x'\;\theta\;z'\;\ker f\; z$. It follows that $(x,z)\in\ker f\circ\theta\circ\ker f$, and thus that $\theta\circ \ker f\circ\theta\subseteq \ker f\circ\theta\circ\ker f $.\newline
Conversely, if $\theta\circ \ker f\circ\theta\subseteq \ker f\circ\theta\circ\ker f $, then a similar argument shows that $\Phi_\theta$ is transitive.
\vskip0.3cm\noindent (c) If $\ker f\subseteq \theta$, then $\theta\circ\ker f\circ \theta = \theta =\ker f\circ \theta\circ\ker f$, so by (b) we have $f(\theta) =\Phi_\theta$. Next, if $(f(x),f(y))\in\Phi_\theta$, then there are $x',y'$ such that $f(x) = f(x')$, $x'\;\theta\;y'$ and $f(y')\;\theta\; f(y)$. It follows that $x\;\theta\; x'\;\theta\;y'\;\theta\; y$ so that $(x,y)\in\theta$.
\vskip0.3cm\noindent (d) Let $\pi:=\ker f$. Clearly $f(\theta)\subseteq f(\theta\lor\pi)$. Now since $\theta\lor\pi\geq\ker f$, we have $f(\theta\lor\pi) = \{(f(x),f(y)):(x,y)\in \theta\lor\pi\}$ by (c). Suppose therefore that $(f(x),f(y))\in f(\theta\lor\pi)$, where $(x,y)\in \theta$. Then there exist $x=a_0,a_1,a_2,\dots ,a_n=y\in A$ such that
\[x=a_0\;\theta\;a_1\;\pi\;a_2\;\theta\;a_3\dots a_n=y\]
and hence
\[f(x)\;f(\theta)\;f(a_1)\quad f(a_1)=f(a_2)\quad f(a_2)\;f(\theta)\;f(a_3)\quad f(a_3)=f(a_4)\dots\] so that also $f(x)\;f(\theta)\; f(y)$. hence $f(\theta\lor\pi)\subseteq f(\theta)$ as required.
\vskip0.3cm\noindent (e) We may assume that $X$ is reflexive and symmetric, and hence so is $\{(f(x),f(y):(x,y)\in X\}$. Suppose that $\theta=\Cg_A(X)$, and that $(b_1,b_2)\in f(\theta)$. Then by (a) there are $y_0, y_1, y_2\dots, y_n\in B$ such that $y_0\;\Phi_\theta\;y_1\;\Phi_\theta\;y_2\dots\;\Phi_\theta\;y_n$ and $y_0 = b_1$, $y_n = b_2$. Hence there are $a_0, a_1, a_2,\dots, a_n\in A$ such that $f(a_i)= y_i$ and $a_i\;\theta\;a_{i+1}$. Thus $(a_0,a_n)\in\theta= \Cg_A(X)$, so there are unary polynomials $p_j$ and pairs $(x_j, y_j)\in X$ (for $j=0,\dots, n$) such that
\[\aligned p_0(x_0)&= a_0\\
p_j(y_j)&=p_{j+1}(x_{j+1})\quad\text{ for } 0\leq j<m\\
p_{n}(y_{n})&=a_{n}\endaligned\]
Applying $f$ to the above Mal'tsev chain from $a_0$ to $a_{n}$ via pairs from $X$ yields a Mal'tsev chain in $B$ from $b_1=y_0=f(a_0)$ to $b_2=y_{m}=f(a_n)$ via pairs from $\{(f(x),f(y)):(x,y)\in X\}$. Hence $f(\theta)\subseteq\Cg_B(\{(f(x),f(y)):(x,y)\in X\})$. The reverse inclusion is obvious.
\eproof
\begin{proposition} \label{propn_generated_con_backward} Let $f:A\twoheadrightarrow B$ be a surjective homomorphism.
\begin{enumerate}[(a)]\item For $\Theta\in\Con(B)$, define
\[f^{-1}(\Theta):=\{(x,y)\in A^2: (f(x),f(y))\in \Theta\}\]
Then $f^{-1}(\Theta)\in\Con(A)$. Moreover $f(f^{-1}(\Theta))=\Theta$.
\item If $\theta\in\Con(A)$, then $f^{-1}(f(\theta))=\theta\lor\ker f$.
\end{enumerate}
\end{proposition}
\bproof (a) It is straightforward to shoe that $f^{-1}(\Theta)$ is a congruence relation on $A$ and that $f^{-1}(\Theta)\geq \ker f$. Hence by Proposition \ref{propn_generated_con_forward}(c) we have
\[f(f^{-1}(\Theta)) = \{(f(x),f(y)): (x,y)\in f^{-1}(\Theta)\}=\Theta\]
(b) Let $\pi:=\ker f$. We have $(x,y)\in f^{-1}(f(\theta))$ if and only if $(f(x),f(y))\in f(\theta)$. But $f(\theta) = f(\theta\lor\pi)$, and $\theta\lor\pi\geq\pi$, so by Proposition \ref{propn_generated_con_forward}, $f(\theta\lor\pi) = \{(f(x),f(y)):(x,y)\in\theta\lor\pi\}$. It follows that if $(f(x),f(y))\in\theta$, then $(f(x),f(y)) = (f(x'),f(y'))$ for some $(x',y')\in\theta\lor\pi$. Hence $x\;\pi\;x'\;\theta\;y'\;\pi\;y$, so $(x,y)\in \theta\lor\pi$. Thus $(x,y)\in f^{-1}(f(\theta))$ implies $(x,y) \in\theta\lor\pi$, i.e. $f^{-1}(f(\theta))\subseteq\theta\lor\pi$.
Conversely, clearly $\theta,\pi\leq f^{-1}(f(\theta))$.
\eproof
\begin{proposition}\label{propn_quotient_con} Suppose that $f:A\twoheadrightarrow B$ is a surjective homomorphism, and that $\pi:=\ker f$. Identifying $B$
with $A/\pi$ and $\Con(B)$ with $\{\theta/\pi: \theta\in\Con(A), \theta\geq\pi\}$, we have:\begin{enumerate}[(a)]\item $f(\theta) = (\theta\lor\pi)/\pi$
\item $f^{-1}(\theta/\pi)= \theta$
\end{enumerate}
\end{proposition}
\bproof (a) If $\theta\in\Con(A)$, then $f(\theta) = f(\theta\lor\pi) $, and $\theta\lor\pi\geq \pi=\ker f$, so by Proposition \ref{propn_generated_con_forward} (c), (d) we
see that \[\tfrac{x}{\pi}\; \tfrac{\theta\lor\pi}{\pi}\;\tfrac{y}{\pi}\quad\Leftrightarrow \quad x\;(\theta\lor\pi)\; y \quad\Leftrightarrow \quad f(x)\;f(\theta\lor\pi)\;f(y)\quad\Leftrightarrow \quad f(x)\;f(\theta)\;f(y)\quad\Leftrightarrow \quad\tfrac{x}{\pi}\;f(\theta)\;\tfrac{y}{\pi}\]
hence $\frac{\theta\lor\pi}{\theta}=f(\theta)$.
(b) is straightforward.
\eproof
| 36,722
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\begin{document}
\title[Multiplicity one theorems]{Multiplicity one theorems for Fourier-Jacobi models}
\author{Binyong Sun}
\address{Academy of Mathematics and Systems Science\\
Chinese Academy of Sciences\\
Beijing, 100190, P.R. China} \email{sun@math.ac.cn}
\subjclass[2000]{22E35, 22E46}
\keywords{Fourier-Jacobi model, classical group, irreducible
representation}
\begin{abstract}
For every genuine irreducible admissible smooth representation $\pi$
of the metaplectic group $\widetilde{\Sp}(2n)$ over a p-adic field,
and every smooth oscillator representation $\omega_\psi$ of
$\widetilde{\Sp}(2n)$, we prove that the tensor product $\pi\otimes
\omega_\psi$ is multiplicity free as a smooth representation of the
symplectic group $\Sp(2n)$. Similar results are proved for general
linear groups and unitary groups.
\end{abstract}
\thanks{Supported by NSFC Grants 10801126 and 10931006.}
\maketitle
\section{Introduction and main results}
\label{sec:intro}
Fix a non-archimedean local field $\rk$ of characteristic zero. The
following multiplicity one theorem for general linear groups,
unitary groups and orthogonal groups, which has been expected since
1980's, is established recently by
Aizenbud-Gourevitch-Rallis-Schiffmann in \cite{AGRS}.
\begin{introtheorem}
Let $G$ denote the group $\GL(n)$, $\oU(n)$, or $\oO(n)$, define
over $\rk$, and let $G'$ denote $\GL(n-1)$, $\oU(n-1)$, or
$\oO(n-1)$, respectively, regarded as a subgroup of $G$ as usual.
Then for any irreducible admissible smooth representation $\pi$ of
$G$, and $\pi'$ of $G'$, one has that
\[
\dim \Hom_{G'}(\pi\otimes\pi',\C)\leq 1.
\]
\end{introtheorem}
As will be clear later, the groups $\GL(n)$, $\oU(n)$ and $\oO(n)$
are automorphism groups of ``Hermitian modules". Therefore, we
consider Theorem A the multiplicity one theorem in the ``Hermitian
case". It is the first step towards the famous Gross-Prasad
Conjecture (\cite{GP92,GP94,GR06,GGP}).
In \cite{GGP}, W. T. Gan, B. Gross and D. Prasad formulate an analog
of the Gross-Prasad Conjecture in the ``skew-Hermitian case". The
corresponding multiplicity one theorem, whose proof is the main goal
of this paper, is the following:
\begin{introtheorem}
Let $G$ denote the group $\GL(n)$, $\oU(n)$, or $\Sp(2n)$, defined
over $\rk$, and regarded as a subgroup of the symplectic group
$\Sp(2n)$ as usual. Let $\widetilde G$ be the double cover of $G$
induced by the metaplectic cover $\widetilde{\Sp}(2n)$ of $\Sp(2n)$.
Denote by $\omega_\psi$ the smooth oscillator representation of
$\widetilde{\Sp}(2n)$ corresponding to a non-trivial character
$\psi$ of $\rk$. Then for any irreducible admissible smooth
representation $\pi$ of $G$, and any genuine irreducible admissible
smooth representation $\pi'$ of $\widetilde G$, one has that
\[
\dim \Hom_{G}(\pi\otimes \pi'\otimes \omega_\psi,\C)\leq 1.
\]
\end{introtheorem}
Recall that an irreducible admissible smooth representation of
$\widetilde G$ is said to be genuine if it does not descent to a
representation of $G$. For symplectic groups, Theorem B is
conjectured by D. Prasad in \cite[Page 20]{Pr96}. \vsp
The ``$\Hom$"-spaces in Theorem A and Theorem B are extreme cases of
their very important generalizations, namely, Bessel models and
Fourier-Jacobi models, respectively. For definitions of these
models, see \cite[Part 3]{GGP}, for example. As explained in
\cite{GPSR97}, uniqueness of Bessel models is the basic starting
point to study L-functions for orthogonal groups by the
Rankin-Selberg method. Similarly, uniqueness of Fourier-Jacobi
models is basic to study L-functions for symplectic groups and
metaplectic groups (\cite{GJRS09}). The importance of Theorem A and
Theorem B lies in the fact that they imply uniqueness of these
models in general (c.f., \cite[Part 3]{GGP}, \cite[Theorem
4.1]{GJRS09} and \cite{JSZ09}). Various special cases of these
uniqueness are obtained in the literature (see \cite{Nov76,GPSR,
BFG92, GRS,BR00,GGP} for example).
\vsp
In order to prove our main results uniformly, we introduce the
following notation. By a commutative involutive algebra, we mean a
finite product of finite field extensions of $\rk$, equipped with a
$\rk$-algebra involution on it. Let $(A,\tau)$ be a commutative
involutive algebra. Let $E$ be an $A$-module which is
finite-dimensional as a $\rk$-vector space. For $\epsilon=\pm 1$,
recall that a $\rk$-bilinear map
\[
\la\,,\,\ra_E:E\times E\rightarrow A
\]
is called an $\epsilon$-Hermitian form if it satisfies
\[
\la u,v\ra_E=\epsilon\la v,u\ra_E^\tau, \quad \la au,v\ra_E=a\la u,
v\ra_E,\quad a\in A,\, u,v\in E.
\]
Assume that $E$ is an $\epsilon$-Hermitian $A$-module, namely it is
equipped with a non-degenerate $\epsilon$-Hermitian form
$\la\,,\,\ra_E$. Denote by $\oU(E)$ the group of all $A$-module
automorphisms of $E$ which preserve the form $\la\,,\,\ra_E$.
Depending on $\epsilon=1$ or $-1$, it is a finite product of general
linear groups, unitary groups, and orthogonal or symplectic groups
(see \cite[Section 3]{SZ} for more details).
View $A^2$ as a standard hyperbolic plane, i.e., it is equipped with
the $\epsilon$-Hermitian form $\la\,,\ra_{A^2}$ so that both $e_1$
and $e_2$ are isotropic, and that
\[
\la e_1, e_2 \ra_{A^2}=1,
\]
where $e_1$, $e_2$ is the standard basis of $A^2$. The orthogonal
direct sum $E\oplus A^2$ is again an $\epsilon$-Hermitian
$A$-module. Note that $\oU(E)$ is identified with the subgroup of
$\oU(E\oplus A^2)$ fixing both $e_1$ and $e_2$. Denote by $\oJ(E)$
the subgroup of $\oU(E\oplus A^2)$ fixing $e_1$.
To be explicit, view $E\oplus A^2$ as the space of column vectors of
$A\oplus E\oplus A$, then $\oJ(E)$ consists of all matrices of the
form
\[
j(x,u,t):= \left[\begin{array}{ccc}
1&-u^\tau \,x& t-\frac{\la u,u\ra_E}{2}\\
0 &x&u\\
0&0&1\\
\end{array}\right],
\]
where
\[
x \in \oU(E),\,\,u\in E,\,\, t\in A^{\tau=-\epsilon}:=\{t\in A\mid t^\tau=-\epsilon
t\},
\]
and $u^\tau$ is the map
\[
E\rightarrow A, \quad v\mapsto\la v, u\ra_E.
\]
The unipotent radical of $\oJ(E)$ is
\[
\oH(E):=\{j(1,u,t)\mid u\in E,\,t\in A^{\tau=-\epsilon}\}=E\times
A^{\tau=-\epsilon},
\]
with multiplication
\[
(u,t)(u',t')=(u+u', t+t'+\frac{\la u,u'\ra_E}{2}-\frac{\la
u',u\ra_E}{2}).
\]
By identifying $j(x,u,t)$ with $(x,(u,t))$, we have
\[
\oJ(E)=\oU(E)\ltimes \oH(E).
\]
\vsp
We will see that Theorem B is a consequence of the following theorem
in the skew-Hermitian case, namely when $\epsilon=-1$.
\begin{introtheorem}
For every irreducible admissible smooth representation $\pi_{\oJ}$
of $\oJ(E)$, and every irreducible admissible smooth representation
$\pi_{\oU}$ of $\oU(E)$, one has that
\[
\dim \Hom_{\oU(E)}(\pi_{\oJ}, \pi_{\oU})\leq 1.
\]
\end{introtheorem}
\vsp
To prove Theorem C by the method of Gelfand-Kazhdan, we extend
$\oU(E)$ to a larger group, which is denoted by $\breve{\oU}(E)$,
and is defined to be the subgroup of $\operatorname{Aut}(E)\times
\{\pm 1\}$ consists of pairs $(g,\delta)$ such that either
\[
\delta=1 \quad\textrm{and}\quad \la gu,gv\ra_E=\la v,u\ra_E,\,\,\, u,v\in E,
\]
or
\[
\delta=-1 \quad\textrm{and}\quad
\la gu,gv\ra_E=\la v,u\ra_E,\,\,\, u,v\in E.
\]
Here $\operatorname{Aut}(E)$ is the automorphism group of the
underlying additive group of $E$. Note that for every element
$(g,\delta)\in \breve{\oU}(E)$, if $\delta=1$, then $g$ is
automatically $A$-linear, and if $\delta=-1$, then $g$ is
$\tau$-conjugate linear. This extended group is first introduced
implicitly by Moeglin-Vigneras-Waldspurger in \cite[Proposition
4.I.2]{MVW87}. It contains $\oU(E)$ as a subgroup of index two. Let
$\breve{\oU}(E)$ act on $\oH(E)$ as group automorphisms by
\begin{equation}\label{gtaction}
(g,\delta).(u,t):=(gu, \delta t).
\end{equation}
This extends the adjoint action of $\oU(E)$ on $\oH(E)$. The
semidirect product
\[
\breve{\oJ}(E):=\breve{\oU}(E)\ltimes\oH(E)
\]
contains
\[
\oJ(E)=\oU(E)\ltimes\oH(E)
\]
as a subgroup of index two.
The main body of this paper is devoted to a proof of the following
\begin{introtheorem}
Let $f$ be a generalized function on $\oJ(E)$. If it is invariant
under the adjoint action of $\oU(E)$, i.e.,
\[
f(g j g^{-1})=f(j),\quad \textrm{for all } g\in\oU(E),
\]
then
\[
f(\breve{g}j^{-1}\breve{g}^{-1})=f(j),\quad \textrm{for all } \breve{g}\in
\breve{\oU}(E)\setminus \oU(E).
\]
\end{introtheorem}
The usual notion of generalized functions will be explained in the
next section. As we will see in the proof, when $\oU(E)$ is
$\GL(n)$, $\oU(n)$, or $\oO(n)$, Theorem D is an easy consequence of
\cite[Theorem 2 and Theorem 2']{AGRS}, and even when $\oU(E)$ is
$\Sp(2n)$, the proof of Theorem D depends on the cases of general
linear groups and unitary groups. \vsp
\vsp
The author thanks Gerrit van Dijk, Dipendra Prasad, Lei Zhang and
Chen-Bo Zhu for helpful comments. He is grateful to Dihua Jiang for
teaching him the method of Gelfand-Kazhdan.
\section{Reduction to the null cone}
We first recall some basic notions and facts about distributions and
generalized functions. By a t.d. space, we mean a topological space
which is Hausdorff, secondly countable, locally compact and totally
disconnected. By a t.d. group, we mean a topological group whose
underlying topological space is a t.d. space. For a t.d space $M$,
denote by $\con^\infty_0(M)$ the space of compactly supported,
locally constant (complex valued) functions on $M$. Denote by
$\oD^{-\infty}(M)$ the space of linear functionals on
$\con^\infty_0(M)$. Such functionals are called distributions on
$M$. If $M$ is furthermore a locally analytic $\rk$-manifold or a
t.d. group (see \cite[Part II]{Sc08} for the notion of locally
analytic manifolds), denote by $\oD^\infty_0(M)$ the space of
compactly supported distributions on $M$ which are locally scalar
multiples of Haar measures. In this case, denote by
$\con^{-\infty}(M)$ the space of linear functionals on
$\oD^\infty_0(M)$. Such functionals are called generalized functions
on $M$.
Let $\varphi: M\rightarrow N$ be a continuous map of t.d. spaces. If
it is proper, then we define the push forward map
\[
\varphi_*: \oD^{-\infty}(M)\rightarrow \oD^{-\infty}(N)
\]
as usual. Recall that $\varphi$ is said to be proper if
$\varphi^{-1}(C)$ is compact for every compact subset $C$ of $N$.
When $\varphi$ is a closed embedding, $\varphi_*$ identifies
$\oD^{-\infty}(M)$ with distributions in $\oD^{-\infty}(N)$ which
are supported in $\varphi(M)$. If $\varphi$ is not proper, then the
push forward map is still defined, but only for distributions with
compact support.
If $\varphi: M\rightarrow N$ is a submersion of locally analytic
$\rk$-manifolds, then the push forward map sends $\oD^\infty_0(M)$
into $\oD^\infty_0(N)$, and its transpose defines the pull back map
\[
\varphi^*: \con^{-\infty}(N)\rightarrow \con^{-\infty}(M).
\]
When $\varphi$ is a surjective submersion, $\varphi^*$ is injective.
If $G$ is an (abstract) group acting continuously on a t.d. space
$M$, then for any group homomorphism $\chi_G:G\rightarrow
\C^\times$, put
\[
\oD^{-\infty}_{\chi_G}(M):=\{\omega\in \oD^{-\infty}(M)\mid (T_g)_*\,\omega=\chi_G(g)\,\omega,\quad g\in
G\},
\]
where $T_g:M\rightarrow M$ is the map given by the action of $g\in
G$. If furthermore $M$ is a locally analytic $\rk$-manifold, and
the action of $G$ on it is also locally analytic, denote by
\[
\con^{-\infty}_{\chi_G}(M)\subset \con^{-\infty}(M)
\]
the subspace consisting of all $f$ which are $\chi_G$-equivariant,
i.e.,
\[
f(g.x)=\chi_G(g)f(x),\quad \textrm{for all } g\in
G,
\]
or to be precise,
\[
T_g^*(f)=\chi_G(g)f,\quad \textrm{for all } g\in
G.
\]
\vsp
Now we return to the notation of the last section. Recall that $E$
is an $\epsilon$-Hermitian $A$-module. Denote by
\[
\chi_E: \breve{\oU}(E)\rightarrow \{\pm 1\}, \quad
(g,\delta)\mapsto \delta
\]
the quadratic character projecting to the second factor. Let
$\breve{\oU}(E)$ act on $\oJ(E)$ by
\begin{equation}\label{actg}
\breve{g}.j:=\breve{g} \,j^{\,\chi_E(\breve g)}\,\breve{g}^{-1}.
\end{equation}
Then Theorem D is equivalent to saying that
\begin{equation}\label{vanishg}
\con^{-\infty}_{\chi_E}(\oJ(E))=0.
\end{equation}
Denote by $\gl_A(E)$ the (associative) algebra of all $A$-module
endomorphisms of $E$. The involution $\tau$ on $A$ extends to an
anti-involution on $\gl_A(E)$, which is still denoted by $\tau$, by
requiring that
\[
\la x^\tau u,v\ra_E=\la u, xv\ra_E, \quad u,v\in E.
\]
Write
\[
\u(E):=\{x\in \gl_A(E)\mid x^\tau+x=0\}.
\]
This is the Lie algebra of (the $\rk$-linear algebraic group)
$\oU(E)$. Let $\breve{\oU}(E)$ act on $\u(E)$ and $E$ by
\begin{equation}\label{actioninf}
\left\{
\begin{array}{ll}
(g,\delta).x:=\delta \,gxg^{-1},\quad& x\in \u(E),\medskip\\
(g,\delta).u:=\delta \,gu,\quad & u\in E,
\end{array}
\right.
\end{equation}
and act on $\u(E)\times E$ diagonally. The infinitesimal analog of
(\ref{vanishg}) which we need is
\begin{equation}\label{vanishl}
\con^{-\infty}_{\chi_E}(\u(E)\times E)=0.
\end{equation}
\vsp
Write
\[
\oU(A):=\{a\in A\mid a a^\tau=1\},
\]
and denote by $\oZ(E)$ the image of the map
\[
\oU(A)\rightarrow \oU(E)
\]
given by scalar multiplications. Note that $\oZ(E)$ coincides with
the center of $\oU(E)$. The Lie algebra of $\oU(A)$ is
\[
\u(A):=\{a\in A\mid a+a^\tau=0\},
\]
and the Lie algebra $\z(E)$ of $\oZ(E)$ is the image of the map
\[
\u(A)\rightarrow \u(E)
\]
given by scalar multiplications. Note that $\z(E)$ may not coincide
with the center of $\u(E)$. Denote by $\CU_E$ the set of unipotent
elements in $\oU(E)$, and by $\CN_E$ the set of nilpotent (as
$\rk$-linear operators on $E$) elements in $\u(E)$.
\vsp If the commutative involutive algebra $A$ is simple, i.e., if
it is a field or a product of two isomorphic fields which are
exchanged by $\tau$, then $E$ is automatically a free $A$-module of
finite rank. In this case, put
\[
\sdim(E):=\max\{\rank_A(E)-1,\,0\}.
\]
In general, $A$ is uniquely of the form
\[
A=A_1\times A_2\times \cdots\times A_n,
\]
where $n\geq 0$, and $A_i$ is a simple commutative involutive
algebra, $i=1,2,\cdots, n$. Note that $A_i\otimes_A E$ is
canonically an $\epsilon$-Hermitian $A_i$-module. Put
\[
\sdim(E):=\sum_{i=1}^n \sdim(A_i\otimes_A E).
\]
We prove the following induction result by Harish-Chandra descent in
the remaining part of this section.
\begin{prp}\label{descent11}
Assume that for all commutative involutive algebras $A^\circ$ and
all $\epsilon$-Hermitian $A^\circ$-modules $E^\circ$,
\begin{equation}\label{vanishep1}
\sdim(E^\circ)<\sdim(E) \quad\textrm{implies}\quad \con^{-\infty}_{\chi_{E^\circ}}(\oJ(E^\circ))=0.
\end{equation}
Then every $f\in \con^{-\infty}_{\chi_{E}}(\oJ(E))$ is supported in
$(\oZ(E)\CU_E)\ltimes \oH(E)$.
\end{prp}
The infinitesimal analog is
\begin{prpp}\label{descent2} Assume that for all
commutative involutive algebras $A^\circ$ and all
$\epsilon$-Hermitian $A^\circ$-modules $E^\circ$,
\[
\sdim(E^\circ)<\sdim(E) \quad\textrm{implies}\quad \con^{-\infty}_{\chi_{E^\circ}}(\u(E^\circ)\times E^\circ)=0.
\]
Then every $f\in \con^{-\infty}_{\chi_{E}}(\u(E)\times E)$ is
supported in $(\z(E)+\CN_E)\times E$.
\end{prpp}
The proof of Proposition \ref{descent11} is close to that of
\cite[Proposition 5.2]{SZ}. Without loss of generality, we assume
that $E$ is faithful as an $A$-module. For every semisimple element
$x\in \oU(E)$, denote by $A_x$ the subalgebra of $\gl_A(E)$
generated by $x$, $x^\tau$ and $A$. Then $(A_x,\tau)$ is again a
commutative involutive algebra. Define an $A$-linear map
$\tr_{A_x/A}: A_x\rightarrow A$ by requiring that the diagram
\[
\begin{CD}
A_x @>\phantom{aaa}\tr_{A_x/A}\phantom{aaa}>> A\\
@VVV @VVV\\
A_0\otimes_A A_x @>\tr_{(A_0\otimes_A A_x)/A_0}>> A_0\\
\end{CD}
\]
commutes for every quotient field $A_0$ of $A$, where the bottom
arrow is the usual trace map for a finite dimensional commutative
algebra over a field. Write $E_x:=E$, viewed as an $A_x$-module.
\begin{lemp}\label{sdim}
There is a unique $\epsilon$-Hermitian form $\la\,,\,\ra_{E_x}$ on
the $A_x$-module $E_x$ such that the diagram
\[
\begin{CD}
E_x \times E_x @>\la\,,\,\ra_{E_x}>> A_x\\
@| @VV \tr_{A_x/A}V \\
E\times E@>\la\,,\,\ra_E>> A\\
\end{CD}
\]
commutes.
\end{lemp}
\begin{proof}
Uniqueness is clear. For existence, we define the form by requiring
that
\[
\tr_{A_x/\rk}(a \la u,v\ra_{E_x})= \tr_{A/\rk}(\la au,v\ra_{E}),\quad a\in A_x,\,u,v\in E.
\]
Then all the desired properties are routine to check.
\end{proof}
Now $E_x$ is an $\epsilon$-Hermitian $A_x$-module. Note that
$\breve{\oU}(E_x)$ is a subgroup of $\breve{\oU}(E)$, and the
homomorphism
\[
\begin{array}{rcl}
\xi_x: \oJ(E_x)=\oU(E_x)\ltimes (E_x\times A_x^{\tau=-\epsilon}) &\rightarrow& \oJ(E)=\oU(E)\ltimes (E\times A^{\tau=-\epsilon}) ,\\
(y,(u,t))&\mapsto & (y,(u,\tr_{A_x/A}(t))
\end{array}
\]
is $\breve{\oU}(E_x)$-intertwining.
We omit the proof of the following elementary lemma.
\begin{lemp}\label{sdim}
With the notation as above. If $x\notin \oZ(E)$, then
\[
\sdim(E_x)< \sdim(E).
\]
\end{lemp}
Now assume that $x\notin \oZ(E)$. For any
\[
j=(y,h)\in \oJ(E_x)=\oU(E_x)\ltimes \oH(E_x),
\]
denote by $J(j)$ the determinant of the $\rk$-linear map
\[
1-\Ad_{y^{-1}}: \u(E)/\u(E_x)\rightarrow \u(E)/\u(E_x).
\]
Note that $\Ad_{y}$ preserves a non-degenerate $\rk$-quadratic form
on $\u(E)/\u(E_x)$, which implies that $J$ is
$\breve{\oU}(E_x)$-invariant. Put
\[
\oJ(E_x)^\circ:=\{j\in \oJ(E_x)\mid J(j)\neq 0\}.
\]
It contains the set $x\CU_{E_x}\ltimes \oH(E_x)$. One easily checks
that the map
\[
\begin{array}{rcl}
\rho_x: \breve{\oU}(E)\times \oJ(E_x)^\circ &\rightarrow& \oJ(E),\\
(\breve{g},j)&\mapsto & \breve{g}.(\xi_x(j))
\end{array}
\]
is a submersion, and we have a well defined map (c.f. \cite[Lemma
2.5]{JSZ})
\begin{equation}\label{reex}
r_x: \con^{-\infty}_{\chi_{E}}(\oJ(E))\rightarrow \con^{-\infty}_{\chi_{E_x}}(\oJ(E_x)^\circ),
\end{equation}
which is specified by the rule
\[
\rho_x^*(f)=\chi _E\otimes r_x(f),\quad f\in \con^{-\infty}_{\chi_{E}}(\oJ(E)).
\]
Lemma \ref{sdim} and the assumption (\ref{vanishep1}) easily imply
the vanishing of the range space of (\ref{reex}) (c.f. \cite[Lemma
2.6]{JSZ}). Thus every $f\in \con^{-\infty}_{\chi_{E}}(\oJ(E))$
vanishes on the image of $\rho_x$. As $x$ is arbitrary, we finish
the proof of Proposition \ref{descent11}. The proof of Proposition
\ref{descent2} is similar. We omit the details.
\section{Linearlization}
With the idea of linearlization by Jaquet-Rallis (\cite{JR96}) in
mind, the goal of this section is to prove the following
\begin{prp}\label{linear} Assume that for all
commutative involutive algebras $A^\circ$ and all
$\epsilon$-Hermitian $A^\circ$-modules $E^\circ$, we have
\[
\con^{-\infty}_{\chi_{E^\circ}}(\u(E^\circ)\times E^\circ)=0.
\]
Then $\con^{-\infty}_{\chi_{E}}(\oJ(E))=0$.
\end{prp}
The Lie algebra of $\oH(E)$ is
\[
\h(E):=E\times A^{\tau=-\epsilon}
\]
with Lie bracket given by
\[
[(u,t),(u',t')]:=(0,\la u,u'\ra_E-\la u',u\ra_E).
\]
The Lie algebra of $\breve{\oJ}(E)$ is
\[
\j(E):=\u(E)\ltimes \h(E),
\]
where the semidirect product is defined by the Lie algebra action
\[
x.(u,t):=(xu, 0).
\]
Let $\breve{\oU}(E)$ act on $\j(E)$ by the differential of its
action on $\oJ(E)$, i.e.,
\[
\breve{g}.j:=\chi_E(\breve{g})\, \Ad_{\breve g}(j),\quad j\in \j(E).
\]
It is easy to see that as a $\breve{\oU}(E)$-space,
\[
\j(E)=\u(E)\times E\times A^{\tau=-\epsilon},
\]
where $A^{\tau=-\epsilon}$ carries the trivial
$\breve{\oU}(E)$-action. Recall the following localization principle
which is due to Bernstein. See \cite[section 1.4]{Be84} or
\cite[Corollary 2.1]{AGRS}.
\begin{lemp}\label{localization}
Let $\varphi: M\rightarrow N$ be a continuous map of t.d. spaces,
and let $G$ be a group acting continuously on $M$ preserving the
fibers of $\varphi$. Then for any group homomorphism
$\chi_G:G\rightarrow \C^\times $, the condition
\[
\oD_{\chi_G}^{-\infty}(\varphi^{-1}(x))=0 \quad \textrm{for all }x\in N
\]
implies that
\[
\oD_{\chi_G}^{-\infty}(M)=0.
\]
\end{lemp}
In particular, the localization principle implies the following
\begin{lemp}\label{loc22} If
\[
\oD^{-\infty}_{\chi_{E}}(\u(E)\times E)=0,
\]
Then $\oD^{-\infty}_{\chi_E}(\j(E))=0$.
\end{lemp}
We need the following obvious fact of exponential maps in the theory
of linear algebraic groups.
\begin{lemp}\label{expn}
The set of unipotent elements in $\oJ(E)$ is $\CU_E\ltimes \oH(E)$,
the set of algebraically nilpotent elements in $\j(E)$ is
$\CN_E\ltimes \h(E)$, and the exponential map is a
$\breve{\oU}(E)$-intertwining homeomorphism from $\CN_E\ltimes
\h(E)$ onto $\CU_E\ltimes \oH(E)$.
\end{lemp}
In all cases that concern us, whenever $M$ is a locally analytic
$\rk$-manifold with a locally analytic $\breve{\oU}(E)$-action,
there is always a canonical choice (up to a scalar) of a positive
smooth invariant measure on $M$. Therefore the space
$\con^{-\infty}_{\chi_{E}}(M)$ is canonically identified with
$\oD^{-\infty}_{\chi_{E}}(M)$. We will use this observation freely.
For every $z\in \oZ(E)$, it is clear that $z\CU_E\ltimes \oH(E)$ is
$\breve{\oU}(E)$-stable.
\begin{lemp}\label{loc222} If
\begin{equation}\label{van11}
\con^{-\infty}_{\chi_{E}}(\u(E)\times E)=0,
\end{equation}
then
\[
\oD^{-\infty}_{\chi_E}(\CU_E\ltimes \oH(E))=0.
\]
\end{lemp}
\begin{proof}
By Lemma \ref{loc22}, we have that
\[
\oD^{-\infty}_{\chi_E}(\j(E))=0,
\]
which implies that
\[
\oD^{-\infty}_{\chi_E}(\CN_E\ltimes \h(E))=0.
\]
The lemma then follows from Lemma \ref{expn}.
\end{proof}
\begin{lemp}\label{loc2} If
\begin{equation}
\con^{-\infty}_{\chi_{E}}(\u(E)\times E)=0,
\end{equation}
then
\[
\oD^{-\infty}_{\chi_E}(\oZ(E)\CU_E\ltimes \oH(E))=0.
\]
\end{lemp}
\begin{proof}
It is easy to see (by using the trace map) that the map
\[
\begin{array}{rcl}
\oZ(E)\CU_E\ltimes \oH(E)&\rightarrow & \oZ(E),\\
(z x, h)&\mapsto& z,
\end{array} \qquad (x\in \CU_E)
\]
is a well defined continuous map. By the localization principle, it
suffices to show that
\begin{equation}\label{van22}
\oD^{-\infty}_{\chi_E}(z\CU_E\ltimes \oH(E))=0,\quad \textrm{for all }z\in\oZ(E).
\end{equation}
Given $z\in \oZ(E)$, denote by $T_z$ the left multiplication by $z$.
One easily checks that the diagram
\[
\begin{CD}
\CU_E\ltimes \oH(E) @>T_z>> z\CU_E\ltimes \oH(E) \\
@V g VV @V g VV\\
\CU_E\ltimes \oH(E) @>T_z>> z\CU_E\ltimes \oH(E) \\
\end{CD}
\]
commutes for all $g\in \oU(E)$, and the diagram
\[
\begin{CD}
\CU_E\ltimes \oH(E) @>T_z>> z\CU_E\ltimes \oH(E) \\
@V \breve{g}z VV @V \breve{g} VV\\
\CU_E\ltimes \oH(E) @>T_z>> z\CU_E\ltimes \oH(E) \\
\end{CD}
\]
commutes for all $\breve{g}\in \breve{\oU}(E)\setminus \oU(E)$,
where all vertical arrows are given by the actions of the indicated
elements. Therefore (\ref{van22}) is a consequence of Lemma
\ref{loc222}.
\end{proof}
We now prove Proposition \ref{linear} by induction on $\sdim(E)$. If
it is zero, then
\[
\oJ(E)=\oZ(E)\CU_E\ltimes \oH(E),
\]
and we are done by Lemma \ref{loc2}. Now assume that it is positive
and we have proved the proposition when $\sdim(E)$ is smaller. Then
Proposition \ref{descent11} implies that every $T\in
\oD^{-\infty}_{\chi_{E}}(\oJ(E))$ is supported in
$(\oZ(E)\CU_E)\ltimes \oH(E)$, and then $T=0$ by Lemma \ref{loc2}.
This finishes the proof.
\section{Reduction within the null cone}
In this section, we treat the case of symplectic groups only.
Througout this section, assume that $\epsilon=-1$, $A$ is a field,
and that the involution $\tau$ on $A$ is trivial.
Let
\[
\CN_E=\CN_0\supset \CN_1\supset \cdots \supset \CN_r=\{0\}\supset
\CN_{r+1}=\emptyset
\]
be a filtration of $\CN_E$ by its closed subsets so that each
difference
\[
\CO_i:=\CN_i\setminus \CN_{i+1},\quad 0\leq i\leq r,
\]
is a $\breve{\oU}(E)$-orbit. The goal of this section is to prove
the following
\begin{prp}\label{indn} If every distribution in $\oD^{-\infty}_{\chi_E}(\u(E)\times
E)$ is supported in $\CN_i\times E$, for some fixed $0\leq i\leq r$,
then every distribution in $\oD^{-\infty}_{\chi_E}(\u(E)\times E)$
is supported in $\CN_{i+1}\times E$.
\end{prp}
\vsp
Without loss of generality, in the remaining part of this section we
further assume that $A=\rk$. Fix a point $\mathbf{e}$ of $\CO_i$.
Recall that $\CO_i$ is said to be distinguished if $\mathbf{e}$
commutes with no nonzero semisimple element in $\u(E)$ (c.f.
\cite[Section 8.2]{CM}). This definition is independent of the
choice of $\mathbf{e}$. We first treat the case when $\CO_i$ is not
distinguished. View $\u(E)$ as a quadratic space over $\rk$ under
the trace form
\[
\la x,y\ra_{\u(E)}:=\tr(xy).
\]
Recall the following elementary fact (c.f., \cite[Lemma 4.2]{SZ}).
\begin{lemp}\label{metricp}
If $\CO_i$ is not distinguished, then there is a non-isotropic
vector in $\u(E)$ which is perpendicular to the tangent space
$\oT_{\mathbf{e}}(\CO_i)\subset \u(E)$.
\end{lemp}
Recall the action (\ref{actioninf}) of $\breve{\oU}(E)$ on $\u(E)$
and $E$. Write $E':=E$ as an $\epsilon$-Hermitian $A$-module, but
equipped with the action of $\breve{\oU}(E)$ given by
\[
(g,\delta).u:=gu.
\]
Let $\breve{\oU}(E)$ act on $\u(E)\times E'$ diagonally. Define a
non-degenerate $\breve{\oU}(E)$-invariant bilinear map
\[
\la\,,\,\ra_\j: (\u(E)\times E)\times (\u(E)\times E')\rightarrow
\rk
\]
by
\[
\la (x,u),(x',u')\ra_\j:=\la x, x'\ra_{\u(E)}+\la u,u'\ra_E.
\]
Fix a nontrivial character $\psi$ of $\rk$. As in the appendix, for
every distribution $T\in \oD^{-\infty}(\u(E)\times E)$, define its
Fourier transform $\widehat{T}\in \con^{-\infty}(\u(E)\times E')$ by
\[
\widehat{T}(\omega):=T(\hat{\omega}), \quad \omega\in \oD_0^\infty(\u(E)\times
E'),
\]
where $\hat{\omega}\in \con_0^{\infty}(\u(E)\times E)$ is given by
\[
\hat{\omega}(j):=\int_{\u(E)\times E'} \psi(\la j,j'\ra_\j)\,
d\omega(j'),\quad j\in \u(E)\times E.
\]
\begin{lemp}\label{funcert}
Assume that $\CO_i$ is not distinguished. Let $T\in
\oD^{-\infty}(\u(E)\times E)$. If $T$ is supported in $\CN_i\times
E$, and its Fourier transform
$\widehat{T}\in\con^{-\infty}(\u(E)\times E')$ is supported in the
null cone
\begin{equation}\label{nulcone}
\{(x,u)\in \u(E)\times E'\mid \tr(x^2)=0\},
\end{equation}
then $T$ is supported in $\CN_{i+1}\times E$.
\end{lemp}
\begin{proof}
This is a direct consequence of Lemma \ref{metricp} and Theorem
\ref{uncert} of the appendix.
\end{proof}
\begin{lemp}
Proposition \ref{indn} holds when $\CO_i$ is not distinguished.
\end{lemp}
\begin{proof}
Let $T\in \oD_{\chi_E}^{-\infty}(\u(E)\times E)$. Then by assumption
it is supported in $\CN_i\times E$. It is clear that the Fourier
transform maps $\oD_{\chi_E}^{-\infty}(\u(E)\times E)$ into
$\con^{-\infty}_{\chi_E}(\u(E)\times E')$. By noting that $-1\in
\oU(E)$, we find that the space $\con^{-\infty}_{\chi_E}(\u(E)\times
E')$ is identical with the space
$\con^{-\infty}_{\chi_E}(\u(E)\times E)$. Apply the assumption to
$\widehat{T}$, we find that $\widehat{T}$ is supported in
$\CN_i\times E'$, which is contained in the null cone
(\ref{nulcone}). We finish the proof by Lemma \ref{funcert}.
\end{proof}
\vsp Now we treat the case when $\CO_i$ is distinguished. For all
$v\in E$, put
\[
\phi_v(u):=\la u,v\ra_E \,v,\quad u\in E.
\]
One easily checks that $\phi_v\in \u(E)$. For all $o\in \CO_i$, put
\[
E(o):=\{v\in E\mid \phi_v\in [\u(E),o]\}.
\]
\begin{lemp}\label{support} If every distribution in $\oD^{-\infty}_{\chi_E}(\u(E)\times
E)$ is supported in $\CN_i\times E$, then the support of every
distribution in $\oD^{-\infty}_{\chi_E}(\u(E)\times E)$ is contained
in
\[
(\CN_{i+1}\times E) \cup \bigsqcup_{\mathbf{o}\in \CO_i}
\{o\}\times E(o).
\]
\end{lemp}
\begin{proof} We follow the method of \cite{AGRS}.
Let $T\in \oD^{-\infty}_{\chi_E}(\u(E)\times E)$ and $(o,v)\in
\CO_i\times E$ be a point in the support of $T$. It suffices to
prove that $v\in E(o)$.
For every $t\in \rk$, define a homeomorphism
\[
\begin{array}{rcl}
\eta_t: \u(E)\times E &\rightarrow &\u(E)\times E,\\
(x,u)&\mapsto& (x+t \phi_u,u),
\end{array}
\]
which is checked to be $\breve{\oU}(E)$-intertwining. Therefore
\[
(\eta_t)_* T\in \oD^{-\infty}_{\chi_E}(\u(E)\times E).
\]
Since $(o,v)$ is in the support of $T$, $\eta_t(o,v)$ is in the
support of $(\eta_t)_* T$. Therefore the assumption implies that
\begin{equation}\label{etat}
\eta_t(o,v)=(o+t\phi_v, v)\in \CN_i\times E.
\end{equation}
As $\CO_i$ is open in $\CN_i$, (\ref{etat}) implies that
\[
\phi_v\in \oT_o(\CO_i)=[\u(E),o].
\]
\end{proof}
Extend $\mathbf e$ to a standard triple
$\mathbf{h},\mathbf{e},\mathbf{f}$ in $\u(E)$, i.e., the
$\rk$-linear map from $\sl_2(\rk)$ to $\u(E)$ specified by
\[
\left[
\begin{array}{cc} 1&0\\ 0&-1\\
\end{array}
\right]
\mapsto \mathbf{h},\quad
\left[
\begin{array}{cc} 0&1\\ 0&0\\
\end{array}
\right]
\mapsto \mathbf{e},\quad
\left[
\begin{array}{cc} 0&0\\ 1&0\\
\end{array}
\right]
\mapsto \mathbf{f},
\]
is a Lie algebra homomorphism. Existence of such an extension is
known as Jacobson-Morozov Theorem. Using this homomorphism, we view
$E$ as an $\sl_2(\rk)$-module with an invariant symplectic form. In
the remaining part of this section assume that $\CO_i$ is
distinguished. By the classification of distinguished nilpotent
orbits (\cite[Theorem 8.2.14]{CM}), we know that $E$ has an
orthogonal decomposition
\[
E=E_1\oplus E_2\oplus\cdots\oplus E_s,\quad s\geq 0,
\]
where all $E_k$'s are irreducible $\sl_2(\rk)$-submodules, with
pairwise different even dimensions. Denote by $E^+$ and $E^-$ the
subspaces of $E$ spanned by eigenvectors of $\mathbf h$ with
positive and negative eigenvalues, respectively. Then
\[
E=E^+\oplus E^-
\]
is a complete polarization of $E$.
We omit the proof of the following elementary lemma (c.f.
\cite[Lemma 3.4]{SZ}).
\begin{lemp}\label{subs1}
One has that
\[
E(\mathbf e)\subset E^+.
\]
\end{lemp}
\vsp
Fix a Haar measure $du'$ on $E'$. For any t.d. space $M$, we define
the partial Fourier transform
\[
\CF_E: \oD^{-\infty}(M\times E)\rightarrow \oD^{-\infty}(M\times E')
\]
by
\[
\CF_E(T)(\varphi_M\otimes \varphi'):=T(\varphi_M\otimes \hat{\varphi'}), \quad
\varphi_M\in \con^\infty_0(M), \,\varphi'\in \con^\infty_0(E'),
\]
where $\hat{\varphi'}\in \con^\infty_0(E)$ is given by
\[
\hat{\varphi'}(u):=\int_{E'} \psi(\la u,u'\ra_E)\,\varphi'(u')
\,du'.
\]
For every $o\in \CO_i$, write $E'(o):=E(o)$, viewed as a subset of
$E'$.
\begin{lemp}\label{haarm}
Let $T\in \oD^{-\infty}(E)$. If $T$ is supported in $E(\mathbf e)$,
and $\CF_E(T)\in \oD^{-\infty}(E')$ is supported in $E'(\mathbf e)$,
then $T$ is a scalar multiple of a Haar measure of $E^+$.
\end{lemp}
\begin{proof}Since $\CF_E(T)$ is supported in $E'(\mathbf e)$, $T$ is
invariant under translations by elements of
\[
\{u\in E\mid \la u, u'\ra_E=0, \quad u'\in E'(\mathbf e)\}.
\]
Lemma \ref{subs1} then implies that $T$ is supported in $E^+$ and is
invariant under translations by $E^+$. This proves the lemma.
\end{proof}
Denote by $\breve{\oU}(E,\mathbf e)$ the stabilizer of $\mathbf e\in
\u(E)$ in $\breve{\oU}(E)$, and by $\chi_{E,\mathbf e}$ the
restriction of $\chi_E$ to $\breve{\oU}(E,\mathbf e)$.
\begin{lemp}\label{van111}
Let $T\in \oD^{-\infty}_{\chi_{E,\mathbf e}}(E)$. If $T$ is
supported in $E(\mathbf e)$, and $\CF_E(T)$ is supported in
$E'(\mathbf e)$, then $T=0$.
\end{lemp}
\begin{proof}
By Lemma \ref{haarm}, $T$ is a scalar multiple of a Haar measure of
$E^+$. Note that all eigenvalues of $\mathbf h$ on $E$ are odd
integers. Let $g:E\rightarrow E$ be the linear map which is the
scalar multiplication by $(-1)^n$ on the $\mathbf h$-eigenspace with
eigenvalue $2n+1$, $n\in \Z$. It is clear that $(g,-1)\in
\breve{\oU}(E,\mathbf e)$, and leaves the Haar measure of
$E^+$-invariant. This finishes the proof.
\end{proof}
\vsp Fix a positive $\breve{\oU}(E)$-invariant measure $do$ on
$\CO_i$ (which always exists), and a Haar measure $d\breve{g}$ on
$\breve{\oU}(E)$. Define a submersion
\[
\begin{array}{rcl}
\rho_\mathbf{e}: \breve{\oU}(E)\times E&\rightarrow&\CO_i\times E,\\
(\breve{g}, v)&\mapsto &\breve{g}.(\mathbf e,v),
\end{array}
\]
and define the pull back
\[
\begin{array}{rcl}
\rho_\mathbf{e}^*: \oD^{-\infty}(\CO_i\times E) &\rightarrow& \oD^{-\infty}(\breve{\oU}(E)\times
E),\\
f\,do\otimes du&\mapsto & \rho_{\mathbf e}^*(f)\,d\breve{g}\otimes
du,
\end{array}
\]
where $du$ is any Haar mesure on $E$, $f\in
\con^{-\infty}(\CO_i\times E)$, and $\rho_{\mathbf e}^*(f)$ is the
usual pull back of a generalized function. By Frobenious
reciprocity (c.f. \cite[Section 1.5]{Be84}), there is a well defined
linear isomorphism
\begin{equation}\label{rmathbfe}
r_\mathbf{e}: \oD^{-\infty}_{\chi_E}(\CO_i\times
E)\stackrel{\sim}{\rightarrow}
\oD^{-\infty}_{\chi_{E,\mathbf{e}}}(E),
\end{equation}
specified by
\[
\rho_\mathbf{e}^*(T)=\chi_E\,d\breve{g}\otimes r_\mathbf{e}(T),\quad T\in \oD^{-\infty}_{\chi_E}(\CO_i\times
E).
\]
Similarly, by using the action of $\breve{\oU}(E)$ on $\CO_i\times
E'$, we define a map
\[
\begin{array}{rcl}
{\rho'}_\mathbf{e}^*: \oD^{-\infty}(\CO_i\times E') &\rightarrow& \oD^{-\infty}(\breve{\oU}(E)\times
E'),\\
\end{array}
\]
and a linear isomorphism
\[
r'_\mathbf{e}: \oD^{-\infty}_{\chi_E}(\CO_i\times E')\stackrel{\sim}{\rightarrow}
\oD^{-\infty}_{\chi_{E,\mathbf{e}}}(E').
\]
The routine verification of the following lemma is left to the
reader.
\begin{lemp}\label{cdiag}
The diagram
\[
\begin{CD}
\oD^{-\infty}_{\chi_E}(\CO_i\times E) @>r_{\mathbf e}>> \oD^{-\infty}_{\chi_{E,\mathbf e}}(E) \\
@V \CF_E VV @V \CF_E VV\\
\oD^{-\infty}_{\chi_E}(\CO_i\times E') @>r'_{\mathbf e}>> \oD^{-\infty}_{\chi_{E,\mathbf e}}(E')\\
\end{CD}
\]
commutes.
\end{lemp}
\vsp
Now we are ready to prove Proposition \ref{indn} when $\CO_i$ is
distinguished. Let $T\in \oD^{-\infty}_{\chi_E}(\CN_i\times E)$.
Then Lemma \ref{support} implies that
\[
r_\mathbf{e}(T|_{\CO_i\times E})\in \oD^{-\infty}_{\chi_{E,\mathbf
e}}(E)\quad\textrm{is supported in $E(\mathbf e)$.}
\]
Similarly, since $\CF_E(T)\in\oD^{-\infty}_{\chi_E}(\CN_i\times
E')$, we have
\[
r'_\mathbf{e}(\CF_E(T)|_{\CO_i\times E'})\in \oD^{-\infty}_{\chi_{E,\mathbf
e}}(E') \quad\textrm{is supported in $E'(\mathbf e)$.}
\]
Now Lemma \ref{van111} and Lemma \ref{cdiag} implies that
\[
r_\mathbf{e}(T|_{\CO_i\times E})=0,
\]
which implies that $T|_{\CO_i\times E}=0$. This finishes the proof.
\section{Proof of the theorems}
We return to the general case. Recall that $A$ is a commutative
involutive algebra, and $E$ is an $\epsilon$-Hermitian $A$-module.
\begin{thm}\label{infin}
One has that
\begin{equation}\label{vanliagebra}
\con^{-\infty}_{\chi_E}(\u(E)\times E)=0.
\end{equation}
\end{thm}
\begin{proof}
Write $s:=\sdim(E)\geq 0$, and assume that the theorem is proved
when $s$ is smaller. Without loss of generality, we assume that $A$
is simple (c.f. \cite[Lemma 6.1]{SZ}). When $\epsilon=1$,
(\ref{vanliagebra}) is proved in \cite{AGRS}, which is the
infinitesimal version of the main result of that paper. When
$\epsilon=-1$ and $\tau$ is nontrivial on $A$, take an element
\[
c_A\in \u(A)\cap A^\times.
\]
Then $(E, c_A\la\,,\,\ra_E)$ is a $-\epsilon$-Hermitian $A$-module,
and (\ref{vanliagebra}) reduces to the case when $\epsilon=1$.
Now to finish the proof, we further assume that $\epsilon=-1$ and
$\tau$ is trivial on $A$. Then $A$ is a field. By Proposition
\ref{descent2}, every generalized function in
$\con^{-\infty}_{\chi_{E}}(\u(E)\times E)$ is supported in
\[
(\z(E)+\CN_E)\times E=\CN_E\times E,
\]
and it has to vanish by Proposition \ref{indn}.
\end{proof}
\vsp
Theorem D is now a consequence of Theorem \ref{infin} and
Proposition \ref{linear}. \vsp
The following Lemma is well known. It
is also a direct consequence of Theorem D.
\begin{lemt}\label{invue}
For every $f\in\con^{-\infty}(\oU(E))$, if
\[
f(gx g^{-1})=f(x)\quad \textrm{for all }\, g\in\oU(E),
\]
then
\[
f(\breve{g}x^{-1}\breve{g}^{-1})=f(x)\quad \textrm{for all } \,\breve{g}\in
\breve{\oU}(E)\setminus \oU(E).
\]
\end{lemt}
Theorem C is a consequence of Theorem D, Lemma \ref{invue}, and the
following version of the Gelfand-Kazhdan criterion.
\begin{lemt}\label{gkmo}
Let $G$ be a t.d. group with a closed subgroup $S$. Let $\sigma$ be
a continuous anti-automorphism of $G$ such that $\sigma(S)=S$.
Assume that for every generalized function $f$ on $G$ or on $S$, the
condition
\[
f(sxs^{-1})=f(x)\quad \textrm{ for all } s\in S
\]
implies that
\[
f(x^\sigma)=f(x).
\]
Then for all irreducible admissible smooth representation $\pi_G$ of
$G$, and $\pi_S$ of $S$, we have
\[
\dim \Hom_{S}(\pi_G,\pi_S)\leq 1.
\]
\end{lemt}
\begin{proof}
This is proved for real reductive groups in \cite[Proposition
7.1]{SZ}. The same proof works here. We sketch a proof for the
convenience of the reader.
Denote by $\Delta(S)$ the diagonal subgroup $S$ of $G\times S$. The
assumption on $G$ implies that every bi-$\Delta(S)$ invariant
generalized function on $G\times S$ is $\sigma\times
\sigma$-invariant. Then the usual Gelfand-Kazhdan criterion (c.f.
\cite[Theorem 2.3]{SZ08}) implies that
\begin{equation}\label{multiplication}
\dim \Hom_S(\pi_G \otimes \pi_S^\vee,\C)\cdot \dim \Hom_S(\pi_G^\vee \otimes\pi_S,\C)\leq
1.
\end{equation}
Here and henceforth, ``$\,^\vee$" stands for the contragredient of
an admissible smooth representation.
Denote by $\sigma'$ the automorphism $g\mapsto \sigma(g^{-1})$. By
considering characters of irreducible representations (which are
conjugation invariant generalized functions on the groups), the
assumption implies that
\[
\pi_G^\vee\cong \pi_G^{\sigma'}\quad\textrm{and}\quad \pi_S\cong
(\pi_S^\vee)^{\sigma'}.
\]
Here $\pi_G^{\sigma'}$ is the representation of $G$ which has the
same underlying space as that of $\pi_G$, and whose action is given
by $g\mapsto \pi_G(\sigma'(g))$. The representation
$(\pi_S^\vee)^{\sigma'}$ is defined similarly. Therefore the two
factors in \eqref{multiplication} are equal to each other, and
consequently,
\[
\dim \Hom_S(\pi_G \otimes \pi_S^\vee,\C)\leq 1,
\]
or the same,
\[
\dim \Hom_S(\pi_G, \pi_S)\leq 1.
\]
\end{proof}
\vsp
To prove Theorem B, we assume that $\epsilon=-1$, and
\[
(A,\tau)=\left\{
\begin{array}{l}
(\rk\times \rk, \textrm{the nontrivial
automophism}),\\
(\textrm{a quadratic field extension of $\rk$}, \textrm{the nontrivial
automophism}),\, \textrm{or}\\
(\rk, \textrm{the trivial
automophism}).
\end{array}
\right.
\]
Then $\oU(E)$ is $\GL(n)$, $\oU(n)$, or $\Sp(2n)$, respectively,
with $2n=\dim_\rk (E)$. Write $E_\rk:=E$, viewed as a symplectic
$\rk$-vector space under the form
\[
\la u,v\ra_{E_\rk}:=\tr_{A/\rk}(\la u,v\ra_E).
\]
Denote by $\widetilde{\Sp}(E_\rk)$ the metaplectic cover of the
symplectic group $\Sp(E_\rk)$. It induces a double cover
$\widetilde{\oU}(E)$ of $\oU(E)\subset \Sp(E_\rk)$. For any
non-trivial character $\psi$ of $\rk$, denote by $\omega_\psi$ the
corresponding smooth oscillator representation of
\[
\widetilde{\Sp}(E_\rk)\ltimes \oH(E_\rk).
\]
Up to isomorphism, this is the only genuine smooth representation
which, as a representation of $\oH(E_\rk)$, is irreducible and has
central character $\psi$.
Now it suffices to show the following lemma. This is actually known
(see \cite[Page 222]{AP06}). We provide a proof for the sake of
completeness.
\begin{lemt}
With the notation as above, for every genuine irreducible admissible
smooth representation $\pi_{\widetilde{\oU}}$ of
$\widetilde{\oU}(E)$, the tensor product
$\pi_{\widetilde{\oU}}\otimes \omega_\psi$ is an irreducible
admissible smooth representation of $\oU(E)\ltimes \oH(E_\rk)$.
\end{lemt}
\begin{proof}
The smoothness and admissibility are clear. We prove that
$\pi_{\widetilde{\oU}}\otimes \omega_\psi$ is irreducible as a
smooth representation of $\oU(E)\ltimes \oH(E_\rk)$. The space
\[
\oH_0:=\Hom_{\oH(E_\rk)}(\omega_\psi, \pi_{\widetilde{\oU}}\otimes
\omega_\psi)
\]
is a smooth representation of $\widetilde{\oU}(E)$ under the action
\[
(\tilde{g}.\phi)(v):=g.(\phi(\tilde{g}^{-1}.v)),
\]
where
\[
\quad
\tilde{g}\in\widetilde{\oU}(E), \,\phi\in \oH_0, \, v\in \omega_\psi,
\]
and $g$ is the image of $\tilde{g}$ under the quotient map
$\widetilde{\oU}(E)\rightarrow \oU(E)$. Let $\pi_{\oJ}$ be a nonzero
$\oU(E)\ltimes \oH(E_\rk)$-subrepresentation of
$\pi_{\widetilde{\oU}}\otimes
\omega_\psi$, then
\[
\Hom_{\oH(E_\rk)}(\omega_\psi, \pi_{\oJ})
\]
is a nonzero $\widetilde{\oU}(E)$-subrepresentation of $\oH_0$.
Since the linear map
\[
\pi_{\widetilde{\oU}} \rightarrow \oH_0,\quad v\mapsto v\otimes (\,\cdot\,)
\]
is bijective and $\widetilde{\oU}(E)$-intertwining, $\oH_0$ is
irreducible. Therefore
\[
\Hom_{\oH(E)}(\omega_\psi, \pi_{\oJ})=\oH_0,
\]
and consequently, $\pi_{\oJ}=\pi_{\widetilde{\oU}}\otimes
\omega_\psi$.
\end{proof}
\vsp \vsp
\appendix
\section{An uncertainty theorem for distributions with supports}
Let $\rk$ be a non-archimedean local field of characteristic zero.
Fix a non-trivial character $\psi$ of $\rk$. Let $E$ and $F$ be two
finite-dimensional $\rk$-vector spaces which are dual to each other,
i.e., a non-degenerate bilinear map
\[
\la \,,\,\ra: E\times F\rightarrow \rk
\]
is given. The Fourier transform
\[
\begin{array}{rcl}
\oD_0^\infty(F)&\rightarrow &\con_0^\infty(E)\\
\omega&\mapsto &\hat{\omega}
\end{array}
\]
is the linear isomorphism given by
\[
\hat{\omega}(x):=\int_{F} \psi(\la x,y\ra)\,
d\omega(y),\quad x\in E.
\]
For every $T\in \oD^{-\infty}(E)$, its Fourier transform $\widehat
T\in \con^{-\infty}(F)$ is given by
\[
\widehat{T}(\omega):=T(\hat{\omega}), \quad \omega\in \oD_0^\infty(F).
\]
For every subset $X$ of $E$, a point $x\in X$ is said to be regular
if there is an open neighborhood $U$ of $x$ in $E$ such that $U\cap
X$ is a closed locally analytic submanifold of $U$. In this case,
the tangent space $\oT_x(X)\subset E$ is defined as usual. We define
the conormal space to be
\[
\operatorname{N}^*_x(X):=\{v\in F\mid \la u,v\ra=0, \quad u\in \oT_x(X)\}.
\]
The uncertainty principle says that a distribution and its Fourier
transform can not be simultaneously arbitrarily concentrated. The
purpose of this appendix is to prove the following theorem, which is
a form of the uncertainty principle.
\begin{thm}\label{uncert}
Let $x$ be a regular point in a close subset $X$ of $E$. Let
$f:F\rightarrow \rk$ be a polynomial function of degree $d\geq 1$.
Let $T\in \oD^{-\infty}(E)$ be a distribution supported in $X$, with
its Fourier transform $\widehat T\in \con^{-\infty}(F)$ supported in
the zero locus of $f$. If $f_d$ take nonzero values at some points
of $\operatorname{N}^*_x(X)$, then $T$ vanishes on some open
neighborhood of $x$ in $E$, where $f_d$ is the homogeneous component
of $f$ of degree $d$.
\end{thm}
\noindent {\bf Remark}: The archimedean analog of Theorem
\ref{uncert} also holds. This is a direct consequence of \cite[Lemma
2.2]{JSZ}.
\vsp
Fix a non-archimedean multiplicative norm
\[
\abs{\,\cdot\,}_\rk:\rk\rightarrow [0,+\infty)
\]
which defines the topology of $\rk$. Also fix a non-archimedean norm
(multiplicative with respect to $\abs{\,\cdot\,}_\rk$)
\[
\abs{\,\cdot\,}_{F}:F\rightarrow [0,+\infty),
\]
which automatically defines the topology of $F$.
Let $f$ and $f_d$ be as in Theorem \ref{uncert}, and denote by
$Z_f\subset F$ the zero locus of $f$. Write $f_0:=f-f_d$, which is a
polynomial function of degree $\leq d-1$. Then
\begin{equation}\label{od}
\abs{f_0(y)}_\rk =o(\abs{y}_{F}^d), \quad
\textrm{as }\abs{y}_F\rightarrow +\infty.
\end{equation}
\begin{lemt}\label{intemp}
Let $V_1$ and $V_2$ be two compact open subsets of $F$. If $f_d$ has
no zero in $V_2$, then
\[
(Z_{f}+V_1)\cap \lambda V_2=\emptyset
\]
for all $\lambda\in \rk^\times$ with $\abs{\lambda}_\rk$
sufficiently large.
\end{lemt}
\begin{proof}
Take a positive number $c$ so that
\begin{equation}\label{e1}
\abs{f_d(y)}_\rk\geq c
\abs{y}_{F}^d,\quad\textrm{for all }y\in \rk^\times
V_2.
\end{equation}
It is easy to see that
\begin{equation}\label{e2}
\max_{v\in V_1}{\abs{f_d(y+v)-f_d(y)}}_\rk=o(\abs{y}_{F}^d), \quad
\textrm{as }\abs{y}_F\rightarrow +\infty.
\end{equation}
If $y\in Z_f$, then
\begin{eqnarray}\label{e3}
&&\phantom{=}\abs{f_d(y+v)}_\rk\\ \nonumber
&&\leq \max\{\,\abs{f_d(y)}_\rk,\,\abs{f_d(y+v)-f_d(y)}_\rk\,\}\\ \nonumber
&&=\max\{\,\abs{f_0(y)}_\rk,\,\abs{f_d(y+v)-f_d(y)}_\rk\,\}.
\end{eqnarray}
The inequalities (\ref{od}), (\ref{e2}) and (\ref{e3}) implies that
\begin{equation}\label{e4}
\abs{f_d(y)}_\rk=o(\abs{y}_{F}^d), \quad
\textrm{as }y\in Z_{f}+V_1,\, \textrm{ and } \abs{y}_F\rightarrow +\infty.
\end{equation}
The lemma then follows by comparing (\ref{e1}) and (\ref{e4}).
\end{proof}
Recall the following
\begin{dfnt} (c.f., \cite[Section 2]{He85})
A distribution $T\in \oD^{-\infty}(E)$ is said to be smooth at a
point $(x,y)\in E\times F$ if there is a compact open neighborhood
$U$ of $x$, and a compact open neighborhood $V$ of $y$ such that the
Fourier transform $\widehat{1_U T}$ vanishes on $\lambda V$ for all
$\lambda\in \rk^\times$ with $\abs{\lambda}_\rk$ sufficiently large.
Here $1_U$ stands for the characteristic function of $U$. The wave
front set of $T$ at $x\in E$ is defined to be
\[
\mathrm{WF}_x(T):=\{y\in F\mid T\textrm{ is not smooth at }
(x,y)\}.
\]
\end{dfnt}
Clearly, the wave front set $\mathrm{WF}_x(T)$ is closed in $F$ and
is stable under multiplications by $\rk^\times$.
\begin{lemt}\label{wavef}
If the Fourier transform $\widehat T$ of a distribution $T\in
\oD^{-\infty}(E)$ is supported in $Z_f$, then for every $x\in E$,
the wave front set $\mathrm{WF}_x(T)$ is contained in the zero locus
of $f_d$.
\end{lemt}
\begin{proof}
Let $y\in F$ be a vector so that $f_d(y)\neq 0$. We need to show
that $T$ is smooth at $(x,y)$. Take an arbitrary compact open
neighborhood $U$ of $x$, and an arbitrary compact open neighborhood
$V$ of $y$ so that $f_d$ has no zero in $V$. We claim that
$\widehat{1_U T}$ vanishes on $\lambda V$ for all $\lambda\in
\rk^\times$ with $\abs{\lambda}_\rk$ sufficiently large. The lemma
is a consequence of this claim.
Note that $\widehat{1_U T}$ is a finite linear combination of
generalized functions of the form
\[
(1_{V_1}\,dy)*\widehat{T}, \quad \textrm{$V_1$ is a
compact open subset of $F$}.
\]
Here $dy$ is a fixed Haar measure on $F$. The support of the
convolution $(1_{V_1}\,dy)*\widehat{T}$ is contained in $Z_f+V_1$.
Therefore the claim follows from Lemma \ref{intemp}.
\end{proof}
\begin{lemt}\label{trans}
If a distribution $T\in \oD^{-\infty}(E)$ is supported in a closed
subset $X$ of $E$, and $x\in X$ is a regular point, then the wave
front set $\mathrm{WF}_x(T)$ is invariant under translations by
elements of $\mathrm{N}_x^*(X)\subset F$.
\end{lemt}
\begin{proof}
This is proved in \cite[Therorem 4.1.2]{Ai}. We indicate the main
steps.
Step 1. When we replace a distribution by a translation of it, the
wave front set does not change. Therefore we may assume that $x=0$.
Step 2. Let $\varphi: E\rightarrow E$ be a locally analytic
diffeomorphism which sends $0$ to $0$ and induces the identity map
on the tangent space at $0$. When we replace $T$ by its pushing
forward $\varphi_*(T)$, the wave front set $\operatorname{WF}_0(T)$
does not change. Therefore we may assume that $X\cap U=E_0\cap U$,
for some subspace $E_0$ of $E$, and some open neighborhood $U$ of
$0$.
Step 3. When we replace $T$ by a distribution which coincides with
$T$ on an open neighborhood of $0$, the wave front set
$\operatorname{WF}_0(T)$ does not change. Therefore we may assume
that $X=E_0$.
Step 4. Assume that $T$ is supported in $E_0$. Then $\widehat T$ is
invariant under translations by
\[
\mathrm{N}^*_x(X)=E_0^\perp:=\{v\in F\mid \la u,v\ra=0, \quad u\in E_0\},
\]
which implies that the same holds for $\operatorname{WF}_0(T)$.
\end{proof}
Now we are ready to prove Theorem \ref{uncert}. It is clear that $T$
vanishes on some open neighborhood of $x$ if and only if $0\notin
\operatorname{WF}_x(T)$. If $0\in \operatorname{WF}_x(T)$, then
Lemma \ref{trans} implies that $\mathrm{N}_x^*(X)\subset
\operatorname{WF}_x(T)$. Now Lemma \ref{wavef} further implies that
$\mathrm{N}_x^*(X)$ is contained in the zero locus of $f_d$, which
contradicts the assumption of the theorem.
| 111,040
|
Search
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| 39,815
|
TITLE: How do I integrate this using the Fourier transform?
QUESTION [3 upvotes]: The integral is:$$I=\displaystyle\int_{0}^{\infty}e^{-ax^2}\cos(bx)\,dx$$
What I tried:
I was trying it with Fourier transform, so I have here:
$$f(t)=e^{-at^2}\cos(bt)$$
I know that: $$\mathcal{F}\left(e^{-ax^2} \right)=\dfrac{1}{\sqrt{2a}}e^{-\frac{\omega^2}{4a}}$$
Then used frequency modulation property: $$\cos(bt)=\dfrac{1}{2}\left(e^{-jbt}+e^{jbt}\right)$$
That gives me:
$$\mathcal{F}\left[f(t)\right]=\dfrac{1}{2\sqrt{2a}}\left[e^{\dfrac{-(\omega+b)^2}{4a}}+e^{\dfrac{-(\omega-b)^2}{4a}}\right]$$
For the integral I have to evaluate at $\omega=0$:
$$\mathcal{F}\bigg{|}_{\omega=0}=\dfrac{1}{\sqrt{2a}}\left[e^{-\frac{b^2}{4a}}\right]$$
This is what i got but the answer is:
$$I=\sqrt{\dfrac{\pi}{4a}}\left[e^{-\frac{b^2}{4a}}\right]$$
I don't know from where this $\sqrt{\dfrac{\pi}{2}}$ term is coming?
REPLY [2 votes]: A differentiation trick works well for this problem.
\begin{align}
F(b) & = \int_{0}^{\infty}e^{-ax^2}\cos(bx)dx \\
F'(b) & = -\int_{0}^{\infty}e^{-ax^2}x\sin(bx)dx \\
& = \frac{1}{2a}\int_{0}^{\infty}\left(\frac{d}{dx}e^{-ax^2}\right)\sin(bx)dx \\
& = -\frac{1}{2a}\int_{0}^{\infty}e^{-ax^2}\frac{d}{dx}\sin(bx)dx \\
& = -\frac{b}{2a}\int_{0}^{\infty}e^{-ax^2}\cos(bx)dx \\
& = -\frac{b}{2a}F(b).
\end{align}
Therefore, there is a constant $C$ such that
$$
F(b) = Ce^{-b^2/4a}
$$
The constant is $F(0)=C$, which is
\begin{align}
F(0) & =\int_{0}^{\infty}e^{-ax^2}dx \\
& = \frac{1}{2}\int_{-\infty}^{\infty}e^{-ax^2}dx \\
& = \frac{1}{2}\left[\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-a(x^2+y^2)}dxdy\right]^{1/2} \\
& = \frac{1}{2}\left[\int_{0}^{2\pi}\int_{0}^{\infty}e^{-ar^2}rdrd\theta\right]^{1/2} \\
& = \frac{\sqrt{2\pi}}{2}\left[\frac{1}{2a}\int_{0}^{\infty}e^{-ar^2}(2ar)dr\right]^{1/2} \\
& = \frac{\sqrt{2\pi}}{2}\frac{1}{\sqrt{2a}}
\end{align}
Therefore,
$$
\int_{0}^{\infty}e^{-ax^2}\cos(bx)dx = F(b)= F(0)e^{-b^2/4a}=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-b^2/4a}.
$$
| 37,300
|
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Emily Rasinski Washington DC photographer
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| 89,988
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TITLE: Why is the $\mu_n$ representation rational?
QUESTION [0 upvotes]: In their paper "On the irregularity of cyclic coverings of algebraic surfaces" by F. Catanese and C. Ciliberto, the authors consider the following situation.
Let $A = V/\Lambda$ be a $g$-dimensional complex abelian variety and let $\mathbb{G}_n$ be a cyclic group of order $n$ consisting of automorphisms of $A$. Then, of course, we have a group action of $\mathbb{G}_n$ on $A$. Denote by $\mu_n$ the group of $n$-th roots of unity, we get a representation $\rho \colon \mu_n \to GL(\Lambda \otimes \mathbb{C})$, so we can decompose $\Lambda \otimes \mathbb{C} = \bigoplus H_\chi$, where for a character $\chi \colon \mu_n \to \mathbb{C}^*$, the $\chi$-eigenspace is denoted $H_\chi$ (i.e. $H_\chi = \{ v \in \Lambda \otimes \mathbb{C}| \rho(g)(v) = \chi(g)\cdot v, \, \forall g \in \mu_n\}$).
So far so good, now for the part which I do not understand. The authors write:
Since the $\mu_n$-representation is rational, if $\chi$ and $\chi'$ have the same order then $\dim(H_\chi) = \dim(H_{\chi'})$.
First of all, I don't really know what a rational representation is. According to Serre's book "Linear Representation of Finite Groups" a rational representation of should be a representation over an algebraic closure of $\mathbb{Q}$ which is isomorphic to a representation over $\mathbb{Q}$ (of the same group, of course). Is that the right definition?
If so, my question is: Why is the $\mu_n$-representation rational and why does the conclusion above hold? I reckon there are some general theorems behind this...
I would also be satisfied with a good reference, where these topics are treated nicely.
REPLY [2 votes]: Automorphisms of $A$ are automorphisms of its universal covering space $V$ that preserve the lattice $\Lambda$. So the action of $\mathbf{G}_n$ on $\Lambda$ extended by scalar is defined over $\mathbf{Z}$, hence it is a rational representation.
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\begin{document}
\begin{center}
{\large{\bf Periodic solutions for nonlinear hyperbolic evolution systems}}
Aleksander \'{C}wiszewski\footnote{Corresponding author
\noindent {\bf 2000 Mathematical Subject Classification}: 47J35, 47J15, 37L05 \\
{\bf Key words}: semigroup, evolution system, evolution equation, topological degree, periodic solution}, Piotr Kokocki\\
{\em Faculty of Mathematics and Computer Science\\
Nicolaus Copernicus University\\
ul. Chopina 12/18, 87-100 Toru\'n, Poland }\\
\end{center}
\begin{abstract}
We shall deal with the periodic problem for
nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation along trajectories operator associated to the nonlinear evolution system, expressed in terms of the topological degree. The abstract results shall be applied to the damped hyperbolic partial differential equation.
\end{abstract}
\section{Introduction}
We shall be concerned with $T$-periodic solutions of the nonlinear evolution equation
$$
\dot u (t) = A(t)u(t) + F(t,u(t)), \quad t\in [0,T] \leqno{(P)}
$$
where $T>0$ is fixed, $\{A(t)\}_{t\in [0,T]}$ is a family of linear operators on a separable Banach space $E$ satisfying the so-called hyperbolic conditions and $F:[0,T]\times E\to E$ is
a $T$-periodic in time continuous map satisfying the local Lipschitz condition with respect to the second variable and having sublinear growth,
uniformly with respect to time. Moreover, it is also assumed that there is $\omega>0$ such that
$$
\|S_{A(t)}(s)\|\leq e^{-\omega s} \quad \mbox{ for \ } t\in [0,T] \mbox{ and } s\geq 0,
$$
where $S_{A(t)}$ stands for the $C_0$ semigroup generated by the operator $A(t)$,
and that there is $k\in [0,\omega)$ such that
$$
\beta (F([0,T]\times Q)) \leq k \beta (Q) \quad \mbox{ for any bounded } Q\subset E,
$$
where $\beta$ denotes the Hausdorff measure of noncompactness.
Under these assumptions, the translation along trajectories operators
$\Phi_t:E\to E$, $t\in [0,T]$, given by $\Phi_t(x):=u(t;x)$, $x\in E$, where $u(\cdot;x)$ stands for the solution of $(P)$ with the initial condition $u(0)=x$, are well-defined and continuous. Moreover, for $t\in [0,T]$, one has $\beta(\Phi_t(Q))\leq e^{-(\omega-k)t} \beta (Q)$ for any bounded $Q\subset E$. This enables us to consider the topological degree of $I-\Phi_t$ and search $T$-periodic solutions corresponding to the fixed points of $\Phi_T$.
Our approach is based on the averaging idea, which says that if increasing the frequency in $(P)$, i.e. considering equations $\dot u(t) = A(t/\lma)u(t) +F(t/\lma,u(t))$ with $\lma\to 0^+$,
then their solutions converge to solutions of the averaged equation
$$
\dot u(t) = \widehat A u(t) + \widehat F (u(t))
$$
where $\widehat A + \widehat F$ is the time averaged right-hand
side of $(P)$ (the precise meaning is explained in the sequel --
see Theorem \ref{18092008-1450}). Therefore, after rescaling time,
we study $T$-periodic solutions of equations
$$
\dot u(t) = \lma A(t) u(t) + \lma F(t,u(t)), \quad t\in [0,T] \leqno{(P_\lma)}
$$
by means of the associated translation along trajectories operator $\Phi_{T}^{(\lma)}$.
We prove that, for small $\lma>0$, the topological degree of $I-\Phi_{T}^{(\lma)}$,
with respect to a proper open bounded $U\subset E$, is equal to the topological degree
$\Deg(\widehat A + \widehat F, U)$ -- see Theorem \ref{18092008-2220}.
This formula will imply the existence of $T$-periodic solutions provided $\Deg(\widehat A+\widehat F,U)\neq 0$.
In some natural cases the geometry of the right-hand side allows concluding the
nontriviality of the topological degree and get some a priori bounds estimates,
which provide effective criteria for the existence of $T$-periodic solutions -- see Theorem \ref{24102008-0927}.\\
\indent The abstract hyperbolic type linear or semi-linear systems
and their applications to partial differential equations were
developed by Kato (see e.g. \cite{Kato}) and Tanabe (see e.g.
\cite{Tanabe} and references therein). Some existence results for
initial value problems associated with nonlinear perturbations of
evolution systems are standard and can be found e.g. in
\cite{Pazy}. As we need the continuity of translation along
trajectories and some related homotopies, we have to verify the
continuity and compactness of solutions as functions of initial
data and parameters. Moreover, due to some infinitesimal passages
related to the averaging method used in the paper, a parameterized
version of the representation formula must be derived. As a tool
we use the topological degree for so called $k$-set contraction
vector fields due to Sadovskii (see \cite{Akhmerov-etal} and
references therein) and Nussbaum (see \cite{Nussbaum}). The
topological degree for maps of the form $A+F$, where an invertible
operator $A$ generates a $C_0$ semigroup and $F$ is a continuous
$k$-set contraction is obtained as the degree of vector field
$I+A^{-1} F$, which is a standard -- see e.g. \cite{Kartsatos} and
some comments on the specific properties that we use are in
\cite{Cwiszewski-Kokocki}. Averaging methods combined with
topological degree and fixed point index were used in
\cite{Furi-Pera} to find periodic solutions for time dependent
vector fields on finite dimensional manifolds. Analogues of this
method, in the case of infinite dimensional Banach spaces was
stated in \cite{Cwiszewski-1}, \cite{Cwiszewski-2}, where periodic
solutions for the equations of the form $\dot u(t) = Au(t) +
F(t,u(t))$, with $A$ generating compact semigroups, were derived.
Also averaging methods together with Rybakowski's version of the
Conley index were used in \cite{Prizzi}, where the existence of
so-called recurrent solutions is studied for nonautonomous
parabolic equations. Periodic solutions for nonautonomous damped
hyperbolic equations has been also thoroughly studied in
\cite{Ortega1} and \cite{Ortega}. The present paper is a
continuation of \cite{Cwiszewski-Kokocki} where the periodic
problem is considered in the case where $A$ generates a
$C_0$-semigroup
of strict contractions and $F$ is a perturbation, i.e. the situation applicable to damped hyperbolic equations.\\
\indent The paper is organized as follows. In Section 2, we prove
a parameterized version of the representation theorem, which is a
useful framework for limit passages concerned with evolution
systems at all and also those considered in the next sections.
Section 3 is devoted to the properties of the translation along
trajectories operator such as the existence, continuity with
respect to the parameter and compactness. In Section 4 we deal
with the main result of the paper, that is the averaging method
for periodic solutions of $(P)$. Section 5 provides an example of
application to second order hyperbolic partial differential
equations.
\section{General Representation Theorem}
We start with a parameterized version of Theorem 3.5 from \cite[Ch. 3]{Pazy}.
\begin{Th}\label{29082008-1821}
Let $L:(0,+\infty)\times [0,1]\to {\cal L} (E,E)$, where $E$ is a Banach space, be a mapping such that
\begin{equation}\label{29082008-1144}
\|L(\lma,\mu)\|\leq 1\quad\mbox{ for \ } \ \lma> 0, \ \mu\in [0,1]
\end{equation}
and there is a dense subspace $V$ of $E$ such that
\begin{equation}\label{29082008-1145}
\lim_{\lma\to 0^+,\mu \to \mu_0} \lma^{-1}(L(\lambda,\mu)v - v) = A^{(\mu_0)}v \quad\mbox{ for \ } \ v\in V, \ \mu_0 \in [0,1],
\end{equation}
where, for each $\mu\in [0,1]$, $A^{(\mu)}:D(A^{(\mu)})\to E$ is a linear operator
such that $V\subset D(A^{(\mu)})$ and $(a_\mu I - A^{(\mu)})V$ is dense in $E$ for some $a_{\mu} >0$.\\
Then\\
\pari{(i)}{ for any $\mu\in [0,1]$, the operator $A^{(\mu)}$ is closable and its closure
$\overline{A^{(\mu)}}$
generates a $C_0$ semigroup of contractions $\{S_{\overline{A^{(\mu)}}}(t):E\to E\}_{t\geq 0}$;}\\[2mm]
\pari{(ii)}{for any sequence of a positive integers $(k_n)$ and a sequence
$(\lambda_n)$ in $(0,+\infty)$ such that $k_n\to\infty$, $k_n\lambda_n\to t$ as $n\to +\infty$, for some $t\geq 0$,
and any $(\mu_n)$ in $[0,1]$ with $\mu_n\to \mu_0$,
\begin{equation}\label{29082008-1157}
\lim_{n\to\infty}L(\lambda_n,\mu_n)^{k_n}x = S_{\overline{A^{(\mu_0)}}}\, (t)x \quad \mbox{ for each } x\in E;
\end{equation}}
\pari{(iii)}{for sequences $(k_n)$, $(\lma_n)$, $(\mu_n)$ and $t\geq 0$ as in {\em (ii)}
\begin{equation}\label{29082008-2233}
\lim_{n\to\infty}\lambda_n(I+L(\lambda_n, \mu_n)+L( \lambda_n,
\mu_n)^2+\ldots +L( \lambda_n, \mu_n)^{k_n-1})x\to \int_0^t
S_{\overline{A^{(\mu_0)}}}\,(\tau) x \d \tau
\end{equation}
for each $x\in E$.}
\end{Th}
In the proof we shall use the following two Lemmata.
\begin{Lem}\label{29082008-1833}{\em(see \cite[Ch. 3, Theorem 4.5]{Pazy})}
If $(A_n)_{n\ge 1}$ is a sequence of operators generating $C_0$
semigroups $\{S_{A_n}(t)\}_{t\geq 0}$, $n\geq 1$, and $A:V\to E$ is a linear operator, where $V$ is a dense subspace of $E$, with the following properties\\[1mm]
\pari{(a)}{ there are $M\geq 1$ and $\omega\in\R$ such that
$\|S_{A_n}(t)\|\le Me^{\omega t}$ for any $n\geq 1$;} \\[1mm]
\pari{(b)}{ for every $v\in V$, $A_n v \to Av$ as $n\to\infty$;}\\[1mm]
\pari{(c)}{ there exists $\mu_0 >\omega$ such that
$(\mu_0 I -A)V$ is dense in $E$,} \\[1mm]
then the closure $\overline A$ of $A$ generates a $C_0$ semigroup $\{S_{\cl A}(t)\}_{t\ge 0}$ such that $$\|S_{\cl A}(t)\|\le Me^{\omega t} \quad\mbox{ for \ } t\geq 0 $$
and
$$\lim_{n\to\infty} S_{A_n}(t)x = S_{\cl A}(t)x \quad\mbox{ for \ }t\geq 0, \ x\in E.$$
The above convergence is uniform with respect to $t$ from bounded intervals.
\end{Lem}
\begin{Lem}\label{29082008-2156}{\em (see \cite[Ch. 3, Corollary 5.2]{Pazy})}
If $T\in {\cal L}(E,E)$ and $\|T\|\le 1$, then for any integer $n\geq 0$
and $x\in E$
$$
\|e^{(T - I)n}x - T^n x\|\leq \sqrt{n}\|x - Tx\|.
$$
\end{Lem}
\noindent{\bf Proof of Theorem \ref{29082008-1821}.}
(i) Define $A_{\lma}^{(\mu)}:E\to E$ by $A_{\lambda}^{(\mu)}:=\lambda^{-1}(L(\lambda,\mu)-I)$
and for any $\lma>0$, $\mu\in [0,1]$
and $t\geq 0$, put $S_{\lambda}^{(\mu)} (t):=\exp(tA_{\lma}^{(\mu)})$. Clearly, in view of (\ref{29082008-1144}), for any $\lma>0$ and $\mu\in [0,1]$
\begin{equation}\label{16072009-1816}
\|S_{\lma}^{(\mu)}(t)\|\le e^{-t/\lma} \sum_{k=0}^\infty(t/\lma)^k\frac{\|L(\lma,\mu)^k\|}{k!}\leq
e^{-t/\lma}\sum_{k=0}^\infty\frac{(t/\lma)^k}{k!} = 1.
\end{equation}
If $\lma_n\to 0^+$ and $\mu_n\to \mu_0$, then due to (\ref{29082008-1145}),
$$
\lim_{n\to \infty} A_{\lma_n}^{(\mu_n)} v = A^{(\mu_0)}v \quad\mbox{ for \ }
v\in V.
$$
By the assumption, there is $a_{\mu_0}>0$ such that $(a_{\mu_0} I - A^{(\mu_0)})V$ is dense in $E$ and, in view of Lemma \ref{29082008-1833},
we infer that $A^{(\mu_0)}$ is closable and
its closure $\overline{A^{(\mu_0)}}$ generates $C_0$ a semigroup $\left\{ S_{\overline{A^{(\mu_0)}}}\, (t)\right\}_{t\geq 0}$ of bounded linear operators on $E$, such that $\|S_{\overline{A^{(\mu_0)}}}\, (t)\|\leq 1$ for any $t\geq 0$ and furthermore
\begin{equation}\label{29082008-2222}
S_{\lma_n}^{(\mu_n)} (t) x\to S_{\overline{A^{(\mu_0)}}}\, (t)x \quad\mbox{ for any $x\in E$}, \mbox{ as } n\to \infty
\end{equation}
uniformly for $t$ from bounded subintervals of $[0,+\infty)$.\\
\indent (ii) Let the sequence of a positive integers $(k_n)$, the
sequence $(\lambda_n)$ in $(0,+\infty)$ and $(\mu_n)$ in $[0,1]$
be such that $k_n\to\infty$, $k_n\lambda_n\to t$ as $n\to
+\infty$, for some $t\geq 0$, and $\mu_n\to \mu_0$ as $n\to
+\infty$. Then for any $v\in V$ and $n\geq 1$
\begin{align}\label{29092008-2220}
& \| L(\lambda_n,\mu_n)^{k_n}v - S_{\overline{A^{(\mu_0)}}}\, (t)v \| \leq
\| L(\lambda_{n}, \mu_{n})^{k_n}v - S_{\lma_n}^{(\mu_n)}(\lma_n k_n)v\| \\ \nonumber
& \quad + \|S_{\lma_n}^{(\mu_n)}(\lma_n k_n)v-S_{\overline{A^{(\mu_0)}}}\, (\lma_n k_n)v\|
+\|S_{\overline{A^{(\mu_0)}}}\, (\lma_n k_n)v-S_{\overline{A^{(\mu_0)}}}\, (t) v\|\nonumber.
\end{align}
By Lemma \ref{29082008-2156} and (\ref{29082008-1145}), for any $v\in V$
\begin{align}\nonumber
\|L(\lambda_n,\mu_n)^{k_n}v - S_{\lma_n}^{(\mu_n)} (\lma_n k_n)v \| & =
\| e^{k_n(L(\lambda_n,\mu_n)-I)}v-L(\lambda_n,\mu_n)^{k_n}v\| \\ \nonumber
& \le \sqrt{k_n}\|v - L(\lambda_n,\mu_n)v\| \\
& = \sqrt{\lma_n} \sqrt{k_n\lma_n} \|\lma_n^{-1}(v-L(\lambda_n, \mu_n)v)\|
\to 0 \quad\mbox{ as } n\to\infty.
\end{align}
Consequently, by (\ref{29082008-1144}), (\ref{16072009-1816}) and the density of $V$ in $E$, we obtain that
\begin{equation}\label{16082008-1832}
\|L(\lambda_n,\mu_n)^{k_n}x - S_{\lma_n}^{(\mu_n)} (\lma_n k_n)x \| \to 0 \quad \mbox{ for each } x\in E, \mbox{ as } n\to \infty .
\end{equation}
Furthermore, in view of the uniform convergence on bounded intervals in (\ref{29082008-2222}), one has
$$
\|S_{\lma_n}^{(\mu_n)}(\mu_n k_n)x-S_{\overline{A^{(\mu_0)}}}\, (\lma_n k_n)x \| \to 0 \quad \mbox{ for any } x\in E, \mbox{ as } n\to \infty.
$$
This, together with (\ref{29092008-2220}), (\ref{16082008-1832}) and the continuity of the semigroup $S_{\overline{A^{(\mu_0)}}}$, gives (\ref{29082008-1157}). \\
\indent (iii) Take any $v\in V$ and observe that
$$
\left\|\lma_n \sum_{k=0}^{k_n-1} L(\lambda_n,\mu_n)^k v -
\int_{0}^{t} S_{\overline{A^{(\mu_0)}}}\,(\tau)v \d \tau
\right\|\leq I_{n}^{(1)} + I_{n}^{(2)} + I_{n}^{(3)},
$$
where
\begin{align*}
I_{n}^{(1)} & :=\left\|\lma_n \sum_{k=0}^{k_n-1} L(\lambda_n,\mu_n)^k v -
\lma_n \sum_{k=0}^{k_n-1} S_{\lma_n}^{(\mu_n)}(k\lma_n)v\right\|,\\
I_{n}^{(2)} & :=\left\| \lma_n \sum_{k=0}^{k_n-1} S_{\lma_n}^{(\mu_n)}(k\lma_n)v -
\lma_n \sum_{k=0}^{k_n-1} S_{\overline{A^{(\mu_0)}}}\,(k\lma_n)v \right\|,\\
I_{n}^{(3)} & :=\left\| \lma_n \sum_{k=0}^{k_n-1}
S_{\overline{A^{(\mu_0)}}}\,(k\lma_n)v - \int_{0}^{t}
S_{\overline{A^{(\mu_0)}}}\, (\tau)v \d \tau\right\|.
\end{align*}
First, in view of Lemma \ref{29082008-2156} and (\ref{29082008-1145}), one has
\begin{align}\label{29082008-2320}
I_{n}^{(1)} & \leq k_n\lma_n \max\{
\|L(\lambda_n,\mu_n)^k-e^{k(L(\lma_n, \mu_n) - I)}\| \mid k=1,\ldots, k_n-1\}\nonumber\\
& \leq k_n\lma_n \max\{\sqrt{k} \|v-L(\lma_n, \mu_n)v \| \mid k=1,\ldots, k_n-1\}\\
& \leq \sqrt{\lma_n} (k_n\lma_n)^{3/2} \|\lma_n^{-1}(v-L(\lma_n,\mu_n)v) \| \to 0 \quad\mbox{ as } n\to \infty\nonumber.
\end{align}
Furthermore, by the uniform convergence in (\ref{29082008-2222}) on the interval $[0,\overline t]$,
where $\overline t := \sup_{n\geq 1} k_n\lma_n$, we get
\begin{equation}\label{29082008-2321}
I_{n}^{(2)} \!\leq\! k_n\lma_n \max\{ \|S_{\lma_n}^{(\mu_n)}(\!k\lma_n\!)v \!-\! S_{\overline{A^{(\mu_0)}}}\,(\!k\lma_n\!)v\|\! \mid \!k=0,\ldots,\! k_n\!-\!1\}\!\to 0 \quad \mbox{ as } n\to\infty.
\end{equation}
It is also clear that $I_{n}^{(3)}\to 0$ as $n\to \infty$, which along with (\ref{29082008-2320}) and (\ref{29082008-2321}) implies that (\ref{29082008-2233}) is satisfied for $x\in V$.
Finally, since for any $n\geq 1$
$$
\left\|\lma_n\sum_{k=0}^{k_n-1} L(\lma_n,\mu_n)^k- \int_{0}^{t}
S_{\overline A^{(\mu_0)}}\,(\tau) \d\tau \right\|\leq k_n\lma_n +
t < C $$ for some constant $C>0$ independent of $n$ and since $V$
is dense in $E$, one has the required convergence for each $x\in
E$. \hfill $\square$
\section{Continuity and compactness properties for solution \\ operator}
A family $\{R(t,s)\}_{0\leq s\leq t\leq T}$, $T>0$ of bounded linear operators on a Banach space $E$ is called an {\em evolution system} provided $R(t,t)=I$ for each $t\in [0,T]$,
$R(t,s)=R(t,r)R(r,s)$ if only $0\leq s\leq t\leq T$ and for any $x\in E$, the map
$(t,s)\mapsto R(t,s)x$ is continuous. A family $\{R^{(\lma)}\}_{\lma\in [0,1]}$ of evolution systems is called {\em continuous} if, for any $x\in E$ and $(\lma_n)$ in $[0,1]$ with $\lma_n\to \lma$, $R^{(\lma_n)}(t,s)x\to R^{(\lma)}(t,s)x$ uniformly with respect to $t,s\in[0,T]$ with $s\leq t$.\\
\indent Evolution systems are naturally determined by time-dependent families of linear operators. Namely,
if $\{ A (t) \}_{t\in [0,T]}$ is a family of linear operators on a Banach space $E$ such that for any $s\in [0,T]$ and $x\in E$, the problem
$$\left\{
\begin{array}{l}
\dot u (t) = A(t)u(t), \quad t\in [s,T]\\
u(s)=x
\end{array}\right.
$$
admits (in some sense) a unique solution $u_{s,x}:[s,T]\to E$, then the corresponding evolution system $\{R(t,s)\}_{0\leq s\leq t\leq T}$ is given by $R(t,s)x:=u_{s,x} (t)$, for $t\in [s,T]$. A particular type of evolution systems -- the so-called {\em hyperbolic evolution systems}, will be discussed in details at the end of this section.
\begin{Prop}\label{30122008-1629}
Suppose that $\left\{R^{(\lma)}\right\}_{\lma\in [0,1]}$ is a continuous family of evolution systems and the operator $\Sigma:E\times L^1([0,T], E)\times [0,1] \to C([0,T],E)$ is given by
$$
\Sigma (x,w,\lma) (t):= R^{(\lma)}(t,0)x+ \int_{0}^{t} R^{(\lma)}(t,s) w(s) \d s.
$$
Then\\
\makebox[10mm][r]{\em (i)} \parbox[t]{138mm}{$\Sigma$ is continuous;}\\
\pari{(ii)}{if $K\subset E$ is relatively compact and $W\subset L^1([0,T],E)$ is such that there is $c\in L^1([0,T])$ with $\|w(t)\|\leq c(t)$ for any $w\in W$ and a.e. $t\in [0,T]$, then
$\Sigma (K\times W\times [0,1])$ is relatively compact if and only if the set
$\{ u (t) \mid u\in \Sigma (K\times W\times [0,1])\}$ is relatively compact for any $t\in [0,T]$.}
\end{Prop}
\begin{Rem}\label{07012009-1059} {\em
\indent (a) Under the above notation, if $t,t+h\in [0,T]$ with $h>0$, then
$$
\Sigma (x,w,\lma) (t+h) = R^{(\lma)}(t+h,t)\Sigma(x,w,\lma)(t) +
\int_{t}^{t+h} R^{(\lma)} (t+h,s)w(s)\ \d s,
$$
which follows directly from the definition of $\Sigma$ and the properties of evolution systems.\\
\indent (b) If $\left\{R^{(\lma)}\right\}_{\lma\in [0,1]}$ is a continuous family of evolution systems, then for any $x\in E$ the set $\{ R^{(\lma)} (t,s)x \ | \ 0\leq s\leq t\leq T, \lma\in [0,1]\}$ is bounded. Hence, in view of the uniform boundedness principle, there exists $M\geq 0$ such that
$$
\|R^{(\lma)}(t,s)\| \leq M \quad\mbox{ for any } t,s\in [0,T] \mbox{ with } s\leq t\mbox{ and } \lma\in [0,1].
$$}
\end{Rem}
{\bf Proof of Proposition \ref{30122008-1629}.} (i) Let $x_n\to x_0$ in $E$, $w_n\to w_0$ in $L^1([0,T],E)$ and $\lma_n\to \lma_0$.
Clearly, by Remark \ref{07012009-1059} one has
\begin{align}\label{07012009-1130} \nonumber
\|R^{(\lambda_n)}(t,0)x_n\! - \! R^{(\lambda_0)}(t,0)x_0 \|
& \leq \| R^{(\lambda_n)}(t,0)x_n\!-\!R^{(\lambda_n)}(t,0)x_0 \| + \\
& \hspace{26mm} + \| R^{(\lambda_n)}(t,0)x_0-R^{(\lambda_0)}(t,0)x_0\| \\
& \leq M\|x_n-x_0\| + \|R^{(\lambda_n)}(t,0)x_0 - R^{(\lambda_0)}(t,0)x_0 \|\nonumber
\end{align}
and hence, by the continuity of the family $\{R^{(\lma)}\}_{\lma\in [0,1]}$, we infer that $\|R^{(\lambda_n)}(t,0)x_n - R^{(\lambda_0)}(t,0)x_0
\| \to 0$ as $n\to +\infty$, uniformly with respect to
$t\in [0,T]$. In a similar manner
\begin{align} \nonumber
\left\| \int_{0}^{t}\!\! R^{(\lma_n)}(t,s) w_n(s) \d s \!-\!\!\int_{0}^{t}\!\! R^{(\lma_0)}(t,s) w_0(s) \d s\right\| \\ \nonumber
& \hspace{-30mm}\leq \int_{0}^{t}\!\! \|R^{(\lma_n)}(t,s) w_n(s) \!-\! R^{(\lma_0)}(t,s) w_0(s)\|\d s \\
& \hspace{-30mm}\leq \int_{0}^{t} \!\!\|R^{(\lma_n)}(t,s) w_n(s) \!-\! R^{(\lma_n)}(t,s) w_0(s)\| \d s \\ \nonumber
& \hspace{-30mm} \quad + \int_{0}^{t}\!\! \|R^{(\lma_n)}(t,s) w_0(s)\!-\! R^{(\lma_0)}(t,s) w_0(s)\| \d s\\ \nonumber
& \hspace{-30mm} \leq M \|w_n-w_0\|_{L^1([0,T],E)} + \int_{0}^{T} \varphi_n (s) \d s, \nonumber
\end{align}
where functions $\varphi_n:[0,T]\to \R$, $n\geq 1$, are given by
$$
\varphi_n (s) := \sup_{\sigma,\tau\in [0,T],\tau\geq \sigma} \| [ R^{(\lma_n)}(\tau,\sigma)- R^{(\lma_0)} (\tau,\sigma)] w_0(s) \|.
$$
It is easy to check that functions $\varphi_n$, $n\ge 1$ are
measurable and, by the continuity of $\{R^{(\lma)}\}_{\lma\in
[0,1]}$ and Remark \ref{07012009-1059}, we infer that, for a.e.
$s\in [0,T]$, $\varphi_n(s)\to 0$ as $n\to+\infty$. On the other
hand $0\leq \varphi_n (s) \leq 2M\|w_0 (s)\|$, for $s\in [0,T]$,
which in view of the Lebesgue dominated convergence theorem
gives $\int_{0}^{T} \varphi_n(s) \d s \to 0$ as $n\to + \infty$ and together with (\ref{07012009-1130}) proves (i).\\
\indent (ii) Suppose that the set $\{ u (t) \mid u\in \Sigma (K\times W\times [0,1])\}$ is relatively compact for any $t\in [0,T]$.
Take any $\eps>0$ and fix $t\in [0,T]$. Let $\delta > 0$ be such that
$\int\limits_{[t-\delta,t+\delta]\cap [0,T]} c(s) \d s <\eps/3 M$.
Suppose that $t\in [0,T)$. Since the set
$Q_t:=\overline{\{ \Sigma(x,w,\lma)(t)\mid x\in K, w\in W, \lma\in [0,1]\}}$
is compact, one may eventually decrease $\delta > 0$ so that
$$
\|R^{(\lma)}(t+h,t)z-z\|<\eps/2, \quad \mbox{ if $t+h \leq T$, $h\in[0,\delta)$, $\lambda\in[0,1]$, $z\in Q_t$.}
$$
Now take $h\in [0,\delta)$ such that $t+h\in [0,T]$. Then, denoting
$\Sigma:=\Sigma(x,w,\lma)$ for any $(x,w,\lma)\in K\times W\times [0,1]$, one has
\begin{align*}
\|\Sigma (t+h)-\Sigma(t)\| & \leq
\| \Sigma (t+h)- R^{(\lma)}(t+h,t)\Sigma (t)\|+\|R^{(\lma)}(t+h,t)\Sigma (t)-\Sigma(t)\|\\
& \leq \int_{t}^{t+h} \|R^{(\lma)}(t+h,s)w(s)\| \d s + \|R^{(\lma)}(t+h,t)\Sigma (t)-\Sigma(t)\|\\
& \leq M \int_{t}^{t+h} c(s) \d s + \eps/2 <\eps.
\end{align*}
If $t\in (0,T]$, then take any $\delta_1 \in (0, \min\{t,\delta\}]$.
Since the set
$$
Q_{t-\delta_1}:=\overline{\{ \Sigma(x,w,\lma)(t-\delta_1)\mid x\in K, w\in W, \lma\in [0,1]\}}
$$
is compact, there exists $\delta' \in (0, \delta_1]$ such that
$$\|R^{(\lambda)}(t-h,t-\delta_1)z - R^{(\lma)}(t,t-\delta_1)z\|\leq \eps/3 \quad \mbox{ for any $h\in [0,\delta')$, $\lambda\in [0,1]$, $z\in Q_{t-\delta_1}$}.$$
In consequence, for any $x\in K$, $w\in W$, $\lma\in [0,1]$ and $h\in [0,\delta')$
\begin{align*}
\|\Sigma (t-h)-\Sigma(t)\| & \leq \|\Sigma(t-h)-R^{(\lma)}(t-h,t-\delta_1)\Sigma(t-\delta_1) \| \\
& \quad + \| R^{(\lambda)}(t-h,t-\delta_1)\Sigma(t-\delta_1) - R^{(\lma)}(t,t-\delta_1)\Sigma(t-\delta_1)\| \\
& \qquad + \|R^{(\lma)}(t,t-\delta_1)\Sigma(t-\delta_1) - \Sigma(t)\|\\
& \leq \left\|\int_{t-\delta_1}^{t-h} R^{(\lma)}(t-h,s) w(s) \d s \right\|+\eps/3+
\left\|\int_{t-\delta_1}^{t} R^{(\lma)}(t,s) w(s) \d s \right\|\\
& \leq M \int_{t-\delta_1}^{t-h} c(s) \d s + \eps/3 + M\int_{t-\delta_1}^{t} c(s) \d s
\leq \eps.
\end{align*}
Hence, the set $\{\Sigma(x,w,\lma) \}_{x\in K, w\in W, \lma\in [0,1]}$ is equicontinuous at any $t\in [0,T]$, which due to the Ascoli-Arzel\`{a} theorem, completes the proof of (ii). \hfill $\square$
We state basic continuity and compactness results for the translation along trajectories operator
for a perturbed (possibly nonlinear) equation
\begin{equation}\label{14092008-0948}
\left\{\begin{array}{l}
\dot u(t)=A(t)u(t)+F(t,u(t)), \quad t\in [t_0,T] \\
u(t_0) = x_0
\end{array}\right.
\end{equation}
where $\{ A(t) \}_{t\in [0,T]}$ is a family of linear operators on Banach space $E$ with the associated evolution system $\{R(t,s)\}_{0\leq s\leq t\leq T}$, $F:[0,T]\times E\to E$ is a continuous map, $t_0\in[0,T)$ and $x_0\in E$. Recall (after \cite{Pazy}) that a continuous function $u:[t_0,T]\to E$ is called a {\em mild solution } of (\ref{14092008-0948}) if and only if
$$
u(t)=R(t,t_0) x_0 + \int_{t_0}^{t} R(t,s)F(s,u(s)) \d s
\quad\mbox{ for any } t\in [t_0,T].
$$
For the purpose of next sections we consider below a parameterized framework.
\begin{Prop} \label{09012009-2320}
For each $\lma\in [0,1]$, let $\{A^{(\lma)}(t)\}_{t\in [0,T]}$ be
family of operators on a separable Banach space $E$ having
associated evolution system $R^{(\lma)}$. If the family
$\{R^{(\lma)}\}_{\lambda\in[0,1]}$ is continuous and
$F:[0,T]\times E\times [0,1]\to E$ is a continuous map being \\[1mm]
\noindent $(F_{1})_{par}$ \parbox[t]{138mm}{locally Lipschitz in the second variable uniformly with respect
to the other variables, i.e. for each $x\in E$, there exists $r_x>0$ and $L_x>0$ such that for each
$t\in [0,T]$, $x_1, \ x_2\in B(x,r_x)$ and $\lma \in [0,1]$
$$\|F(t,x_1, \lma) - F(t,x_2,\lma)\|\le L_x \|x_1 - x_2\|;$$}
\noindent $(F_{2})_{par}$ \parbox[t]{138mm}{ of sublinear growth in the second variable uniformly with respect to the others, i.e. there is $c>0$ such that
$$
\|F(t,x,\lma)\|\le c(1 + \|x\|) \quad \mbox{ for any } x\in E, t\in [0,T], \lma\in [0,1];
$$ } \\[1mm]
\noindent $(F_{3})_{par}$ \parbox[t]{138mm}{a $k$-set contraction, i.e. there exists $k\geq 0$ such that
$$
\beta(F([0,T]\times Q\times [0,1]))\le k\beta(Q)
\quad \mbox{ for any bounded } Q\subset E,$$}\\[1em]
\indent then\\
\pari{(i)}{ {\em (Existence)} for any $x\in E$ and $\lambda \in [0,1]$,
the initial value problem
\begin{equation}\label{07012009-1523}
\left\{
\begin{array}{l}
\dot u(t) = A^{(\lma)}(t)u(t)+F(t,u(t),\lma),\quad t\in [0,T]\\
u(0)=x
\end{array}
\right.
\end{equation}
admits a unique mild solution $u(\,\cdot\, ;0,T,x,\lambda)$;}\\[2mm]
\pari{(ii)}{{\em (Continuity)} if $( x_n, \lambda_n )\to ( x_0 , \lambda_0 )$
in $E\times [0,1]$, then $$
u(\, \cdot \, ;0,T,x_n,\lambda_n) \to u(\,\cdot\, ;0,T,x_0,\lambda_0) \ \ \ \mbox{ in } C([0,T],E);
$$}
\pari{(iii)}{ {\em (Compactness)} if additionally, there is $\omega >0$ such that
\begin{equation}\label{03082009-1958}
\|R^{(\lma)}(t,s) \| \leq e^{-\omega(t-s)} \quad \mbox{ for \ } 0
\leq s \leq t\leq T
\end{equation}
and, for any $t\in [0,T]$, $\Phi_t:E\times [0,1] \to E$ is given by $\Phi_t (x,\lambda):= u(t;0,T,x,\lambda),$
then, for any bounded $Q\subset E$ and $t\in [0,T]$, the set $\Phi_t (Q\times [0,1])$ is bounded and
\begin{equation*}
\beta \left( \Phi_t (Q\times [0,1]) \right) \leq e^{(k-\omega) t} \beta(Q).
\end{equation*}}
\end{Prop}
\begin{Lem} \label{19092008-0029} {\em (see \cite{Deimling}, \cite{ObuKaZ})}
Suppose that $E$ is a separable Banach space, $W\subset L^1([a,b],E)$ is countable and there is $c\in L^1([a,b])$
such that $\|w(t)\|\leq c(t)$, for all $w\in W$ and a.e. $t\in [a,b]$,
and let $\phi:[a,b]\to\R$ be given by $\phi (t):= \beta(\{w(t)\ | \ w\in W \})$.
Then $\phi\in L^1([a,b])$ and
$$
\beta \left(\left\{
\int_{a}^{b} w(\tau) \d \tau\,|\, w \in W\right\} \right)
\leq \int_{a}^{b} \phi(\tau) \d \tau.
$$
\end{Lem}
\begin{Lem}\label{08012009-2310}
Let $\{R^\lma\}_{\lma\in [0,1]}$ be a continuous family of evolution system such that
\begin{equation}\label{03082009-1907}
\|R^{(\lma)}(t,s) \| \leq M e^{\omega(t-s)} \quad \mbox{ for \ } 0
\leq s \leq t\leq T \mbox{ and }\lma\in [0,1]
\end{equation}
where $M>0$ and $\omega\in\R$ are constants. Then, for any bounded $Q\subset E$ and $s,t \in [0,T]$ with $s\leq t$,
$$
\beta \left( \{ R^{(\lma)}(t,s)x\mid x\in Q, \, \lma\in [0,1] \}\right) \leq M e^{\omega(t-s)} \beta(Q).
$$
\end{Lem}
The proof is analogical to the proof of Lemma 2.1 from \cite{Cwiszewski-Kokocki}.
\noindent {\bf Proof of Proposition \ref{09012009-2320}.}
(i) follows by standard arguments (see e.g. \cite{Pazy} and \cite{Daners}).\\
\indent The proofs of (ii) and (iii) go in analogy to that of Proposition 3 in \cite{Cwiszewski-Kokocki}.
(ii) corresponds also to results from \cite{Daners} (here we need a version with locally Lipschitz nonlinearity).
To see (ii), observe that if $x_n\to x_0$ in $E$ and $\lma_n\to \lma_0$, then putting
$u_n(s):=\Phi_s(x_n, \lma_n)$ for $s\in [0,T]$, one has
\begin{eqnarray*}\beta\left(\{ u_n(t)\}_{n\ge 1}\right)
\leq \beta(\{R^{(\lma_n)}(t,0)x_n\}_{n\ge 1}) + \beta\left( \left\{ \int_{0}^{t} h_{t,n} (s) \d s \ | \ n\ge 1 \right\}\right)
\end{eqnarray*}
where $h_{t,n} (s):=R^{(\lma_n)}(t,s)F(s,u_n(s),\lma_n)$ for
$s\in[0,t]$. In view of the $(F_2)_{par}$ and the Gronwall
inequality, there is $M_0\geq 0$ such that $\|u_n(s)\| \leq M_0$,
for all $s\in [0,T]$ and $n\geq 1$, which again by $(F2)_{par}$
gives $M_1\geq 0$ such that $\|h_{t,n} (s)\| \leq M_1$, for $s\in
[0,T]$ and $n\geq 1$. This allows applying Remark
\ref{07012009-1059} and Lemmata \ref{19092008-0029} and
\ref{08012009-2310} and as a result, for any $t\in[0,T]$,
\begin{align*}
\beta\left(\{u_n(t)\}_{n\geq 1}\right) & \leq \beta\left( \left\{ \int_{0}^{t} h_{t,n}(s) \d s \ | \ n\ge 1 \right\}\right) \\
& \leq \int_{0}^{t} \beta(\{h_{t,n} (s) \ | \ n\ge 1 \}) \d s
\leq k M \int_{0}^{t}\beta\left(\{u_n(s)\}_{n\ge 1}\right)
\end{align*}
By use of the Gronwall inequality, one gets $\beta\left(\{u_n(t)\}_{n\geq 1}\right)=0$ for any $t\in [0,T]$.
Hence, due to the fact that $u_n= \Sigma (x_n, w_{n}, \lma_n)$ with $w_n(s):=F(s,u_n (s),\lma_n)$
and Proposition \ref{30122008-1629} (ii) any subsequence of $(u_n)$ contains a subsequence
converging to some $u_0$. By Proposition \ref{30122008-1629} (i),
$u_0 = \Sigma (x_0, w_0, \lma_0)$ with $w_0(s):=F(s,u_0(s), \lma_0)$ and therefore
$u_0$ is a unique mild solution of (\ref{07012009-1523}). This completes the proof of (ii).\\
\indent (iii) Let $Q$ be an arbitrary bounded subset of $E$
and let $Q_0$ be a countable subset of $Q$ such that
$\overline{Q_0} \supset Q$ and $\Lambda_0$ a countable
dense subset of $[0,1]$. First observe that using $(F_2)_{par}$ and the Gronwall inequality as before,
we infer that the sets $\Phi_t (\overline{Q_0} \times [0,1])$, $t\in [0,T]$, are contained in a ball,
and clearly, by use of Lemmata \ref{08012009-2310} and
\ref{19092008-0029},
\begin{align*}
\beta(\Phi_t(Q_0\times \Lambda_0)) & \leq \beta \left( \{ R^{(\lma)}(t,0)x\mid x\in Q_0, \, \lma\in \Lambda_0 \}\right) + \\
& \quad +\beta\left( \left\{ \int_{0}^{t}w_{x,\lma, t}(s) \d s \,\mid\,
x\in Q_0, \lma\in \Lambda_0 \right\}\right)\\
& \leq e^{-\omega t}\beta(Q_0) + \int_{0}^{t} \beta (\{w_{x,\lma,t}(s) \,\mid x\in Q_0, \lma\in \Lambda_0 \}) \d s,
\end{align*}
where $w_{x,\lma,t}(s):=R^{(\lma)}(t,s) F(s,\Phi_s(x,\lma),\lma)$
for $s\in [0,t]$. Further observe that, by (\ref{03082009-1958})
and Lemma \ref{08012009-2310}, for any $s\in [0,t]$, one has
\begin{align*}
\beta (\{w_{x,\lma,t}(s) \!\mid\! x\!\in\! Q_0, \lma\!\in\! \Lambda_0 \}) \\
& \hspace{-20mm}\leq \beta (\{R^{(\lma)}(t,s)z \, \mid \, z\!\in\! F([0,T]\!\times\! \Phi_s(Q_0\!\times\! \Lambda_0)\!\times\! \Lambda_0), \ \lma\!\in\!\Lambda_0\}) \\
& \hspace{-20mm}\leq e^{-\omega(t-s)} \beta\left( F([0,T]\!\times\! \Phi_s(Q_0\!\times\! \Lambda_0)\!\times\! \Lambda_0)\right)\\
& \hspace{-20mm}\leq k e^{-\omega(t-s)} \beta(\Phi_s(Q_0\!\times\! \Lambda_0)).
\end{align*}
Combining the previous two inequalities together and applying the
Gronwall inequality give $\beta(\Phi_t(Q_0\times \Lambda_0)) \leq
e^{(k-\omega)t} \beta(Q_0)$ and finally $\beta(\Phi_t(Q\times
[0,1])) \le \beta(\Phi_t(\overline{Q_0}\times
\overline{\Lambda_0}))\leq \beta(\overline{\Phi_t(Q_0\times
\Lambda_0)}) =\beta(\Phi_t(Q_0\times \Lambda_0)) \leq
e^{(k-\omega)t} \beta(Q_0) \leq e^{(k-\omega)t} \beta(Q)$. \hfill
$\square$
Now we pass to the hyperbolic case. We shall assume in the rest of this section that
$V$ is a Banach space which is densely and continuously embedded into $E$.
Given a linear operator $A:D(A)\to E$ generating
a $C_0$ semigroup $\{S_A(t)\}_{t\geq 0}$ of bounded linear operators on $E$,
$V$ is said to be {\em $A$-admissible} provided $V$ is an invariant subspace for each $S_A(t)$ for $t\geq 0$ and the family of restrictions $\{S_{A}(t)_V:V\to V\}_{t\geq 0}$ ($S_A(t)_V x:=S_A(t)x$, $x\in V$)
is a $C_0$ semigroup on $V$. Define the \emph{part of $A$ in the space $V$} as a linear operator $A_V:D(A_V)\to V$ given by $
D(A_V):=\{v\in D(A)\cap V\,\mid\, A v\in V\}$, $A_V v:=A v$ for $v\in D(A_V)$.
In view of \cite[Ch. 4, Theorem 5.5]{Pazy}, if $V$ is $A$-admissible then $A_V$ is the generator of the $C_0$ semigroup $\{S_A(t)_V\}_{t\geq 0}$.
\vspace{-5mm}
\begin{Prop}{\em (see \cite[Ch. 5, Theorem 3.1]{Pazy})}\label{17092008-1611}
Let $\{A(t)\}_{t\in [0,T]}$ be a family of linear operators on a Banach space $E$ satisfying the following conditions\\[2mm]
\noindent $(Hyp_1)$ \parbox[t]{140mm}{$\{A(t)\}_{t\in [0,T]}$ is
a stable family of infinitesimal generators of $C_0$ semigroups, i.e. there are $M\geq 1$ and $\omega\in\R$ such that
$$
\|S_{A(t_1)}(s_1)\ldots S_{A(t_n)}(s_n)\|_{{\cal L}(E,E)}\leq M e^{\omega(s_1+\ldots+s_n)},
$$
whenever $0\leq t_1 \leq \ldots \leq t_n\leq T$ and $s_1,\ldots, s_n\geq 0$, where $\{S_{A(t)}(s)\}_{s\geq 0}$ is the $C_0$ semigroup generated by $A(t)$;} \\[2mm]
\noindent $(Hyp_2)$ \parbox[t]{140mm}{$V$ is $A(t)$-admissible for each $t\in [0,T]$ and the family $\{ A_V (t)\}_{t\in [0,T]}$ is a stable family of generators of
$C_0$ semigroups with constants $M_V\geq 1$ and $\omega_V \in\R$;} \\[1mm]
\noindent $(Hyp_3)$ \parbox[t]{140mm}{$V\subset D(A(t))$ and $A(t)\in{\cal L}(V,E)$ for $t\in [0,T]$ and the mapping $[0,T]\ni t\mapsto A(t)\in {\cal L}(V,E)$ is continuous.}\\[1mm]
Then there exists a unique evolution system $\{R(t,s)\}_{0\leq s\leq t\leq T}$ in $E$ with the following properties\\[1mm]
\pari{(i)}{ $\|R(t,s)\| \leq M e^{\omega(t-s)}$ \quad for \ $0\leq s\leq t\leq T$;}\\[1mm]
\pari{(ii)}{ $\left.\frac{\partial^+}{\partial t} R(t,s)v\right|_{t=s}= A(s)v$
\quad for \ $v\in V$, $s\in [0,T)$;}\\[1mm]
\pari{(iii)}{ $\frac{\partial}{\partial s} R(t,s)v = -R(t,s)A(s)v$ \quad for \ $v\in V$, $0\leq s\leq t\leq T$.}
\end{Prop}
\vspace{-3mm}
Using homotopy invariants will require the continuity of linear
evolution systems with respect to parameters.
\vspace{-5mm}
\begin{Prop}\label{14012008-1812}
Let, for each $\lma\in [0,1]$, a family $\{A^{(\lma)}(t)\}_{0\leq t\leq T}$ satisfy conditions $(Hyp_1)-(Hyp_3)$ with constants $M, M_V, \omega, \omega_V$ independent of $\lambda$ and let $R^{(\lma)}=\{R^{(\lma)}(t,s)\}_{0\leq s\leq t\leq T}$ be the corresponding evolution systems in $E$ determined by Proposition \ref{17092008-1611}. If, for any $\lma_0\in [0,1]$,
\begin{equation}\label{13072009-1426}
\int_{0}^{T} \|A^{(\lma)}(\tau) - A^{(\lma_0)} (\tau)\|_{{\cal L}(V,E)} d \tau\to 0 \quad
\mbox{ as } \lma\to \lma_0,
\end{equation}
then $\{ R^{(\lma)}\}_{\lma\in [0,1]}$ is a continuous family of evolution systems in $E$.
\end{Prop}
{\bf Proof.} We use the construction from \cite[Ch. 5, Theorem 3.1]{Pazy}.
Recall that for any $\lma\in [0,1]$ and $x\in E$
$$
R^{(\lma)}(t,s)x := \lim_{n\to +\infty} R_{n}^{(\lma)}(t,s)x \quad\mbox{ for \ } 0\leq s\leq t\leq T,
$$
where, for each $n\geq 1$, the operator $R_{n}^{(\lma)} (t,s):E\to
E$ is given by (\footnote{Here we adopt the convention that
$\prod_{k=1}^n T_k := T_n\circ T_{n-1}\circ\ldots \circ T_1$, for
the sequence $T_1, T_2, \ldots, T_n$ of bounded operators on
$E$.})
$$
R_{n}^{(\lma)}(t,s)\!:= \!\!\left\{
\begin{array}{ll}
\!\! S^{(\lma)}_j(t\!-\!s) & \mbox{ if } s,t\in [t_{j}^{n}, t_{j+1}^{n}], \, s \le t,\\
\!\! S^{(\lma)}_k(t\!-\!t_{k}^{n})\!\left(\prod\limits_{j=l+1}^{k-1} \!\!S^{(\lma)}_{j}(T/n)\! \right)\!S^{(\lma)}_{l}(t_{l+1}^{n}\!-\!s)&
\mbox{ if } s\in [t_{l}^{n}, t_{l+1}^{n}], \, t\in [t_{k}^{n}, t_{k+1}^{n}],\\
&\mbox{ and } k>l \ge 0,
\end{array}
\right.
$$
with $t_{j}^{n}:=(j/n)T$, $S_{j}:=S_{A^{(\lma)}(t_{j}^n)}$, for $j=0,1,\ldots, n$.
Moreover recall (after \cite{Pazy}) that $\{R_{n}^{(\lma)}(t,s)\}_{0\leq s\leq t\leq T}$ are evolution systems such that
\begin{align}\label{14012009-1920}
\|R_{n}^{(\lma)}(t,s)\|_{{\cal L}(E,E)}\leq Me^{\omega(t-s)},
\quad \|R_{n}^{(\lma)}(t,s)\|_{{\cal L}(V,V)}\leq M_V e^{\omega_V
(t-s)}, \quad R_n^{(\lma)}(t,s)V\!\subset \! V,
\end{align}
for $0\leq s\leq t\leq T$ and for any $v\in V$
\begin{align}\label{14012009-1921}
\frac{\partial}{\partial t} R_{n}^{(\lma)}(t,s)v &
=A_{n}^{(\lma)}(t)R_n^{(\lma)}(t,s)v && \hspace{-10mm} \mbox{ for
\ } t\not\in \{t_{0}^{n},t_{1}^{n},\ldots, t_{n}^{n}\}, \ s\le t,
\\ \label{14012009-1922} \frac{\partial}{\partial s}
R_{n}^{(\lma)}(t,s)v & = - R_n^{(\lma)}(t,s) A_{n}^{(\lma)}(s)v
&&\hspace{-10mm}\mbox{ for \ }s\not\in
\{t_{0}^{n},t_{1}^{n},\ldots, t_{n}^{n}\}, \ s\le t,
\end{align}
with $A_{n}^{(\lma)}(t):=A^{(\lma)}(t_{k}^{n})$ if $t_k^n \leq t <
t_{k+1}^{n}$ for $k=0,\ldots,n-1$ and
$A_n^{(\lma)}(T):=A^{(\lma)}(T)$. Observe that in view of
$(Hyp_3)$ for any $\lma\in [0,1]$, one has $\|A_{n}^{(\lma)}(t) -
A^{(\lma)}(t)\|_{{\cal L}(V,E)} \to 0$ as $n\to +\infty$
uniformly with respect to $t\in [0,T]$. Fix any $v\in V$, $\lma,
\mu\in [0,1]$, $n\geq 1$ and $s,t \in [0,T]$ with $s<t$ and define
$\phi:[s,t]\to E$ by $\phi(r):=
R_n^{(\lma)}(t,r)R_n^{(\mu)}(r,s)v$. In view of
(\ref{14012009-1920}), (\ref{14012009-1921}) and
(\ref{14012009-1922}) the map $\phi$ is differentiable on $[s,t]$
except finite number of points and
\begin{eqnarray*}
R_n^{(\mu)}(t,s)v- R_n^{(\lma)}(t,s)v = \phi(t)-\phi(s)
=\int_{s}^{t} \phi'(r) dr, \\
= \int_{s}^{t} \left(R_n^{(\lma)}(t,r)(A_n^{(\mu)}(r))-A_n^{(\lma)}(r))R_n^{(\mu)}(r,s)v \right)dr.
\end{eqnarray*}
Hence, by (\ref{14012009-1920}),
$$
\| R_n^{(\mu)}(t,s)v- R_n^{(\lma)}(t,s)v \|
\leq MM_Ve^{(\omega +\omega_V)T}\|v\|_{V} \int_{0}^{T} \|A_n^{(\mu)}(r)-A_n^{(\lma)}(r)\|_{{\cal L}(V,E)} dr.
$$
Passing to the limit with $n\to +\infty$, we get
$$
\|R^{(\mu)} (t,s)v - R^{(\lma)}(t,s)v\| \leq
MM_Ve^{(\omega +\omega_V)T}\|v\|_{V} \int_{0}^{T} \|A^{(\mu)}(r)-A^{(\lma)}(r)\|_{{\cal L}(V,E)} dr
$$
and in consequence, $R^{(\lma)}(t,s)v\to R^{(\lma_0)}(t,s)v$ for any $v\in V$, as $\lma\to \lma_0$ and the convergence is uniform with respect to $s,t$.
Using the density of $V$ in $E$, we complete the proof since $\|R^{(\lambda)}(t,s)\| \leq M e^{\omega(t - s)}$ for $\lambda\in[0,1]$ and $0\leq s\leq t\leq T$. \hfill $\square$
The following criterion for verification of conditions $(Hyp_1)-(Hyp_3)$ is useful in applications.
\begin{Prop} {\em (\cite[Ch. 5, Theorem 4.8]{Pazy})}
\label{27022009-1214} Suppose that a family $\{A(t) \}_{t\in
[0,T]}$, where $D(A(t))=D$ for any $t\in [0,T]$ and some $D\subset
E$, is stable and for each $v\in D$ the mapping $[0,T]\ni t\mapsto
A(t)v\in E$ is continuously differentiable. Then, the family $\{
A(t)\}_{t\in [0,T]}$ satisfies conditions $(Hyp_1)-(Hyp_3)$ with
$V:=D$ equipped with the norm given by $\|v\|_V:=\|A(0)v\| +
\|v\|$ for $v\in V$.
\end{Prop}
\section{Averaging method for periodic solutions}
We shall deal with the periodic problem
$$
\left\{\begin{array}{ll}
\dot u(t) = A(t)u(t) + F(t,u(t)), \quad t\in [0,T] \\
u(0) = u(T)
\end{array}\right.\leqno{(P)}
$$
where the family $\{A(t)\}_{t\in [0,T]}$ of linear operators on a separable Banach space $E$ satisfies
a more restrictive variant of $(Hyp_1)$ (from Proposition \ref{17092008-1611})\\ [1mm]
\noindent $(Hyp'_1)$ \ \parbox[t]{140mm}{there is $\omega > 0$, such that
$$
\|S_{A(t_1)}(s_1)\ldots S_{A(t_n)}(s_n)\|_{{\cal L}(E,E)}\leq
e^{-\omega(s_1+\ldots+s_n)};
$$
whenever $0\leq t_1 \leq \ldots \leq t_n\leq T$ and $s_1,\ldots, s_n\geq 0$,} \\[2mm]
conditions $(Hyp_2)$, $(Hyp_3)$ and, additionally, \\[1mm]
$(Hyp_4)$ \parbox[t]{135mm}{ there is $\mu_0 > -\omega$ such that the space $(\mu_0 I - A_0)V$ is dense in $E$,
where $$A_0:= \frac{1}{T}\int_{0}^{T} A(\tau)\, d\tau \in {\cal L}(V,E);$$}\\
$(Hyp_5)$ \parbox[t]{135mm}{ $A(0)x = A(T)x$ for $x\in D(A(0)) = D(A(T))$.}\\[2mm]
\noindent Furthermore, we assume that a continuous mapping $F:[0,T] \times E\to E$\\
\noindent $(F_{1})$ \parbox[t]{138mm}{is locally Lipschitz with respect to the second variable uniformly with respect to the first one;}\\
\noindent $(F_{2})$ \parbox[t]{138mm}{ has sublinear growth uniformly with respect to
the first variable, i.e. there is a constant $c>0$ such that
$$\|F(t,x)\|\le c(1 + \|x\|)\quad\mbox{ for $x\in E$, $t\in [0,T]$};$$}\\
\noindent $(F_{3})$ \parbox[t]{138mm}{ there is $k\in[0,\omega)$
such that
$$\beta(F([0,T]\times Q))\le k\beta(Q) \quad\mbox{ for any bounded
$Q\subset E$};$$}\\
\noindent $(F_{4})$ \parbox[t]{138mm}{ $F(0,x) = F(T,x)$ for $x\in E$.}\\[2mm]
The existence of periodic solutions will be obtained by means of a
continuation principle for a parameterized family of periodic
problems
$$
\left\{\begin{array}{ll}
\dot u(t) = \lambda A(t)u(t) + \lambda F(t,u(t)),\quad t\in [0,T] \\
u(0) = u(T)
\end{array}\right.\leqno{(P_{T,\lambda})}
$$
with the parameter $\lambda\geq 0$.
For any $x\in E$ and $\lma\in [0,T]$, by $u(\,\cdot\,;x,\lambda)$ denote the unique mild solution of
\begin{eqnarray}\label{29082008-2340}
\dot u(t)=\lambda A(t)u(t)+\lambda F(t,u(t)), \quad t\in [0,T]
\end{eqnarray}
satisfying the initial condition $u(0;x,\lma) = x$. The translation along trajectories operator
for $(\ref{29082008-2340})$ is denoted by $\Phi_{t}^{(\lambda)}:E \to E$ where $t\in [0,T]$.
A point $(x,\lambda)\in E\times [0,+\infty)$ is a {\em $T$--periodic point}
for (\ref{29082008-2340}) if $\Phi_{T}^{(\lambda)}(x)=x$.
We say that $x_0\in E$ is a {\em branching point } (or a {\em cobifurcation point}) for $(P_{T,\lambda})$, $\lma\geq 0$,
if there exists a sequence of $T$-periodic points $(x_n,\lambda_n)\in E \times(0,+\infty)$
such that $\lambda_n\to 0$ and $x_n\to x_0$ as $n\to +\infty$.
\begin{Th}\label{18092008-1450}
If $x_0\in E$ is a branching point of $(P_{T,\lambda})$, then
$\widehat{A} x_0 + \widehat{F} (x_0) = 0$
where
$\widehat A:D(\widehat A) \to E$ is the closure of $A_0$
and $\widehat F:E\to E$ is given by $\widehat F(x):=(1/T)\int_{0}^{T} F(\tau,x)\,d\tau$.
\end{Th}
\begin{Rem}\label{19072009-2137}{\em
(a) Recall that due to \cite[Ch. 1, Th. 4.3]{Pazy}, if $A:D(A)\to E$ generates a $C_0$ semigroup of contractions,
then the dissipativity condition
\begin{equation}\label{rr1}
\|x-\lma Ax\| \geq \|x\| \quad \mbox{ for any \ } x\in D(A), \ \lambda > 0
\end{equation}
is equivalent to
$$
(p,Ax)\leq 0 \quad\mbox{ for any \ } x\in D(A), \ p\in J(x),
$$
where $J(x):=\{ p \in E^* \mid \la p, x\ra = \|x\|^2 = \|p\|^{2}\}$ is the dual set of $x$.\\
\indent (b) Hence, if $(Hyp'_1)$ and $(Hyp_3)$ hold, then $\omega
I + A_0= \omega I + (1/T)\int_{0}^{T} A(\tau) d\tau$ in ${\cal
L}(V,E)$ is a dissipative operator. This implies that the closure
$\widehat A:D(\widehat A) \to E$ of $A_0$ is a well-defined linear
operator and, by (\ref{rr1}), the operator $\widehat
A_\omega:=\omega I + \widehat A$ is also dissipative, hence
$\lambda I - \widehat A_\omega$ has closed range whenever $\lambda
> 0$. If condition $(Hyp_4)$ holds, then the operator $(\mu_0 +
\omega)I - \widehat A_\omega = \mu_0 I - \widehat A$ has closed
and dense range, since $\mu_0 + \omega > 0$.
It means that $(\mu_0 + \omega)I - \widehat A_\omega$ is $m$--dissipative, since its range is the whole $E$ and, by the Lumer-Phillips theorem, $\widehat A$ generates a $C_0$ semigroup such that $\|S_{\widehat A}(t)\|\le e^{-\omega t}$ for $t\geq 0$.\\
\indent (c) In particular, $(-\omega,+\infty)\subset\varrho(\widehat A)$ and, for each $\mu > -\omega$,
$$
(\mu I - A_0)V = (\mu I - \widehat A)V = (\mu I - \widehat A)(\mu_0 I - \widehat A)^{-1}V_0
$$
with $V_0 := (\mu_0 I - \widehat A)V$ being a dense subset of $E$.
Since $(\mu I - \widehat A)(\mu_0 I - \widehat A)^{-1}:E\to E$
is a bounded bijection, we infer that $(\mu I - A_0)V$ is dense in $E$ for any $\mu > -\omega$.\\
\indent (d) If $\{A(t)\}_{t\in [0,T]}$ satisfies conditions $(Hyp'_1)$, $(Hyp_2)$ and $(Hyp_3)$, then, in view of Proposition \ref{17092008-1611}
and point (b), one has
$$\|R(t,s)\|_{{\cal L}(E,E)}\leq e^{-\omega(t-s)}
\quad \mbox{ for any } s,t\in [0,T], s\leq t.
$$}
\end{Rem}
In the proof of Theorem \ref{18092008-1450} and later on in the section we use the following lemma.
\begin{Lem}\label{14092008-1339}
Let $\{A(t)\}_{t\in [0,T]}$ satisfy $(Hyp'_1)$, $(Hyp_2)$ -- $(Hyp_4)$ and let, for each $\mu\in [0,1]$, the family $\{A^{(\mu)}(t)\}_{t\in [0,T]}$ be defined by $A^{(\mu)}(t):=-\mu I + (1-\mu)A(t)$ for $t\in [0,T]$.
Then, for each $\lma\geq 0$ and $\mu\in [0,1]$, the family of operators $\{\lma A^{(\mu)}(t)\}_{t\in [0,T]}$
satisfies $(Hyp_1)'$, $(Hyp_2)$ -- $(Hyp_4)$ and the corresponding evolution systems $\{R^{(\mu,\lambda)}(t,s)\}_{0\leq s\leq t\leq T}$, $\lambda\ge 0$, $\mu\in [0,1]$ have the following properties \\
\pari{(i)}{for any $x\in E$, $t,s\in [0,T]$ with $s\leq t$,
$(\lambda_n)$ in $(0,+\infty)$ and $(\mu_n)$ in $[0,1]$ such that $\lma_n\to 0$, $\mu_n \to \mu_0$, one has
$$R^{(\mu_n,\lambda_n)}(t,s)x \to x \quad \mbox{ as } n\to +\infty,$$
uniformly with respect to $t,s\in [0,T]$ with $s \le t$;}\\[2mm]
\pari{(ii)}{if $(k_n)$ is a sequence of positive integers and sequences $(\lma_n)$ in $(0,+\infty)$ and $(\mu_n)$ in $[0,1]$ are such that $k_n\to +\infty$, $k_n\lambda_n\to \eps$ for some $\eps>0$ and $\mu_n\to \mu_0$ for some $\mu_0\in [0,1]$, then for any $x\in E$
$$
R^{(\mu_n,\lambda_n)}(T,0)^{k_n} x\to S_{\widehat{A^{(\mu_0)}}}(\eps T)x \quad \mbox{ as } n\to\infty,
$$
where $\widehat{A^{(\mu_0)}}$ is the closure of the operator
$\frac{1}{T}\int_{0}^{T} A^{(\mu_0)}(\tau)\,d\tau$;}\\[2mm]
\pari{(iii)}{if $(k_n)$, $(\lma_n)$ and $(\mu_n)$ are as in {\em (ii)}, then for any $x\in E$
$$\lambda_n(I+R^{(\mu_n,\lambda_n)}(T,0)+\ldots+R^{(\mu_n,\lambda_n)}(T,0)^{k_n-1})x\to
\frac{1}{T}\int\limits_{0}^{\eps T}\!\! S_{\widehat{A^{(\mu_0)}}}(\tau)x \,d\tau \quad \mbox{ as } n\to+\infty.$$} \\[-10mm]
\end{Lem}
\noindent\textbf{Proof.} (i) It is easy to check that, for each
$\lma > 0$ and $\mu\in [0,1]$, the family $\{\lma
A^{(\mu)}(t)\}_{t\in [0,T]}$ satisfies $(Hyp'_1)$ with constant
$\omega:=\lambda \min\{1,\omega\}$ and conditions $(Hyp_2)$ --
$(Hyp_3)$ as well. From now on we write $A_{0}^{(\mu)} := -\mu I +
(1-\mu) A_0$ for $\mu\in [0,1]$. We claim that also $(Hyp_4)$
holds. Indeed if $\mu = 1$, then $A^{(\mu)}(t) = I$ for $t\in
[0,T]$ and $(a_{\mu, \lambda} I - A^{(\mu)}_0)V$ with $a_{\mu,
\lambda} = 0$ is dense in $E$, for $\lambda > 0$. If $\mu \neq 1$,
then putting $a_{\mu, \lambda} := (1 - \mu)\lambda\mu_0 -
\lambda\mu$ we see that $a_{\mu, \lambda} > - \omega$, since
$\mu_0 > -\omega$, and $(a_{\mu, \lambda} I - \lambda
A^{(\mu)}_0)V = \lambda(1 - \mu)(\mu_0 I - A_0)V$ is dense in $E$
and thus $\lambda A^{(\mu)}_0$ satisfies $(Hyp_4)$. For the
corresponding evolution system $R^{(\mu,\lma)}$, one gets, for any
$(\lma,\mu)\in (0,+\infty)\times [0,1]$,
\begin{equation}\label{18072009-2214}
\|R^{(\mu,\lambda)}(t,s)\|\leq e^{-\lambda\overline{\omega} (t - s)} \leq 1 \quad\mbox{ for \ }t,s\in [0,T], \, s\le t,
\end{equation}
with $\overline\omega:=\min\{1,\omega\}$, and
$\frac{\partial}{\partial s} R^{(\mu,\lma)}(t,s)v = -\lma R^{(\mu,\lma)}(t,s)A^{(\mu)}(s)v$, for $v\in V$, $0\leq s\leq t\leq T$.
In consequence, for any $v\in V$, $t,r\in [0,T]$, $r \le t$, $\mu\in [0,1]$ and $\lma>0$, one has
$$
R^{(\mu,\lma)}(t,r)v-v= R^{(\mu,\lma)}(t,r)v -
R^{(\mu,\lma)}(t,t)v=\lambda\int_{r}^{t}
R^{(\mu,\lma)}(t,s)A^{(\mu)}(s)v \d s.
$$
Since, for any $s\in [r,t]$,
$$
\|R^{(\mu,\lma)}(t,s)A^{(\mu)}(s)v\| \leq \|R^{(\mu,\lma)}(t,s)\|\|A^{(\mu)}(s)\|_{{\cal L}(V,E)} \|v\|_{V}
\leq \|A^{(\mu)}(s)\|_{{\cal L}(V,E)} \|v\|_{V},
$$
we infer that $\|R^{(\mu,\lma)}(t,r)v-v\| \leq \lma C\|v\|_{V}$
with $C:=\sup_{\mu\in[0,1]} \int_{0}^{T} \|A^{(\mu)}(s)\|_{{\cal
L}(V,E)} \d s<+\infty$. This, due to the density of $V$ in $E$,
means that
$$
\lim_{\lma\to 0^+,\, \mu\to \mu_0 } R^{(\mu,\lma)}(t,r)x = x \quad\mbox{ for any } x\in E
$$
uniformly with respect to $t,r\in[0,T]$ with $r\le t$, which implies (i).\\
\indent Define a map $L:(0,+\infty)\times [0,1]\to {\cal L}(E,E)$ by
$$
L(\lambda,\mu):= R^{(\mu,\lambda)}(T,0) \quad \mbox{ for any } \lambda>0 \mbox{ and }\mu\in [0,1].
$$
Clearly, by (\ref{18072009-2214}), $\|L(\lambda,\mu)\| \leq 1$ for each $(\lma,\mu)\in (0,+\infty)\times [0,1]$.
Observe also that, for each $\mu\in[0,1]$, $a_\mu := a_{\mu, \lambda} /\lambda$ is such that $(a_\mu I - A_{0}^{(\mu)})V$ is dense in $E$. Further, for any $v\in V$,
\begin{eqnarray}
\lma^{-1}(L(\lma,\mu)v-v)=\lma^{-1}(R^{(\mu,\lma)}(T,0)v-v) =
\int_{0}^{T} R^{(\mu,\lma)}(T,s) A^{(\mu)}(s)v \d s
\end{eqnarray}
and
\begin{align*}
&\| \lma^{-1}(L(\lma,\mu)v-v) - T A^{(\mu_0)}_0v\|\leq
\int_{0}^{T} \|R^{(\mu,\lma)}(T,s)A^{(\mu)}(s)v-A^{(\mu_0)}(s)v\| \d s\\
&\quad \leq \int_{0}^{T} \|R^{(\mu,\lma)}(T,s) A^{(\mu)}(s)v-A^{(\mu)}(s)v\| \d s +
\|v\|_{V}\int_{0}^{T} \|A^{(\mu)} (s)-A^{(\mu_0)}(s)\|_{{\cal L}(V,E)} \d s,
\end{align*}
which by use of point (i) of this lemma and $(Hyp_3)$ gives
$$
\lim_{\lma\to 0^+,\, \mu\to \mu_0} \lma^{-1}(L(\lma,\mu)v-v) = TA^{(\mu_0)}_0v \quad \mbox{ for \ } v\in V.
$$
Hence, applying Theorem \ref{29082008-1821} and changing the time variable we get (ii)
and (iii) as the closure of $A^{(\mu_0)}_0$ is equal to $\widehat {A^{(\mu_0)}}$. \hfill $\square$
\noindent {\bf Proof of Theorem \ref{18092008-1450}.}
Let sequences $(\lambda_n)$ in $(0,+\infty)$ and $(x_n)$ in $E$ be
such that $\lambda_n \to0$, $x_n\to x_0$ as $n\to \infty$ and $\Phi_T^{(\lambda_n)}(x_n)=x_n$ for each $n\geq 1$. Then, by definition
\begin{equation}\label{14092008-1236}
\Phi_t^{(\lambda_n)} (x_n) = R^{(\lambda_n)}(t,0)x_n + \lambda_n
\int_{0}^{t} R^{(\lambda_n)}(t,s)F(s,\Phi_s^{(\lambda_n)}(x_n)) \d
s
\end{equation}
for any $t\in [0,T]$, where $\{R^{(\lambda)}(t,s)\}_{0\leq s\leq t\leq T}$ denotes the evolution system generated by the family $\{\lambda A(t)\}_{t\in [0,T]}$.
This yields
\begin{align*}
\|x_n-\Phi_t^{(\lambda_n)}(x_n)\| & = \|R^{(\lambda_n)}(t,0)x_n - x_n\| +
\left\|\lambda_n \int_{0}^{t} R^{(\lambda_n)}(t,s)F(s,\Phi_s^{(\lambda_n)}(x_n)) \d s \right\|\\
& \le \|R^{(\lambda_n)}(t,0)x_n - x_n\|+ \lambda_nc \int_0^t (1 +
\|\Phi_s^{(\lambda_n)}(x_n)\|)\d s.
\end{align*}
In view of Lemma \ref{14092008-1339} (i) with $\mu_n := 0$ for $n\ge 1$ and the boundedness of
$\{\Phi_s^{(\lambda_n)}(x_n)\mid s\in [0,T], n\geq 1 \}$, we infer that
$\Phi_t^{(\lambda_n)}(x_n)\to x_0$, uniformly with respect to $t\in [0, T]$.
Further, by (\ref{14092008-1236}), one has
\begin{equation}
x_n = \Phi_T^{(\lambda_n)}(x_n) = R^{(\lambda_n)}(T,0)x_n +
\lambda_n\int_{0}^{T}R^{(\lambda_n)}(T,s)F(s,\Phi_s^{(\lambda_n)}(x_n))
\d s
\end{equation}
and consequently, for each $k\geq 0$
$$
R^{(\lambda_n)}(T,0)^{k} x_n = R^{(\lambda_n)}(T,0)^{k+1} x_n +
\lambda_n R^{(\lambda_n)}(T,0)^{k}\int_0^T R^{(\lambda_n)}(T,s)F(s,\Phi_s^{(\lambda_n)}(x_n))
d s.
$$
Let $\eps>0$ be arbitrary and let $(k_n)$ be a sequence
of positive integers such that $k_n\lambda_n\to \eps$.
Summing up the above equalities with $k=0,1,\ldots, k_n-1$ for any $n\geq 1$, we obtain
$$
\begin{array}{l}
x_n = R^{(\lambda_n)}(T,0)^{k_n}x_n + \lambda_n\left[\sum\limits_{k=0}^{k_n-1} R^{(\lambda_n)}(T,0)^{k}\right]
\left(\int\limits_{0}^{T} R^{(\lambda_n)}(T,s)F(s,\Phi_s^{\lambda_n} (x_n) )
\d s\right)
\end{array}
$$
and, by use of Lemma \ref{14092008-1339} (ii) and (iii) with $\mu_n := 0$ for $n\ge 1$,
$$x_0 = S_{\widehat{A}} (\eps T) x_0 + \left[\frac{1}{T}\int_0^{\eps T} S_{\widehat{A}}(\tau) d\tau \right] \left(\int_{0}^{T} F(s,x_0)\d s\right).$$
In consequence
$$
-\frac{1}{\eps T} \left( S_{\widehat{A}} (\eps T) x_0 - x_0\right) =
\frac{1}{\eps T}\int_0^{\eps T}
S_{\widehat{A}}(\tau)\widehat{F} (x_0) d\tau.
$$
Thus, since $\eps\!>\!0$ was arbitrary, letting $\eps\!\to\! 0^+$, one has
$-\widehat A x_0 \!=\! \widehat F(x_0)$. \hfill $\square$
\begin{Th}\label{18092008-2220} {\em (Averaging principle)}
Let $\{A(t)\}_{t\in [0,T]}$ be a family of generators of $C_0$ semigroups satisfying
$(Hyp'_1)$, $(Hyp_2)$ -- $(Hyp_5)$ and let $F:[0,T] \times E \to E$
be a continuous map with properties $(F_1)$ -- $(F_4)$.
If $U\subset E$ is an open bounded set such that $\widehat{A}x + \widehat{F}(x)\neq 0$ for any
$x \in\partial U\cap D(\widehat{A})$, then there exists $\lambda_0>0$
such that for all $\lambda\in (0,\lambda_0]$, $\Phi_T^{(\lambda)} (x)\neq x$
for all $x\in\partial U$ and
\begin{equation}
\Deg (\widehat{A}+\widehat{F}, U) = \deg (I-\Phi_T^{(\lambda)}, U).
\end{equation}
Here $\deg$ stands for the topological degree for condensing vector fields (see {\em \cite{Akhmerov-etal}} or {\em \cite{Nussbaum}})
and $\Deg(\widehat{A}+\widehat{F}, U):= \deg(I+{\widehat A}^{-1} \widehat F, U)$ (see {\em \cite{Cwiszewski-Kokocki}}).
\end{Th}
In the proof we shall need the following lemmata.
\begin{Lem}\label{19092008-0012}{\em (see \cite[Lemma 5.4]{Cwiszewski-Kokocki})}
Let $T_n:E\to E$, $n\geq 1$, be bounded linear operators, such that, for any $x\in E$,
$(T_n x)$ is a Cauchy sequence {\em(}\footnote{This is actually equivalent to the existence of a bounded operator
$T:E\to E$ being the strong limit of $(T_n)$.}{\em)}. Then, for any bounded set $\{x_n\}_{n\geq 1} \subset E$
$$
\beta\left( \{T_n x_n\}_{n\geq 1} \right) \leq \left( \limsup_{n\to +\infty}
\|T_n\|\right) \beta\left( \{x_n\}_{n\geq 1}\right).
$$
\end{Lem}
\begin{Lem}{\em(cf. Step 2 in the proof of Theorem 5.1 in \cite{Cwiszewski-Kokocki})} \label{22092008-1201}
Let $A$ be a generator of a $C_0$ semigroup $S_A$ such that $\|S_A(t)\|\leq e^{-\omega t}$ for $t\geq 0$
and $F:E \to E$ be a continuous map with $k\in [0,\omega)$
such that $\beta(F(Q))\leq k\beta(Q)$ for any bounded $Q$.
If an open bounded $U\subset E$ is such that $Ax+F(x)\neq 0$ for each $x\in \partial U\cap D(A)$,
then there exists a locally Lipschitz compact mapping $F_L:E\to E$ such that
\begin{equation}\label{19072009-2232}
Ax+(1-\mu)F(x) + \mu F_L(x)\neq 0 \quad \mbox{ for \ } x\in \partial U\cap D(A), \ \mu\in [0,1].
\end{equation}
\end{Lem}
\begin{Lem}\label{19092008-0014}
Let $\{A^{(\mu)}(t)\}_{t\in [0,T]}$ for $\mu\in [0,1]$, satisfy
$(Hyp'_1)$, $(Hyp_2)-(Hyp_5)$ with the common, independent of
$\mu$, constants $\omega > 0$, $\omega_V$, $M_V$ and let
$F:[0,T]\times E \times [0,1]\to E$ be a continuous mapping
satisfying $(F_1)_{par}$ -- $(F_3)_{par}$ and the periodicity
condition
$$F(0,x,\lma)=F(T,x,\lma) \quad \mbox{ for \ }(x,\lma) \in E
\times [0,1].$$ Suppose that $U\subset E$ is open bounded and
$\widehat{A^{(\mu)}}x+\widehat{F}(x,\mu)\neq 0$ for $x\in \partial
U \cap D(\widehat{A^{(\mu)}})$ and $\mu\in [0,1]$, where
$\widehat{A^{(\mu)}}$ is the closure of $(1/T)\int_{0}^{T}
A^{(\mu)}(s) \d s$ and $\widehat{F}:E\times [0,1]\to E$ is given
by $\widehat{F}(x,\mu):=(1/T)\int_{0}^{T} F(s,x,\mu) \d s$. Then,
there exists $\lma_0>0$, such that, for any $\lma\in (0,\lma_0]$,
$$
\Psi_T^{(\lma)}(x,\mu)\neq x \quad \mbox{ for all } x\in \partial U, \ \mu\in [0,1],
$$
where $\Psi_T^{(\lma)}: \overline U\times [0,1]\to E$ is given by
$\Psi_{T}^{(\lma)}(x,\mu):=u(T;x,\mu,\lma)$ for $(x,\mu)\in \overline U\times [0,1]$, $\lma>0$
and $u(\,\cdot \,;x,\mu,\lma):[0,T]\to E$ is the unique mild solution of
$$\left\{
\begin{array}{l}\dot u(t)=\lma A^{(\mu)}(t) u(t)+ \lma F(t,u(t),\mu), \quad t\in [0,T] \\
u(0)=x.\end{array}\right.$$
\end{Lem}
{\bf Proof. }Suppose to the contrary that there exist sequences $(\lma_n)$ in $(0,+\infty)$ with $\lma_n \to 0^+$, $(x_n)$ in $\partial U$ and $(\mu_n)$ in $[0,1]$ such that $\Psi_{T}^{(\lma_n)}(x_n,\mu_n)=x_n$ for $n\geq 1$.
Without loss of generality, we may assume that $\mu_n\to \mu_0$ as
$n\to +\infty$, for some $\mu_0\in [0,1]$. Let $\{\overline
A^{(\mu)}(t)\}_{t\in [0,2T]}$, $\mu\in [0,1]$ and a mapping $\overline F: [0,2T]\times E\times [0,1]\to E$ be given by
\begin{align*}
\overline A^{(\mu)}(t) & :=A^{(\mu)}(t-[t/T]T) && \hspace{-15mm}\mbox{ for \ } (t,\mu)\in [0,2T]\times [0,1], \\
\overline F (t,x,\mu) & :=F(t-[t/T]T,x,\mu) && \hspace{-15mm}\mbox{ for \ } (t,x,\mu)\in [0,2T]\times E\times [0,1].
\end{align*}
where $[s]$ stands for the integer part of $s\in\R$. It is easy to
check that, for each $\lambda\in(0,\infty)$ and $\mu\in [0,1]$,
the family $\{\lambda \overline A^{(\mu)}(t)\}_{t\in [0,2T]}$ and
the mapping $\lambda F$ satisfies $(Hyp'_1)$, $(Hyp_2)$ --
$(Hyp_4)$ and $(F1)_{par}$ -- $(F3)_{par}$. Denote by $\{\overline
R^{(\mu,\lma)}\}_{0\le s\le t\le 2T}$ the corresponding evolution
system obtained by Proposition \ref{17092008-1611}. From the very
construction of hyperbolic evolution systems (see \cite[Ch. 5,
Theorem 3.1]{Pazy} and the proof of Proposition
\ref{14012008-1812}), we see that, for all $\lma > 0$, and $\mu\in
[0,1]$,
\begin{equation}\label{19072009-1921}
\overline R^{(\mu,\lma)}(T+t,T+s) = \overline R^{(\mu,\lma)}(t,s) = R^{(\mu,\lma)}(t,s) \quad\mbox{ for \ } t,s\in[0,T], \ s\le t.
\end{equation}
For each $t\in [0,2T]$, define $\overline{\Psi}_{t}^{(\lma)}:E\times
[0,1]\to E$, by
$\overline{\Psi}_{t}^{(\lma)}(x,\mu):=\overline u(t; x,\mu,\lma)$,
where $\overline u(\, \cdot \,;x,\mu,\lma):[0,2T]\to E$ is a solution of
$$\left\{\begin{array}{l}
\dot u(t)=\lma\overline A^{(\mu)}(t)u(t)+\lma\overline F(t,u(t),\mu),\quad t\in [0,2T] \\
u(0) = x.
\end{array}\right.$$
It clearly follows from (\ref{19072009-1921}) that, for any $n\geq 1$ and $t\in [0,T]$,
$$
\Psi_{t}^{(\lma_n)} (x_n, \mu_n)=
\overline{\Psi}_{t+T}^{(\lma_n)}(x_n, \mu_n ) = \overline
R^{(\mu_n, \lma_n)}(T+t,t)\Psi_{t}^{(\lma_n)}(x_n,\mu_n) +\lma_n
\int_{t}^{T+t} w_{n,t}(s) \d s,
$$
with $w_{n,t}(s):=\overline R^{(\mu_n,\lma_n)}(T+t,s) \overline F(s,\overline{\Psi}_{s}^{(\lma_n)}(x_n,\mu_n),\mu_n)$
for $s\in [t,T+t]$. Therefore, for any integer $k\geq 0$, one has
\begin{align*}
\overline R^{(\mu_n,\lma_n)}(T+t,t)^{k}\Psi_{t}^{(\lma_n)}(x_n,\mu_n) & = \overline R^{(\mu_n,\lma_n)}(T+t,t)^{k+1}
\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\\
& \qquad + \lma_n \overline R^{(\mu_n, \lma_n)}(T+t,t)^{k}
\int_{t}^{T+t} w_{n,t}(s) \d s.
\end{align*}
Putting $k_n:=[1/\lma_n]$ and summing up the above equalities with $k=0,\ldots, k_n-1$,
we find that
\begin{equation}\label{19072009-1956}
\Psi_{t}^{(\lma_n)}(x_n,\mu_n) = \overline
R^{(\mu_n,\lma_n)}(T+t,t)^{k_n}\Psi_{t}^{(\lma_n)}(x_n,\mu_n) +
K_n \left(\int_{t}^{T+t} w_{n,t}(s) \d s \right)
\end{equation}
where
$$
K_n:=\lma_n \sum_{k=0}^{k_n - 1}\overline
R^{(\mu_n,\lma_n)}(T+t,t)^{k} \quad\mbox{ for \ }n\geq 1.$$ By
(\ref{19072009-1921}) and the fact that $\lambda_n k_n \to 1$ as
$n\to +\infty$, going along the lines of the proof of Lemma
\ref{14092008-1339}, we infer that, for any $x\in E$,
$$
\overline R^{(\mu_n,\lma_n)}(T+t,t)^{k_n}x\to
S_{\widehat{A^{(\mu_0)}}}(T) x \quad \mbox{ and } \quad K_n x \to
\frac{1}{T}\int_{0}^{T} S_{\widehat{A^{(\mu_0)}}}(s)x \d s.
$$
This, along with Lemmata \ref{19092008-0012} and \ref{19092008-0029},
gives
\begin{align*}
\beta\left(\left\{\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right) \le \\
& \hspace{-30mm}\leq e^{-\omega T} \beta\left(\left\{\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right)
+ \frac{1-e^{-\omega T}}{\omega T} \beta\left(\left\{\int_{t}^{T+t} w_{n,t}(s) \d s \right\}_{n\ge 1}\right) \\
& \hspace{-30mm}\leq e^{-\omega T} \beta\left(\left\{\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right) +
\frac{1-e^{-\omega T}}{\omega T} \int_{t}^{T+t} \beta(\{w_{n,t}(s)\}_{n\ge 1}) \d s\\
& \hspace{-30mm} \leq e^{-\omega T} \beta\left(\left\{\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right) +
\frac{1-e^{-\omega T}}{\omega T} \int_{t}^{T+t} k\beta\left(\left\{\overline{\Psi}_{s}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right)ds
\end{align*}
and, in consequence,
\begin{equation}
\beta\left(\left\{\Psi_{t}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right) \leq \frac{k}{\omega T}\int_{t}^{T+t} \beta\left(\left\{\overline{\Psi}_{s}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\ge 1}\right) ds.
\end{equation}
Define $\phi:[0,2T]\to \R$ by
$$
\phi(s):= \beta\left(\left\{\overline{\Psi}_{s}^{(\lma_n)}(x_n,\mu_n)\right\}_{n\geq 1}\right) \quad \mbox{ for \ } s\in [0,2T].
$$
We claim that $\phi \equiv 0$. Indeed, otherwise $M:=\sup_{s\in [0,2T]} \phi(s) \in (0,+\infty)$ and by its $T$-periodicity, for $\eps\in (0,(1-k/\omega)M)$ there exists $t_\eps\in [0,T]$ such that
$$
M-\eps <\phi(t_\eps) \leq \frac{k}{\omega T}
\int_{t_\eps}^{T+t_\eps}\phi(s) \d s \leq (k/\omega) M<M-\eps,
$$
which is a contradiction. Hence, in particular $\beta(\{x_n \}_{n\ge 1})=0$ and without loss of generality we may assume that $x_n\to x_0$ as $n\to +\infty$, for some $x_0\in \partial U$.\\
\indent Further, observe that $\Psi_{t}^{(\lma_n)}(x_n,\mu_n) \to x_0$ as $n\to +\infty$, uniformly with respect to $t\in [0,T]$, which follows from the inequality
$$
\|\Psi_{t}^{(\lma_n)}\! (x_n,\mu_n)-x_0\|\!\leq\!
\|R^{(\mu_n,\lma_n)}\! (t,0) x_n \!- \!x_0\| \!+ \!\lma_n \!\int_{0}^{t}\!\!\! \|R^{(\mu_n,\lma_n)}(t,s)F(s,\!\Psi_{s}^{(\lma_n)}\!(\!x_n, \mu_n),\!\mu_n) \| \d s
$$
and Lemma \ref{14092008-1339} (i). Note that, for any $k\geq 0$
\begin{eqnarray*}
R^{(\mu_n,\lma_n)}(T,0)^{k}x_n = R^{(\mu_n,\lma_n)}(T,0)^{k+1}x_n+
\lma_n R^{(\mu_n,\lma_n)}(T,0)^{k} \left(\int_{0}^{T} h_n(s) \d s
\right),
\end{eqnarray*}
where $h_n(s):=R^{(\mu_n,\lma_n)}(T,s) F(s,\Psi_{s}^{(\lma_n)}(x_n,\mu_n),\mu_n)$ for $s\in [0,T]$ and
$n\geq 1$. Let $\eps > 0$ be arbitrary and a sequence $(k_n)$ of positive integers be such that
$k_n \to +\infty$ and $k_n \lma_n\to \eps$ as $n\to +\infty$. Then, reasoning as before, one obtains
\begin{equation}\label{rr2}
x_n = R^{(\mu_n,\lma_n)}(T,0)^{k_n}x_n +J_n \left(\int_{0}^{T}
h_n(s) \d s\right) \quad \mbox{ for \ }n\ge 1,
\end{equation}
where $J_n:=\lma_n\sum_{k=0}^{k_n-1} R^{(\mu_n,\lma_n)}(T,0)^{k}$.
Note that, in view of Lemma \ref{14092008-1339},
\begin{align*}
h_n(s)\to F(s,x_0,\mu_0) & \quad \mbox{ as } n\to+\infty, \mbox{ uniformly for } s\in [0,T], \\
R^{(\mu_n,\lma_n)}(T,0)^{k_n}x_n \to S_{\widehat{A^{(\mu_0)}}} (\eps T) x_0 & \quad\mbox{ as } n\to +\infty \mbox{ and }\\
J_n x\to \frac{1}{T}\int_{0}^{\eps T} S_{\widehat{A^{(\mu_0)}}}(s)
x\d s & \quad \mbox{ as } n\to +\infty, \mbox{ for any } x\in E.
\end{align*}
Thus, after passing in (\ref{rr2}) to the limit with $n\to +\infty$, one has
$$
x_0=S_{\widehat{A^{(\mu_0)}}} (\eps T) x_0 + \left[\frac{1}{T}\int_{0}^{\eps T} S_{\widehat{A^{(\mu_0)}}}(\tau) d\tau\right]\left( \int_{0}^{T} F(s,x_0,\mu_0) \d s\right),
$$
which rewritten as
$$
-\frac{1}{\eps T}\left( S_{\widehat{A^{(\mu_0)}}} (\eps T) x_0 -
x_0\right) = \frac{1}{\eps T} \int_{0}^{\eps T}
S_{\widehat{A^{(\mu_0)}}}(\tau)\widehat{F}(x_0,\mu_0) \d\tau.
$$
Letting $\eps\to 0^+$ yields $-\widehat{A^{(\mu_0)}}x_0= \widehat
F(x_0,\mu_0)$, a contradiction completing the proof. \hfill
$\square$
\begin{Lem}\label{25092008-1149} {\em (see \cite[Proposition 4.3]{Cwiszewski-1})}
Let $F:E\to E$ be a completely continuous locally Lipschitz with sublinear growth and let $\Xi_t:E\to E$ be the translation along trajectories operator by time $t > 0$ for the equation $$\dot u(t) = - u(t) +F(u(t)), \quad t\in[0,T].$$
Then, for each $t > 0$, the mapping $\Xi_t$ is a $k$-set contraction and if an open bounded $U\subset E$ is such that
$0\not\in (I-F)(\partial U)$, then there exists $t_0>0$ such that for any $t\in (0,t_0]$, $\Xi_t(x)\neq x$ and
$$
\deg(I-F, U) = \deg(I-\Xi_t, U).
$$
\end{Lem}
\noindent\textbf{Proof of Theorem \ref{18092008-2220}.}
First we reduce the proof to the case where the nonlinear perturbation is compact. By Remark \ref{19072009-2137} (b) we infer that the operator $\widehat{A}$ and mapping $\widehat F$ satisfy assumptions of Lemma \ref{22092008-1201}. Therefore there is locally Lipschitz compact mapping $\widehat F_L :E \to E$ such that
\begin{equation}\label{13092009-2214}
\widehat A x + (1-\mu)\widehat F(x) + \mu \widehat F_L(x) \neq 0 \quad \mbox{ for \ } x\in
\partial U \cap D(\widehat A), \ \mu\in [0,1].
\end{equation}
Thus, applying Lemma \ref{19092008-0014} to equations associated
to
$$\dot u(t)=\lma A(t) u(t)+ \lma ((1 - \mu)\widehat F(u(t)) + \mu \widehat F_L(u(t))), \quad t\in [0,T], $$
and the homotopy invariance of the topological degree, provide \\
\noindent {\bf Claim A.} {\em There exists $\lma_1>0$ such that for any $\lma\in (0,\lma_1]$
$$
\deg(I-\Phi_{T}^{(\lma)}, U) = \deg(I-\widetilde\Phi_{T}^{(\lma)}, U),
$$
where $\widetilde \Phi_{T}^{(\lma)}:\overline U\to E$ is the
translation along trajectories operator by the time $T$ for the
equation
$$
\dot u(t) = \lma A(t)u(t) + \lma \widehat F_L (u(t)).
$$}
\indent Next we prove
\noindent {\bf Claim B.} {\em There exists $\lma_2\in (0, \lma_1]$ such that for any $\lma\in (0,\lma_2]$
\begin{equation} \label{25092008-1143}
\deg(I-\widetilde\Phi_{T}^{(\lma)}, U) = \deg(I-\overline\Phi_{T}^{(\lma)}, U),
\end{equation}
where $\overline \Phi_{T}^{(\lma)}$ is the translation along trajectories operator
by the time $T$ for the equation
$$\dot u(t) = - \lma u(t) - \lma \widehat A^{-1}\widehat F_L(u(t)), \quad t\in [0,T].$$}
To this end, consider a differential problem given by
\begin{equation}\label{22092008-2350}
\dot u(t) = \lma \widetilde A^{(\mu)}(t)u(t) + \lma \widetilde F(u(t),\mu) \quad\mbox{ on } [0,T],
\end{equation}
where
\begin{align*}
\widetilde A^{(\mu)}(t) & := -\mu I + (1-\mu) A(t) &&\hspace{-20mm} \mbox{ for \ } t\in [0,T], \\
\widetilde F(x,\mu) & := [(1-\mu)I - \mu \widehat A^{-1}]\widehat F_L(x) &&\hspace{-20mm} \mbox{ for \ } x\in E, \ \mu\in [0,1].
\end{align*}
Lemma \ref{14092008-1339} shows that the family
$\{\lambda\widetilde A^{(\mu)}(t)\}_{t\in [0,T]}$ satisfies
$(Hyp'_1)$, $(Hyp_2)$ -- $(Hyp_4)$ and so the family $\{\widetilde
A^{(\mu)}(t)\}_{t\in [0,T]}$ fulfills the assumptions of Lemma
\ref{19092008-0014}. It is also clear that $\widetilde F$ is
locally Lipschitz in $x$ uniformly with respect to $\mu$ and
compact, which, in particular, means that it has sublinear growth
uniformly with respect to $\mu$. For any $\lambda\in(0,\infty)$,
let $\Psi:\overline U\times [0,1]\to E$ be given by
$$
\Psi_{T}^{(\lma)}(x,\mu):=u(T;x,\mu,\lma), \quad\mbox{ for \ }x\in E, \ \mu\in[0,1],
$$
where $u(\, \cdot\, ;x,\mu,\lma)$ stands for the mild solution of (\ref{22092008-2350}) starting at $x$.\\
\indent Observe that
\begin{equation}\label{22092008-2359}
[(1 - \mu) \widehat A - \mu I]x + \widetilde F(x,\mu)\neq 0 \quad\mbox{ for \ }
\mu\in [0,1],\, x\in \partial U\cap D(\widetilde A^{(\mu)}).
\end{equation}
Indeed, suppose that for some $\mu\in [0,1]$ and $x\in \partial U\cap D(\widehat A)$ we have
$[(1-\widetilde \mu) \widehat A -\mu I]x + \widetilde F(x,\mu)= 0.$
If $\mu=1$ then $-x-\widehat A^{-1}\widehat F_L(x)=0$, which contradicts (\ref{13092009-2214})
and if $\mu\in [0,1)$ then
$$
x=-R(\widehat A;\mu/(1-\mu)) (I-\mu/(1-\mu)\widehat A^{-1})\widehat F_L(x),
$$
which due to the resolvent identity gives $x=-\widehat{A}^{-1} \widehat F_L(x)$,
again a contradiction proving (\ref{22092008-2359}). Thus, applying Lemma \ref{19092008-0014}
and the homotopy invariance of the topological degree to $\Psi_{T}^{(\lma)}$, we find $\lma_2\in (0, \lma_1]$ such that
for any $\lma\in (0,\lma_2]$, (\ref{25092008-1143}) holds, which ends the proof of Claim B.\\
\indent Finally, by applying Lemma \ref{25092008-1149}, one gets $\lma_0\in (0,\lma_2]$ such that for any $\lma\in (0,\lma_0]$
\begin{equation*}
\deg(I+ \widehat {A}^{-1} \widehat F_L, U) = \deg(I-\overline \Phi_{\lma T}^{(1)},U).
\end{equation*}
Combining this with (\ref{13092009-2214}), we infer that, for
$\lambda\in(0,\lambda_0]$,
\begin{align*}
\Deg(\widehat A+ \widehat F,U) & = \Deg(\widehat A + \widehat F_L, U)
= \deg(I+ \widehat {A}^{-1} \widehat F_L, U)\\
& = \deg(I-\overline \Phi_{\lma T}^{(1)},U) = \deg(I-\overline \Phi_{T}^{(\lma)}, U),
\end{align*}
which together with Claims A and B completes the proof.\hfill $\square$
As an immediate consequence of Theorem \ref{18092008-2220}, we get the following result.
\begin{Cor}
If $0\not\in (\widehat A+\widehat F) (\partial U \cap D(\widehat A))$
and $\deg(\widehat A+\widehat F, U)\neq 0$, then $(P_{T,\lambda})$
admits a solution for small $\lma>0$.
\end{Cor}
By means of {\em a priori bounds} type assumption, we get the existence criterion for periodic solutions.
\begin{Th} {\em (Continuation principle)} \label{30092008-1514}
Let a family $\{A(t)\}_{t\in [0,T]}$ and a mapping $F:[0,T]\times E\to E$ satisfy $(Hyp'_1)$, $(Hyp_2)$ -- $(Hyp_5)$ and $(F_1)$ -- $(F_3)$, respectively. If $(P_{T,\lambda})$ has no $T$-periodic points in $\partial U\times (0,1)$ and $\Deg(\widehat{A} + \widehat{F}, U)\neq 0$, then $(P)$ admits a mild solution $u:[0,T]\to E$ such that $u(0) = u(T)\in\overline U$.
\end{Th}
\noindent\textbf{Proof.} If $\Phi_T^{(1)}(x) = x$ for some $x\in\partial
U$, then the assertion holds. Hence, assume that $\Phi_T^{(1)}(x)\neq x$
for $x\in\partial U$. By Theorem \ref{18092008-2220}, there
exists $\lambda_0 \in (0,1)$ such that, for any $\lma\in (0,\lma_0]$, $\Phi_T^{(\lambda)}(x)\neq x$ and
\begin{equation}\label{22092008-2340}
\deg (I - \Phi_T^{(\lambda)}, U) = \Deg ( \widehat{A} + \widehat{F}, U).
\end{equation}
Then the mapping $\overline U\times [\lambda_0,1]\ni (x,\lambda)\mapsto \Phi_T^{(\lambda)}(x)$
provides an admissible homotopy (in the degree theory of $k$-set contraction vector fields)
and by the homotopy invariance
$$
\deg(I - \Phi_T^{(1)}, U) = \deg(I - \Phi_T^{(\lambda_0)},U),
$$
which together with (\ref{22092008-2340}) and the assumption implies the existence of
$x\in U$ such that $\Phi_{T}^{(1)}(x)=x$.\hfill $\square$
The above Continuation Principle can be useful when studying asymptotically linear evolution systems.
\begin{Th}\label{24102008-0927}
Let a family $\{A(t)\}_{t\in [0,T]}$ satisfy $(Hyp'_1)$ and $(Hyp_2)$ -- $(Hyp_5)$
and let $F:[0,T]\times E\to E$ be a completely continuous mapping satisfying $(F_1)$, $(F_2)$ and $(F_4)$. Assume also that $\{F_\infty (t):E\to E \}_{t\in [0,T]}$ is a family of compact linear operators such that the mapping $t\mapsto F_\infty (t)\in {\cal L} (E,E)$ is continuous on $[0,T]$, $F_\infty (0) = F_\infty (T)$ and
\begin{equation}\label{13072009-2241}
\lim_{\|x\|\to +\infty} \frac{\| F(t,x)-F_\infty(t)x\|}{\|x\|} = 0 \quad\mbox{ uniformly with respect to } t\in [0,T].
\end{equation}
If, for each $\lambda \in (0,1]$, the parameterized linear periodic problem
\begin{equation}\label{23012009-1323}
\left\{ \begin{array}{l}
\dot u(t)=\lambda (A(t)+F_\infty (t))u(t), \quad t\in [0,T]\\
u(0)=u(T)
\end{array}\right.
\end{equation}
has no nontrivial solution and $\Ker ({\widehat A}+ {\widehat F_\infty }) =\{ 0 \}$, then $(P)$ admits a $T$-periodic mild solution.
\end{Th}
\textbf{Proof.} We begin with proving that there exists $R_1>0$ such that
$0\not\in (\widehat A + \widehat F) ((E\setminus B(0,R_1)) \cap D(\widehat A))$ and
\begin{equation}\label{26102008-2220}
\left|\Deg(\widehat A+\widehat F, B(0,R_1) ) \right| = 1.
\end{equation}
Define $H:[0,T] \times E \times [0,1] \to E$ by
$$
H(t,x,\lma):= \left\{\begin{array}{ll}
\lambda F(t,\lambda^{-1} x) & \mbox{ for \ } t\in [0,T], \ x\in E, \ \lambda\in (0,1],\\
F_\infty (t)x & \mbox{ for \ } t\in [0,T], \ x\in E, \ \lambda=0.
\end{array}\right.
$$
Standard arguments show that both $H$ and $\widehat H$ are
completely continuous and for any $\lambda_0\in [0,1]$
\begin{equation}\label{26102008-2214}
\lim_{\|x\|\to + \infty,\, \lma\to \lma_0} \frac{\|\widehat H(x,\lma)-\widehat F_\infty x \|}{\|x\|}=0.
\end{equation}
Observe that there is $R_1>0$ such that
\begin{equation}
\widehat A x + \widehat H(x,\lambda)\neq 0
\ \ \mbox{ for \ } x\in (E\setminus B(0,R_1))\cap D(\widehat A), \ \lambda \in [0,1].
\end{equation}
Otherwise there are $(x_n)$ in $E$ and $(\lma_n)$ in $[0,1]$ such that
$\widehat A x_n + \widehat H (x_n, \lma_n)=0$ and $\|x_n\|\to +\infty$.
Put $z_n:=x_n/\|x_n\|$. If $\lma_n=0$ for some $n\geq 1$, then
$z_n = - \widehat A^{-1} \widehat F_\infty z_n$,
a contradiction to the assumption. If $(\lma_n)$ in $(0,1]$, then
\begin{equation}\label{13072009-2237}
z_n= - \|x_n\|^{-1}{\widehat A}^{-1}\widehat H(x_n, \lma_n) = - {\widehat A}^{-1} (\lma_n^{-1}
\|x_n\|)^{-1} \widehat F(\lma_{n}^{-1} \|x_n\| z_n).
\end{equation}
Since $\lim_{\|x\|\to +\infty} \|\widehat F(x)-\widehat F_\infty x\|/\|x\| = 0$ and $\rho_n := \lma_n^{-1} \|x_n\| \to +\infty$ as $n\to\infty$,
\begin{equation}\label{20072009-0116}
\|z_n + \widehat A^{-1} \widehat F_\infty z_n\| \le \|\widehat A^{-1}\|
\|\widehat F(\rho_n z_n) - \widehat F_\infty (\rho_n z_n)\| / \rho_n \to \infty
\quad \mbox{ as \ }n\to \infty.
\end{equation}
By Lemma \ref{19092008-0029} the linear operator $\widehat
F_\infty$ is compact, which together with (\ref{20072009-0116})
means that $(z_n)$ contains a convergent subsequence. Hence, we
may assume that $z_n\to z_0$ for some $z_0\in E$ and by
(\ref{20072009-0116}) we infer that $z_0= -\widehat A^{-1}
\widehat F_\infty z_0$, which is again a contradiction
meaning that (\ref{26102008-2220}) holds for sufficiently large $R_1>0$.\\
\indent Now we claim that
\begin{equation} \label{23012009-1328}\mbox{\parbox[t]{130mm}{
there is $R\geq R_1$ such that, for any $\lma\in (0,1)$, the problem $(P_{T,\lma})$
has no periodic solutions starting at points from $\partial B(0,R)$.} }
\end{equation}
Otherwise there exist
$(u_n)$ and $(\lma_n)$ in $(0,1)$ such that, for each $n\geq 1$, $u_n$ is a solution of $(P_{T, \lma_n})$ and $\|u_n(0)\| \to +\infty$ as $n\to +\infty$. Putting $v_n:=u_n / \|u_n\|_{\infty}$, where
$\|u_n\|_\infty:=\max_{t\in [0,T]} \|u_n(t)\|$, one has
\begin{equation}\label{25012009-2112}
v_n (t)= R^{(\lma_n)}(t,0) v_n(0) + \lma_n \|u_n\|_{\infty}^{-1}
\int_{0}^{t} R^{(\lma_n)} (t,s) F ( s , \|u_n\|_{\infty} v_n (s) ) \d s.
\end{equation}
Note that, by (\ref{13072009-2241}) and the fact that $F$ is a completely continuous mapping,
for any $\eps>0$, there is $m_\eps\geq 0$ such that
$\| F(t,x)-F_\infty(t)x\| \leq \eps\|x\| + m_\eps$ for $x\in E$.
Consequently, for each $\eps > 0$, there exists $n_\eps\geq 1$ such that, for any $n\geq n_\eps$ and $s\in [0,T]$,
\begin{equation}\label{05022009-1319}
\ \ \left\| \|u_n\|_\infty^{-1} F(s, \|u_n\|_{\infty} v_n(s))- F_\infty(s) v_n(s) \right\| \leq \|u_n\|_{\infty}^{-1} (\eps \| u_n \|_\infty \|v_n(s)\| + m_\eps) \leq 2\eps.
\end{equation}
For each $n\geq 1$, put $h_n(s):= \|u_n\|_{\infty}^{-1} F ( s , \|u_n\|_{\infty} v_n (s) )$ for $s\in[0,T]$. If $s\in [0,T]$, then using (\ref{05022009-1319}) and the compactness of $F_\infty(s)$, for arbitrary $\eps > 0$, we deduce that
$$
\beta(\{ h_n(s) \}_{n\ge 1})\le \beta(F_\infty(s)(\{v_n(s)\}_{n\ge 1})) + 2\eps = 2\eps\quad\mbox{ for \ }s\in[0,T],
$$
which, by passing to the limit with $\varepsilon\to 0$, gives $\beta(\{ h_n(s) \}_{n\ge 1}) = 0$. Obviously, we may also assume that $\lma_n\to\lma_0$ for some $\lma_0\in [0,1]$. \\
\indent If $\lma_0=0$, then note that
\begin{equation}\label{20072009-0255}
v_n(0)=R^{(\lma_n)}(T,0)^{k_n} v_n(0) + \left[\lma_n \sum_{k=0}^{k_n-1} R^{(\lma_n)} (T,0)^{k}\right] \left( \int_{0}^{T} R^{(\lma_n)}(T,s)h_n(s) \d s\right),
\end{equation}
where $(k_n)$ is an arbitrary sequence of positive integers.
If we put $k_n:=[T/\lma_n]$ for $n\geq 1$, then
$k_n\lambda_n \to T$ as $n\to +\infty$ and, in view of Theorem \ref{29082008-1821} and Lemma \ref{19092008-0012}, we get
\begin{eqnarray*}
\beta(\{ v_n(0) \}_{n\ge 1}) \leq e^{-\omega T}\beta(\{ v_n(0) \}_{n\ge 1})
+ \frac{1-e^{-\omega T}}{\omega} \beta\left(\left\{ \int_{0}^{T} R^{(\lma_n)}(T,s)h_n(s) \d s\right\}_{n\ge 1}\right) \\
\le e^{-\omega T}\beta(\{ v_n(0) \}_{n\ge 1}) +
(1-e^{-\omega T})\omega^{-1}\int_{0}^{T} \beta(\{ h_n(s) \}_{n\ge 1}) \d s
= e^{-\omega T}\beta(\{ v_n(0) \}_{n\ge 1}).
\end{eqnarray*}
In consequence $\beta(\{v_n(0)\}_{n\ge 1})=0$. Furthermore, by (\ref{25012009-2112}) and Lemma \ref{19092008-0012}, for any $t\in [0,T]$, one has
\begin{align*}
\beta(\{v_n (t)\}_{n\ge 1}) & \le \beta(\{R^{(\lma_n)}(t,0) v_n(0)\}_{n\ge 1})
+ \lma_n \int_{0}^{t} \beta(\{R^{(\lma_n)} (t,s) h_n(s)\}_{n\ge 1}) \d s \\
& \le \beta(\{v_n(0)\}_{n\ge 1}) +
\int_{0}^{t} \beta\left(\left\{\lma_n h_n(s)\right\}_{n\ge 1}\right) ds = 0,
\end{align*}
i.e. $\beta(\{ v_n (t)\}_{n\ge 1})=0$ for $t\in[0,T]$.
Since, by (\ref{05022009-1319}), the set $\{h_n\}_{n\ge 1}$ is bounded in $C([0,T],E)$, applying Proposition \ref{30122008-1629} (ii), we infer that $\{ v_n\}_{n\ge 1}$ is relatively compact in
$C([0,T],E)$ and without loss of generality, we assume that $v_n\to v_0$ in $C([0,T],E)$
and $v_n(0) \to x_0:=v_0 (0)$.
Furthermore, by (\ref{05022009-1319}), for arbitrary $\eps > 0$, there exists $n_\eps \geq 1$ such that, for any $n\geq n_\eps$ and $t\in [0,T]$,
\begin{align*}\label{20072009-0249}
\|v_n (t) - R^{(\lma_n)}(t,0) v_n(0)\| & \leq
\lma_n \int_{0}^{t} \|R^{(\lma_n)} (t,s) F_\infty(s) v_n (s)\| \d s+ 2\lma_n \eps T \\
& \leq \lma_n K \int_{0}^{t} \|v_n (s)\| \d s +2\lma_n \eps T \to 0 \quad\mbox{ as }n\to \infty,
\end{align*}
where $K:=\sup_{\tau\in[0,T]}\|F_\infty(\tau)\|$. This together with Lemma \ref{14092008-1339} (i) imply that $v_0(t) = x_0$ for any $t\in E$ and in particular $x_0 \neq 0$, since $\|v_0\|_\infty = 1$. Hence, in view of (\ref{05022009-1319}) we find that
\begin{equation}\label{20072009-0305}
h_n(s) \to F_{\infty} (s)x_0 \quad \mbox{ as }n\to \infty \mbox{ uniformly for } s\in [0,T].
\end{equation}
Now fix an arbitrary $\eps>0$ and take any sequence $(k_n)$ of positive integers such that
$k_n \lma_n \to \eps$.
Applying Lemma \ref{14092008-1339} (ii), (iii) and (\ref{20072009-0305}) and passing to the limits in (\ref{20072009-0255}), we obtain
$$
x_0 = S_{\widehat A} (\eps T) x_0 + \left[ \frac{1}{\eps T} \int_{0}^{\eps T} S_{\widehat A}(\tau) d\tau \right] \left( \int_{0}^{T} F_\infty(s) x_0 \d s \right),
$$ i.e.
$$
-\frac{1}{\eps T} \left(S_{\widehat A} (\eps A) x_0 - x_0\right) = \frac{1}{\eps T}\int_{0}^{\eps T} S_{\widehat A }(\tau) \widehat F_\infty x_0 d\tau,
$$
Hence, a passing to the limit with $\eps\to 0$ yields $x_0\in D(\widehat A)$ and $(\widehat A + \widehat F_\infty)x_0=0$, a contradiction proving (\ref{23012009-1328}) in the case $\lma_0 = 0$.\\
\indent If $\lma_0\in (0,1]$, then, by (\ref{25012009-2112}) and Lemma \ref{19092008-0012},
\begin{align*}
& \beta(\{v_n(0)\}_{n\ge 1}) \le e^{-\lma_0 \omega T}\beta(\{v_n(0)\}_{n\ge 1}) +
\lambda_0\beta\left(\left\{ \int_{0}^{T} R^{(\lma_n)}(T,s)h_n(s) \d s\right\}_{n\ge 1}\right) \\
& \le e^{-\lma_0 \omega T}\beta(\{v_n(0)\}_{n\ge 1}) +
\int_{0}^{T} \lma_0 e^{-\lma_0 \omega (T-s)}\beta\left(\left\{ h_n(s)\right\}_{n\ge 1}\right) ds = e^{-\lma_0 \omega T}\beta(\{v_n(0)\}_{n\ge 1})
\end{align*}
and, consequently, $\beta(\{v_n(0)\}_{n\geq 1})=0$. Using again
(\ref{25012009-2112}), for all $t\in [0,T]$,
\begin{equation*}
\beta(\{v_n(t)\}_{n\ge 1}) \le e^{-\lma_0 \omega t}\beta(\{v_n(0)\}_{n\ge 1}) +
\int_{0}^{t} \lma_0 e^{-\lma_0 \omega (t-s)}\beta\left(\left\{ h_n(s)\right\}_{n\ge 1}\right) ds = 0,
\end{equation*}
which gives $\beta(\{ v_n(t)\}_{n\ge 1})=0$ for any $t\in [0,T]$.
Hence, due to the boundedness of $\{h_n\}_{n\ge 1}$ in $C([0,T],E)$ and Proposition \ref{30122008-1629} (ii), it follows that $\{ v_n\}_{n\ge 1}$ is relatively compact in $C([0,T],E)$ and, without loss of generality, we assume that $v_n\to v_0$ in $C([0,T],E)$.
Then, using (\ref{25012009-2112}), (\ref{05022009-1319}) and Proposition \ref{14012008-1812},
we infer, that for any $t\in [0,T]$,
$$
v_0(t)= R^{(\lma_0)} (t,0)v_0 (0) + \lma_0 \int_{0}^{t} R^{(\lma_0)} (t,s) F_\infty (s) v_0(s) \d s,
$$
i.e. $v_0$ is a nontrivial mild solution of (\ref{23012009-1323}) with $\lma=\lma_0\in(0,1]$,
which is a contradiction proving (\ref{23012009-1328}). \\
\indent Finally, (\ref{26102008-2220}) and (\ref{23012009-1328})
allow us to apply Theorem \ref{30092008-1514} to finish the proof.
\hfill $\square$
\section{An application to hyperbolic partial differential equations}
We end the paper with an example of a periodic problem for the
hyperbolic evolution equation with a time-dependent damping term.
Suppose that $\Omega$ is an open bounded subset of $\R^N$ and
$A:D(A)\to E$ is a positive self-adjoint linear operator with
compact resolvents defined on a Hilbert space $X:=L^2(\Omega)$
with the scalar product and the corresponding norm denoted
by $(\cdot,\cdot)_0$ and $|\cdot|_0$, respectively. It is well known that
such $A$ determines its fractional power space $X^{1/2}$ being a Hilbert
space as well. If we denote the scalar product and the corresponding norm by $(\cdot,\cdot)_{1/2}$ and
$|\cdot|_{1/2}$, respectively, then it is known that
$$
|u|_{1/2} \ge \lambda_1^{1/2}|u|_0 \quad \mbox{ for any } u\in X^{1/2}
$$
where $\lma_1>0$ is the smallest eigenvalue of $A$. Typical examples of $A$ satisfying
these conditions is $-\Delta_D$, where $\Delta_D $ is the Laplacian operator with zero the Dirichlet
boundary conditions or $-\Delta_N + \alpha I$, where $\Delta_N$ is the Laplacian
operator with the zero Neumann boundary conditions and $\alpha >0$.\\
\indent Consider a periodic problem
\begin{equation}\label{08082009-0110}
\left\{
\begin{array}{ll}
u_{tt} (x,t) + \beta (t)u_t (x,t) + (A u)(x, t) + f(t,u(x,t)) = 0 & \quad \mbox{ in } \Omega\times(0,T] \\
u(x,0)=u(x,T), \ u_t (x,0) = u_t (x,T) & \quad \mbox{ on }
\partial\Omega,
\end{array}
\right.
\end{equation}
where $\beta:[0,T]\to \R$ is a $T$-periodic continuously
differentiable function such that $\beta(t)>0$, for $t\in [0,T]$ and
$f:[0,T]\times \R \to \R$ is a continuous function satisfying the following properties
\begin{align}\label{z1}
& \mbox{there is } L>0 \mbox{ such that } |f(t,s_1)\! -\! f(t,s_2)| \leq L|s_1 \!-\! s_2| \mbox{ for \ }t\!\in\! [0,T], \ s_1, s_2\in\mathbb{R}, \\ \label{z2}
& \mbox{there is } c>0 \mbox{ such that } |f(t,s)| \leq c (1+|s|) \mbox{ for \ } t\in[0,T], \ s\in\mathbb{R},\\ \label{z3}
& f(0,s)=f(T,s) \mbox{ for \ } s\in\mathbb{R}, \\ \label{z4}
& \lim_{|s| \to +\infty} \displaystyle{\frac{f(t, s)}{s}} =
f_{\infty} \mbox{ uniformly with respect to } t\in [0,T],
\end{align}
for some $f_{\infty}\in \R\setminus \sigma (A)$. If we define
$N_f:[0,T] \times X\to X$ by $N_f (t,u)(x):= f(t,u(x))$ for a.e.
$x\in \Omega$ and $t\in[0,T]$, then (\ref{08082009-0110}) can be
rewritten as a system
$$
\left\{
\begin{array}{ll}
\dot u(t) = v(t)\\
\dot v(t) = - A u(t) - \beta (t) v(t) - N_f (t,u(t)), \quad\mbox{ for \ }t\in [0,T]
\end{array}
\right.
$$
and in a matrix form as
\begin{equation}\label{30062009-1633}
\dot z(t) = {\bold A} (t) z(t)+ {\bold F} (t,z(t)), \quad\mbox{ for \ }t\in [0,T]
\end{equation}
with operators ${\bold A}(t):D({\bold A}(t))\to {\bold E}$, $t\in [0,T]$, on the separable Banach space ${\bold E}:=X^{1/2} \times X$,
defined by
\begin{align}\label{06082009-1922}
D({\bold A} (t)) & := X^{1}\times X^{1/2} && \hspace{-15mm}\mbox{ for \ } t\in [0,T],\\
{\bold A} (t) (u, v) & :=(v,-Au-\beta (t)v) && \hspace{-15mm}\mbox{ for \ } t\in [0,T], \ (u, v) \in D({\bold A} (t)) \label{06082009-1923}
\end{align}
and ${\bold F}:[0,T] \times {\bold E}\to {\bold E}$ given by
${\bold F}\left(t, (u, v )\right) := (0, -N_f (t,u) )$ for $t\in [0,T]$, $(u, v) \in \textbf{E}$. \\
\indent We claim that the family $\{ {\bold A}(t)\}_{t\in [0,T]}$ and the map ${\bold F}$ satisfy the assumptions of
Theorem \ref{24102008-0927} provided ${\bold E}$ is endowed with a proper norm.
To this end, for $\eta > 0$, define a new scalar product
$(\cdot,\cdot)_{{\bold E},\eta}:{\bold E}\times {\bold E} \to\R$, by
$$
\left( (u_1,v_1), (u_2, v_2)\right)_{{\bold E},\eta}: = (u_1,u_2)_{1/2} + (v_1+\eta u_1, v_2 +\eta u_2)_0.
$$
Clearly it is a well-defined scalar product and the corresponding
norm $\| \cdot \|_{{\bold E},\eta}$ is equivalent to the usual
product norm $\|\cdot\|$ in ${\bold E}=X^{1/2}\times X$. Let
$\beta_0 > 0$ be such that $\beta(t)\ge \beta_0$ for $t\in [0,T]$.
Putting $\gamma:=\max_{t\in [0,T]} \lambda_{1}^{-1/2}(\beta(t) +
1)$ for $0<\eta\leq 1$, one has
\begin{align*}
\left( {\bold A}(t)(u,v),(u, v)\right)_{{\bold E},\eta}
& = (v,u)_{1/2}+(- A u - \beta(t)v+\eta v, v + \eta u)_0 \\
& \hspace{-35mm} = (v,u)_{1/2} -(Au,v)_0 - (\beta(t)v,v)_0 + \eta|v|_{0}^2 - \eta (Au,u)_0 - \eta(\beta(t)v,u)_0 + \eta^2 (v,u)_0\\
& \hspace{-35mm} \leq -\eta|u|_{1/2}^{2} - (\beta_0-\eta)|v|_0^2 + \eta(\beta(t) + 1)|(v,u)|_0 \\
& \hspace{-35mm}\leq -\eta|u|_{1/2}^{2} - (\beta_0 - \eta)|v|_0^2 + \eta\gamma |u|_{1/2}|v| \\
& \hspace{-35mm}\leq -\eta|u|_{1/2}^{2} - (\beta_0 - \eta)|v|_0^2 + (\eta/2) |u|_{1/2}^{2}+ (\eta\gamma^2/2)|v|_{0}^{2}\\
& \hspace{-35mm} = -(\eta/2)|u|_{1/2}^{2} -(\beta_0 - \eta - \eta \gamma^2/2)|v|_{0}^{2},
\end{align*}
and therefore, decreasing $\eta>0$ if necessary, there exists
$\omega=\omega(\eta)>0$ such that, $({\bold
A}(t)(u,v),(u,v))_{{\bold E},\eta} \leq -\omega \|(u,v)\|_{{\bold
E},\eta}^{2}$ for any $(u,v)\in D({\bold A} (t)) = X^{1}\times
X^{1/2}$. Since $\Im \, {\bold A}(t) = {\bold E}$, for $t\in
[0,T]$, we infer that, for $t\in [0,T]$, the operator ${\bold
A}(t)$ is a generator of $C_0$ semigroup satisfying
$$
\|S_{{\bold A}(t)}(s)\|_{{\bold E},\eta} \le e^{-\omega s}
\quad\mbox{ for } s \ge 0
$$
and, in particular, condition $(Hyp'_1)$ holds.
Moreover, observe that, for each $(u,v)\in X^{1}\times X^{1/2}$, the map $t \mapsto {\bold A}(t)(u,v)\in {\bold E}$ is continuously differentiable on $[0,T]$ as $\beta$ is so. Hence, in view of Proposition \ref{27022009-1214}, the family
$\{{\bold A}(t)\}_{t\geq 0}$ satisfies also conditions $(Hyp_2)$ and $(Hyp_3)$ with
${\bold V}:=X^{1}\times X^{1/2}$ equipped with
the norm given by $\|(u,v)\|_{{\bold V}}:= \|{\bold A(0)}(u,v)\|_{{\bold E},\eta}
+ \|(u,v)\|_{{\bold E},\eta}$ for $(u,v)\in {\bold V}$. By the periodicity of $\beta$, $(Hyp_5)$ holds.
Furthermore, observe that for $(u,v)\in {\bold V}=X^{1}\times X^{1/2}$
$$
{\bold A}_0 (u,v):= \frac{1}{T}\int_{0}^{T} {\bold A}(\tau)(u,v) d \tau = (v,-A u - \widehat \beta v)
$$
where $\widehat\beta:=(1/T)\int_{0}^{T}\beta(\tau)d\tau$ and $\Im \, {\bold A}_0 = {\bold E}$ which implies $(Hyp_4)$.
It can be easily verified that ${\bold A}_0$ is closed and consequently, $\widehat {\bold A} = {\bold A}_0$.
Since $\beta$ is $T$-periodic function, we conclude that $(Hyp_5)$ is also satisfied.\\
\indent It may be checked that ${\bold F}$ is continuous and, by
(\ref{z1}), (\ref{z2}) and (\ref{z2}), satisfies conditions
$(F_1)$, $(F_2)$ and $(F_4)$. Since the operator $A$ has compact
resolvents, the inclusion $X^{1/2}\subset X$ is compact and
therefore, both ${\bold F}$ and ${\bold F}_\infty:{\bold E}\to
{\bold E}$, given by ${\bold F}_{\infty} (u,v):=(0, - f_\infty
u)$, are completely continuous. Furthermore observe that
\begin{equation}\label{30072009-2032}
\limsup_{\|(u,v) \|\to \infty,\, t\to t_0} \frac{\|{\bold F}(t,(u,v)) -
{\bold F}_\infty (u,v)\|}{\|(u,v)\|} \leq \limsup_{|u|_{1/2}\to
\infty,\, t\to t_0} \frac{|N_f(t,u)-f_{\infty}u|_{0}}{|u|_{1/2}}.
\end{equation}
Now suppose that $(u_n)$ is a sequence in $X^{1/2}$ such that
$|u_n|_{1/2} \to +\infty$ as $n\to+\infty$ and $(t_n)$ in $[0,T]$
is such that $t_n\to t_0$ as $n\to +\infty$. If we put
$z_n:=u_n/|u_n|_{1/2}$ for $n\geq 1$, then by the compactness of
the inclusion $X^{1/2}\subset X$, there exists subsequence
$(z_{n_k})$ and $z_0\in X$ such that $z_{n_k} \to z_0$ in $X$ as
$k\to +\infty$. Without lost of generality we may assume that
$z_{n_k}(x) \to z_0(x)$, a.e. on $\Omega$, and there is $g\in X$
such that, for each $k\ge 1$, $|z_{n_k}(x)| \leq g(x)$ a.e. on
$\Omega$. Then, putting $\mu_n:= |u_n|_{1/2}$, by (\ref{z4}), we
find that
$$
|\mu_{n_k}^{-1} f(t_{n_k},x,\mu_{n_k} z_{n_k}(x)) - f_{\infty}z_{n_k} (x)|^{2} \to 0 \quad
\mbox{ a.e. on }\Omega, \ \mbox{ as }k\to \infty$$
and, for each $k\geq 1$,
\begin{align*}
|\mu_{n_k}^{-1} f(t_{n_k},x,\mu_{n_k} z_{n_k}(x)) -
f_{\infty}z_{n_k} (x)|^{2} \le g_0(x) \quad \mbox{ a.e. on } \Omega
\end{align*}
with $g_0 := (c(m + g) + f_\infty g)^2$ where $m:=\sup \{
\mu_{n}^{-1} \mid n\geq 1 \}$. Since $\Omega$ is a bonded set,
$g_0$ is integrable and by the Lebesgue dominated convergence
theorem
$$
|N_f(t_{n_k},u_{n_k})-f_\infty u_{n_k}|_0/|u_{n_k}|_{1/2} =
\int_{\Omega} |\mu_{n_k}^{-1} f(t_{n_k},x,\mu_{n_k} z_{n_k}(x)) -
f_{\infty}z_{n_k} (x)|^{2} dx \to 0
$$
as $k\to +\infty$, which together with (\ref{30072009-2032}), implies that
$$
\lim_{\|z \|\to \infty,\, t\to t_0} \frac{\|{\bold F}(t,z) - {\bold
F}_\infty z\|}{\|z\|} = 0
$$
and condition (\ref{13072009-2241}) is satisfied.\\
The following lemma will be helpful in verifying that (\ref{23012009-1323}) has no
nontrivial $T$-periodic solutions for $\lma\in (0,1)$.
\begin{Lem}\label{06082009-2014}
Let $A:D(A)\to E$ be a positive self-adjoint operator with compact
resolvents on a Hilbert space $X$ and, for fixed $\bar \beta>0$,
${\bold A}:D({\bold A})\to {\bold E}$ be an linear operator on ${\bold E}:= X^{1/2} \times X^{0}$ given by
$D({\bold A}):= X^{1}\times X^{1/2}$ and
$$
{\bold A}(u,v):= (v, - Au-\bar \beta v) \quad\mbox{ for \ } (u,v)\in D({\bold A}).
$$
Let ${\bold E}_k:= X_k \times X_k$, $k\geq 1$, where $X_k$ is the space spanned by the first $k$ eigenvectors of $A$ (corresponding to the first $k$ smallest eigenvalues of $A$), and ${\bold A}_k:{\bold E}_k\to {\bold E}$ be given by
$$
{\bold A}_k(u,v):={\bold A}(u,v) \quad\mbox{ for \ } (u,v)\in {\bold E}_k.
$$
Then\\
\pari{(i)}{${\bold A} ({\bold E_k})\subset {\bold E}_k$ for each $k\geq 1$;}\\
\pari{(ii)}{$R(\mu;{\bold A}_k)(u,v)= R(\mu;{\bold A}) (u,v)$ for any $k\geq 1$ and $(u,v)\in {\bold E}_k$ and $\mu>0$;}\\
\pari{(iii)}{$S_{{\bold A}_k}(t)(u,v) = S_{{\bold A}}(t)(u,v)$ for any $k\geq 1$ and $(u,v)\in {\bold E}_k$.}
\end{Lem}
{\bf Proof. } (i) comes straightforwardly from the fact that $A(X_k)\subset X_k$ for $k\geq 1$.\\
\indent (ii) If $(p,q)\in {\bold E}_k$ and $(u,v):=R(\mu;{\bold A}_k)(p,q)$ for some $\mu>0$, then
$\mu u - v = p$ and $Au+(\mu+\bar \beta)v=q$, i.e. $\mu(\mu+\bar \beta)u+Au = (\mu+\bar \beta)p+ q\in X_k$, which shows that
$u\in X_k$ and $v=\mu u-p\in X_k$. Therefore, $\mu(u,v)-{\bold A}_k(u,v)= \mu (u,v)-{\bold A}(u,v)=(p,q)$,
that is $R(\mu;{\bold A})(p,q)=(u,v)= R(\mu;{\bold A}_k)(p,q)$.\\
\indent (iii) follows from the Euler formula $S_{\bold A}(t)(u,v) = \lim_{n\to +\infty} (n/t)^{n}R(n/t; {\bold A})^{n} (u,v)$, $(u,v)\in {\bold E}_k$,
and (ii). \hfill $\square$
\begin{Lem}\label{07082009-1207}
Let $\{ {\bold A}(t)\}_{t\in [0,T]}$ be given by {\em
(\ref{06082009-1922}), (\ref{06082009-1923})} and $\{{\bold A}_k
(t)\}_{t\in [0,T]}$, $k\geq 1$, be given by ${\bold A}_k (t)
(u,v):= {\bold A}(t)(u,v)$ for $(u,v)\in {\bold E}_k$. If
$\{{\bold R}(t,s) \}_{0\leq s \leq t\leq T}$ and $\{{\bold
R}^{(k)} (t,s)\}_{0\leq s\leq t\leq T}$, $k\geq 1$ are the
evolution systems determined by $\{ {\bold A}(t)\}_{t\in [0,T]}$
and $\{{\bold A}_k (t):{\bold E}_k\to {\bold E}_k\}_{t\in [0,T]}$,
$k\geq 1$, respectively, then, for any $k\geq 1$ and $(u,v)\in
{\bold E}_k$,
$$
{\bold R}^{(k)}(t,s) (u,v) = {\bold R}(t,s)(u,v) \quad\mbox{ for \ } 0\leq s\leq t\leq T.
$$
\end{Lem}
{\bf Proof}. By the construction of evolution systems (see \cite[Ch. 5, Theorem 3.1]{Pazy}), for any $(u,v)\in {\bold E}$,
\begin{equation}\label{06082009-2044}
{\bold R}(t,s) (u,v) = \lim_{n\to +\infty} {\bold R}_{n} (t,s) (u,v) \quad\mbox{ for \ } 0\leq s\leq t\leq T,
\end{equation}
where ${\bold R}_{n} (t,s):{\bold E}\to {\bold E}$, $n\geq 1$ are given by
$$ {\bold R}_{n} (t,s)\!:= \!\!\left\{
\begin{array}{ll}
\!\! S_{{\bold A}(t_j^{n})}(t\!-\!s) & \mbox{ if } s,t\in [t_{j}^{n}, t_{j+1}^{n}], s \le t,\\
\!\! S_{{\bold A}(t_r^{n})}(t\!-\!t_{r}^{n})\!\left
(\prod\limits_{j=l+1}^{r-1} \!\!S_{{{\bold A}(t_j^{n})}}(T/n)\!
\right)\!S_{ {\bold A}(t_l^{n}) }( t_{l+1}^{n}\!-\!s)& \mbox{ if }
l < r \mbox{ and } s\in [t_{l}^{n}, t_{l+1}^{n}], \\ & \ \ \ \
t\in [t_{r}^{n}, t_{r+1}^{n}]
\end{array}
\right.$$
with $t_{j}^{n}:=(j/n)T$ for $j=0,1,\ldots, n$.
Similarly, for any $k\geq 1$ and $(u,v)\in {\bold E}_k$,
\begin{equation}\label{06082009-2045}
{\bold R}^{(k)} (t,s) (u,v) = \lim_{n\to +\infty} {\bold R}_{n}^{(k)} (t,s) (u,v) \quad\mbox{ for \ } 0\leq s\leq t\leq T.
\end{equation}
where
$$
{\bold R}_{n}^{(k)} (t,s)\!:= \!\!\left\{
\begin{array}{ll}
\!\! \!\!S_{{\bold A}_k (t_j^{n})}(t\!-\!s) &\!\!\!\!\mbox{if } s,t\in [t_{j}^{n}, t_{j+1}^{n}], \, s\le t,\\
\!\!\! \!S_{{\bold A}_k
(t_r^{n})}(t\!-\!t_{r}^{n})\!\left(\prod\limits_{j=l+1}^{r-1}
\!\!S_{{{\bold A}_k(t_j^{n})}}(T/n)\! \right)\!S_{ {\bold
A}_k(t_l^{n}) }( t_{l+1}^{n}\!\!-\!s) & \!\!\!\!\mbox{if } l < r
\mbox{ and } s\in [t_{l}^{n}, t_{l+1}^{n}], \\ & \ t\in
[t_{r}^{n}, t_{r+1}^{n}].
\end{array}
\right.
$$
Lemma \ref{06082009-2014} (iii) states that $S_{{\bold
A(t_{n}^{j})}} (s) (u,v) = S_{{\bold A_k(t_{n}^{j})}} (s)(u,v)$
for $k\geq 1$, $(u,v)\in {\bold E}_k$, $s\geq 0$, $n\geq 1$ and
$j\in \{1,\ldots, n \}$. Hence, using the formulae
(\ref{06082009-2044}) and (\ref{06082009-2045}) completes the
proof. \hfill $\square$
\begin{Lem} {\em (cf. \cite{Lad}, \cite{Rybakowski})}\label{30062009-1545}
If $f:[0,T]\to X^{0}$ is continuous and
$(u,v):[0,T]\to X^{1/2}\times X^{0}$ is a mild solution of
$$
(u(t),v(t))' = {\bold A}(t) (u(t),v(t)) + (0,f(t)), \quad
t\in[0,T],
$$
then
\begin{eqnarray}\label{07082009-1309}
& & \frac{1}{2} \frac{d}{d t} |u(t)|^2_0 = (u(t),v(t))_0 \quad \mbox{ for \ } t\in [0,T], \\
\label{06082009-2221} & & \frac{1}{2} \frac{d}{d t} \left( |u
(t)|_{1/2}^{2} + |v(t)|_{0}^{2}\right) = - \beta(t) |v
(t)|_{0}^{2} + (f(t),v(t))_0 \quad \mbox{ for \ } t\in [0,T].
\end{eqnarray}
\end{Lem}
{\bf Proof}. Let $(u,v):[0,T]\to {\bold E}$ be a mild solution of
$$
\dot z (t) = {\bold A}(t) z(t) + (0, f(t)), \quad t\in [0,T].
$$
Put $(\overline u_k, \overline v_k):= (\widetilde P_k u(0), P_k v(0))$ for $k\geq 1$,
where $\widetilde P_k:X^{1/2}\to X_k$ and $P_k:X^0 \to X_k$ are the orthogonal projections.
Furthermore, let $(\widetilde u_k, \widetilde v_k):[0,T]\to {\bold E}_k$ be the mild solution of
\begin{equation}\label{06082009-2212}
\left\{
\begin{array}{ll}
(\dot u(t),\dot v(t))={\bold A}_k(t)(u(t),v(t)) + (0, P_k f(t)), \quad t\in [0,T]\\
(u(0),v(0))= ( \overline u_k, \overline v_k )
\end{array}\right.
\end{equation}
and for each $k\geq 1$ define $( u_k, v_k):[0,T]\to {\bold E}$ by
$(u_k(t), v_k(t)):=(\widetilde u_k(t), \widetilde v_k(t))$ for
$t\in [0,T]$. Then
$$
(\widetilde u_k(t), \widetilde v_k(t))= {\bold R}^{(k)}(t,0)
(\overline u_k, \overline v_k) + \int_{0}^{t} {\bold R}^{(k)}
(t,s) (0,P_k f(s)) \d s \quad \mbox{ for } t\in [0,T],
$$
and by Lemma \ref{07082009-1207}, one gets
$$
(u_k(t), v_k(t)) = {\bold R} (t,0) (\overline u_k, \overline v_k)
+ \int_{0}^{t} {\bold R}(t,s) (0, P_k f(s))\d s \quad\mbox{ for }
t\in [0,T].
$$
Further, since $(\overline u_k, \overline v_k)\to (u(0),v(0))$ in
${\bold E}$ and $P_k f(t)\to f(t)$ in $X^0$ as $k\to +\infty$
uniformly for $t\in [0,T]$, by Proposition \ref{30122008-1629} (i)
we infer that $(u_k, v_k)\to (u,v)$ in $C([0,T], {\bold E})$.
Finally, treating (\ref{06082009-2212}) as a system of ordinary
differential equations with the family $\{{\bold
A}_k(t)\}_{t\in[0,T]}$ of bounded operators we see that
$(\widetilde u_k, \widetilde v_k)$ is, in particular, a classical
solution. Therefore, we obtain
\begin{eqnarray*}
& & (u_k(t),\dot u_k(t))_{1/2} = (u_k(t), v_k(t))_{1/2}\\
& & (v_k(t), \dot v_k(t))_{0} = - (u_k(t), v_k(t))_{1/2} -
\beta(t) |v_k(t)|_{0}^{2} + (v_k(t), P_k f(t))_0,
\end{eqnarray*}
for any $t\in [0,T]$ and, as a result,
$$
\frac{1}{2} \frac{d}{d t}(|u_k(t)|_{1/2}^{2} + |v_k(t)|_{0}^{2}) =
- \beta(t) |v_k(t)|_{0}^{2} + (v_k(t), P_k f(t))_0 \quad\mbox{ for
\ } t\in [0,T].
$$
Thus, we see that both, the functions $|u_k|_{1/2}^{2} + |v_k|_{0}^{2}$, $k\geq 1$ and their derivatives converge uniformly on $[0,T]$, which gives
(\ref{06082009-2221}). To see (\ref{07082009-1309}) observe that
\begin{equation}\label{07082009-1340}
\frac{1}{2} \frac{d}{d t}|u_k(t)|_{0}^{2} = (u_k(t),\dot u_k(t))_{0} = (u_k(t), v_k(t))_{0} \quad\mbox{ for \ } t\in [0,T]
\end{equation}
and $u_k(t) \to u(t)$, $v_k(t) \to v(t)$ is a space $X^0$, as $k\to \infty$, uniformly with respect to $t\in[0,T]$. Hence we see that the functions $|u_k|_0^{2}$, $k\ge 1$, and their derivatives are convergent uniformly and, by (\ref{07082009-1340}), we are done.
\hfill $\square$
Now return to our considerations of (\ref{30062009-1633}) and suppose that, for some, $\lma\in (0,1]$, $(u,v):[0,T]\to E$ is a $T$-periodic mild solution of
$$
(u,v)' = \lma({\bold A} (t) (u(t),v(t))+ {\bold F}_\infty (u(t),v(t))), \quad t\in[0,T].
$$
If view of Lemma \ref{30062009-1545} we get
$$
\frac{1}{2} \frac{d}{d t} \left(|u(t)|_{1/2}^{2}+|v (t)|_{0}^{2}
\right) = - \lma \beta(t) |v (t)|_{0}^{2} - \lma f_{\infty}
(u(t),v(t))_0
$$
and, after integrating and using (\ref{07082009-1309}), one has
$$
0 < \int_{0}^{T} \lma \beta(t)|v(t)|_{0}^{2} \d t = -
\frac{1}{2}\int_{0}^{T} \lambda f_{\infty} (|u(t)|_{0}^2)' \d t =
0,
$$
a contradiction proving that (\ref{23012009-1323}) has no nontrivial $T$-periodic solutions.\\
Finally, by a direct calculation, we see that $\Ker (\widehat {\bold A} + {\bold F}_\infty)=\{0\}$ since $f_\infty\not\in \sigma(A)$. Thus, in view of Theorem \ref{24102008-0927}, problem (\ref{08082009-0110}) admits a $T$-periodic solution in the sense that (\ref{30062009-1633}) has a $T$-periodic mild solution.
| 124,898
|
\begin{document}
\parskip 3pt
\parindent 8pt
\begin{center}
{\Large \bf Non-commutative waves for gravitational anyons }
\baselineskip 20 pt
\vspace{.2cm}
{ \bf Sergio Inglima and Bernd~J.~Schroers} \\
Department of Mathematics and Maxwell Institute for Mathematical Sciences \\
Heriot-Watt University,
Edinburgh EH14 4AS, United Kingdom \\
\tt{sfi1@hw.ac.uk} and \tt{b.j.schroers@hw.ac.uk}
\vspace{0.3cm}
{December 2018}
\baselineskip 16 pt
\end{center}
\begin{abstract}
\noindent
We revisit the representation theory of the quantum double of the universal cover of the Lorentz group in 2+1 dimensions, motivated by its role as a deformed Poincar\'e symmetry and symmetry algebra in (2+1)-dimensional quantum gravity. We express the unitary irreducible representations in terms of covariant, infinite-component fields on curved momentum space satisfying algebraic spin and mass constraints. Adapting and applying the method of group Fourier transforms, we obtain covariant fields on (2+1)-dimensional Minkowski space which necessarily depend on an additional internal and circular dimension. The momentum space constraints turn into differential or exponentiated differential operators, and the group Fourier transform induces a star product on Minkowski space and the internal space which is essentially a version of Rieffel's deformation quantisation via convolution.
\end{abstract}
\section{Introduction}
The possibility of anyonic statistics in two spatial dimensions lies at the root of the peculiarity and intricacy of planar phenomena in quantum physics, ranging from the quantum Hall effect to potential uses of anyons in topological quantum computing \cite{AnyonBook}. The mathematical origin of anyonic statistics is the infinite connectedness of the planar rotation group $\mathrm{SO}(2)$ which is topologically a circle. The goal of this paper, in brief, is to explore the consequences of this fact for quantum gravity in 2+1 dimensions, where the infinite connectedness is doubled and appears in both momentum space and the rotation group.
Our strategy in pursuing this goal is to extend and generalise the method developed in \cite{SemiDual,SchroersWilhelm}, which proceeds from a definition of the symmetry algebra, via a covariant formulation of its irreducible unitary representations (UIRs) in momentum space to, finally, covariant fields in spacetime obeying differential or finite-difference equations. One of the upshots of our work is a construction of non-commutative plane waves for gravitational anyons via a group Fourier transform. While this transform arose in relatively recent literature in quantum gravity \cite{FL,FM,Raasakka,GOR} we note that it is essentially an example and extension of Rieffel's deformation quantisation of the canonical Poisson structure on the dual of a Lie algebra via convolution \cite{Rieffel}.
Anyonic behaviour occurs in both non-relativistic and relativistic physics, and for the same topological reason. The proper and orthochronous Lorentz group $\Lor$ in 2+1 dimensions retracts to $\mathrm{SO}(2)$, and is therefore infinitely connected. Its universal cover, which cannot be realised as a matrix group, governs the properties of relativistic anyons. As we shall review, the double cover of $\Lor$ is the matrix group $\mathrm{SU}(1,1)$ or, equivalently, $ \mathrm{SL}(2,\RR)$. In this paper, we write the universal cover as $\widetilde{\mathrm{SU}}(1,1)$.
In the spirit of Wigner's classification of particles \cite{Wigner},
relativistic anyons are classified by the UIRs of the universal cover of the (proper, orthochronous) Poincar\'e group in 2+1 dimensions, which is the semidirect product of $\widetilde{\mathrm{SU}}(1,1)$ with the group of spacetime translations. It is natural to identify the latter with the dual of the Lie algebra of $\mathrm{SU}(1,1)$. Then the universal cover of the Poincar\'e group is $\widetilde{\mathrm{SU}}(1,1)\ltimes \mathfrak{su}(1,1)^*$, with the first factor acting on the second by the co-adjoint action.
While the UIRs of the Poincar\'e group in 2+1 dimensions are labelled by a real mass parameter and an integer spin parameter, the corresponding mass and spin parameters for the universal cover both take arbitrary real values \cite{Grigore}. In other words, going to the universal cover puts mass and spin on a more equal footing.
The canonical construction of the UIRs of the Poincar\'e group in 2+1 dimensions leads to states being realised as functions on momentum space, obeying constraints \cite{Binegar}. Writing these constraints in a Lorentz-covariant way, and Fourier transforming leads to standard wave equations of relativistic physics like the Klein-Gordon or Dirac equation. Relativistic wave equations for anyons have also been constructed, but for spins which are not half-integers they require infinite-component wave functions, and the derivation of the equations is not straightforward \cite{Gitman,JackiwNair,Plyushchay1,Plyushchay2}.
Our treatment will naturally lead us to an equation derived via a different route by Plyushchay in \cite{Plyushchay1,Plyushchay2}. It makes use of the discrete series UIR of $\widetilde{\mathrm{SU}}(1,1)$, but, as pointed out by Plyushchay, it is essentially a dimensional reduction of an equation already studied by Majorana \cite{Majorana,ST}.
The focus of this paper is a deformation of the universal cover of the Poincar\'e group to a quantum group which arises in (2+1)-dimensional quantum gravity. In 2+1 dimensions, there are no propagating gravitational degrees of freedom, and the phase space of gravity interacting with a finite number of particles and in a universe where spatial slices are either compact or accompanied by suitable boundary conditions at spatial infinity is finite-dimensional. In those cases where the resulting phase space could be quantised, the Hilbert space of the quantum theory can naturally be constructed out of unitary representations of the quantum double of $\Lor$ or one of its covers \cite{MeusburgerSchroers1,CMBSCombQuant,LQGQDouble}. In a sense that can be made precise, this double is a deformation of the Poincar\'e group in 2+1 dimensions \cite{EuclidSchroers,BaisScatt}, with the linear momenta in $ \mathfrak{su}(1,1)^*$ (the generators of translations) being replaced by functions on $\Lor$ or one of its covers.
In the quantum double of $\Lor$, Lorentz transformations and translation are implemented via Hopf algebras which are in duality, namely the group algebra of $\Lor$ and the dual algebra of functions on $\Lor$. The quantum double is a ribbon Hopf algebra whose $R$-matrix can be given explicitly. The unitary irreducible representations describing massive particles are labelled by an integer spin and a mass parameter taking values on a circle.
In analogy with the treatment of the Poincar\'e group, one could define the universal cover of the double of $\Lor$ by replacing the group algebra of the Lorentz group with the group algebra of the universal cover $\widetilde{\mathrm{SU}}(1,1)$. There are indications that this is required when applying the quantum double to quantum gravity in 2+1 dimensions. In particular, several independent arguments lead to the conclusion that, in (2+1)-dimensional gravity, the spin $s$ is quantised in units which depends on its mass $m$ according to
\begin{equation}
s=\frac{n}{1-\frac{\mu}{2\pi}},
\label{SpinQuant}
\end{equation}
where $n \in \ZZ$, $\mu =8\pi mG$ and $G$ is Newton's gravitational constant, see \cite{BaisScatt,Matschull}.
Covering the Lorentz transformations without changing the momentum algebra would destroy the duality between the two.
It is therefore more natural to also consider a universal covering of the momentum algebra, i.e., to identify the momentum algebra with the function algebra on $\widetilde{\mathrm{SU}}(1,1)$. The resulting quantum double of the universal cover of the Lorentz group, called Lorentz double in \cite{BaisScatt}, is a ribbon-Hopf algebra and has UIRs describing massive particles which are labelled by a spin parameter $s$ and a mass parameter $\mu$ for which \eqref{SpinQuant} makes sense.
In this paper, we consider the Lorentz double and derive a new formulation of its UIRs in terms of infinite-component functions on momentum space $\widetilde{\mathrm{SU}}(1,1)$ obeying Lorentz-covariant constraints. We extend and use the notion of group Fourier transforms \cite{FL,FM,Raasakka,GOR} to derive covariant wave equations on Minkowski space equipped with a $\star$-product.
Our method extends earlier work in \cite{SchroersWilhelm} where analogous relativistic wave equations for massive particles were obtained from the UIRs of the quantum double of the Lorentz group. The transition to the universal cover poses two separate challenges. Our wave functions in momentum space now live on $\widetilde{\mathrm{SU}}(1,1)$, and they take values in infinite-dimensional UIRs of $\widetilde{\mathrm{SU}}(1,1)$ called the discrete series. The change in momentum space leads to a particularly natural version of the group Fourier transform, essentially because group-valued momenta can be parametrised bijectively via the exponential map and one additional integer label. In order to include the integer in our Fourier transform we are forced to introduce a dual circle on the spacetime side. The emergence of a compact additional dimension is a remarkable and intriguing aspect of our construction.
Fourier transforming the algebraic spin and mass constraints from momentum space to Minkowski space produces equations on non-commutative Minkowski space which involve either differential operators or the exponential of a first-order differential operator. We end the paper with a short discussion of these non-commutative wave equations for gravitational anyons, leaving a detailed study for future work. Non-commutative waves for anyons have previously been discussed in the literature \cite{HP}. However, the discussion there is in the context of non-relativistic limits rather than the inclusion of gravity, and the mathematics is rather different.
The paper is organised as follows.
In Sect.~2, we introduce our notation and review the definition and parametrisation of the universal cover $\widetilde{\mathrm{SU}}(1,1)$ as well as the discrete series UIRs. We also revisit the UIRs of the Poincar\'e group and briefly summarise the covariantisation of the UIRs and their Fourier transform, following \cite{SchroersWilhelm}. In Sect.~3, we generalise the covariantisation procedure to the universal cover of the Poincar\'e group, thus obtaining wave equations for infinite-component anyonic wave functions. Our version of these equations is essentially that considered by Plyushchay \cite{Plyushchay1,Plyushchay2} but our derivation of them appears to be new. Sect.~4 extends the analysis to the Lorentz double. We derive a Lorentz-covariant form of the UIRs, and point out that one of the defining constraints, called the spin constraint, can be expressed succinctly in terms of the ribbon element of the Lorentz double. The group Fourier transform on
$\widetilde{\mathrm{SU}}(1,1)$ requires a parametrisation of this group via the exponential map, and we discuss this in some detail.
We use the Fourier transform to derive non-commutative wave equations, and then use the short final Sect.~5 to discuss our results and to point out avenues for further research.
\section{Poincar\'e symmetry and massive particles in 2+1 dimensions}
\label{conventions}
We review the symmetry group of (2+1)-dimensional Minkowski space: its Lie algebra, its various covering groups and their representation theory. Unfortunately there is no single book or paper which covers these topics in conventions which are convenient for our purposes. We therefore adopt a mixture of the conventions in the papers \cite{BaisScatt} and \cite{SchroersWilhelm} and the book \cite{Sally}, which are key references for us.
\subsection{Minkowski space and the double cover of the Poincar\'e group}
\label{poincareconventions}
We denote $(2+1)$ dimensional Minkowski space by $\mathbb{R}^{2,1}$, with the convention that the Minkowski metric is mostly minus. Vectors in $\mathbb{R}^{2,1}$ will be denoted by $\bx=(x^0,x^1,x^2)$, with Latin indices for components and the inner product given by
\begin{equation}
\eta(\bx,\by)=\eta_{ab}x^ay^b=x^0y^0-x^1y^1-x^2y^2.
\end{equation}
The group of linear transformations that leave $\eta$ invariant is the Lorentz group $L_3=O(2,1)$. This group has four connected components, but we are mainly interested in the component connected to the identity, i.e., the subgroup of proper orthochronous Lorentz transformations, denoted $\Lor$.
The group of affine transformations that leave the Minkowski metric invariant is the semi-direct product $L_3\ltimes \RR^3$ of the Lorentz group with the abelian group of translations. We call its identity component Poincar\'{e} group and denote it as
\begin{equation}
P_3=\Lor \ltimes \RR ^3.
\end{equation}
The Lie algebra $\mathfrak{p}_3$ of the Poincar\'e group is spanned by rotation and boost generators $J^0,J^1,J^2$, and time and space translation generators $P_0,P_1,P_2$. They can be chosen so that the commutators are
\bee
\label{poincom}
[J^a,J^b]=\epsilon^{abc}J_c, \quad [J_a,P_b]=\epsilon_{abc} P^c, \quad [P_a,P_b]=0.
\eee
Here and in the following $\epsilon_{abc}$ is the totally antisymmetric tensor with $\epsilon_{012}=1$, and indices are raised and lowered with the Minkowski metric $\eta=\text{diag} (1,-1,-1)$. Note that this means in particular that $-J^0$ is the generator of mathematically positive rotations in momentum space. This affects our conventions for the sign of the spin later in this paper.
In quantum mechanics, classical symmetries described by a Lie group $G$ are implemented by projective representations, which, in the case of the Lorentz group, may equivalently be described by unitary representations of the universal covering group. In the case of relativistic symmetries in 3+1 dimensions the universal cover of the Lorentz group is its double cover and is isomorphic to $\mathrm{SL(2,\CC)}$. In 2+1 dimensions the double cover of $\Lor$ is isomorphic to $\mathrm{SL}(2,\RR)$ or, equivalently, $\mathrm{SU}(1,1)$, but this is not the universal cover. Therefore a choice has to be made as to which covering group one should implement in the quantum theory. The paper \cite{SchroersWilhelm}, to which we will refer frequently for details, works with the double cover in the realisation $\mathrm{SL}(2,\RR)$.
We will consider the universal cover here. For our purposes it is more convenient to first consider the double cover $ \mathrm{SU}(1,1)$ because it allows for an easy transition to the universal cover. In order to translate results from \cite{SchroersWilhelm}, the reader will need to apply the unitary matrix
\bee
\label{htraf}
h=\frac 12 \bpm 1+i & -1-i \\ 1-i & \phantom{-}1-i \epm,
\eee
which conjugates $\mathrm{SL}(2,\RR)$ into $ \mathrm{SU}(1,1)$ within $\mathrm{SL}(2,\CC)$, i.e., $h^{-1}\mathrm{SL}(2,\RR)h= \mathrm{SU}(1,1)$.
We use the following basis of the Lie algebra $\mathfrak{su}(1,1)$
\bee
\label{su(1,1) algebra}
s^0 =-\frac{i}{2}\sigma_3=\bpm -\frac{i}{2} & 0 \\ \phantom{-} 0 & \frac{i}{2} \epm, \quad
s^1 =-\frac{1}{2}\sigma_1=\bpm \phantom{-} 0 & -\frac{1}{2} \\ -\frac{1}{2} & \phantom{-} 0 \epm, \quad
s^2 =\frac 12 \sigma_2=\bpm 0 & -\frac{i}{2} \\ \frac{i}{2} & \phantom{-} 0 \epm,
\eee
which are normalised to have the commutation relations of the Lorentz part of \eqref{poincom}:
\bee
\label{su11coms}
[s_a,s_b]=\epsilon_{abc}s^c.
\eee
The basis $t^a$, $a=0,1,2,$ of $\mathfrak{sl}(2,\RR)$ used in \cite{SchroersWilhelm}
is related to our basis of $\mathfrak{su}(1,1)$ via $
s^a=h^{-1}t^ah.$
An arbitrary matrix $M \in \mathrm{SU}(1,1)$ can be parametrised in terms of two complex numbers $a,b$ which satisfy $|a|^2 -|b|^2 =1$ via
\bee
\label{Mab}
M = \bpm a & \bar{b} \\ b & \bar{a} \epm .
\eee
However, a more convenient parametrisation for the extension to the universal cover is obtained
by introducing an angular coordinate $ \omega \in [0,4\pi)$ and a complex number $\gamma=b/a$ of modulus $|\gamma|<1$
so that
\bee
\label{abangles}
a=\frac{1}{\sqrt{1-|\gamma|^2}}e^{i \frac{\omega} {2}} , \qquad
b=\frac{\gamma}{ \sqrt{1-|\gamma|^2} }e^{ i\frac{\omega} {2} }.
\eee
Using this parametrisation, one can see that $\mathrm{SU}(1,1)$ is topologically the open solid torus $S^1 \times D$, where $D$ is the open unit disk.
Note that an $\mathrm{SU}(1,1)$ element parametrised via \eqref{abangles} can be written as
\bee
\label{Momegamma}
M(\omega, \gamma) =\frac{1}{\sqrt{1-|\gamma|^2}}\bpm e^{i\frac{\omega }{2} } & \bar{\gamma} e^{-i \frac{\omega }{2}} \\
\gamma e^{i \frac{\omega }{2}} & e^{-i \frac{\omega }{2}} \epm= \frac{1}{\sqrt{1-|\gamma|^2}}
\bpm 1 & \bar{\gamma} \\
\gamma & 1 \epm \bpm e^{i\frac{\omega }{2} } & 0 \\
0 & e^{-i \frac{\omega }{2}} \epm.
\eee
We will need to understand the conjugacy classes of $\mathrm{SU}(1,1)$ for various applications in this paper. These are obtained from the well-known conjugacy classes of $\mathrm{SL}(2,\RR)$ (see \cite{SchroersWilhelm}, for example) by conjugation with $h$ and therefore determined by the trace (which, for determinant one, fixes the eigenvalues up to ordering) plus additional data.
Elements $u\in \mathrm{SU}(1,1)$ with absolute value of the trace less than 2 are called elliptic elements. They are conjugate to rotations of the form
\bee
\label{elliptic}
\begin{pmatrix} e^{i\frac{\alpha}{2}} & 0 \\
0 & e^{-i\frac{\alpha}{2}}
\end{pmatrix}, \quad \alpha \in (0,2\pi) \cup (2\pi, 4\pi).
\eee
Each value of $\alpha$ labels one conjugacy class (so there are two conjugacy classes for each value of the trace).
Elements with absolute value of the trace equal to 2 include $\pm \text{id}$ (forming a conjugacy class each) and parabolic elements, which are conjugate to
\bee
\label{parabolic}
\begin{pmatrix} \pm1-i\zeta & \zeta \\
\zeta & \pm 1+i\zeta
\end{pmatrix}, \quad \zeta \in (-\infty, 0)\cup (0,\infty).
\eee
Each choice of sign on the diagonal and each value of $\zeta$ labels one conjugacy class.
Element with absolute value of the trace greater than two are called hyperbolic. They are conjugate to
\bee
\label{hyperbolic}
\pm \begin{pmatrix} \cosh \xi & \sinh\xi \\
\sinh\xi & \cosh \xi
\end{pmatrix}, \quad \xi \in (0,\infty).
\eee
Each choice of sign and each value of $\xi$ determines one conjugacy class (negative values are not needed since conjugation with $\sigma_3\in \mathrm{SU}(1,1)$ flips the sign).
The double cover of the Poincar\'e group is
\begin{equation}
\tilde{P}_3 \simeq \mathrm{SU}(1,1) \ltimes \RR^3,
\end{equation}
but for our purposes it is natural to identify the translation subgroup with the vector space $\mathfrak{su}(1,1)^*$,
and to view $\tilde{P}_3$ as
\bee
\label{P3Iso}
\tilde{P}_3 \simeq \mathrm{SU(1,1)} \ltimes \mathfrak{su}(1,1)^*,
\eee
with $\mathrm{SU(1,1)}$ acting on translations via the co-adjoint action $\Ad^*$. More precisely, the product of elements $(g_1,a_1), (g_2,a_2) \in \tilde{P}_3$ in the conventions of \cite{SchroersWilhelm,SemiDual} is
\bee
\label{product}
(g_1,a_1) (g_2,a_2) = (g_1g_2,\Ad^*_{g_2}a_1 +a_2).
\eee
The motivation for the interpretation \eqref{P3Iso} of the double cover of the Poincar\'e group comes from
the formulation of 3d gravity as a Chern-Simons theory with the Poincar\'e group as a gauge group. This requires an invariant and non-degenerate pairing on $\mathfrak{p}_3$. The, up to scale, a unique such pairing is the dual pairing between the Lorentz generators and the translation generators, i.e., the canonical pairing on the Lie algebra
\bee
\label{p3algebra}
\mathfrak{p}_3= \mathfrak{su}(1,1)\oplus \mathfrak{su}(1,1)^*.
\eee
As remarked in \cite{SchroersWilhelm} and for the Euclidean case in \cite{SemiDual}, the dual basis of the translation generators $P_a$ (now viewed as a basis of $ \mathfrak{su}(1,1)^*$) may have a different normalisation to the basis ${s^a}$ which is fixed by the commutation relations \eqref{su(1,1) algebra} .
Adapting the conventions of \cite{SemiDual,SchroersWilhelm}, and working with the generators $s^a$ for $ \mathfrak{su}(1,1)$, we
write the pairing in our basis as
\bee
\label{pairing}
\langle s_a, P_b\rangle = -\frac 1 \lambda \eta_{ab},
\eee
where the scale $\lambda $ is a constant of dimension inverse mass. In the context of 2+1 gravity it is related to Newton's gravitational constant $G$ via $\lambda=8\pi G$.
With the usual identification of momentum space as the dual of spacetime translations, momenta $ p$ in 2+1 dimensions now naturally live in $(\mathfrak{su}(1,1)^*)^*\simeq \mathfrak{su}(1,1)$.
For consistency with \eqref{pairing} we set
\begin{equation}
P^{*a}=-\lambda s^a,
\end{equation}
so that a general element $p$ of momentum space may be expressed as
\begin{align}
\label{pexpand}
p=p_a P^{*a}=- \lambda p_as^a.
\end{align}
The reason for inserting a minus sign here is that $- s^0$ generates mathematically positive rotation, as mentioned after \eqref{poincom}.
While the pairing between Lorentz and translation generators is canonical (up to scale), the inner product between two momenta $p$ and $q$ in $\mathfrak{su}(1,1)$ requires the trace. We define the inner product as
\begin{equation}
-\frac{2}{\lambda ^2}\text{tr}(pq)=p_aq^a.
\end{equation}
This is invariant under the (adjoint) action of $\mathrm{SU(1,1)}$. Using the convention to mark coordinate vectors by bold letters we
write
\bee
\bp\cdot \bq = p_aq^a,
\eee
and occasionally also
\bee
\bp\cdot \bs = p_as^a.
\eee
We also require a notation for the absolute value of the norm of $\bp$. We write this as
\bee
|\bp| = \sqrt{ |\bp\cdot \bp|}.
\eee
As pointed out in the Introduction, $\mathrm{SU(1,1)}$ and its universal cover both play two mutually dual roles in this paper as Lorentz symmetry and as curved momentum space. In the latter context we will require one further parametrisation of ${\mathrm{SU}}(1,1)$, namely the one obtained via the exponential map. By `conjugating' the detailed discussion of the corresponding parametrisation of $\mathrm{SL}(2,\RR)$ in \cite{SchroersWilhelm} with \eqref{htraf}, one checks that any element $u\in {\mathrm{SU}}(1,1)$ can, up to a sign, be written as the exponential of the Lie-algebra valued momentum $p$, i.e.,
\bee
\label{exponential}
u =\pm \exp(p).
\eee
We will revisit this fact and its geometrical interpretation in Sect.~\ref{anywave}. In that context we will also need the following explicit form of the exponential:
\bee
\label{expoformula}
\exp(-\lambda p_as^a)= c(|\bp|)\,\text{id} - 2 s(|\bp|)\hat{p}\cdot\bs,
\eee
where we introduced the function
\bee
s(|\bp|) = \begin{cases} \sin\left(\lambda \frac{ |\bp| }{2}\right) \; & \text{if} \; \bp^2 >0
\\ \lambda \; & \text{if} \; \bp^2 =0 \\ \sinh\left(\lambda\frac{|\bp| }{2}\right)\; & \text{if} \; \bp^2 <0
\end{cases} \quad \text{and} \quad
c(|\bp|) = \begin{cases} \cos\left(\lambda \frac{|\bp| }{2}\right) \; & \text{if} \; \bp^2 >0
\\ 1 \; & \text{if} \; \bp^2 =0 \\ \cosh\left(\lambda\frac{|\bp|}{2}\right)\;& \text{if} \; \bp^2 <0
\end{cases},
\eee
as well as the generalised unit-vector
\bee
\hat{p} = \begin{cases}\frac{\bp}{|\bp|}\; & \text{if} \; \bp^2 \neq 0
\\ \bp \; & \text{if} \; \bp^2 =0
\end{cases}.
\eee
The functions $s$ and $c$ satisfy a generalised version of the Pythagorean formula:
\bee
\label{pythagoras}
c^2(|\bp|) + \hat{p}^2 s^2(|\bp|)=1.
\eee
\subsection{The universal cover the Lorentz group and the discrete series representation } \label{SL2 Coverings}
In this paper we shall consider anyonic particles and for this we will need to work with the universal cover of the Poincar\'{e} group which we denote by $P^\infty_3$:
\begin{equation}
\label{UCovPoinc}
P^\infty_3 =\widetilde{\mathrm{SU}}(1,1) \ltimes \mathfrak{su}(1,1)^*,
\end{equation}
where $\widetilde{\mathrm{SU}}(1,1)$ is the universal cover of $\mathrm{SU}(1,1)$. The group $\widetilde{\mathrm{SU}}(1,1)$ is not a matrix group, but, as we will explain in some detail below, its elements are conveniently parametrised by a real number $\omega$ and complex number $\gamma$ of modulus less than $1$. Thus we write elements of $\widetilde{\mathrm{SU}}(1,1)$ as pairs $(\omega,\gamma)$ and elements of $P^\infty_3$
as pairs $((\omega, \gamma),a)$, with $a\in\mathfrak{su}(1,1)^*$, so that the product is
\bee
\label{productcover}
((\omega_1,\gamma_1),a_1) ((\omega_2,\gamma_2),a_2) = ((\omega_1,\gamma_1)(\omega_2,\gamma_2),\Ad^*_{(\omega_2,\gamma_2)}a_1 +a_2).
\eee
The universal cover of $\mathrm{SU}(1,1)\simeq S^1\times D$, denoted $\widetilde{\mathrm{SU}}(1,1)$, is diffeomorphic to the product $\RR \times D$, where the circle factor $S^1$ has been covered by the real line. We extend our parametrisation \eqref{Momegamma} of $\mathrm{SU}(1,1)$ to the universal cover by simply allowing the angular variable $\omega$ to take values in $\RR$. We then identify elements of $\widetilde{\mathrm{SU}}(1,1)$ with pairs $(\omega, \gamma) \in \RR \times D$. The product $(\omega_1,\gamma_1)(\omega_2,\gamma_2)$ is the element $(\omega,\gamma)$ given by an analytic extension of the formulae one obtains when writing the matrix product of elements of the form \eqref{Momegamma} in terms of the parameters\footnote{The group product is essentially the one given in \cite{Sally} except that we have parametrised the $S^1$ in $\mathrm{SU}(1,1)$ with $[0,4\pi)$ rather than $[0,2\pi)$. This ensures that a final projection to the Lorentz group results in a complete spatial rotation having an angle of $2\pi$.}:
\begin{align}
\label{SU(1,1)tildeproduct}
\gamma &= (\gamma_1+ \gamma_2e^{-i\omega_1})(1+ \bar{\gamma}_1 \gamma_2e^{-i\omega_1})^{-1} \nonumber \\
\omega &= \omega_1 + \omega_2 + \frac 1 i \ln\left(\frac{1+ \bar{\gamma}_1 \gamma_2e^{-i \omega_1} }{ 1+ \gamma_1 \bar{\gamma_2}e^{i\omega_1}}\right).
\end{align}
Here the logarithms of the form $\ln (1+w)$ are defined in terms of the usual (Mercator) power series, which returns the principal value of $\ln(1+w)$. With $\gamma_1$ and $\gamma_2$ inside the unit disk, this means in particular
\bee
\frac 1 i \ln\left(\frac{1+ \bar{\gamma}_1 \gamma_2e^{-i \omega_1} }{ 1+ \gamma_1 \bar{\gamma_2}e^{i\omega_1}}\right) \in \left(- \pi , \pi \right).
\eee
We also note that
\bee
(\omega,\gamma)^{-1}=(-\omega, -e^{i\omega} \gamma).
\eee
In terms of the coordinates $(\omega,\gamma)$, the canonical projection
\bee
\label{canproj}
\pi : \widetilde{\mathrm{SU}}(1,1) \rightarrow \mathrm{SU}(1,1)
\eee
is simply the map
\bee
\label{coverproject}
\pi: (\omega, \gamma) \mapsto M( \omega, \gamma),
\eee
where it is clear from \eqref{Momegamma} that the right hand side only depends on $\omega$ mod $4\pi$. This is a homomorphism whose kernel is the central subgroup generated by the element $(4\pi,0)$. It is, however, convenient to introduce a special name for rotations by $2\pi$. With
\bee
\label{Omegadef}
\Omega=(2\pi,0) \in \widetilde{\mathrm{SU}}(1,1),
\eee
we have $\pi(\Omega) = -\text{id}$ and
\bee
\label{Omegaker}
\text{ker} \;\pi =\{\Omega^{2n} | n\in \ZZ\}.
\eee
When working with the Lie algebra of $\widetilde{\mathrm{SU}}(1,1) $ we will continue to use the notation $s^a$, $a=0,1,2$, for the generators, i.e., we identify the Lie algebra of $\widetilde{\mathrm{SU}}(1,1) $ with its image under the differential of the projection \eqref{canproj}. However, the exponential of these generators $\widetilde{\mathrm{SU}}(1,1) $ cannot be computed via matrix exponentials; instead one has to use the differential geometric definition in terms of the geodesic flow with respect to the bi-invariant (Lorentzian) metric on $\widetilde{\mathrm{SU}}(1,1) $. We will need to exponentiate timelike, lightlike and spacelike generators at various points in this paper, and therefore note the relevant expressions here. Since the geometric definition coincides with the matrix exponential for $\mathrm{SU}(1,1) $, the results are essentially analytic continuations of the expressions for $\mathrm{SU}(1,1) $ in the variable $\omega$. With the notation
\bee
\label{exponetialtilde}
\exp: \mathfrak{su}(1,1) \rightarrow \mathrm{SU}(1,1), \qquad \widetilde{\exp}: \mathfrak{su}(1,1) \rightarrow \widetilde{\mathrm{SU}}(1,1),
\eee
for the exponential maps, we note exponentials of typical timelike, spacelike and lightlike elements in the Lie algebra. They give rise to, respectively, elliptic, hyperbolic and parabolic elements in the group, as exhibited in \eqref{elliptic}, \eqref{parabolic} and \eqref{hyperbolic}.
\begin{align}
\label{expcover}
\exp(-\alpha s^0) &= \begin{pmatrix} e^{i\frac{\alpha}{2}} & 0 \\
0 & e^{-i\frac{\alpha}{2}}
\end{pmatrix}, & \widetilde{\exp}(-\alpha s^0) &= (\alpha,0),
\\
\exp(-2\xi s^1 ) &= \begin{pmatrix} \cosh \xi & \sinh\xi \\
\sinh\xi & \cosh \xi
\end{pmatrix},
&
\widetilde{\exp}(-2\xi s^1 )&= \left(0,\tanh \xi \right),
\nonumber \\
\exp (2\zeta(s^0-s^1))&= \begin{pmatrix} 1-i\zeta & \zeta \\
\zeta & 1+i\zeta
\end{pmatrix}, &
\widetilde{\exp}(2\zeta(s^0-s^1)) &= \left(-2\tan^{-1}(\zeta) , \frac{\zeta }{1-i \zeta}\right).\nonumber
\end{align}
Bearing in mind the comment made after \eqref{poincom}, we note in particular that the adjoint action of $(\alpha,0)$ on momentum space induces a mathematically positive rotation.
The unitary irreducible representations (UIRs) of $\widetilde{\mathrm{SU}}(1,1)$ have been classified into a number of infinite families and are given in detail in \cite{Sally} and also \cite{Grigore}. The particular class that is used to model anyonic particles in \cite{JackiwNair,Plyushchay1,Plyushchay2} is called the discrete series. We briefly review this here, using \cite{Sally} as a main reference but adapting notation used in \cite{Grigore}. In particular, we label the UIRs in the discrete series by $l \in \RR^+$ and a sign.
The carrier spaces for the discrete series are given by the space of suitably completed holomorphic ($+$) or anti-holomorphic ($-$) functions on the open unit disk. These spaces will be denoted $\mathcal{H}_{l\pm}$ or simply $\mathcal{H}$ when no confusion can arise. The completion is with respect to the appropriate Hilbert space inner product given below. For the family ($l+$), the inner product between two holomorphic functions $f=\sum_{n=0}\alpha_nz^n$ and $g=\sum_{n=0}\beta_nz^n$ is
\bee
\label{innerprodl}
\left( f,g \right)_l= \sum_{n=0}^{\infty}\frac{\Gamma(2l)\Gamma(n+1)}{\Gamma(2l+n)}\alpha_n\bar{\beta}_n,
\eee
where $\Gamma$ is the Gamma function. The inner product above may be given an integral expression for the case $l>\frac{1}{2}$ as
\bee
\left( f,g \right)_{l\,>\frac{1}{2}}= \frac{2l-1}{\pi}\int_D (1-\abs{z}^2)^{2(l-1)}f(z)\bar{g(z)}\;\frac i 2 \dd z \wedge \dd \bar{z}.
\eee
For the representations labelled by $l-$ the inner products is as above except that the functions $f$ and $g$ are anti-holomorphic and therefore given as power series of the form $f=\sum_{n=0}\alpha_n\bar{z}^n$ and $g=\sum_{n=0}\beta_n\bar{z}^n$.
In order to express the action of $\widetilde{\mathrm{SU}}(1,1)$ on $\mathcal{H}$ for these representations we recall the projection map \eqref{coverproject} and \eqref{Momegamma}, and combine it
with a right action of $\mathrm{SU}(1,1)$ on the open unit disk:
\begin{equation}
z \cdot \bpm a & \bar{b} \\ b & \bar{a} \epm = \frac{az+b}{\bar{b}z+\bar{a}} = e^{i\omega} \frac{z+\gamma}{\bar{\gamma}z+1}.
\end{equation}
The action of $\widetilde{\mathrm{SU}}(1,1)$ on the carrier space $\mathcal{H}_{l+}$ is
\bee
\label{PlusAnyonRep}
(D_{l+}(\omega, \gamma)f)(z)=e^{il\omega}(1-\abs{\gamma}^2)^{l}(1+\bar{\gamma}z)^{-2l}f(z\cdot \pi (\omega, \gamma)).
\eee
On $\mathcal{H}_{l-}$ it is
\bee
\label{MinusAnyonRep}
(D_{l-}(\omega, \gamma)f)(\bar{z})=e^{-il\omega}(1-\abs{\gamma}^2)^{l}(1+\gamma \bar{z})^{-2l}f(\overline{z\cdot \pi (\omega, \gamma)}).
\eee
The above action simplifies in the case $l \in \frac{1}{2}\NN$, where one recovers a genuine representation of the group $\mathrm{SU}(1,1)$. In this case, one can use \eqref{abangles} to write
\begin{equation}
(D_{l+}(\omega, \gamma)f)(z)=(\bar{a}+\bar{b}z)^{-2l}f(z\cdot \pi (\omega, \gamma)),
\end{equation}
and
\begin{equation}
(D_{l-}(\omega, \gamma)f)(\bar{z})=(a+b\bar{z})^{-2l}f(\overline{z\cdot \pi (\omega, \gamma)}).
\end{equation}
A canonical choice of basis in $\mathcal{H}_{l +}$ is given by the orthonormal functions
\begin{equation}
e^+_{n,l}(z)=\left( \frac{\Gamma(2l+n)}{\Gamma(2l)\Gamma(n+1)} \right)^{\frac{1}{2}} z^n,\;\;n \in \NN,
\end{equation}
and for $\mathcal{H}_{l -}$ we have the basis
\begin{equation}
e^-_{n,l}(\bar{z})=\left( \frac{\Gamma(2l+n)}{\Gamma(2l)\Gamma(n+1)} \right)^{\frac{1}{2}} \bar{z}^n,\;\;n \in \NN.
\end{equation}
The state for $n=0$ in both $(l,+)$ and $(l,-)$ is the constant function. It plays an important role in constructing a covariant descriptions of UIRs, so we introduce the notation
\bee
\label{groundstate}
\Ket {0}_l= e^+_{0,l}= e^-_{0,l}
\eee
for the map $z\mapsto 1$.
In these representations the infinitesimal generators of the Lie algebra, denoted $d_{l\pm}(s^{a})$, can be realised as differential operators acting on the carrier space $\mathcal{H}$. Starting with the usual definition
\begin{equation}
\label{dDrel}
d_{l\pm}(s^{a})= \left. \frac{\dd}{\dd \epsilon} \right \rvert_{\epsilon=0} D_{l\pm}(\widetilde{\exp}{\epsilon s^a}),
\end{equation}
we compute for the positive discrete series
\begin{align}
\label{InfAnyonRep+}
d_{l+}(s^{0})&=-il-iz\frac{\dd}{\dd z}, \nonumber \\
d_{l+}(s^{1})&= lz-\frac{1}{2}(1-z^2)\frac{\dd}{\dd z}, \nonumber \\
d_{l+}(s^{2})&= ilz+\frac{i}{2}(1+z^2)\frac{\dd}{\dd z} ,
\end{align}
and for the negative series
\begin{align}
\label{InfAnyonRep-}
d_{l-}(s^{0})&=il +i\bar{z}\frac{\dd}{\dd \bar{z}}, \nonumber \\
d_{l-}(s^{1})&=l\bar{z}-\frac{1}{2}(1-\bar{z}^2)\frac{\dd}{\dd\bar{z}}, \nonumber\\
d_{l-}(s^{2})&=-il\bar{z}-\frac{i}{2}(1+\bar{z}^2)\frac{\dd}{\dd\bar{z}}.
\end{align}
For later use, we note that the linear combination $s^{a}p_a$ for an arbitrary vector $\bp$ acts according to
\begin{equation}
\label{dl+}
d_{l+}(s^{a})p_a=-il p_0 +i(p_2-ip_1)lz +i\left(\frac{1}{2}(p_2+ip_1)- p_0z+\frac{1}{2}(p_2-ip_1)z^2\right)\frac{\dd}{\dd z},
\end{equation}
and
\begin{equation}
\label{dl-}
d_{l-}(s^{a})p_a=il p_0 -i(p_2+ip_1)l\bar{z} -i\left(\frac{1}{2}(p_2-ip_1)- p_0\bar{z}+\frac{1}{2}(p_2+ip_1)\bar{z}^2\right)\frac{\dd}{\dd\bar{z}}.
\end{equation}
The vector fields appearing in the action of the $s^{a}p_a$ have a natural geometrical interpretation which is familiar in the context of the mini-twistor correspondence between points in Euclidean 3-space and the set of all the lines through that point. We explain briefly how this point of view fits into our Lorentzian setting.
We can parametrise the set of all timelike lines in 2+1 dimensional Minkowski space in terms of a timelike vector $\bq$, normalised so that $\bq^2=1$ and giving the direction of the line, and a vector $\bk$ which lies on the line and which can be chosen to satisfy $\bk\cdot \bq=0$ without loss of generality. Geometrically, $\bq$ lies on the two-sheeted hyperboloid, and $\bk$ lies in the tangent space at $\bq$.
In terms of an orthonormal basis $\be^{0},\be^{1},\be^{2}$ of Minkowski space, and the complex linear combination $\be=\be^{1}+i\be^{2}$, we can parametrise $\bq$ in terms of a complex variable $z$ in the unit disk via
\[
\bq=\frac{1}{1-|z|^2}\left( (1+|z|^2)\be^{0} +z\bar{\be} + \bar{z} \be\right),
\]
and the tangent vector $\bk$ in terms of a complex number $w$ via
\[
\bk = w \frac{\partial \bq}{\partial z} + \bar{w} \frac{\partial \bq}{\partial \bar z} = \frac{1}{(1-|z|^2)^2} \left( 2(w \bar z + \bar w z) \be^{0} + (\bar w + w \bar z^2)\be + (w + \bar w z^2)\bar \be\right).
\]
Then one checks that the point $\bp=p_0\be^{0} + p_2\be^{1} +p_1\be^{2}$ lies on the line through $\bk$ and in the direction $\bq$ if and only if
\bee
w= \frac 12 (p_2+ip_1) - p_0z + \frac 12 (p_2-ip_1) z^2.
\eee
In other words, the derivatives appearing in \eqref{dl+} and \eqref{dl-} are precisely the holomorphic and anti-holomorphic part of the tangent vector which characterises a line containing $\bp$ and in the direction $\bq$ determined by $z$.
\subsection{Massive $\tilde{P}_3$ representations}
\label{P3 Reps}
There are various ways of getting from the UIRs of the Poincar\'e group to the covariant wave equations of relativistic physics.
In \cite{SemiDual,SchroersWilhelm}, a procedure was developed which is also effective when the Poincar\'e group is deformed to the quantum double of the Lorentz group or one of its covers. We briefly review the method here in a convenient form for extension to the anyonic case. However, relative to \cite{SchroersWilhelm} we change the sign convention for mass and spin to agree with the one used in \cite{BaisScatt}.
Using the isomorphism $\tilde{P}_3 \simeq \mathrm{SU}(1,1) \ltimes \mathfrak{su}(1,1)^*$ in \eqref{P3Iso}, UIRs of $\tilde{P}_3$ may be classified by the adjoint orbits of $\mathrm{SU(1,1)}$ in momentum space $\mathfrak{su}(1,1)$ together with an UIR of associated stabiliser groups. For massive particles, the former encodes the mass, and the latter its spin. In the case of a particle of mass $m\neq 0$, we denote the adjoint orbit by $ O^T_m$. It elements are obtained by boosting the representative momentum $-\lambda ms^0$ (which generates a mathematically positive rotation) to obtain the typical element
\bee
\label{signconventions}
p= -v\lambda ms^0v^{-1}=-\lambda \bp\cdot\bs.
\eee
The coordinate vector $\bp$ satisfies $\bp^2 =m^2$ and the sign constraint $mp_0>0$ so that $\bp$ lies in the $p_0>0$ half space for positive $m$, and in the $p_0<0$ half space for negative $m$. Thus we have two equivalent characterisations of the adjoint orbit corresponding to massive particles:
\bee
O^T_m = \Set{ \Ad_v(-\lambda ms^0) | v\in \mathrm{SU}(1,1)} = \Set{- \lambda p_as^a \in \mathfrak{su}(1,1) | \bp^2=m^2, mp_0>0 }.
\eee
Geometrically, this is the upper ($m>0$) or lower ($m<0$) sheet of the two-sheeted hyperboloid. Other types of adjoint orbits include the trivial orbit $\{ 0\}$ describing the vacuum, the forward or backward lightcone describing massless particles and single-sheeted hyperboloids describing tachyons, but we will not consider these here.
The associated stabiliser group $N^T$ of $O^T_m$ is given by
\bee
N^T=\Set{ v \in \mathrm{SU}(1,1) | \Ad_v(s^0)= s^0)} = \Set{\exp (-\phi s^0) | \phi \in [0,4\pi) } \nonumber \\
\simeq U(1).
\eee
The UIRs of $U(1)$ are one-dimensional and labelled by $s\in \frac{1}{2}\mathbb{Z}$ in our conventions.
The carrier space of the UIRs of $\tilde{P}_3$ for massive particles with spin $s$ can be given in two equivalent ways. Either one considers functions on $\mathrm{SU}(1,1)$ which satisfy an equivariance condition or sections of associated vector bundles over the homogeneous space $\mathrm{SU}(1,1)/N^T \simeq O^T_m$. We focus on the former method here but, refer the reader to \cite{BaisScatt} for a discussion of their equivalence in the context of 3d gravity and to \cite{Barut} for a general reference. Adopting the conventions of \cite{BaisScatt}, we define the carrier space as
\begin{align}
\label{P3 Carrier Sp}
V_{ms}= \left \lbrace \psi \colon \mathrm{SU}(1,1) \rightarrow \CC \vert \psi \left(ve^{-\alpha s^0}\right)=e^{-is\alpha}\psi(v) \; \forall \,\alpha \in [0,4\pi), v \in \mathrm{SU(1,1)}, \right. \nonumber \\
\left. \int_{\mathrm{SU}(1,1)/N^T}\abs{\psi}^2 d \nu < \infty \right\rbrace,
\end{align}
where $d \nu$ is the invariant measure on the coset $\mathrm{SU}(1,1)/N^T$.
The action of an element $(g,a)\in \tilde{P}_3$ on $\psi \in V_{ms}$ is
\begin{equation}
\label{pi ms rep}
(\pie_{ms}(g,a)\psi )(v)=\exp\left(i\langle a,\Ad_{g^{-1}v}(-\lambda ms^0)\rangle \right)\psi(g^{-1}v),
\end{equation}
where, in accordance with \eqref{P3Iso}, $a$ is interpreted as an element of $\mathfrak{su}(1,1)^*$, and $\langle \cdot , \cdot\rangle$ is the pairing between elements of $\mathfrak{su}(1,1)^*$ and $\mathfrak{su}(1,1)$ introduced and discussed in Sect.~\ref{poincareconventions}. We have attached the superscript `eq' to distinguish this equivariant formulation from the later covariant version.
\subsection{Covariant field representations}
In field theory we do not usually work with the space of equivariant functions as just described. Instead we use covariant fields
\begin{equation}
\phi \colon \RR^{2,1} \rightarrow V,
\end{equation}
where $V$ is a carrier space for a (usually finite dimensional) representation of the Lorentz group. In general such fields do not form irreducible representations of the Poincar\'e group and, as a result, additional constraints need to be imposed to achieve this. For fields defined on momentum space these constraints are algebraic, but after Fourier transform they yield the familiar wave equations for a field of definite spin.
Following \cite{SchroersWilhelm} for the method, but changing the sign convention to agree with \cite{BaisScatt}, we construct a covariant field
\begin{align}
\tilde{\phi} \colon O^T_m \rightarrow \CC^{2\abs{s}+1}
\end{align}
from a given $\psi \in V_{ms}$ via
\begin{equation}
\tilde{\phi}(p)=\psi(v) \rho ^{\abs{s}}(v)\Ket {\abs{s},-s}, \qquad \text{with} \quad
p=-\lambda m vs^0v^{-1}.
\end{equation}
Here $\{ \Ket{\abs{s},k}: k = -|s|,-|s|+1, \ldots |s|-1, |s|\}$ is a basis of the (non-unitary) $(2|s|+1)$-dimensional representation of $\mathfrak{su}(1,1)$, satisfying
\bee
\rho^{|s|}(s^0) \Ket{\abs{s},k}=ik \Ket{\abs{s},k},
\eee
see \cite{SchroersWilhelm} for details.
To check that $\tilde{\phi}(p)$ is well-defined, one needs to show that
\begin{equation}
\psi(v) \rho ^{\abs{s}}(v)\Ket {\abs{s},-s}=\psi(vn) \rho ^{\abs{s}}(vn)\Ket {\abs{s},-s}\;\forall n \in N^T,
\end{equation}
but this is true because the phase picked up by $\psi$ under the action of an element $n$ of the stabiliser is precisely cancelled by the action of $\rho ^{\abs{s}}(n)$ on the state $\Ket {\abs{s},-s}$. This construction works for massive particles since $\rho ^{\abs{s}}(s^0)$ has imaginary eigenvalues. However, this is not the case for the momentum representatives on massless and tachyonic orbits and hence the above procedure is limited to particles with timelike momentum.
Adapting the results in \cite{SchroersWilhelm} to our conventions, the covariant field $\tilde{\phi}$ necessarily satisfies the condition
\begin{equation}
\label{SpinConstraint}
\left(i\rho^{\abs{s}}(s^a)p_a-ms\right)\tilde{\phi}(p)=0,
\end{equation}
which we call the spin constraint. In order to carry out the envisaged Fourier transform, we would like to extend $\tilde\phi$ to a function on all of the linear momentum space $\mathfrak{su}(1,1)$. However, we then need to impose the mass constraint $\bp^2=m^2$ and the sign constraint $mp_0 >0$
to ensure that $\tilde \phi$ has support on the orbit $ O^T_m$. The sign constraint makes sense when the mass constraint is enforced since $m\neq 0$ by assumption and therefore $(p_0)^2 >0$. To restrict the support of $\tilde \phi$ to the `forward' mass shell when $m>0$ and to the `backward' mass shell when $m<0$, we use the Heaviside function $\Theta$ and
define the carrier space
\begin{equation}
\label{covcarrier}
W_{ms}=\Set{ \tilde{\phi} \colon \mathfrak{su}(1,1) \rightarrow \CC^{2\abs{s}+1} | (i\rho^{\abs{s}}(s^a)p_a-ms)\tilde{\phi}(p)=0, (\bp^2-m^2)\tilde{\phi}(p)=0, \Theta( -mp_0)\tilde \phi=0}.
\end{equation}
In the corresponding definition in \cite{SchroersWilhelm}, the sign constraint was not included, but for us this inclusion is convenient because we directly obtain a UIR of $\tilde{P}_3$ without adding further conditions.
The action of $(g,a)\in \tilde{P}_3$ on this space is given by
\begin{equation}
(\pic(g,a) \tilde{\phi})(p)=\exp(i\langle a,\Ad_{g^{-1}}p\rangle)\rho ^{\abs{s}}(g)\tilde{\phi}(\Ad_{g^{-1}}p),
\end{equation}
which we call the covariant formulation.
In \cite{SchroersWilhelm} it is also shown that the above covariant fields produce UIRs of $\tilde{P}_3$ for the familiar cases of spin $s=0,\frac{1}{2}, 1$ and that the mass constraint for spin zero and the spin constraints for $s=\frac{1}{2}$ and $s=1$ produce the momentum space versions of the Klein-Gordon equation, Dirac equation and of field equations which square to the Proca equation\footnote{The spin 1 equation was simply called Proca equation in \cite{SchroersWilhelm} but it is more precisely a first order equation which implies the Proca equation. Its relation to self-dual massive field theory is discussed in \cite{Gitman}}.
We refer the reader to \cite{SchroersWilhelm} for details of the Fourier transform of the spin constraints to relativistic field equations in spacetime. We now turn to the anyonic case, where we will discuss both the covariant formulation of the UIRs and the Fourier transform.
\section{Anyonic wave equations}
Anyons are quantum particles with fractional spin which occur in systems confined to two spatial dimensions. In the relativistic case, the theoretical possibility of anyonic particles is a consequence of the infinite connectedness of the Lorentz group $\Lor$.
To describe relativistic anyons we need to consider the representation theory of the universal cover of the Poincar\'{e} group $P^\infty_3$. The UIRs are classified in \cite{Grigore} using the method of induced representations. The action of $ (\omega, \gamma)\in \widetilde{\mathrm{SU}}(1,1)$ on momentum space is
the adjoint action $\Ad_{(\omega, \gamma)}$.
The stabiliser group for a massive particle, with standard momentum $-\lambda ms^0$, is therefore
\bee
\tilde{N}^T = \Set{(\omega, \gamma) \in \widetilde{\mathrm{SU}}(1,1) | \Ad_{(\omega, \gamma)}s^0=s^0 } = \left\lbrace (\omega, 0) \in \widetilde{\mathrm{SU}}(1,1) \right\rbrace \simeq \RR.
\eee
The one-dimensional UIRs of the stabiliser are labelled by $s \in \RR$ which represents the spin of the massive particle. With our results \eqref{expcover} for the exponential map into $\widetilde{\mathrm{SU}}(1,1)$ and using \eqref{Momegamma}, we note that
\bee
(\omega,\gamma)\widetilde{\exp}(-\alpha s^0)= (\omega+\alpha, \gamma).
\eee
Thus, we have the following equivariant description of the carrier space:
\begin{align}
\label{AnyonicEquivariance}
V_{ms}^A= \left\lbrace \psi \colon \widetilde{\mathrm{SU}}(1,1) \rightarrow \CC \vert \psi (\omega+\alpha, \gamma)=e^{-is\alpha} \psi(\omega,\gamma)\; \forall \alpha \in \RR, \forall (\omega,\gamma) \in \widetilde{\mathrm{SU}}(1,1),
\right. \nonumber \\
\left. \int_{\widetilde{\mathrm{SU}}(1,1)/\tilde{N}^T}\abs{\psi}^2 d\nu < \infty \right\rbrace.
\end{align}
The action of $((\omega,\gamma), a)\in P
^\infty_3$ on the space $V_{ms}^A$ is
\begin{equation}
\label{AnyonicUIR}
\left(\pie_{ms}((\omega, \gamma),a)\psi \right)(v)=\exp\left(i\langle a,\Ad_{((\omega, \gamma)^{-1}v)}(-\lambda ms^0)\rangle \right)\psi \left((\omega, \gamma)^{-1}v\right).
\end{equation}
We now follow the procedure of the previous section to construct anyonic covariant fields.
\begin{definition}[Anyonic Covariant Field] The anyonic covariant field $\tilde{\phi}_{\pm}$ associated to an equivariant field $\psi\in V^A_{ms}$ is the map
\begin{align}
\tilde{\phi}_{\pm} \colon O^T_m \rightarrow \mathcal{H}_{l \pm},
\end{align}
where $\mathcal{H}_{l \pm}$ is the carrier space for the discrete series representations of $\widetilde{\mathrm{SU}}(1,1)$ given in \eqref{SL2 Coverings},
defined via
\begin{equation}
\label{AnyonicCovField}
\tilde{\phi}_{\pm}(p)=\psi(\omega, \gamma) D_{l\pm}(\omega, \gamma)\Ket {0}_l.
\end{equation}
Here $(\omega,\gamma)\in\widetilde{\mathrm{SU}}(1,1)$ is chosen so that $p=-\lambda m \,\Ad_{(\omega, \gamma)}(s^0)$ and
$s=l$ for $\tilde{\phi}_+$ and $s=-l$ for $\tilde{\phi}_-$.
As before, $D_{l\pm}$ are the discrete series representations of $\widetilde{\mathrm{SU}}(1,1)$ and $\Ket {0}_l$ is defined in \eqref{groundstate}.
\end{definition}
Thus we should use $\tilde{\phi}_+$ to describe positive spin particles and $\tilde{\phi}_-$ for negative spin.
\begin{lemma} The anyonic covariant fields \eqref{AnyonicCovField} are well defined.
\end{lemma}
\begin{proof}
One needs to check that this definition is independent of the choice of $(\omega,\gamma)$, i.e., we require
\begin{equation}
\psi(\omega,\gamma) D_{l\pm}(\omega,\gamma)\Ket {0}_l=\psi((\omega,\gamma)(\alpha,0)) D_{l\pm}\left((\omega,\gamma)(\alpha,0)\right)\Ket {0}_l \qquad \text{for all} \quad \alpha \in \RR.
\end{equation}
Expanding the right hand side and using the equivariance of $\psi$, and the action of the stabiliser subgroup elements in the representations $D_{l\pm}$ on the vacuum state one obtains
\begin{align}
\label{invariance}
\psi((\omega,\gamma) (\alpha,0)) D_{l\pm}\left((\omega,\gamma)(\alpha,0)\right)\Ket {0}_l &=\psi((\omega+\alpha,\gamma) D_{l\pm}\left((\omega,\gamma)(\alpha,0)\right)\Ket {0}_l \nonumber \\
&=e^{-is\alpha} \psi(\omega,\gamma)D_{l+}(\omega,\gamma)D_{l+}(\alpha, 0))\Ket {0}_l\nonumber \\
&=e^{-is\alpha} \psi(\omega,\gamma)D_{l+}(\omega,\gamma)e^{il\alpha}\Ket {0}_l \nonumber \\
&=e^{i(l-s)\alpha}\psi(\omega,\gamma)D_{l+}(\omega,\gamma)\Ket {0}_l .
\end{align}
Hence we obtain invariance if $l-s=0$, as claimed. An analogous argument applied to $\tilde{\phi}_-$ shows that $l+s=0$ is required in that case.
\end{proof}
The anyonic covariant field carries a unitary representation of the Poincar\'e group, which we again denote $\pic$:
\begin{equation}
\label{AnyonicCovAction}
(\pic_{ms}((\omega, \gamma), a) \tilde{\phi}_{\pm})(p)=\exp(i\langle a,\Ad_{(\omega, \gamma))^{-1}}p\rangle)D_{l\pm}((\omega, \gamma))\tilde{\phi}_{\pm}(\Ad_{(\omega, \gamma)^{-1}}p).
\end{equation}
Without further condition, this representation is not irreducible. To achieve irreducibility, we need
anyonic versions of the spin, mass and sign constraints. Our version of the spin constraint is the equation considered by Plyushchay in \cite{Plyushchay1,Plyushchay2}. As pointed out in those papers, it is also essentially a dimensional reduction of the Majorana equation \cite{Majorana}.
\begin{lemma}[Anyonic Spin Constraint]
The anyonic covariant fields $\tilde \phi_{\pm}$ satisfy the following constraints
\bee
\label{AnyonSpinCon}
(d_{l\pm}(p)-i\lambda ms) \tilde{\phi}_{\pm}(p)=0,
\eee
where $d_{l\pm}$ is the Lie algebra representation associated to the discrete series representation $D_{l\pm}$ via \eqref{dDrel}, with $s=l$ for the positive series and $s=-l$ for the negative series.
\end{lemma}
\begin{proof} With
$p=-\lambda p_as^a=-\lambda m\Ad_{(\omega, \gamma)}s^0 $, we compute for $\tilde{\phi}_+$
\begin{align}
d_{l+}(p) \tilde{\phi}_+(p)&=-\lambda md_{l+}(\Ad_{(\omega, \gamma)}s^0)\tilde{\phi}_+(p) \nonumber \\
&=-\lambda m\psi(\omega, \gamma)D_{l+}(\omega, \gamma)d_{l+}(s^0)D_{l+}((\omega, \gamma)^{-1})D_{l+}(\omega, \gamma)\Ket {0}_l \nonumber \\
&= -\lambda m\psi(\omega, \gamma)D_{l+}(\omega, \gamma)d_{l+}(s^0)\Ket {0}_l \nonumber \\
&=i\lambda ml \psi(\omega, \gamma)D_{l+}(\omega, \gamma)\Ket {0}_l \nonumber \\
&= i \lambda ms\tilde{\phi}_+(p).
\end{align}
An analogous computation for $\tilde{\phi} _-$, noting that $s=-l$, completes the proof.
\end{proof}
Note that, in components and with the sign convention \eqref{signconventions}, the spin constraint takes the form
\bee
\label{convenspin}
(i d_{l\pm}(s^a)p_a - ms) \tilde{\phi}_{\pm}(p)=0,
\eee
which has the same form as the finite-component version \eqref{SpinConstraint}.
In order to construct UIRs of $P^\infty_3$ in terms of covariant fields we define
the carrier space which generalises \eqref{covcarrier} to infinite-component fields:
\begin{equation}
\label{WAms}
W_{ms}^A=\Set{ \tilde{\phi}_{\pm} \colon \mathfrak{su}(1,1) \rightarrow \mathcal{H}_{l\pm} | \left( d_{l\pm}(p) - i\lambda ms\right)\tilde{\phi}_{\pm}(p)=0, (\bp^2-m^2)\tilde{\phi}_{\pm}(p)=0 },
\end{equation}
where we choose the upper sign if $s>0$ and the lower sign if $s<0$.
Note that, unlike in the finite-dimensional case, we do not also need to impose the constraint that $p^0$ and $m$ have the same sign. This follows from our conventions \eqref{signconventions} and from the fact that $id_{l+}(s^0)$ has only positive eigenvalues and $i d_{l-}(s^0)$ only negative eigenvalues. This property was one of the motivations for Majorana to construct his infinite-component fields in \cite{Majorana}.
The space $W_{ms}^A$ is a Hilbert space with the inner product
\bee
(\tilde{\phi}_1, \tilde{\phi}_2) = \int_{O^T_{m}}(\tilde \phi_1(p),\tilde\phi_2(p))_l\, d \nu,
\eee
where $d\nu$ is the invariant measure on the hyperboloid $O^T_m$, and $(\cdot,\cdot)_l$ is the inner product \eqref{innerprodl} on $\mathcal{H}_{l+}$, with an analogous expression for fields taking values in $\mathcal{H}_{l-}$.
We now show that the representations $V^{A}_{ms}$ and $W^{A}_{ms}$ are isomorphic. This implies that the covariant fields subject to the mass and spin constraints form UIRs of the universal cover $P^\infty_3$ of the Poincar\'e group.
\begin{theorem}[Irreducibility of the carrier space $W^{A}_{ms}$]
\label{intertwining} The covariant representation $\pic_{ms}$ of $P^\infty_3$ on $W^{A}_{ms}$ defined in \eqref{AnyonicCovAction} is unitarily equivalent to the equivariant representation $\pie_{ms}$ on $V^{A}_{ms}$ defined in \eqref{AnyonicUIR}. In particular, it is therefore irreducible.
\end{theorem}
\begin{proof}
We claim that the following maps are intertwiners:
\begin{equation}
L_{\pm} \colon V^{A}_{ms} \rightarrow W^{A}_{ms},\qquad (L_{\pm}(\psi))(p)= \psi(\omega, \gamma) D_{l\pm}(\omega, \gamma)\Ket {0}_l,
\end{equation}
where, as before, $(\omega,\gamma)\in\widetilde{\mathrm{SU}}(1,1)$ is chosen so that $p=-\lambda m \Ad_{(\omega, \gamma)}(s^0)$ and $s=l$ for the positive series and $s=-l$ for the negative series. We have already shown that
$\psi(\omega, \gamma) D_{l\pm}(\omega, \gamma)\Ket {0}_l$ only depends on $p$, satisfies the spin constraint and has support entirely on the orbit $O^T_m$, so that the mass constraint is also satisfied. Thus the maps $L_\pm$ are well-defined.
The maps $L_{\pm}$ are injective because of the unitarity of $D_{l\pm}$. To show that they are surjective, we pick $\tilde \phi \in W^{A}_{ms}$ and construct a preimage. Focusing on $D_{l+}$, and noting that
\begin{equation}
d_{l+}(p)=-\lambda m D_{l+}(\omega, \gamma)d_{l+}(s^0)D_{l+}((\omega, \gamma)^{-1}),
\end{equation}
the spin constraint \eqref{AnyonSpinCon} is equivalent to
\bee
d_{l+}(s^0)D_{l+}((\omega, \gamma)^{-1})\tilde \phi (p) = -is D_{l+}((\omega, \gamma)^{-1})\tilde \phi (p).
\eee
Recalling that $l=s$, comparing with \eqref{InfAnyonRep+}, and recalling that $\ket{0}_l$ is, up to a factor, the unique solution of
$ d_{l+}(s^0)f = -il f$, we deduce the proportionality
\bee
D_{l+}((\omega, \gamma)^{-1})\tilde \phi (p) = \psi (\omega,\gamma) \ket{0}_l,
\eee
where the proportionality factor $\psi$ may depend on $(\omega,\gamma)$. Moreover, it must have the property
\bee
\psi (\omega+\alpha,\gamma) = e^{-i\alpha s} \psi(\omega,\gamma),
\eee
to ensure independence of the choice of $(\omega,\gamma)$ for given $p$, since
\begin{align}
\psi (\omega+\alpha,\gamma) \ket{0}_l &= D_{l+}((\omega+\alpha, \gamma)^{-1})\tilde \phi (p) \nonumber \\
& = D_{l+}(-\alpha,0)D_{l+}((\omega, \gamma)^{-1})\tilde \phi (p) \nonumber \\
&= D_{l+}(-\alpha,0)\psi (\omega,\gamma) \ket{0}_l \nonumber \\
&=e^{-i\alpha s} \psi (\omega,\gamma) \ket{0}_l.
\end{align}
Thus $\psi \in V^A_{ms}$ and $L_+(\psi) =\tilde \phi$, as required to show that $L_+$ is surjective.
An analogous argument for $L_{-}$ shows that both $L_{\pm}$ are bijections.
The intertwining property is equivalent to the commutativity of the diagram
\bee
\begin{tikzcd}[row sep=huge, column sep = large]
V^A_{ms} \arrow{r}{\quad \pie_{ms}((\omega, \gamma),a) \quad} \arrow[swap]{d}{L_{\pm}} & V^A_{ms} \arrow{d}{L_{\pm}} \\
W^A_{ms} \arrow{r}{\quad \pic_{ms} ((\omega, \gamma),a)\quad } & W^A_{ms}
\end{tikzcd},
\eee
for $((\omega, \gamma),a)\in P^\infty_3$. This is a straightforward calculation based upon the maps $L_{\pm}$, and the actions given in \eqref{AnyonicUIR} and \eqref{AnyonicCovAction}.
The unitarity of $L_{\pm}$ follows from the unitarity of $D_{l\pm}$, since
\begin{align}
(L_\pm\psi_1, L_\pm \psi_2) &= \int_{O^T_{m}}(\psi_1(\omega,\gamma) D_{l\pm} (\omega,\gamma)\ket{0}_l, \psi_2(\omega,\gamma) D_{l\pm} (\omega,\gamma)\ket{0}_l)_l
\, d \nu \nonumber \\
& = \int_{O^T_{m}} \psi_1(\omega,\gamma) \bar{\psi}_2(\omega,\gamma)\, d \nu.
\end{align}
\end{proof}
Finally, we Fourier transform the spin constraint in the form \eqref{convenspin} to obtain the anyonic wave equation promised in the title of this section. Since the field $\phi$ lives on the Lie algebra $\mathfrak{su}(1,1)$, its Fourier transform should live on $\mathfrak{su}(1,1)^*$, i.e., the Fourier transform is a map
\bee
\label{flatfourier}
L^2(\mathfrak{su}(1,1), \mathcal{H}_{l\pm})\rightarrow L^2(\mathfrak{su}(1,1)^*, \mathcal{H}_{l\pm}).
\eee
Using the terminology introduced after equation \eqref{p3algebra}, we expand $x\in \mathfrak{su}(1,1)^*$ and $p\in \mathfrak{su}(1,1)$ as
\bee
x=x^aP_a, \quad p= -\lambda p_as^a.
\eee
Then, with the pairing given in \eqref{pairing},
we define the Fourier transform of $\tilde \phi \in W_{ms}^A$ by $\phi$
as
\bee
\phi_{\pm}(x)=\int_{\mathfrak{su(1,1)}}
e^{i\langle x,p\rangle}\tilde \phi_{\pm}(p)\; d^3 \bp= \int_{\RR^3}
e^{i\bx\cdot\bp}\tilde \phi_{\pm}(p)\; d^3 \bp.
\eee
The field $\phi_\pm$ satisfies the Klein-Gordon equation
\bee
\label{KG}
(\partial_0^2 -\partial_1^2-\partial_2^2+m^2)\phi_{\pm} =0,
\eee
by virtue of the mass constraint. The spin constraint implies the following first order equation:
\bee
\label{anywave1}
\left(d_{l\pm}(s^a)\partial_a - ms \right)\phi_{\pm}=0,
\eee
where we wrote $\partial_a=\partial/\partial x^a$.
Using the explicit forms \eqref{dl+} and \eqref{dl-} of $d_{l\pm}$ with $l=s$ for the positive series and $l=-s$ for the negative series,
and with the abbreviations
\bee
\partial=\frac 12\left(\partial_2 -i \partial_1\right), \quad
\bar{\partial}=\frac 12\left(\partial_2 +i \partial_1\right),
\eee
the equation \eqref{anywave1} can also be written as
\bee
\label{anywave2+}
(-is\partial_0 +2isz\partial +i\left(\bar{\partial}- z\partial_0+ z^2 \partial \right)\frac{\dd}{\dd z} -ms)\phi_+=0,
\eee
and
\bee
\label{anywave2-}
(-is\partial_0 +2is\bar{z}\bar{\partial} -i\left(\partial- \bar{z}\partial_0+ \bar{z}^2 \bar{\partial }\right)\frac{\dd}{\dd\bar z} -ms)\phi_-=0.
\eee
The anyonic relativistic wave equation we have constructed for arbitrary spin $s \in \RR$ can be viewed in two ways: either as a partial differential equation in Minkowski space for a field taking values in any infinite-dimensional Hilbert space, as suggested by the formulation \eqref{anywave1}, or as a partial differential equation for a field on the product of Minkowski space and the hyperbolic disk, as emphasised in the formulation \eqref{anywave2+} and \eqref{anywave2-}.
It is worth stressing that, for $s \in \frac{1}{2} \NN$, our anyonic equation \eqref{anywave1} does not reduce to the equation \eqref{SpinConstraint} for an equivariant field with finitely many components. These equations are not equivalent as they are characterised by different irreducible representations.
\section{Gravitising anyons}
\label{anywave}
\subsection{The Lorentz double and its representations}
We now extend and apply our method for deriving wave equations from Lorentz covariant UIRs of the Poincar\'e group to a deformation of the Poincar\'e symmetry to the quantum double of the universal cover of the Lorentz group, or Lorentz double for short.
As reviewed in our Introduction, this is motivated by results from the study of 3d gravity and a general interest in understanding possible quantum deformations of standard wave equations. Referring to \cite{Lessons3DGrav} and \cite{NonCommutSch} for reviews,
we sum up evidence for the emergence of quantum doubles in the quantisation of 3d gravity.
\noindent {\em Deformation of Poincar\'e symmetry:} As explained in \cite{EuclidSchroers} and \cite{BaisScatt} for the, respectively, Euclidean and Lorentzian case, the quantum double of the rotation and Lorentz group is a deformation of the group algebra of, respectively, the Euclidean and Poincar\'e group.
\noindent {\em Gravitational scattering:} The $R$-matrix of the Lorentz double can be used to derive a universal scattering cross section for massive particles with spin by treating gravitational scattering in 2+1 dimensions as a non-abelian Aharonov-Bohm scattering process \cite{BaisScatt}. This universal scattering cross section agrees with previously computed special cases, like the quantum scattering of a light spin 1/2 particle on the conical spacetime generated by a heavy massive particle, in suitable limits - see \cite{tHooft,DeserJackiw}.
\noindent {\em Combinatorial quantisation:} The quantum double of the Lorentz group arises naturally in the combinatorial quantisation of the Chern-Simons formulation of 3d gravity with vanishing cosmological constant. The classical limit of the quantum $R$-matrix is a classical $r$-matrix which is compatible with the non-degenerate bilinear symmetric and invariant pairing used in the Chern-Simons action \cite{EuclidSchroers,BaisScatt}, and the Hilbert space of the quantised theory can be constructed from unitary representations of the Lorentz double \cite{MeusburgerSchroers1,CMBSCombQuant}.
\noindent {\em Independent derivations:} Quantum doubles also emerges in approaches to 3d quantum gravity which do not rely on the combinatorial quantisation programme. In \cite{LQGQDouble} the quantum double is shown to play the role of quantum symmetry in 3d loop quantum gravity. In \cite{FL} it appears in a path integral approach to 3d quantum gravity.
In analogy with our treatment of the Poincar\'e group in 2+1 dimensions, we consider the double cover $\mathrm{SU}(1,1)$ and the universal cover $\widetilde{\mathrm{SU}}(1,1)$ of the identity component of the Lorentz group. Our goal is to obtain a deformation of the wave equation by covariantising and then Fourier transforming, in a suitable sense, the UIRs of the quantum double of $\widetilde{\mathrm{SU}}(1,1)$. This extends the results obtained in \cite{SchroersWilhelm} for the double cover $\mathrm{SU}(1,1)$. As we shall see, the universal cover is technically more involved but also conceptually more interesting.
The quantum double of a Lie group can be defined in several ways. We follow \cite{QuantDoubBaisMull,QDLocComp} with the conventions used in \cite{SchroersWilhelm,SemiDual}. In this approach we view the quantum double $\mathcal{D}(G)$ as the Hopf algebra which, as a vector space, is the space of continuous complex valued functions $C(G\times G)$. In order to exhibit the full Hopf algebra structure we need to adjoin singular $\delta$-distributions.
The Hopf algebra structure for $\mathcal{D}(G)$ with product $\bullet$, co-product $\Delta$, unit $1$, co-unit $\epsilon$, antipode $S$, $*$-structure and ribbon element $c$ is then as follows:
\begin{align}
&(F_1 \bullet F_2)(g,u) =\int_{G}F_1(v,vuv^{-1})F_2(v^{-1}g,u)d v, \nonumber \\
&1(g,u)= \delta_e(g), \nonumber \\
&(\Delta F)(g_1,u_1;g_2,u_2)=F(g_1,u_1u_2)\delta_{g_1}(g_2), \nonumber \\
&\epsilon(F)=\int_G F(g,e) d g, \nonumber \\
&(SF)(g,u)=F(g^{-1}, g^{-1}u^{-1}g), \nonumber \\
&F^*(g,u)=\overline{F(g^{-1}, g^{-1}ug)}. \nonumber \\
&c(g,u)=\delta_g(u),
\end{align}
where we write $dg$ for the left Haar measure on the group.
The representation theory of the double is given in \cite{QDLocComp}. In the case of $\mathcal{D}(\mathrm{\mathrm{SU(1,1)})}$, the UIRs are classified by conjugacy classes in $\mathrm{SU(1,1)}$ and UIRs of associated stabiliser subgroups. This should be viewed as a deformation of the discussion of $\tilde{P}_3$, where we had adjoint orbits in the linear momentum space. In the gravitational case, the group itself is interpreted as momentum space and orbits are conjugacy classes. Here we encounter the idea of curved momentum space discussed in the outline.
As we are interested in the case of massive particles we will only give the analogue of the massive representations of $\tilde{P}_3$, and refer to \cite{QDLocComp} and \cite{SchroersWilhelm} for the complete list. By definition, elements of $\mathrm{SU(1,1)}$ have eigenvalues which multiply to $1$. In the case where these eigenvalues are complex conjugates, one has two disjoint families of elliptic conjugacy classes labelled by an angle $\theta$:
\begin{equation}
E(\theta)=\Set{v\exp \left(-\theta s^0\right)v^{-1}|v\in \mathrm{\mathrm{SU(1,1)}},\;\theta \in (0,2\pi)\cup (2\pi,4\pi)}.
\end{equation}
The stabiliser subgroup of the representative element $\exp (-\theta s^0)$ in $E(\theta)$ is
\begin{equation}
N^T=\Set{\exp (-\alpha s^0)|\alpha \in [0,4\pi)}\simeq U(1).
\end{equation}
In (2+1)-dimensional gravity, the variable $\theta$ parametrising the conjugacy classes gives the mass of a particle in units of the Planck mass, or in our convention $\theta = \lambda m$. Geometrically, it gives the deficit (or surplus) angle of the conical spatial geometry surrounding the particle's worldline.
The carrier spaces of UIRs of $\mathcal{D}(\mathrm{SU(1,1)})$ can, as with the Poincar\'{e} group, be given in terms of functions on $\mathrm{SU(1,1)}$ subject to an equivariance condition. The equivariance condition only depends upon the stabiliser subgroup, and in fact the carrier space is undeformed. The action of $\mathcal{D}(\mathrm{SU(1,1)})$ on $V_{ms}$ is a deformed version of \eqref{pi ms rep} and is given by
\begin{equation}
\label{doubleequivariant}
(\Pie_{ms}(F)\psi )(v)=\int_{\mathrm{SU(1,1)}}F\left(g,g^{-1}ve^{-m\lambda s^0}v^{-1}g\right) \psi(g^{-1}v) \, d g,
\end{equation}
where $F \in \mathcal{D}(\mathrm{SU(1,1)})$ and $g,\:v \in \mathrm{SU}(1,1)$. To relate this formula to the representation of the Poincar\'e group, it is useful to consider singular elements
\bee
F=\delta_h\otimes f, \quad h \in \mathrm{SU}(1,1), \; f\in C(\mathrm{SU}(1,1)).
\eee
Its action on $V_{ms}$ now more closely resembles that of $(h,a) \in \tilde{P}_3$ in the UIRs $\pie_{ms}$:
\bee
(\Pie_{ms}(\delta_h\otimes f)\psi ](v)=f(h^{-1}ve^{-m\lambda s^0}v^{-1}h)\psi(h^{-1}v),
\eee
with $f$ generalising the function $\exp(i\langle a, \cdot\rangle)$.
In \cite{SchroersWilhelm}, local covariant fields are introduced for $\mathcal{D}(\mathrm{SU(1,1)})$ and deformed momentum space (spin) constraints are derived. After Fourier transform these constraints are interpreted as deformed relativistic wave equations. In \cite{SchroersWilhelm} this is explicitly done for particles of spin $s=0, \frac{1}{2}$ and $1$. We will not review these results here, but derive the analogues for the anyonic case, where we need to consider the quantum double $\mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$. The UIRs are discussed in \cite{BaisScatt}. We only recall the UIRs describing massive particles at this point, though we will need all conjugacy classes when we consider the Fourier transform in the next section.
Massive particles are described by UIRs with conjugacy classes obtained by exponentiating timelike generators of $\mathfrak{su}(1,1)$. Recalling from \eqref{expcover} that mathematically positive rotations by an angle $\mu$ have the form $\exp (-\mu s^0)=(\mu,0)$, we define
\begin{equation}
\label{TimelikeConClass}
E(\mu)=\Set{v(\mu, 0)v^{-1}|v \in \widetilde{\mathrm{SU}}(1,1),\;\mu \in (\RR \setminus 2\pi \ZZ)}.
\end{equation}
Elements in this class project to elliptic elements in $\mathrm{SU}(1,1)$ and our notation is chosen to reflect this.
The interpretation of the unbounded parameter $\mu$ in terms of 3d gravity is something we will discuss in detail in Sect.~\ref{wavesect}. For now we again identify it with the mass in Planck units, i.e.,
\bee
\mu = \lambda m,
\eee
but note that in the decomposition
\bee
\label{decompmu}
\mu = \mu_0 + 2\pi n, \quad n\in \ZZ, \quad \mu_0 \in (0,2\pi),
\eee
only the `fractional part' $\mu_0$ has a classical geometrical interpretation as a deficit angle. The integer parameter
\bee
n=\text{int}\left( \frac {\mu}{2\pi} \right)
\eee
would affect gravitational Aharonov-Bohm scattering, as mentioned in the Introduction, but has no obvious classical meaning. In that sense, it is a `purely quantum' aspect of the particle.
We will need a fairly detailed understanding of the elliptic conjugacy classes $E(\mu)$ later in this paper, so we note that a generic element can be written without loss of generality by choosing $v=(0,\beta)$, $\beta\in D$, so that
\bee
(\omega,\gamma) =(0,\beta)(\mu,0)(0,-\beta),
\eee
and therefore, using \eqref{SU(1,1)tildeproduct},
\bee
\omega= \mu +\frac 1 i \ln \frac{1-|\beta|^2e^{i\mu_0}}{1-|\beta|^2 e^{-i\mu_0}}.
\eee
It is an elementary exercise to check that, if $\mu_0 < \pi$, then $\omega < \mu +\pi$, and if $\mu_0>\pi$ then
$\omega < \mu +(2\pi-\mu_0)$. It follows that
\bee
\label{interparts}
\text{int} \left( \frac {\omega}{2\pi}\right)
=\text{int}\left(\frac {\mu}{2\pi}\right),
\eee
and this will be useful later.
The stabiliser subgroup of the representative element $(\mu,0)$ is given by
\begin{equation}
\tilde{N}^T=\Set{(\omega, 0) \in \widetilde{\mathrm{SU}}(1,1)} \simeq \RR.
\end{equation}
The stabiliser subgroup is the same as for UIRs of $P^\infty_3$ describing massive particles, and its UIRs are labelled by a real-valued spin $s$.
The carrier space of UIRs labelled by the mass parameter $m$ and spin $s$ is $V^A_{m s}$ as defined in \eqref{AnyonicEquivariance}. Elements $F$ of the double $\mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$ act on $\psi \in V^A_{ms}$ according to
\begin{equation}
\label{coverdoubleequivariant}
(\Pie_{ms}(F)\psi )(v)=\int_{\widetilde{\mathrm{SU}}(1,1) }F\left(g,g^{-1}v(\lambda m, 0)v^{-1}g\right) \psi(g^{-1}v)d g,
\end{equation}
As in the previous section, we now use the equivariant UIRs to construct a covariant formulation.
\begin{definition}[Deformed Anyonic Covariant Field] The deformed anyonic covariant field $\tilde{\phi}_{\pm}$
associated to an equivariant field $\psi\in V^A_{ms}$ is the map
\begin{align}
\label{gravancov}
\tilde{\phi}_{\pm} \colon E(\lambda m) \rightarrow \mathcal{H}_{l\pm},
\end{align}
where $\mathcal{H}_{l \pm}$ is the carrier space for the discrete series representations of $\widetilde{\mathrm{SU}}(1,1)$ \eqref{SL2 Coverings},
defined via
\begin{align}
\tilde{\phi}_{\pm}(u)=\psi(v)D_{l\pm}(v)\Ket {0}_l.
\end{align}
Here $u,v\in \widetilde{\mathrm{SU}}(1,1)$, $v$ is chosen so that $u=v(\lambda m,0)v^{-1}$ and $s=l$ for $\tilde{\phi}_+$ and $s=-l$ for $\tilde{\phi}_-$.
\end{definition}
\begin{lemma}
The covariant fields $\tilde{\phi}_\pm$ are well-defined.
\end{lemma}
\begin{proof} One needs to check that the definition is independent of the choice of $v$, i.e., that
\begin{equation}
\psi(v) D_{l\pm}(v)\Ket {0_l}=\psi(v(\alpha,0)) D_{l\pm}(v(\alpha,0))\Ket {0}_l \quad \forall \alpha \in \RR.
\end{equation} This follows by the calculation \eqref{invariance} carried out for the universal cover of the Poincar\'e group.
\end{proof}
The anyonic covariant field carries a unitary representation of $\mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$
which we denote $\Pic$:
\begin{equation}
\label{GravAnyonicCovAction}
(\Pic_{ms}(F) \tilde{\phi}_{\pm})(u)=\int_{\widetilde{\mathrm{SU}}(1,1)} F(g,g^{-1}ug) D_{l\pm}(g)\tilde\phi_{\pm}(g^{-1}ug) \; d g.
\end{equation}
Without further condition, this representation is not irreducible. We need gravitised,
anyonic versions of the spin and mass constraints.
\begin{lemma} [Deformed Anyonic Spin Constraint]
\label{ribbonlemma}
The anyonic fields \eqref{gravancov} satisfy the following spin constraint
\bee
\label{DefAnyCon}
\left( D_{l\pm}(u)-e^{i\lambda m s} \right) \tilde{\phi}_{\pm}(u)=0.
\eee
This constraint can be expressed in terms of the ribbon element as
\bee
\Pic_{ms}(c)\tilde{\phi}_{\pm} = e^{i\lambda m s} \tilde{\phi}_{\pm}.
\eee
\end{lemma}
\begin{proof} Let $u=v(\lambda m,0)v^{-1}$ where $v \in \widetilde{\mathrm{SU}}(1,1)$. Then, focusing on the positive series for simplicity, we compute
\begin{align}
D_{l+}(u)\tilde{\phi}_+(u)&= D_{l+}(u)\psi(v)D_{l\pm}(v)\Ket {0}_l \nonumber \\
&= \psi(v)D_{l+}(v)D_{l+}((\lambda m,0))\Ket {0}_l \nonumber \\
&= \psi(v)D_{l+}(v)e^{ il\lambda m}\Ket {0}_l \nonumber \\
&= e^{is\lambda ms}\tilde{\phi}_{+}(u) .
\end{align}
Using the action $\Pic_{ms}$ given in \eqref{GravAnyonicCovAction}, we also compute
\begin{align}
\label{ribbonact}
(\Pic_{ms}(c)\tilde{\phi}_+ )(u)&= \int_{\widetilde{\mathrm{SU}}(1,1)} \delta_g(g^{-1}ug) D_{l+}(g)\tilde\phi_{+}(g^{-1}ug) \, dg \nonumber \\
&= D_{l+}(u)\tilde\phi_{+}(u),
\end{align}
thus confirming the second claim. The calculation for $\phi_-$ is entirely analogous.
\end{proof}
The mass constraint can be formulated using the projection map $\pi:\widetilde{\mathrm{SU}}(1,1) \rightarrow \mathrm{SU}(1,1)$ defined in \eqref{coverproject} and the relation \eqref{expoformula}. A necessary condition for an element $u\in \widetilde{\mathrm{SU}}(1,1)$ to be in the conjugacy class $E(\lambda m) $ is
\bee
\label{massconstraint}
\frac 12 \tr (\pi(u)) = \cos\left(\frac{\lambda m}{2}\right).
\eee
This condition only sees the fractional part in the decomposition \eqref{decompmu} of $\mu=\lambda m$. To ensure that $u \in
E(\lambda m) $ we also need to impose the condition \eqref{interparts}. Writing $u=(\omega_u, \gamma_u)$ this reads
\bee
\label{integerconstraint}
\text{int} \left( \frac {\omega_u}{2\pi}\right)
=\text{int}\left(\frac {\lambda m}{2\pi}\right).
\eee
This is an analogue of the constraint relating the sign of the energy to that of the mass in the representation theory of $P^\infty_3$. However, it resolves an infinite instead of a two-fold degeneracy and is not generally implied by the spin constraint \eqref{DefAnyCon}.
We thus arrive at the following carrier space for the anyonic covariant representation $\Pic$
of the double $\mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$:
\begin{align}
\label{WGAms}
W_{ms}^{GA}= & \left\{ \tilde{\phi}_{\pm} \colon \widetilde{\mathrm{SU}}(1,1) \rightarrow \mathcal{H}_{l\pm} | \left( D_{l\pm}(u) - e^{i\lambda ms}\right)\tilde{\phi}_{\pm}(u)=0, \right.
\nonumber \\
& \left.
\left( \frac 12 \tr (\pi(u)) - \cos\left(\frac{\lambda m}{2}\right)\right) \tilde{\phi}_{\pm}(u)=0 ,
\left( \text{int} \left( \frac {\omega_u}{2\pi}\right)
-\text{int}\left(\frac {\lambda m}{2\pi}\right)\right) \tilde \phi
(u)=0\right\},
\end{align}
where we choose the upper sign for $s>0$ and the lower sign for $s<0$.
\begin{theorem}[Irreducibility of the carrier space $W^{GA}_{ms}$] The covariant representation $\Pic_{ms}$ of $\mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$ on $W^{GA}_{ms}$ defined in \eqref{GravAnyonicCovAction} is unitarily equivalent to the equivariant representation $\Pie_{ms}$ on $V^{A}_{ms}$ defined in \eqref{coverdoubleequivariant}. In particular, it is therefore irreducible.
\end{theorem}
\begin{proof}
We claim that the following maps are intertwiners:
\begin{equation}
L_{\pm} \colon V^{A}_{ms} \rightarrow W^{GA}_{ms},\qquad (L_{\pm}(\psi))(u)= \psi(\omega, \gamma) D_{l\pm}(\omega, \gamma)\Ket {0}_l,
\end{equation}
where $(\omega,\gamma)\in\widetilde{\mathrm{SU}}(1,1)$ is chosen so that $u=(\omega, \gamma)(\lambda m, 0) (\omega,\gamma)^{-1}$.
This follows the steps in the proof of Theorem \ref{interparts}, but requires replacing each statement for Lie algebras by the corresponding statement for groups.
Injectivity of the maps $L_\pm$ is immediate. To show surjectivity,
we write the spin constraint of a given state $\tilde \phi \in W^{GA}_{ms}$, $s >0$ without loss of generality, as
\bee
D_{l+}(\lambda m,0 )D_{l+}((\omega, \gamma)^{-1})\tilde \phi (u) = e^{i \lambda m s} D_{l+}((\omega, \gamma)^{-1})\tilde \phi (u).
\eee
Recalling that $l=s$, comparing with \eqref{InfAnyonRep+}, and also recalling that $\ket{0}_l$ is, up to a factor, the unique solution of
$ D_{l+}(\mu,0 )f = e^{i\mu l } f$, we deduce the proportionality
\bee
D_{l+}((\omega, \gamma)^{-1})\tilde \phi (u) = \psi (\omega,\gamma) \ket{0}_l,
\eee
where the proportionality factor $\psi$ may depend on $(\omega,\gamma)$. Moreover, it must have the property
\bee
\psi (\omega+\alpha,\gamma) = e^{-i\alpha s} \psi(\omega,\gamma),
\eee
to ensure independence of the choice of $(\omega,\gamma)$ for given $u$. Thus $\psi \in V^A_{ms}$.
The intertwining property is equivalent to the commutativity of the diagram
\bee
\begin{tikzcd}[row sep=large, column sep = large]
V^A_{ms} \arrow{r}{\Pie_{ms}(F)} \arrow[swap]{d}{L_{\pm}} & V^A_{ms} \arrow{d}{L_{\pm}} \\
W^{GA}_{ms} \arrow{r}{\Pic_{ms} (F)} & W^{GA}_{ms}
\end{tikzcd},
\eee
for $F\in \mathcal{D}(\widetilde{\mathrm{SU}}(1,1))$.
This is a straightforward calculation based upon the maps $L_{\pm}$, and the actions given in \eqref{doubleequivariant} and \eqref{GravAnyonicCovAction}.
\end{proof}
\subsection{Group Fourier transforms}
We now turn to the promised Fourier transform of the covariant UIRs of the Lorentz double. Our discussion here will be less complete and rigorous than our treatment so far. In particular, we do not survey different approaches to Fourier transforms and differential calculus in the context of Hopf algebras, but note that some relevant references are collected in \cite{SchroersWilhelm}. Instead, we only show how ideas first proposed by Rieffel in \cite{Rieffel} and recently pursued in the quantum gravity community under the heading of group Fourier transforms can be used to translate the algebraic mass and spin constraints in the definition \eqref{WGAms} into differential and difference equations.
It is worth formulating the problem we want to address
for a general Lie group $G$ with Lie algebra $\mathfrak{g}$. Concentrating for simplicity on complex-valued (rather than Hilbert space valued) functions, the standard Fourier transform \eqref{flatfourier} is a map
\begin{equation}
\mathrm{L}^2(\mathfrak{g}) \rightarrow \mathrm{L}^2(\mathfrak{g}^*).
\end{equation}
However, in order to deal with the `gravitised' anyons, we require a Fourier transform
\begin{equation}
\mathrm{L}^2(G) \rightarrow \mathrm{L}_\star^2(\mathfrak{g}^*),
\end{equation}
where the $\star$ indicates the space has been equipped with a (generally non-commutative) $\star$-product.
This is precisely the situation considered by Rieffel in \cite{Rieffel}, where he observed that, if the exponential map can be used to identify the Lie group with the Lie algebra, one can transfer the convolution product of functions on $G$ to functions on $\mathfrak{g}$ and then, by Fourier transform, to functions on $\mathfrak{g}^*$. This induces a non-commutative $\star$-product on functions on $\mathfrak{g}^*$ which is a strict deformation quantisation of the canonical Poisson structure on $\mathfrak{g}^*$. This works globally for nilpotent groups, but, as explained in \cite{Rieffel}, still makes sense, in an appropriate way, more generally. For details we refer the reader to Rieffel's excellent exposition in the paper \cite{Rieffel} which also contains comments on the relation to other quantisation methods, such as Kirillov's coadjoint orbit method.
Ideas very similar to Rieffel's have, more recently and apparently independently, been considered by a number of authors in the context of quantum gravity \cite{FL,FM,Raasakka,GOR}. This work has resulted in a general framework called
group Fourier transforms. In developing our Fourier transform for gravitised anyons we essentially need to adapt and extend the ideas of Rieffel and the concept of a group Fourier transforms to $G=\widetilde{\mathrm{SU}}(1,1)$.
We have found it convenient to use the terminology and notation used in the discussion of group Fourier transforms, particularly in \cite{Raasakka,GOR}, which we review briefly.
The starting point of the group Fourier transform is the existence of non-commutative plane waves
\bee
E : G \times \mathfrak{g}^* \rightarrow \CC,
\eee
satisfying the following normalisation and completeness relations
\begin{align}
\label{completeness}
E(e;x)&=1, \nonumber \\
E(u^{-1};x)&= \bar{E} ( u;x) = E(u;-x), \nonumber \\
\delta_e(u)&= \frac{1}{(2\pi)^d}\int_{\mathfrak{g}^*}E(u; x) \; dx,
\end{align}
where $d$ is the dimension of $G$ and $\delta_e$ is the Dirac $\delta$-distribution at the group identity element $e$ with respect to the left Haar measure $dg$.
Such non-commutative plane waves induce a $\star$-product on a suitable set of functions on $\mathfrak{g}^*$ (to be specified below) via the group multiplication in $G$:
\bee
\label{waveproduct}
E(u_1;x)\star E(u_2;x)=E(u_1u_2;x).
\eee
More precisely, given the non-commutative plane waves, one defines
\begin{align}
\mathcal{F}&: L^2(G) \rightarrow L^2_\star(\mathfrak{g}^*), \nonumber \\
\phi(x)&=\mathcal{F}(\tilde \phi)(x)=\int_{G} E(u;x)\tilde{\phi}(u)\, du,
\end{align}
where $ L^2_\star(\mathfrak{g}^*)$ is the image under $\mathcal{F}$ in $ L^2(\mathfrak{g}^*)$, equipped with the $\star$-product defined by linear extension of \eqref{waveproduct} and with the inner product imported from $ L^2(G)$. One checks that
\bee
\langle \phi_1, \phi_2\rangle = \frac{1}{(2\pi)^d} \int_{\mathfrak{g}^\star} \bar{\phi}_1(x) \star \phi_2(x)\; dx = \int_G \bar{\tilde{\phi}}_1 \tilde{\phi}_2 \; dg .
\eee
By construction, this Fourier transform intertwines the convolution product on $L^2(G)$ with the star product on $ L^2_\star(\mathfrak{g}^*)$.
We also define a candidate for an inverse transform via
\begin{align}
\mathcal{F}^\star&: L^2_\star(\mathfrak{g}^*)\rightarrow L^2(G), \nonumber \\
\tilde{\phi}(u)&=\mathcal{F}^\star(\phi)(u) =\frac{1}{(2\pi)^d} \int_{\mathfrak{g}^*}\bar{E}(u,x)\star \phi(x)\;dx ,
\end{align}
where we emphasise the presence of the $\star$-product.
It is easy to check that completeness ensures that $ \mathcal{F}^\star \circ \mathcal{F}=\text{id}_{L^2(G)}$. However, $\mathcal{F} \circ \mathcal{F}^\star$ generally has a non-trivial kernel, see \cite{GOR}.
In \cite{GOR} it is shown that under certain assumptions, one can find a coordinate map $k:G\rightarrow \mathfrak{g}$ on $G$ and a function $\eta:G \rightarrow \CC$ so that, up to a set of measure zero, the plane waves take the form
\begin{equation}
\label{PlaneWave}
E(u;x)= \eta(u) e^{i \langle x, k(u)\rangle}.
\end{equation}
Our task in the next section is to construct such non-commutative plane waves for $\widetilde{\mathrm{SU}}(1,1)$.
\subsection{Non-commutative waves for $\widetilde{\mathrm{SU}}(1,1)$ and anyonic wave equations}
\label{wavesect}
Our proposal for a Fourier transform on $\widetilde{\mathrm{SU}}(1,1)$ is based on the parametrisation of group elements summarised in the following proposition.
\begin{proposition}[Parametrisation of $\widetilde{\mathrm{SU}}(1,1)$]
Every element $(\omega, \gamma)\in \widetilde{\mathrm{SU}}(1,1)$ can be uniquely expressed in terms of the $2\pi$-rotation $\Omega$ \eqref{Omegadef} which generates the centre of $\widetilde{\mathrm{SU}}(1,1)$ and the exponential map via
\bee
\label{pnparametrisation}
(\omega, \gamma)=\Omega^n\widetilde{\exp}(p), \quad p = -\lambda \bp \cdot \bs \in \mathfrak{su}(1,1), \; n \in \ZZ, \; \lambda^2 \bp^2< (2\pi)^2, \; \text{and} \; \;p^0>0\; \text{if} \; \,\bp^2 >0.
\eee
\end{proposition}
Before we enter the proof, we should point out that, for elements in the elliptic conjugacy class $E(\mu)$ defined in \eqref{TimelikeConClass}, the integer $n$ introduced in the Proposition is the same integer which appears in the decomposition \eqref{decompmu} of the rotation angle $\mu$. This follows since, for $(\omega,\gamma)\in E(\mu) $,
\bee
(\omega,\gamma) = v(\mu,0)v^{-1}=\Omega^n \widetilde{\exp}(p)
\eee
implies
\bee
v^{-1}\widetilde{\exp}(p) v = (\mu_0, 0), \quad 0<\mu_0 = \lambda |\bp| <2\pi,
\eee
so that $(\omega,\gamma)$ belongs to the conjugacy class with label $\mu =\mu_0 + 2\pi n$.
\begin{proof} To construct the claimed representation of a
given $(\omega, \gamma)\in \widetilde{\mathrm{SU}}(1,1)$, we first compute the element
$u=\pi(\omega,\gamma) \in \mathrm{SU}(1,1)$. Then, as reviewed in Sect.~2 and discussed in \cite{SchroersWilhelm}, the element $u$ or the element $-u$ is in the image of the exponential map in $\mathrm{SU}(1,1)$, i.e., there is a
$p \in \mathfrak{su}(1,1)$ and a choice of sign so that
\bee
\label{uexp}
u = \pm \exp(p).
\eee
However, then
\bee
\pi (\widetilde{\exp}(p)) = \pm u = \pm \pi(\omega,\gamma).
\eee
For the positive sign, this means $(\omega,\gamma)\widetilde{\exp}(-p)$ is in the kernel of $\pi$, which is generated by $\Omega^2$. Thus $(\omega,\gamma) = \Omega^{2m}\widetilde{\exp}(p)$ for some $m\in \ZZ$ in this case.
For the negative sign, we recall that $\pi(\Omega) =-\text{id}$, to deduce $(\omega,\gamma) = \Omega^{2m+1}\widetilde{\exp}(p)$ for some $m\in \ZZ$ in that case. Thus we obtain the claimed decomposition \eqref{pnparametrisation}, with even $n$ for the positive sign in \eqref{uexp} and odd $n$ for the negative sign.
In order to establish uniqueness of the decomposition \eqref{pnparametrisation}, consider $p,p'\in \mathfrak{su}(1,1)$ both satisfying the stated assumptions and $n,n'\in \ZZ$ so that
\bee
\widetilde{\exp}(p)\Omega^n = \widetilde{\exp}(p')\Omega^{n'}.
\eee
We need to show $p=p'$ and $n=n'$.
Projecting into $\mathrm{SU}(1,1)$ we deduce
\bee
\label{test}
\exp(p)=\pm \exp(p').
\eee
In particular, $ \exp(p)$ and $\exp(p')$ must be of the same type, i.e, both must be either elliptic, parabolic or hyperbolic.
We first consider the case where either $p$ or $p'$ vanishes. If one, say $p$, did then \eqref{test} would imply $ \exp(p')= \pm\text{id}$, but under the restriction on $p'$, this is only possible if the upper sign holds and $p'=0$, so that $n=n'$ follows.
Both parabolic and hyperbolic elements have the property that, if such an element is in the image of the exponential map, its negative is not. For such elements we must therefore have the upper sign in \eqref{test}. Moreover, one checks from the expressions \eqref{expoformula} that parabolic and hyperbolic elements which are in the image of the exponential map have a unique logarithm, so that we conclude $p=p'$ and hence $n=n'$.
Finally, elliptic elements differ from hyperbolic and parabolic elements in that both the element and its negative are in the image of the exponential map, so that we must consider both signs in \eqref{test}. With either sign, that equation shows that $\exp(p)$ and $ \exp(p')$ commute with each other. Then, the explicit expression \eqref{expoformula} and
\bee
\label{rangelimits}
0 <\lambda |\bp| <2\pi, \quad 0 <\lambda |\bp'| <2\pi,
\eee
imply that $p$ and $p'$ must be multiples of each other. By the assumption that both lie in the forward light cone, we can deduce (recalling the sign conventions \eqref{signconventions}) that
\bee
p= -\lambda |\bp|\hat p\cdot\bs , \quad p'=- \lambda|\bp'|\hat p\cdot\bs
\eee
Then \eqref{test} requires $\exp(p-p')=\pm\text{id}$ which is only possible if $ \lambda |\bp|$ and $\lambda |\bp'|$ are equal or differ by $2\pi$ or $4\pi$. The last two possibilities are not compatible with \eqref{rangelimits}, and so we deduce $p=p'$ and $n=n'$ in this case as well.
\end{proof}
\begin{figure}[h]
\begin{centering}
\includegraphics[width=4truecm,trim= 0 0.3cm 0 0]{Cylinder.pdf}
\hspace{2cm}
\includegraphics[width=3.15truecm ]{Cylinder2.pdf}
\vspace{-0.4cm}
\caption{Conjugacy classes and exponential curves in $\widetilde{\mathrm{SU}}(1,1)$. The diagram on the left shows selected conjugacy classes from the list \eqref{conjugacylist}, including single element (red dots), elliptic (red), parabolic (black) and hyperbolic conjugacy classes (blue). The diagram on the
right shows selected exponential curves \eqref{expocurves} with initial tangent vector $p$ timelike (red), lightlike (black) and spacelike (blue).}
\label{Cylinder}
\end{centering}
\end{figure}
The decomposition \eqref{pnparametrisation} can be be visualised and illustrated by thinking of $\widetilde{\mathrm{SU}}(1,1)$ as an infinite cylinder, with $\omega$ plotted along the vertical axis and $\gamma$ parametrising the horizontal slices. In Fig.~\ref{Cylinder} we show a vertical cross section of this cylinder and display the conjugacy classes and the exponential curves.
A full list of conjugacy classes of $\widetilde{\mathrm{SU}}(1,1)$ is given in the appendix of \cite{BaisScatt}. There are four kinds: single element conjugacy consisting of the elements $\Omega^n$, as well as elliptic, parabolic and hyperbolic conjugacy classes covering the corresponding classification for $\mathrm{SU}(1,1)$ discussed in Sect.~\ref{conventions}. In terms of the decomposition \eqref{pnparametrisation}, they can be described as follows:
\begin{align}
\label{conjugacylist}
O^n&=\{\widetilde{\exp}(p)\Omega^n \in \widetilde{\mathrm{SU}}(1,1) | p=0\}, \nonumber \\
E^n_{\mu_0}& = \{ \widetilde{\exp}(p)\Omega^n \in \widetilde{\mathrm{SU}}(1,1) | p^0>0, \lambda |\bp| = \mu_0, 0< \mu_0 < 2\pi \}, =E(\mu_0+2\pi n)\nonumber \\
P^n_+&= \{ \widetilde{\exp}(p)\Omega^n \in \widetilde{\mathrm{SU}}(1,1) | p^0>0, \bp^2 =0 \}, \nonumber \\
P^n_-&= \{ \widetilde{\exp}(p)\Omega^n \in \widetilde{\mathrm{SU}}(1,1) | p^0<0, \bp^2 =0 \}, \nonumber \\
H^n_\xi&= \{ \widetilde{\exp}(p)\Omega^n \in \widetilde{\mathrm{SU}}(1,1) | \bp^2= -\xi^ 2<0 \}.
\end{align}
The diagram on the right in Fig.~\ref{Cylinder} shows schematic sketches of exponential curves of the form
\bee
\label{expocurves}
\Gamma_{p,n}= \{\widetilde{\exp}(t p)\Omega^n| t\in [0,\infty)]\},
\eee
where $p$ is a fixed element of $\mathfrak{su}(1,1)$, and $n\in \ZZ$. In other words, these are images of the exponential map with chosen initial tangent vector $p$ translated by $\Omega^n$. We stress that the cross section we are showing suppresses the three-dimensional nature of these curves. To illustrate this, we show three-dimensional plots of some exponential curves starting at the identity in Fig.~\ref{SampleGeodesics}. Note that the spacelike and lightlike curves approach the boundary of the cylinder, but that the timelike curve winds round the axis of the cylinder, carrying out a complete rotation when $\omega$ increases by $2\pi$
\begin{figure}
\begin{centering}
\includegraphics[width=11truecm]{SelectedGeodesics.pdf}
\vspace{-6cm}
\caption{Exponential curves in $\widetilde{\mathrm{SU}}(1,1)$, obtained by exponentiating a spacelike (blue), lightlike (black) and timelike (red) tangent vector at the identity.}
\label{SampleGeodesics}
\end{centering}
\end{figure}
In order to compute the group Fourier transform we require an expression for the Haar measure on $\widetilde{\mathrm{SU}}(1,1)$ in the coordinates \eqref{pnparametrisation}.
Using the abbreviations introduced in \eqref{expoformula}, we note that for $
u= \exp(p)$,
the left-invariant Maurer-Cartan form is
\bee
u^{-1} \dd u = \lambda \dd |\bp| \hat{p}\cdot \bs - 2 c s\, \dd \hat{p}\cdot \bs- 2 s^2 \hat{p}\times \dd \hat{p}\cdot{s},
\eee
where we use \eqref{pythagoras} and suppressed the argument $|\bp|$ of the functions $c$ and $s$ for readability. Thus
\bee
[u^{-1} \dd u, u^{-1}\dd u] = 4s^2 \dd \hat{p}\times \dd \hat{p}\cdot \bs
- 4\lambda c s\, \dd |\bp|\wedge (\hat{p}\times \dd \hat{p})\cdot\bs + 4 \hat{p}^2 \lambda s^2 \dd |\bp| \wedge \dd\hat{p}\cdot\bs.
\eee
Thus, multiplying out and using again \eqref{pythagoras}, the Haar measure in exponential coordinates comes out as
\begin{align}
du &=\mathrm{tr}\left(u^{-1}\dd u \wedge u^{-1}\dd u \wedge u^{-1}\dd u\right) \nonumber \\
&= \frac{1}{2} \mathrm{tr}\left([u^{-1}\dd u,u^{-1}\dd u] \wedge u^{-1}\dd u\right)\nonumber \\
&=6 \lambda s^2\, \dd\hat{p}\times \dd\hat{p}\cdot \hat{p}\wedge \dd |\bp|.
\end{align}
Away from the set of measure zero where $\bp^2=0$, we have
\bee
d^3 \bp =\dd p^0\wedge \dd p^1 \wedge \dd p^2 =
\frac 1 6 \dd\bp \times \dd \bp \cdot \dd \bp
= \frac 12 |\bp|^2 \dd\hat{p}\times \dd\hat{p}\cdot \hat{p}\wedge \dd |\bp|,
\eee
so that, again away from the set where $\bp^2=0$,
\bee
du=\rho(p)d^3\bp, \quad \text{with} \quad
\rho(p)= 12 \lambda \frac{ s^2 ( |\bp|) }{ | \bp|^2 }.
\eee
Our parametrisation \eqref{pnparametrisation} of elements in $\widetilde{\mathrm{SU}}(1,1)$ requires both an element $p \in \mathfrak{su}(1,1)$ and an integer $n$. It is clear that a suitable non-commutative wave cannot depend only on a dual variable $x\in \mathfrak{su}(1,1)^*$. It also requires an argument which is dual to the integer $n$. The most natural candidate is an angular coordinate, parametrising a circle $S^1$. The necessity of a fourth and circular dimension to describe the spacetime dual to $\widetilde{\mathrm{SU}}(1,1)$ is a surprise. We will introduce it and explore its consequences at this point, postponing a discussion to our final section.
\begin{definition}We define non-commutative plane waves for $\widetilde{\mathrm{SU}}(1,1)$ as the maps
\begin{align}
\label{planewavedef}
E: \widetilde{\mathrm{SU}}(1,1) \times (\mathfrak{su}(1,1)^* \times S^1) & \rightarrow \CC, \nonumber \\
E(u;x,\varphi) =\frac{1}{\rho (p)} e^{i(\langle x,p\rangle + n\varphi)},
\end{align}
where $p\in \mathfrak{su}(1,1)$ and $n\in \ZZ$ are the parameters determining $u$ via the decomposition \eqref{pnparametrisation}, and $\varphi\in[0,2\pi)$ is an angular coordinate on the circle $S^1$.
\end{definition}
We need to check that the non-commutative waves satisfy a suitable version of the completeness relation \eqref{completeness}. Expressing the $\delta$-function with respect to the left Haar measure on $\widetilde{\mathrm{SU}}(1,1)$ in terms of the parameters $p$ and $n$ (see also B in \cite{GOR}),
\begin{equation}
\delta_e(g)=\frac{\delta_{n,0}}{\rho(p)}\delta^3(\bp),
\end{equation}
we confirm the required condition:
\begin{align}
\frac{1}{(2\pi)^4}\int_{\mathfrak{su}(1,1)^* \times S^1} E(u;x,\varphi)\; d x d\varphi &=\frac{1}{2\pi} \int_{S^1} e^{in\varphi }\,d \varphi \; \frac{1}{(2\pi)^3}\int_{\RR ^3} \,\frac{1}{\rho(p)} e^{i\bp\cdot \bx}\, d^3 \bx \nonumber\\
&=\delta_{n,0}\, \frac{1}{\rho(p)}\delta^3(\bp) \nonumber \\
&=\delta_{e}(u).
\end{align}
Before we use our non-commutative waves to carry out the group Fourier transform, we make some observations and comments.
With the terminology explained after \eqref{flatfourier}, the non-commutative waves
\bee
e^{i(\langle x,p\rangle + n\varphi)}= e^{i(\bx \cdot \bp + n\varphi)}
\eee
look like standard plane waves on the product of Minkowski space with a circle. However, the momentum $\bp$ is constrained by the conditions in \eqref{pnparametrisation}, so that timelike momenta have an invariant mass which is bounded from above by
\bee
\label{Planckmass}
m_p= \frac{2\pi}{\lambda} =\frac{1}{4G},
\eee
and are always in the forward lightcone. The existence of the Lorentz-invariant Planck mass $m_p$, and
hence also of an invariant Planck length, in the Lorentz double is one of its important features. It means in particular that it provides an example of a `doubly special theory of relativity' in 2+1 dimensions which neither deforms nor breaks Lorentz symmetry, see \cite{Lessons3DGrav} and our Summary and Outlook for further comments on this point.
It is natural to interpret the integer $n$ in the spirit of particle physics as a label for different kinds of particles in the theory. Timelike momenta $p$ with $n=-1$ may then be viewed as describing antiparticles. Lightlike and spacelike momenta for $n=0$ have the usual interpretation as momenta of massless or (hypothetical) tachyonic particles. The other values of $n$ describe additional types of massive, massless and tachyonic particles. Their existence is required by the fusion rules obeyed by the plane waves, which follow from the star product
\bee
\label{starwaves}
e^{i(\langle x,p_1\rangle + n_1\varphi)}\star e^{i(\langle x,p_2\rangle + n_2\varphi)} =e^{i(\langle x,p(u_1u_2)\rangle + n(u_1u_2)\varphi)}.
\eee
The general features of this fusion rule can be read off from the picture of the conjugacy classes of $\widetilde{\mathrm{SU}}(1,1)$ on the left in Fig.~\ref{Cylinder}. When multiplying plane waves for particles of types $n_1$ and $ n_2$, the particle type of the combined system is determined by the product, in $\widetilde{\mathrm{SU}}(1,1)$, of the group-valued momenta $u_1$ and $u_2$. This is a generalisation of the well-know Gott-pair in 2+1 gravity, where two ordinary particles ($n=0$) with high relative speed can combine into a particle with tachyonic momentum (and, in our terminology, of type $n=1$).
Thus we think of the plane waves for $\widetilde{\mathrm{SU}}(1,1)$ as describing kinematic states of particles in a theory with an invariant mass scale $m_p$ and with infinitely many different types of particles which combine according to the fusion rules encoded in the star product.
The Fourier transform of a covariant field
$
\tilde \phi: \widetilde{\mathrm{SU}} (1,1) \rightarrow \mathcal{H}_{l\pm}$
is given by
\begin{align}
\label{spacetimefield}
\phi(x,\varphi) =\int E(u;x,\varphi)\tilde\phi(u) \, du
=\sum_{n\in \ZZ} \int_{B_+} e^{i(\langle x,p\rangle + n\varphi)}\tilde \phi (u) \, d^3\bp,
\end{align}
where
\bee
B_+= \{p =-\lambda \bp \cdot \bs \in \mathfrak{su}(1,1)| \lambda^2 \bp^2 <(2\pi)^2, p^0>0 \; \; \text{if} \; \bp^2 >0 \}
\eee
is the region in momentum space required in the parametrisation \eqref{pnparametrisation}.
Since the field $\phi$ takes values in the Hilbert space $\mathcal{H}_{l\pm}$, the extension of the $\star$-product \eqref{starwaves} to products of such fields requires a careful tensor product decomposition. We will not pursue this here, but note that the decomposition of tensor products in the equivariant formulation of the UIRs for quantum doubles of compact Lie groups was studied in detail in \cite{KBM}. A full study of the $\star$-product for covariant fields $\phi$ will require an extension to non-compact groups and an adaptation of the results of that paper to our covariant formulation.
Here, we focus on single fields and apply the group Fourier transform to the deformed anyonic spin constraint \eqref{DefAnyCon} and the mass constraints \eqref{massconstraint} and \eqref{integerconstraint}.
Expanding $u\in \mathrm{SU}(1,1)$ as in \eqref{pnparametrisation}, we first note that
\bee
D_{l\pm}(u)= D_{l\pm}(\Omega^n\widetilde{\exp}(p)) = e^{2\pi i ns } D_{l\pm}(\widetilde{\exp}(p)) = e^{2\pi i ns } e^{d_{l\pm}(p)}.
\eee
But then, with $\lambda m =\mu$ and the decomposition \eqref{decompmu}, we also have
\bee
e^{i\lambda m s} = e^{i\mu_0 s} e^{2\pi ins}.
\eee
Hence, with $p=-\lambda s^ap_a$, the momentum space spin constraint \eqref{DefAnyCon} is equivalent to
\bee
\label{DefAnyConn}
\left(e^{-\lambda d_{l\pm}(s^a)p_a}-e^{i\mu_0 s} \right) \tilde{\phi}_{\pm}(u)=0.
\eee
The mass constraints \eqref{massconstraint} and \eqref{integerconstraint} also take a simple form in the parametrisation \eqref{pnparametrisation}. The former only see the fractional part $\mu_0\in (0,2\pi)$ of $\lambda m$, and is equivalent to
\bee
\label{newmassconstraint}
\bp^2 =\frac{\mu_0^2}{\lambda^2}.
\eee
The condition \eqref{integerconstraint} simply fixes the integer $n$ in the decomposition \eqref{pnparametrisation}.
Applying the Fourier transform \eqref{spacetimefield} turns the algebraic momentum space constraints into differential equations.
The mass constraint \eqref{newmassconstraint} becomes the Klein-Gordon equation for the fractional part of the mass:
\bee
\left( (\partial_0^2-\partial_1^2-\partial_2^2)+\frac{\mu_0^2}{\lambda^2}\right)\phi(x,\varphi)=0.
\eee
The integer constraint \eqref{integerconstraint} fixes the integer part of the mass
via the differential condition on the angular dependence of $\phi$:
\bee
-i\frac{\partial}{\partial \varphi} \phi (x,\varphi)= n\phi(x,\varphi).
\eee
Finally, the spin constraint \eqref{DefAnyConn} becomes
\begin{align}
\label{exponentialspace}
\left( e^{ i\lambda d_{l\pm}(s^a)\partial_a}-e^{i\mu_0 s} \right) \phi_{\pm}(x,\varphi)=0.
\end{align}
This equation involves an exponential of the differential operators \eqref{anywave2+}and \eqref{anywave2-}, combining spacetime derivatives with complex derivatives in the hyperbolic disk.
This is the anyonic generalisation of the exponential Dirac operator $e^{-\frac{\lambda}{2}\gamma^a\partial_a}\phi (x)$ that was obtained in \cite{SchroersWilhelm} for the massive spin $\frac{1}{2}$ particle.
Similar exponential operators have been considered in a more general context in \cite{Atiyah}, where it was stressed that they are essentially finite difference operators. The appearance of difference-differential equations was first mentioned in relation to (2+1) gravity in \cite{Matschullparticle}.
It is clear that further work is required to make sense of the equation \eqref{exponentialspace}. Stripped down to its simplest elements (by reducing all dimensions to one), it is an equation of the form
\bee
e^{\lambda \frac{\dd}{\dd x} }\phi(x)= e^{\lambda k}\phi(x),
\eee
or, assuming analyticity,
\bee
\phi(x+\lambda)= e^{\lambda k} \phi(x).
\eee
For analytic functions, this is equivalent to the infinitesimal version
\bee
\frac{\dd\phi}{\dd x} = k \phi(x),
\eee
obtained by differentiating with respect to $\lambda$. This simple example suggests that the anyonic constraint \eqref{anywave1} and the gravitational anyonic constrained \eqref{exponentialspace} may, suitably defined, be infinitesimal and finite versions of the same condition. However, careful analysis is required to clarify the definition of \eqref{exponentialspace} and its relation to \eqref{anywave1}.
\section{Summary and Outlook }
This paper was motivated by the observation that, in the context of 2+1 dimensional quantum gravity, the spin quantisation \eqref{SpinQuant} forces one to consider the universal cover of the Lorentz group and that, in order to preserve the duality between momentum space and Lorentz transformations in the quantum double, it is natural to take the universal cover in momentum space, too.
We showed how the representation theory of the quantum double of the universal cover
$\widetilde{\mathrm{SU}}(1,1)$ can be cast in a Lorentz-covariant form, and can be Fourier transformed. In this process, the universal covering of the Lorentz group necessitates the use of infinite-component fields, but the universal covering of momentum space has more interesting and far-reaching consequences.
The first of these, exhibited both in the decomposition \eqref{decompmu} and in the parametrisation \eqref{pnparametrisation}, is the extension of the range of the allowed mass. The fractional part $\mu_0$ of $\mu=8\pi Gm$ is the conventional mass of a particle, which manifests itself in classical (2+1)-dimensional gravity as a conical deficit angle in the spacetime surrounding the particle. The integer label $n$ in the decomposition $\mu= \mu_0 +2\pi n$ appears to be a purely quantum observable with no classical analogue.
It manifests itself, for example, in the Aharonov-Bohm scattering cross section of two massive particles, as discussed in \cite{BaisScatt}. It is a rather striking illustration of the concept of `quantum modular observables' as introduced in \cite{APP}, extensively discussed in the textbook \cite{AR} and recently applied to the notion of spacetime in \cite{FLM}.
We have chosen to interpret $n$ as a label of different types of particles or matter in (2+1)-dimensional quantum gravity. These particles can be converted into each other during interactions, according to fusion rules determined by the group product in $\widetilde{\mathrm{SU}}(1,1)$ and the decomposition of factors and products according \eqref{pnparametrisation}.
The second and surprising consequence of the universal covering of momentum space is the appearance of an additional and compact dimension on the dual side, in spacetime. This is needed to define the group Fourier transform, and allows for a simple expression of the constraint determining the particle type $n$ as a differential condition.
Our results raise a number of questions and suggest avenues for future research. As discussed at the end of the previous section, the exponentiated differential operators which generically appear as group Fourier transforms of the spin constraint should be studied using rigorous analysis. One expects these to be natural operators, possibly best defined as difference operators, not least because they are, by Lemma~\ref{ribbonlemma}, essentially Fourier transforms of the ribbon element of the quantum double.
It seems clear that our Hilbert-space valued fields $\phi$ on Minkowski space equipped with a $\star$-product fit rather naturally into the framework of braided quantum field theories, defined in \cite{Oeckl} and studied, in a Euclidean setting, in \cite{FL,SS}. Our paper is only concerned with a single particle and we only looked at simple examples of fusion rules for two spinless particles. However, braided quantum field theory naturally provides the language for discussing the gravitational interactions of several gravitational anyons in a spacetime setting. This provides an alternative viewpoint to existing momentum space discussions, with the non-commutative $\star$-product and the universal $R$-matrix of the quantum double encoding the quantum-gravitational interactions.
It is worth stressing that the $\star$-product on Minkowski space considered here preserves Lorentz covariance, and that any braided quantum field theory constructed from it would similarly be Lorentz covariant. This follows essentially from the Lorentz invariance of the mass scale \eqref{Planckmass} and the associated Planck length scale \cite{NonCommutSch}. It reflects the important fact that the Lorentz double deforms Poincar\'e symmetry by introducing a mass scale while preserving Lorentz symmetry, which is a challenge for any theory of quantum gravity.
Finally, it would be interesting to repeat the analysis of this paper with the inclusion of a cosmological constant. This leads to a $q$-deformation of the quantum double of $\widetilde{\mathrm{SU}}(1,1)$
\cite{NonCommutSch}, and there should similarly be a $q$-deformation of the spacetime picture of the representations. Some remarks on how this might work are made in \cite{SemiDual}, but none of the details have been worked out. In the Lorentzian context, a positive cosmological constant will lead to real deformation parameter $q$ while a negative cosmological constant will lead to $q$ lying on the unit circle \cite{NonCommutSch}. It would clearly be interesting to understand how this change in $q$ captures the radically different physics of the two regimes.
\vspace{0.5cm}
\noindent {\bf Acknowledgements} \; SI acknowledges support through an EPSRC doctoral training grant. Several of the results in this paper were reported by BJS at the Workshop on Quantum Groups in Quantum Gravity, University of Waterloo 2016. BJS thanks the organisers for the invitation, and acknowledges discussions with participants at the workshop. We thank Peter Horvathy for drawing our attention to Plyushchay's work on anyonic wave equations and to their joint work on noncommutative waves.
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Like fall leaves clinging to the rocks in the rushing water, so are the middle class fighting to just hang on and not be swept away by the headlong tide of governmental control and greed! Are you a middle-class American? How does it feel to be an endangered species?!)
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\begin{document}
\title{A class of conserved currents for linearized gravity in the Kerr spacetime}
\author{Alexander M. Grant, \'{E}anna \'{E}. Flanagan}
\affiliation{Department of Physics, Cornell University, Ithaca, NY 14853, USA}
\begin{abstract}
We construct a class of conserved currents for linearized gravity on a Kerr background.
Our procedure, motivated by the current for scalar fields discovered by Carter (1977), is given by taking the symplectic product of solutions to the linearized Einstein equations that are defined by symmetry operators.
We consider symmetry operators that are associated with separation of variables in the Teukolsky equation, as well as those arising due the self-adjoint nature of the Einstein equations.
In the geometric optics limit, the charges associated with these currents reduce to sums over gravitons of positive powers of their Carter constants, much like the conserved current for scalar fields.
We furthermore compute the fluxes of these conserved currents through null infinity and the horizon and identify which are finite.
\end{abstract}
\maketitle
\tableofcontents
\section{Introduction and Summary} \label{section:intro}
In the Kerr spacetime, freely falling point particles possess a constant of motion, distinct from the energy $E$ and the $z$ component of angular momentum $L_z$, known as the Carter constant $K$~\cite{PhysRev.174.1559}.
Much like $E$ and $L_z$, which are associated with Killing vectors, this constant of motion can be written in terms of a symmetric rank two \emph{Killing tensor} $K_{ab}$ as~\cite{1970CMaPh..18..265W}
\begin{equation}
K = K_{ab} p^a p^b,
\end{equation}
where $p^a$ is the four-momentum of the particle and $K_{ab}$ satisfies
\begin{equation}
\nabla_{(a} K_{bc)} = 0.
\end{equation}
This Killing tensor is not associated with any isometry of the Kerr spacetime, although the Carter constant reduces to the particle's total squared angular momentum (which is associated with spherical symmetry) in the Schwarzschild limit.
We fix our conventions for $K_{ab}$ in equation~\eqref{eqn:carter_tensor} below.
In addition to point particles, one can also consider test fields on the Kerr background, that is, fields whose magnitudes are small enough that their gravitational backreaction can be neglected.
In the Kerr spacetime, scalar, spin-$1/2$, and electromagnetic test fields possess conserved charges that generalize the Carter constant:
\begin{itemize}
\item For a sourceless complex scalar field $\Phi$, the conserved charge is the Klein-Gordon inner product of $\Phi$ with $\pb{0} \mathcal{D} \Phi$~\cite{PhysRevD.16.3395}:
\begin{equation} \label{eqn:carter_charge}
\pb{0} K \equiv \frac{1}{2i} \int_\Sigma \ud^3 \Sigma^a \left[(\pb{0} \mathcal{D} \Phi) \nabla_a \overline{\Phi} - \overline{\Phi} \nabla_a \pb{0} \mathcal{D} \Phi\right],
\end{equation}
where $\Sigma$ is any spacelike hypersurface, the differential operator $\pb{0} \mathcal{D}$ is defined by
\begin{equation} \label{eqn:carter_operator}
\pb{0} \mathcal{D} \Phi \equiv \nabla_a (K^{ab} \nabla_b \Phi),
\end{equation}
and bars denote complex conjugation.
The operator $\pb{0} \mathcal{D}$ commutes with the d'Alembertian, and so maps the space of solutions into itself.
The charge $\pb{0} K$ is associated with the Carter constant in the following sense: for a solution of the form $\Phi \propto e^{-i\vartheta/\epsilon}$, which represents a collection of scalar quanta with Carter constants $\{K_\alpha\}$, the charge is given by (in the geometric optics limit $\epsilon \to 0$)
\begin{equation}
\pb{0} K = \frac{1}{\hbar} \sum_\alpha K_\alpha.
\end{equation}
That is, the charge is proportional to the sum of the Carter constants of each scalar quantum.
In the case of real scalar fields, the charge vanishes in the geometric optics limit.
\item A similar result holds for any spin-$1/2$ field $\psi$ satisfying the Dirac equation~\cite{PhysRevD.19.1093}.
In Kerr, there exists an antisymmetric \emph{Killing-Yano tensor} $f_{ab}$, which satisfies $\nabla_{(a} f_{b)c} = 0$ and $K_{ab} = f_{ac} f^c{}_b$, with our particular choice of $K_{ab}$ in equation~\eqref{eqn:carter_tensor}.
An operator $\pb{1/2} \mathcal{D}$, which is defined in terms of $f_{ab}$ and commutes with the Dirac operator, is given by
\begin{equation}
\pb{1/2} \mathcal{D} = i\gamma_5 \gamma^a \left(f_a{}^b \nabla_b - \frac{1}{6} \gamma^b \gamma^c \nabla_c f_{ab}\right),
\end{equation}
where $\gamma^a$ is the usual gamma matrix and, in terms of the Levi-Civita tensor $\epsilon_{abcd}$, $\gamma_5 \equiv i \epsilon_{abcd} \gamma^a \gamma^b \gamma^c \gamma^d$.
The charge which generalizes the charge in equation~\eqref{eqn:carter_charge} is proportional to the following integral over a spacelike hypersurface $\Sigma$:
\begin{equation}
\pb{1/2} K \propto \int_\Sigma \ud^3 \Sigma_a\; \overline{(\pb{1/2} \mathcal{D} \psi)} \gamma^a \pb{1/2} \mathcal{D} \psi.
\end{equation}
As in the scalar field case, this charge is proportional to the sum of the Carter constants of the individual quanta in the geometric optics limit.
This construction works for massive as well as massless spin-$1/2$ particles, and even charged spin-$1/2$ particles in the case of the Kerr-Newman spacetime~\cite{PhysRevD.19.1093}.
\item For electromagnetic fields, there are several conserved charges which satisfy the requirement of reducing, in the geometric optics limit, to a sum of (some power) of the Carter constants of the photons; some examples are given by~\cite{Andersson:2015xla}, which we have considered in~\cite{Grant:2019qyo} (along with additional examples).
\end{itemize}
It would be interesting to find similar conserved currents in the case of linearized gravity.
One application of such a conserved current would be to gravitational wave astronomy, in the form of further advances in the so-called \emph{extreme mass-ratio inspiral problem}.
The gravitational waves radiated during the inspiral of compact objects into supermassive black holes will be an important signal for LISA~\cite{AmaroSeoane:2012km}.
There is therefore a major effort currently underway to accurately compute gravitational waveforms that these sources would produce (see, for example,~\cite{Wardell:2015kea} and the references therein).
As there is a great separation of scales in the masses of the inspiralling object and the supermassive black hole, this is known as the extreme mass-ratio inspiral (EMRI) problem.
The compact object is treated as a point particle, and given an orbit, which on short timescales is geodesic, the radiation can be computed using black hole perturbation theory.
However, on long timescales, the orbital parameters change due to the effects of radiation reaction, and so on these timescales the computed radiation must be corrected.
Special classes of orbits, such as circular or equatorial orbits, can be evolved in the adiabatic limit by using the fluxes of energy and angular momentum to infinity and down the horizon to evolve the orbital energy and angular momentum, since for these orbits the Carter constant is completely determined by the energy and angular momentum (see, for example,~\cite{Hughes:1999bq}).
Generic orbits require a method of obtaining time-averaged rates of change of an orbit's Carter constant.
A formula for this quantity to leading adiabatic order has been derived directly from the self-force~\cite{Mino:2003yg} (see~\cite{Isoyama:2018sib} for recent efforts in this problem, including extensions of this result to the resonant case).
It is qualitatively similar to the formulae for energy and angular momentum fluxes, having terms corresponding to infinity and to the horizon~\cite{Sago01052006}.
There is, however, no known derivation of this formula from a conserved current.
Such a derivation would provide a unified framework with which to understand these results, and may be necessary to obtain results at higher order.
These higher-order results may be necessary for parameter estimation, or perhaps even simply detection, of signals from EMRIs.
Unfortunately, no conserved currents generalizing the Carter constant for general stress-energy tensors exist.
More precisely, we have shown that, given a general, conserved stress-energy tensor in Kerr, there is no functional of the stress-energy tensor and its derivatives on a spacelike hypersurface $\Sigma$ that a) reduces to the Carter constant for a point particle and b) is independent of the choice of hypersurface $\Sigma$ when the stress-energy tensor is of compact spatial support~\cite{Grant:2015xqa}.
This implies that there can be no generic derivation of a flux formula for a ``Carter constant'' that applies to arbitrary fields and sources.
It is still possible, however, that such derivations could exist for specific types of fields.
In particular, it may be possible to derive a flux formula for determining the evolution of an orbit's Carter contant in linearized gravity from an appropriate conserved current.
Motivated by this possibility, in this paper we construct four conserved currents, denoted $\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs g]$, $\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$, and $\pb{\pb{\pm 2} \Omega} j^a [\var \bs g]$, that generalize the Carter constant in Kerr, in the sense that each of their charges reduce to the sum of some positive power of the Carter constants of the gravitons in the geometric optics limit.
Moreover, we show that these currents have the further property that their fluxes at null infinity and the horizon are finite for well-behaved solutions that describe radiation.
While these currents themselves are new, their construction involves symmetry operators which have been studied extensively in the literature (see, for example,~\cite{PhysRevLett.41.203, Chrzanowski:1975wv, Aksteiner:2016mol}).
The organization of this paper is as follows.
Section~\ref{section:kerr_perturbation} is a review of the theory of linearized gravity in Kerr, using both the spinor and Newman-Penrose formalisms, and fixes conventions which we use throughout.
It also reviews the Teukolsky formalism and separation of variables in the Kerr spacetime.
Section~\ref{section:symmetry} defines symmetry operators, which are the maps from the space of solutions into itself, such as the operator $\pb{0} \mathcal{D}$ in equation~\eqref{eqn:carter_operator} above.
We give particular examples of symmetry operators for linearized gravity in Kerr, and show how they act on expansions that arise in the Teukolsky formalism.
In section~\ref{section:currents}, we first define the symplectic product, a generalization of the Klein-Gordon inner product used in the scalar case, which we then use to generate the conserved currents that we consider in this paper.
In section~\ref{section:geometric_optics}, we review the geometric optics limit of solutions in linearized gravity on a curved background and use it to deduce the limits of currents defined in section~\ref{section:currents}.
In section~\ref{section:fluxes}, we compute fluxes of these currents through the horizon and null infinity.
We conclude in section~\ref{section:discussion} with general discussion and a summary of the properties of these currents in table~\ref{table:summary}.
Appendices~\ref{appendix:fluxes_integration} and~\ref{appendix:asymptotics} contain details of the calculations in section~\ref{section:fluxes}.
We use the following conventions in this paper: we follow most texts on spinors by using the $(+, -, -, -)$ sign convention for the metric and bars to denote complex conjugation.
We denote tensors with indices removed by bold face.
For any linear operator $T_{a_1 \cdots a_p}{}^{b_1 \cdots b_q}$ which maps tensors of rank $q$ to those of rank $p$, we write $T_{a_1 \cdots a_p}{}^{b_1 \cdots b_q} S_{b_1 \cdots b_q}$ as $\boldsymbol{T} \cdot \boldsymbol{S}$ when indices have been removed.
Furthermore, we will leave explicit the soldering forms $\sigma_a{}^{AA'}$ which form the isomorphism between the tangent vector space and the space of Hermitian spinors~\cite{penrose1987spinors}.
\section{Kerr perturbations: review and definitions} \label{section:kerr_perturbation}
\subsection{Spinor formalism} \label{section:spinors}
In this paper, we will be using a combination of the spinor and Newman-Penrose formalisms in order to describe linearized gravity about some arbitrary vacuum solution of the Einstein equations.
In general, we follow the notation of Penrose and Rindler~\cite{penrose1987spinors, penrose1988spinors}.
The spinor formalism is particularly convenient in Kerr, since not only is there a rank two Killing tensor $K_{ab}$ as discussed in section~\ref{section:intro}, but also a rank two symmetric spinor $\zeta_{AB}$ which satisfies the \emph{Killing spinor equation}~\cite{penrose1988spinors}:
\begin{equation} \label{eqn:killing_spinor}
\nabla^{A'}{}_{(A} \zeta_{BC)} = 0.
\end{equation}
This Killing spinor generates the related \emph{conformal Killing tensor} $\Sigma_{ab}$ given by
\begin{equation} \label{eqn:carter_tensor}
\Sigma_{ab} \equiv \sigma_a{}^{AA'} \sigma_b{}^{BB'} \zeta_{AB} \bar{\zeta}_{A'B'} \equiv \frac{1}{2} K_{ab} - \frac{1}{4} \Re\left[\zeta_{CD} \zeta^{CD}\right] g_{ab},
\end{equation}
which we use to define our Killing tensor $K_{ab}$~\cite{1970CMaPh..18..265W}.
Note that, given a Killing spinor $\zeta_{AB}$, equation~\eqref{eqn:carter_tensor} fixes the ambiguity in $K_{ab}$, which is otherwise only defined only up to terms of the form $\lambda g_{ab}$, for constant $\lambda$, or up to terms that are products of Killing vectors.
Petrov type D spacetimes possess a Killing spinor intimately connected with the Weyl spinor $\Psi_{ABCD}$~\cite{1970CMaPh..18..265W}, the symmetric spinor constructed from the Weyl tensor:
\begin{equation}
C_{abcd} \equiv \sigma_a{}^{AA'} \sigma_b{}^{BB'} \sigma_c{}^{CC'} \sigma_d{}^{DD'} \left(\epsilon_{AB} \epsilon_{CD} \overline{\Psi}_{A'B'C'D'} + \epsilon_{A'B'} \epsilon_{C'D'} \Psi_{ABCD}\right).
\end{equation}
Since $\Psi_{ABCD}$ is symmetric, it can be written as a symmetric product of four spinors
\begin{equation}
\Psi_{ABCD} = \alpha_{(A} \beta_B \gamma_C \delta_{D)}.
\end{equation}
For spacetimes of Petrov type D, there is a choice of these spinors such that $\alpha_A = \beta_A$ and $\gamma_A = \delta_A$ (this is one of many equivalent definitions of a type D spacetime).
Normalizing $\alpha_A$ and $\gamma_A$ to be a spin basis $(o, \iota)$ (that is, setting $o_A \iota^A = 1$), one finds
\begin{equation} \label{eqn:type_d}
\Psi_{ABCD} = 6 \Psi_2 o_{(A} o_B \iota_C \iota_{D)}.
\end{equation}
We are using the following notation for contractions of spinors with a given spin basis~\cite{penrose1987spinors}: given a symmetric spinor field $S_{B_1 \cdots B_n}$ and a spin basis $(o, \iota)$, we define (for any integer $i$ with $0 \leq i \leq n$)
\begin{equation}
S_i = S_{B_1 \cdots B_n} \iota^{B_1} \cdots \iota^{B_i} o^{B_{i + 1}} o^{B_n}.
\end{equation}
Thus, in equation~\eqref{eqn:type_d} $\Psi_2$ means the Weyl scalar $\Psi_{ABCD} \iota^A \iota^B o^C o^D$.
The spin basis $(o, \iota)$ is called a \emph{principal spin basis} for the Weyl spinor if it satisfies equation~\eqref{eqn:type_d}.
On such a basis, we define the Killing spinor $\zeta_{AB}$ by
\begin{equation}
\zeta_{AB} \equiv \zeta o_{(A} \iota_{B)},
\end{equation}
where $\zeta \sqrt[3]{\Psi_2}$ is constant~\cite{1970CMaPh..18..265W}.
For the remainder of the paper, we will restrict ourselves (generally) to a principal spin basis of the background Weyl spinor.
With these definitions in hand, we turn to the construction of linearized gravity in Kerr.
We fix the background Kerr metric $g_{ab}$, and consider a one-parameter family of metrics $g_{ab} (\lambda)$, with $g_{ab} (0) = g_{ab}$.
In general, we will use a notational convention where, for any quantity $Q$, $Q(\lambda)$ will denote the quantity at an arbitrary value of $\lambda$, and $Q$ without an argument will denote $Q(0)$, the background value.
The \emph{linearization} $\var Q$ of $Q(\lambda)$ is defined by\footnote{We are using $\var$, instead of the more conventional $\delta$, in order to avoid confusion with the Newman-Penrose operator $\delta$.}
\begin{equation}
\var Q = \left.\frac{\ud Q}{\ud \lambda}\right|_{\lambda = 0}.
\end{equation}
The linearized Einstein equations take the form
\begin{equation} \label{eqn:linearized_efe}
\pb{2} \mathcal{E}^{abcd} \var g_{cd} = 8\pi \var T^{ab},
\end{equation}
where
\begin{equation} \label{eqn:einstein_operator}
\pb{2} \mathcal{E}^{abcd} \equiv -\nabla^{(c} g^{d)(a} \nabla^{b)} + \frac{1}{2} (g^{cd} \nabla^{(a} \nabla^{b)} + g^{ac} g^{bd} \Box) - \frac{1}{2} g^{ab} (g^{cd} \Box - \nabla^{(c} \nabla^{d)}).
\end{equation}
is the linearized Einstein operator and $\var T^{ab}$ is the linearized stress-energy tensor.
Here the covariant derivative $\nabla_a$ is that associated with $g_{ab}$; the covariant derivative associated with $g_{ab} (\lambda)$ is denoted $\nabla_a (\lambda)$.
The prepended subscript 2 in $\pb{2} \mathcal{E}^{abcd}$ refers to the fact that linearized gravity is a spin-2 field.
To describe linearized perturbations using spinors, we consider the following quantity:
\begin{equation}
(\var g)_{AA'BB'} \equiv \sigma^a{}_{AA'} \sigma^b{}_{BB'} \var g_{ab}.
\end{equation}
Note that this is \emph{not} the variation of a spinor; we are performing the variation \emph{first}, and then computing a spinor field using the soldering forms $\sigma^a{}_{AA'}$ that are associated with the background spacetime\footnote{We note that there have been recent developments on a variational formalism for spinors~\cite{Backdahl:2015yua} which we will not be using.
We instead follow the traditional approach of~\cite{penrose1987spinors}.}.
In general, the placement of parentheses around a quantity that we are varying implies that we take the variation first, and then perform the operation, such as raising or lowering indices: for example, $(\var g)^{ab} = g^{ac} g^{bd} \var g_{cd}$, whereas $\var g^{ab}$ would be the variation of the raised metric, and in fact $\var g^{ab} = -(\var g)^{ab}$.
In a similar manner, one can define a spinor $(\var \Psi)_{ABCD}$ that is frequently called the perturbed Weyl spinor~\cite{penrose1987spinors} (although it is also not the variation of a spinor), again using the background soldering forms:
\begin{equation} \label{eqn:perturbed_weyl_spinor}
(\var \Psi)_{ABCD} \equiv \frac{1}{4} \sigma^a{}_{AE'} \sigma^b{}_B{}^{E'} \sigma^c{}_{CF'} \sigma^d{}_D{}^{F'} \var C_{abcd}.
\end{equation}
Using the form of the perturbed Riemann tensor, one finds that~\cite{penrose1987spinors}
\begin{equation} \label{eqn:perturbed_weyl_spinor_metric}
(\var \Psi)_{ABCD} = \frac{1}{2} \nabla^{A'}{}_{(C} \nabla^{B'}{}_{D} (\var g)_{AB)A'B'} + \frac{1}{4} (\var g)_e{}^e \Psi_{ABCD}.
\end{equation}
The equations of motion for the perturbed Weyl spinor are derived from the Bianchi identity, and are~\cite{penrose1987spinors}
\begin{equation} \label{eqn:massless_spin_2}
\nabla^{AA'} (\var \Psi)_{ABCD} = \frac{1}{2} (\var g)^{EFA'B'} \nabla_{BB'} \Psi_{EFCD} - \Psi_{EF(BC} \nabla_{D)}{}^{B'} (\var g)^{EFA'}{}_{B'} - \frac{1}{2} \Psi_{EF(BC} \nabla^{EB'} (\var g)_{D)}{}^{FA'}{}_{B'}.
\end{equation}
Thus, the equations of motion depend explicitly on the metric perturbation as well as the perturbed Weyl spinor.
Note further that equation~\eqref{eqn:massless_spin_2} reduces to the spin-2 massless spinor field equation $\nabla^{AA'} (\var \Psi)_{ABCD} = 0$ only when the manifold is conformally flat ($\Psi_{ABCD} = 0$).
The perturbed Weyl spinor, moreover, is not gauge invariant: under a gauge transformation $\var g_{ab} \to \var g_{ab} + 2 \nabla_{(a} \xi_{b)}$~\cite{penrose1987spinors},
\begin{equation}
(\var \Psi)_{ABCD} \to (\var \Psi)_{ABCD} + \xi^{EE'} \nabla_{E'(A} \Psi_{BCD)E} + 2 \Psi_{E(ABC} \nabla_{D)E'} \xi^{EE'}.
\end{equation}
For type D spacetimes, however, $(\var \Psi)_0$ and $(\var \Psi)_4$ \emph{are} gauge invariant, and they are the pieces that correspond to gravitational radiation~\cite{Stewart:1974uz}.
Moreover, as is well known, the equations of motion for $(\var \Psi)_0$ and $(\var \Psi)_4$ can be ``decoupled'' from those for $(\var \Psi)_1$, $(\var \Psi)_2$, and $(\var \Psi)_3$, and each other~\cite{1973ApJ...185..635T}, as we will discuss in section~\ref{section:teukolsky}.
It suffices to use either $(\var \Psi)_0$ or $(\var \Psi)_4$ to describe a generic, well-behaved perturbation, up to $l = 0, 1$ modes~\cite{1973JMP....14.1453W}, and therefore we can describe such perturbations in terms of gauge invariant variables.
\subsection{Newman-Penrose formalism} \label{section:np}
We will also be using the Newman-Penrose notation: given a spin basis $(o, \iota)$, the null basis $\{l^a, n^a, m^a, \bar{m}^a\}$ is defined by
\begin{equation} \label{eqn:soldering}
l^a = \sigma^a{}_{AA'} o^A \bar{o}^{A'}, \quad n^a = \sigma^a{}_{AA'} \iota^A \bar{\iota}^{A'}, \quad m^a = \sigma^a{}_{AA'} o^A \bar{\iota}^{A'},
\end{equation}
such that
\begin{equation}
g_{ab} = 2(l_{(a} n_{b)} - m_{(a} \bar{m}_{b)}).
\end{equation}
Using these four vectors, one can define the Newman-Penrose operators by $D = l^a \nabla_a$, $\npDelta = n^a \nabla_a$, and $\npdelta = m^a \nabla_a$, as well as the twelve spin coefficients via the following eight equations:
\begin{equation} \label{eqn:spin_coefficients}
\begin{aligned}
D o_A &= \epsilon o_A - \kappa \iota_A, \qquad &D \iota_A &= \pi o_A - \epsilon \iota_A, \\
\npDelta o_A &= \gamma o_A - \tau \iota_A, \qquad &\npDelta \iota_A &= \nu o_A - \gamma \iota_A, \\
\npdelta o_A &= \beta o_A - \sigma \iota_A, \qquad &\npdelta \iota_A &= \mu o_A - \beta \iota_A, \\
\bar{\npdelta} o_A &= \alpha o_A - \rho \iota_A, \qquad &\bar{\npdelta} \iota_A &= \lambda o_A - \alpha \iota_A.
\end{aligned}
\end{equation}
The five Weyl scalars $\Psi_0$, $\Psi_1$, $\Psi_2$, $\Psi_3$, and $\Psi_4$, in Newman-Penrose notation, take the form~\cite{Newman:1961qr}
\begin{equation} \label{eqn:weyl_scalars}
\Psi_i = -C_{abcd} \begin{cases}
l^a m^b l^c m^d & i = 0 \\
l^a n^b l^c m^d & i = 1 \\
\frac{1}{2} l^a n^b (l^c n^d - m^c \bar{m}^d) & i = 2 \\
l^a n^b \bar{m}^c n^d & i = 3 \\
n^a \bar{m}^b n^c \bar{m}^d & i = 4
\end{cases}.
\end{equation}
A null tetrad such that $\Psi_0 = \Psi_1 = \Psi_3 = \Psi_4 = 0$ and $\Psi_2 \neq 0$, for a Petrov type D spacetime, is called a \emph{principal tetrad} (as it is a tetrad associated with a principal spin basis).
Furthermore, at certain points throughout this paper, we will be using the notion of $'$ and $*$ transformations (reviewed in~\cite{Geroch:1973am}) to simplify the presentation.
These are defined by replacing, in some expression, the members of the spin basis via the following rules:
\begin{equation}
\begin{aligned}
': &\; o_A \mapsto i\iota_A,& &\iota_A \mapsto io_A,& &\bar{o}_{A'} \mapsto -i\bar{\iota}_{A'},& &\bar{\iota}_{A'} \mapsto -i \bar{o}_{A'},& \\
*: &\; o_A \mapsto o_A,& &\iota_A \mapsto \iota_A,& &\bar{o}_{A'} \mapsto -\bar{\iota}_{A'},& &\bar{\iota}_{A'} \mapsto -\bar{o}_{A'}.&
\end{aligned}
\end{equation}
The $'$ and $*$ transformations elucidate certain symmetries that appear in Newman-Penrose notation.
The $'$ transformation, which merely switches $l^a \leftrightarrow n^a$ and $m^a \leftrightarrow \bar{m}^a$, is particularly important in Kerr, since it preserves $(o, \iota)$ as a principal spin basis.
As an example, applying the transformations to equation~\eqref{eqn:spin_coefficients} yields
\begin{equation}
\begin{aligned}
\epsilon' &= -\gamma, &\kappa' &= -\nu, &\pi' &= -\tau, \\
\beta' &= -\alpha, &\sigma' &= -\lambda, &\mu' &= -\rho, \\
\epsilon^* &= -\beta, &\kappa^* &= -\sigma, &\pi^* &= -\mu, \\
\gamma^* &= -\alpha, &\tau^* &= -\rho, &\nu^* &= -\lambda.
\end{aligned}
\end{equation}
As another example, consider the following equations, in Newman-Penrose notation, that the scalar $\zeta$ obeys in Kerr:
\begin{equation}
D \zeta = -\zeta \rho, \qquad \Delta \zeta = \zeta \mu, \qquad \delta \zeta = -\zeta \tau, \qquad \bar{\delta} \zeta = \zeta \pi.
\end{equation}
The second equation can be derived from the first via a $'$ transformation, and likewise the fourth from the third, while the third follows from the first via a $*$ transformation.
In the future, we will only list one of the equations, and specify that the others can be obtained by the appropriate transformations.
\subsection{Teukolsky formalism} \label{section:teukolsky}
The Teukolsky formalism is a choice of variables for test fields in Kerr such that the equations of motion decouple, yielding equations that describe radiation, and furthermore, as we will discuss later in this section, separate in Boyer-Lindquist coordinates.
It builds off of the Newman-Penrose formalism: in the case of linearized gravity, the variables involve variations of the Weyl scalars.
Note that, taking variations of the Weyl scalars, we find that
\begin{equation} \label{eqn:np_vs_spinor}
\var \Psi_0 = (\var \Psi)_0, \qquad \var \Psi_4 = (\var \Psi)_4.
\end{equation}
On the left-hand sides of these equations, there is a variation of the null tetrad as well as the Weyl tensor; on the right, only the Weyl tensor is varied, according to equation~\eqref{eqn:perturbed_weyl_spinor}.
Note that equation~\eqref{eqn:np_vs_spinor} only holds for $\var \Psi_0$ and $\var \Psi_4$, and only because the background is type D, as the tetrad is varied when varying equation~\eqref{eqn:weyl_scalars}.
This result is rather convenient, since we will have reason to use $\var \Psi_0$ and $(\var \Psi)_0$, for example, interchangeably.
The choice of variables that are employed here are the so-called ``master variables'' $\pb{s} \Omega$, defined by~\cite{1973ApJ...185..635T}
\begin{equation} \label{eqn:decoupled_variables}
\pb{s} \Omega \equiv \begin{cases}
\zeta^4 \var \Psi_4 & s = -2 \\
\Phi & s = 0 \\
\var \Psi_0 & s = 2
\end{cases}.
\end{equation}
The value of $s$ is known as the \emph{spin-weight} of the particular variable.
Moreover, for $s > 0$, one can write these variables in terms of an operator $\pb{s} \boldsymbol{M}$, which maps from the space of gauge fields (such as the metric perturbation $\var g_{ab}$) to the corresponding master variable $\pb{s} \Omega$.
For example, for $|s| = 2$,
\begin{equation} \label{eqn:M_def}
\pb{s} \Omega = \pb{s} M^{ab} \var g_{ab}.
\end{equation}
From equations~\eqref{eqn:perturbed_weyl_spinor},~\eqref{eqn:decoupled_variables}, and~\eqref{eqn:M_def} (see, for example,~\cite{Chrzanowski:1975wv}),
\begin{subequations} \label{eqn:M}
\begin{align}
& \phantom{_-} \begin{aligned}
\pb{2} M^{ab} = - \frac{1}{2} \Big\{(\npdelta &+ \bar{\pi} - 3\beta - \bar{\alpha}) (\npdelta + \bar{\pi} - 2\beta - 2\bar{\alpha}) l^a l^b + (D - \bar{\rho} - 3\epsilon + \bar{\epsilon}) (D - \bar{\rho} - 2\epsilon + 2\bar{\epsilon}) m^a m^b \\
&- \big[(D - \bar{\rho} - 3\epsilon + \bar{\epsilon}) (\npdelta + 2\bar{\pi} - 2\beta) + (\npdelta + \bar{\pi} - 3\beta - \bar{\alpha}) (D - 2\bar{\rho} - 2\epsilon)\big] l^a m^b\Big\}, \\
\end{aligned} \\
& \begin{aligned}
\pb{-2} M^{ab} = - \frac{1}{2} \zeta^4 \Big\{(\bar{\npdelta} &- \bar{\tau} + 3\alpha + \bar{\beta}) (\bar{\npdelta} - \bar{\tau} + 2\alpha + 2\bar{\beta}) n^a n^b + (\npDelta + \bar{\mu} + 3\gamma - \bar{\gamma}) (\npDelta + \bar{\mu} + 2\gamma - 2\bar{\gamma}) \bar{m}^a \bar{m}^b \\
&- \big[(\npDelta + \bar{\mu} + 3\gamma - \bar{\gamma}) (\bar{\npdelta} - 2\bar{\tau} + 2\alpha) + (\bar{\npdelta} - \bar{\tau} + 3\alpha + \bar{\beta}) (\npDelta + 2\bar{\mu} + 2\gamma)\big] n^a \bar{m}^b\Big\}.
\end{aligned}
\end{align}
\end{subequations}
In terms of these variables, and in a type D spacetime, the equations of motion for the scalar field $\Phi$ ($s = 0$) and linearized gravity ($s = \pm 2$) may be written in the form~\cite{1973ApJ...185..635T}
\begin{equation} \label{eqn:teukolsky}
\pb{s} \Box \pb{s} \Omega = 8\pi \pb{s} \boldsymbol{\tau} \cdot \pb{|s|} \boldsymbol{T},
\end{equation}
known as the \emph{Teukolsky equation}.
Here, $\pb{s} \Box$ is a second-order differential operator (the \emph{Teukolsky operator}) that equals, for $s \geq 0$,
\begin{subequations}
\begin{align} \label{eqn:teukolsky_operator}
&\phantom{_-} \begin{aligned}
\pb{s} \Box = 2 \{[&D - (2s - 1) \epsilon + \bar{\epsilon} - 2s\rho - \bar{\rho}] (\npDelta - 2s\gamma + \mu) - [\npdelta - \bar{\alpha} - (2s - 1) \beta - 2s\tau + \bar{\pi}] (\bar{\npdelta} - 2s\alpha + \pi) \\
&- 2 (2s - 1) (s - 1) \Psi_2\},
\end{aligned} \\
&\begin{aligned}
\pb{-s} \Box = 2 \{[&\npDelta + (2s - 1) \gamma - \bar{\gamma} + \bar{\mu}] [D + 2s\epsilon + (2s - 1) \rho] - [\bar{\npdelta} + (2s - 1) \alpha + \bar{\beta} - \bar{\tau}] [\npdelta + 2s\beta + (2s - 1) \tau] \\
&- 2 (2s - 1) (s - 1) \Psi_2\}.
\end{aligned}
\end{align}
\end{subequations}
On the right-hand side of equation~\eqref{eqn:teukolsky}, $\pb{s} \boldsymbol{\tau}$ is an operator which converts $\pb{s} \boldsymbol{T}$, the source term for the equations of motion (for example, $\pb{2} T^{ab}$ is the stress-energy tensor $\var T^{ab}$), into the source term for the Teukolsky equation~\eqref{eqn:teukolsky}.
For example, one choice of $\pb{\pm 2} \tau_{ab}$ is given by inspection of equations~(2.13) and~(2.15) of~\cite{1973ApJ...185..635T}:
\begin{subequations} \label{eqn:tau}
\begin{align}
&\phantom{_-} \begin{aligned}
\pb{2} \tau_{ab} &= \left[(\npdelta + \bar{\pi} - \bar{\alpha} - 3\beta - 4\tau) l_{(a|} - (D - 3\epsilon + \bar{\epsilon} - 4\rho - \bar{\rho}) m_{(a|}\right] \\
&\qquad\qquad\qquad\times \left[(D - \epsilon + \bar{\epsilon} - \bar{\rho}) m_{|b)} - (\npdelta + \bar{\pi} - \bar{\alpha} - \beta) l_{|b)}\right],
\end{aligned} \\
&\begin{aligned}
\pb{-2} \tau_{ab} &= \zeta^4 \left[(\npDelta + 3\gamma - \bar{\gamma} + 4\mu + \bar{\mu}) \bar{m}_{(a|} - (\bar{\npdelta} - \bar{\tau} + \bar{\beta} + 3\alpha + 4\pi) n_{(a|}\right] \\
&\qquad\qquad\qquad\times \left[(\bar{\npdelta} - \bar{\tau} + \bar{\beta} + \alpha) n_{|b)} - (\npDelta + \gamma - \bar{\gamma} + \bar{\mu}) \bar{m}_{|b)}\right].
\end{aligned}
\end{align}
\end{subequations}
A freedom in $\pb{\pm 2} \tau_{ab}$ is discussed in section~\ref{section:gauge} below.
One can also rewrite Teukolsky's original result as an operator equation~\cite{PhysRevLett.41.203}, as we will find useful in section~\ref{section:tsi}.
In terms of $\pb{s} \boldsymbol{M}$,
\begin{equation} \label{eqn:decoupling}
\pb{s} \boldsymbol{\tau} \cdot \pb{|s|} \boldsymbol{\mathcal{E}} = \pb{s} \Box \pb{s} \boldsymbol{M},
\end{equation}
where, for $|s| = 2$, $\pb{|s|} \boldsymbol{\mathcal{E}}$ is the linearized Einstein operator~\eqref{eqn:einstein_operator}.
Applying equation~\eqref{eqn:decoupling} to a metric perturbation and using equation~\eqref{eqn:M_def} and the linearized Einstein equation~\eqref{eqn:linearized_efe} yields the Teukolsky equation~\eqref{eqn:teukolsky} for $|s| = 2$.
Since all of the operations just described are $\mathbb{C}$-linear, equation~\eqref{eqn:decoupling} holds for complexified metric perturbations as well.
So far, we have not tied our discussion to a particular coordinate system, nor a particular tetrad (other than enforcing that we use a principal null tetrad), since we have only required the background metric to be Petrov type D.
We now work in Kerr, and in Boyer-Lindquist coordinates $(t, r, \theta, \phi)$, where the metric takes the form
\begin{equation}
\ud s^2 = \ud t^2 - \Sigma \left(\frac{\ud r^2}{\Delta} + \ud \theta^2\right) - (r^2 + a^2) \sin^2 \theta \ud \phi^2 - \frac{2Mr}{\Sigma} \left(a \sin^2 \theta \ud \phi - \ud t\right)^2,
\end{equation}
where $\Delta = r^2 - 2Mr + a^2$ and $\Sigma = r^2 + a^2 \cos^2 \theta = |\zeta|^2$, and where we have chosen
\begin{equation} \label{eqn:zeta_bl}
\zeta = r - ia\cos \theta.
\end{equation}
This choice of $\zeta$ has the property that $\bs{t} \equiv \partial_t$ can be defined in terms of $\zeta_{AB}$~\cite{penrose1988spinors}:
\begin{equation} \label{eqn:time}
t^{AA'} = -\frac{2}{3} \nabla_B{}^{A'} \zeta^{AB}.
\end{equation}
Using the Kinnersley tetrad (a principal tetrad of the background Weyl tensor), which is given by
\begin{equation} \label{eqn:kinnersley}
\begin{gathered}
\bs{l} = \frac{(r^2 + a^2) \partial_t + a \partial_\phi}{\Delta} + \partial_r, \quad \bs{n} = \frac{(r^2 + a^2) \partial_t + a \partial_\phi}{2\Sigma} - \frac{\Delta}{2\Sigma} \partial_r, \\
\bs{m} = \frac{1}{\sqrt{2} \bar{\zeta}} \left(ia \sin \theta \partial_t + \partial_\theta + \frac{i}{\sin \theta} \partial_\phi\right),
\end{gathered}
\end{equation}
we find that $\Psi_2 = -M/\zeta^3$.
Furthermore, the non-zero spin coefficients are given by
\begin{equation} \label{eqn:kinnersley_spin_coefficients}
\begin{gathered}
\rho = -\frac{1}{\zeta}, \quad \mu = -\frac{\Delta}{2\Sigma \zeta}, \quad \gamma = \mu + \frac{r - M}{2\Sigma}, \\
\beta = \frac{\cot \theta}{2\sqrt{2} \bar{\zeta}}, \quad \pi = \alpha + \bar{\beta} = \frac{ia}{\sqrt{2} \zeta^2} \sin \theta, \quad \tau = -\frac{ia}{\sqrt{2} \Sigma} \sin \theta.
\end{gathered}
\end{equation}
We now review how the source-free version of the Teukolsky equation~\eqref{eqn:teukolsky} separates in these coordinates.
Consider, for integers $s$ and $n$, the operators~\cite{1974ApJ...193..443T, chandrasekhar1983mathematical}
\begin{equation} \label{eqn:D_L_operators}
\mathscr{D}_n = \partial_r + \frac{r^2 + a^2}{\Delta} \partial_t + \frac{a}{\Delta} \partial_\phi + 2n \frac{r - M}{\Delta}, \quad \mathscr{L}_s = \partial_\theta - i\left(a \sin \theta \partial_t + \frac{1}{\sin \theta} \partial_\phi\right) + s \cot \theta.
\end{equation}
Note that these operators satisfy
\begin{equation} \label{eqn:DL_raising}
\Delta^{-m} \mathscr{D}_n \Delta^m = \mathscr{D}_{n + m}, \qquad \sin^{-r} \theta \mathscr{L}_s \sin^r \theta = \mathscr{L}_{r + s}.
\end{equation}
We also define the operators $\mathscr{D}_n^+$ and $\mathscr{L}_s^+$, by taking $\mathscr{D}_n$ and $\mathscr{L}_s$ and setting $\partial_t \to -\partial_t$ and $\partial_\phi \to -\partial_\phi$; note that $\mathscr{L}_s^+ = \overline{\mathscr{L}_s}\;$~\footnote{Note that here, and below, our definition of the complex conjugate $\overline{\mathcal O}$ of an operator $\mathcal{O}$ is $\overline{\mathcal O} (f) = \overline{\mathcal{O} (\bar f)}$, where $f$ is the argument of this operator.
This is consistent with the standard notation for the Newman-Penrose operator $\bar{\delta}$.}.
Equations analogous to equations~\eqref{eqn:DL_raising} hold for $\mathscr{D}_n^+$ and $\mathscr{L}_s^+$.
We will also need a way to express these operators in terms of Newman-Penrose operators; using equations~\eqref{eqn:kinnersley} and~\eqref{eqn:kinnersley_spin_coefficients}, we find
\begin{equation} \label{eqn:np_reverse_engineer}
\mathscr{L}_s = \sqrt{2} \zeta \left(\bar{\npdelta} + 2s \bar{\beta}\right), \qquad \mathscr{D}_n = D + 2n \rho \mu^{-1} (\gamma - \mu), \qquad \mathscr{D}_n^+ = -\rho \mu^{-1} [\npDelta - 2n (\gamma - \mu)].
\end{equation}
Note that these formulae are only valid for the Kinnersley tetrad.
For real frequencies $\omega$ and integers $m$, we further define operators $\mathscr{D}_{nm\omega}$ and $\mathscr{L}_{sm\omega}$ by the requirement that, for any function $f(r, \theta)$,
\begin{equation} \label{eqn:DL_fourier}
\mathscr{D}_n \left[e^{i(m\phi - \omega t)} f(r, \theta)\right] \equiv e^{i(m\phi - \omega t)} \mathscr{D}_{nm\omega} f(r, \theta), \quad \mathscr{L}_s \left[e^{i(m\phi - \omega t)} f(r, \theta)\right] \equiv e^{i(m\phi - \omega t)} \mathscr{L}_{sm\omega} f(r, \theta).
\end{equation}
This equation yields the formulae
\begin{equation}
\mathscr{D}_{nm\omega} \equiv \partial_r + \frac{iK_{m\omega}}{\Delta} + 2n \frac{r - M}{\Delta}, \quad \mathscr{L}_{sm\omega} \equiv \partial_\theta + Q_{m\omega} + s \cot \theta,
\end{equation}
where
\begin{equation}
K_{m\omega} \equiv am - \omega (r^2 + a^2), \quad Q_{m\omega} \equiv m \csc \theta - a \omega \sin \theta
\end{equation}
(note that the conventions for $K_{m\omega}$ in~\cite{chandrasekhar1983mathematical} and~\cite{1973ApJ...185..635T} differ by a sign; here, we use the convention of~\cite{chandrasekhar1983mathematical}).
The operator on the left-hand side of the Teukolsky equation~\eqref{eqn:teukolsky} takes the following simple form:
\begin{equation} \label{eqn:RS_decomp}
\pb{s} \Box = \pb{s} \mathcal{R} + \pb{s} \mathcal{S},
\end{equation}
where
\begin{subequations} \label{eqn:RS}
\begin{align}
\pb{s} \mathcal{R} &\equiv \begin{cases}
\Delta \mathscr{D}_1 \mathscr{D}^+_s - 2 (2s - 1) r \partial_t & s \geq 0 \\
\Delta \mathscr{D}_{1 + s}^+ \mathscr{D}_0 - 2 (2s + 1) r \partial_t & s \leq 0
\end{cases}, \label{eqn:R} \\
\pb{s} \mathcal{S} &\equiv \begin{cases}
\mathscr{L}^+_{1 - s} \mathscr{L}_s + 2i (2s - 1) a \cos \theta \partial_t & s \geq 0 \\
\mathscr{L}_{1 + s} \mathscr{L}^+_{-s} + 2i (2s + 1) a \cos \theta \partial_t & s \leq 0
\end{cases}, \label{eqn:S}
\end{align}
\end{subequations}
where it can be readily shown that either the top or bottom lines of equations~\eqref{eqn:R} and~\eqref{eqn:S} yield equal results for $s = 0$; that is, $\pb{+0} \mathcal{R} = \pb{-0} \mathcal{R}$ and $\pb{+0} \mathcal{S} = \pb{-0} \mathcal{S}$.
Note that $\pb{s} \mathcal{R}$ is a differential operator that only depends on $r$, $t$, and $\phi$, while $\pb{s} \mathcal{S}$ only depends on $\theta$, $t$, and $\phi$.
As such, it is clear that the \emph{sourceless} Teukolsky equation~\eqref{eqn:teukolsky} separates in $r$ and $\theta$, and so one can write~\cite{1973ApJ...185..635T}
\begin{equation} \label{eqn:mode_expansion}
\pb{s} \Omega (t, r, \theta, \phi) = \int_{-\infty}^\infty \ud \omega \sum_{l = |s|}^\infty \sum_{|m| \leq l} \pb{s} \widehat{\Omega}_{lm\omega} (r) \pb{s} \Theta_{lm\omega} (\theta) e^{i(m\phi - \omega t)}.
\end{equation}
Inserting this expansion into the sourceless Teukolsky equation~\eqref{eqn:teukolsky}, followed by using equations~\eqref{eqn:RS_decomp}, \eqref{eqn:RS}, \eqref{eqn:DL_raising}, and~\eqref{eqn:DL_fourier}, one finds that (for $s \geq 0$), the functions $\pb{\pm s} \widehat{\Omega}_{lm\omega}$ and $\pb{\pm s} \Theta_{lm\omega}$ satisfy~\cite{chandrasekhar1983mathematical}
\begin{subequations} \label{eqn:separated_teukolsky}
\begin{align}
\left[\mathscr{L}_{(1 - s)(\mp m)(\mp \omega)} \mathscr{L}_{s(\pm m)(\pm \omega)} \pm 2(2s - 1) \omega a \cos \theta\right] \pb{\pm s} \Theta_{lm\omega} &= -\pb{\pm s} \lambda_{lm\omega} \pb{\pm s} \Theta_{lm\omega}, \label{eqn:angular_teukolsky} \\
\left[\Delta \mathscr{D}_{(1 - s)(\pm m)(\pm \omega)} \mathscr{D}_{0(\mp m)(\mp \omega)} \pm 2i (2s - 1) \omega r\right] \Delta^{(s \pm s)/2} \pb{\pm s} \widehat{\Omega}_{lm\omega} &= \Delta^{(s \pm s)/2} \pb{\pm s} \lambda_{lm\omega} \pb{\pm s} \widehat{\Omega}_{lm\omega}, \label{eqn:radial_teukolsky}
\end{align}
\end{subequations}
where $\pb{\pm s} \lambda_{lm\omega}$ is a separation constant.
This constant reduces to $(l + s)(l - s + 1) = l(l + 1) - s(s - 1)$ in the Schwarzschild limit~\cite{1973ApJ...185..649P, chandrasekhar1983mathematical}.
The functions $\pb{s} \Theta_{lm\omega}$ are regular solutions to a Sturm-Liouville problem on $[0, \pi]$ with eigenvalues $\pb{s} \lambda_{lm\omega}$.
Thus, there is only one solution for each value of $l$, $m$, and $\omega$, up to scaling.
Note, moreover, that the differential operator on the left-hand side of equation~\eqref{eqn:angular_teukolsky} commutes with the following three operations: complex conjugation, $(s, m, \omega) \to (-s, -m, -\omega)$, and $(s, \theta) \to (-s, \pi - \theta)$.
As such, we can simultaneously diagonalize this operator with each of these operations, choosing $\pb{s} \lambda_{lm\omega}$ and $\pb{s} \Theta_{lm\omega}$ to be real, as well as choosing
\begin{equation} \label{eqn:Theta_flips}
\pb{s} \Theta_{lm\omega} (\theta) = (-1)^{m + s} \pb{-s} \Theta_{l(-m)(-\omega)} (\theta), \qquad \pb{s} \Theta_{lm\omega} (\pi - \theta) = (-1)^{l + m} \pb{-s} \Theta_{lm\omega} (\theta)
\end{equation}
(a convention which is used by~\cite{1982JPhA...15.3737G}), as well as
\begin{equation} \label{eqn:lambda_flips}
\pb{s} \lambda_{lm\omega} = \pb{-s} \lambda_{lm\omega} = \pb{s} \lambda_{l(-m)(-\omega)}.
\end{equation}
Finally, the scaling freedom in $\pb{s} \Theta_{lm\omega}$ is fixed by imposing the following normalization condition~\cite{1973ApJ...185..635T}
\begin{equation} \label{eqn:Theta_normalization}
\int_0^\pi \pb{s} \Theta_{lm\omega} (\theta) \pb{s} \Theta_{l'm\omega} (\theta) \sin \theta \ud \theta = \delta_{ll'}.
\end{equation}
The functions
\begin{equation}
\pb{s} Y_{lm\omega} (\theta, t, \phi) \equiv e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta)
\end{equation}
are the so-called \emph{spin-weighted spheroidal harmonics}, and are orthogonal for different $l$, $m$, and $\omega$.
We now define another expansion for $\pb{s} \Omega$, subtly different from that in equation~\eqref{eqn:mode_expansion}, which results in a convenient way of expanding $\overline{\pb{s} \Omega}$ as well.
To do so, note that the differential operator on the right-hand side of equation~\eqref{eqn:radial_teukolsky} commutes with taking $(m, \omega) \to (-m, -\omega)$ followed by complex conjugation.
As such, we can construct two linearly independent solutions labelled by $p = \pm 1$ [their eigenvalue under this operation, multiplied by a conventional factor of $(-1)^{m + s}$]:
\begin{equation} \label{eqn:p_def}
\pb{s} \widehat{\Omega}_{lm\omega p} (r) \equiv \frac{1}{2} \left[\pb{s} \widehat{\Omega}_{lm\omega} (r) + p(-1)^{m + s} \overline{\pb{s} \widehat{\Omega}_{l(-m)(-\omega)} (r)}\right],
\end{equation}
and so
\begin{equation}
\pb{s} \widehat{\Omega}_{lm\omega} (r) = \sum_{p = \pm 1} \pb{s} \widehat{\Omega}_{lm\omega p} (r).
\end{equation}
It is occasionally more convenient to re-express the expansion~\eqref{eqn:mode_expansion} in terms of $\pb{s} \widehat{\Omega}_{lm\omega p} (r)$, instead of $\pb{s} \widehat{\Omega}_{lm\omega} (r)$:
\begin{equation} \label{eqn:decoupled_expansion}
\pb{s} \Omega (t, r, \theta, \phi) = \int_{-\infty}^\infty \ud \omega \sum_{l = |s|}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta) \pb{s} \widehat{\Omega}_{lm\omega p} (r).
\end{equation}
A simple consequence of equations~\eqref{eqn:Theta_flips} and~\eqref{eqn:p_def} is that
\begin{equation} \label{eqn:decoupled_bar_expansion}
\overline{\pb{s} \Omega (t, r, \theta, \phi)} = \int_{-\infty}^\infty \ud \omega \sum_{l = |s|}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} p e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} (\theta) \pb{s} \widehat{\Omega}_{lm\omega p} (r),
\end{equation}
and so this is a convenient expansion of the complex conjugate of the master variables.
Note, however, that these expansions are different in status from the expansion~\eqref{eqn:mode_expansion}, as the coefficients in this expansion must satisfy
\begin{equation} \label{eqn:eigenvalue_condition}
\overline{\pb{s} \widehat{\Omega}_{l(-m)(-\omega)p} (r)} = p (-1)^{m + s} \pb{s} \widehat{\Omega}_{lm\omega p} (r).
\end{equation}
\section{Symmetry operators} \label{section:symmetry}
As defined by Kalnins, McLenaghan, and Williams~\cite{1992RSPSA.439..103K}, a \emph{symmetry operator} is an $\mathbb{R}$-linear operator that maps the space of solutions to the equations of motion, which must be linear, into itself.
For the space of complexified solutions to real equations of motion, there exists a trivial symmetry operator mapping solutions to their complex conjugates.
In his original paper, Carter constructed the symmetry operator for scalar fields in equation~\eqref{eqn:carter_operator}, which commutes with the d'Alembertian~\cite{PhysRevD.16.3395}.
If an operator commutes with the operators in the sourceless equations of motion, then it must be a symmetry operator: if a field $\phi$ satisfies $\mathcal{L} \phi = 0$, and $[\mathcal{D}, \mathcal{L}] = 0$, then
\begin{equation}
\mathcal{L} \mathcal{D} \phi = \mathcal{D} \mathcal{L} \phi = 0,
\end{equation}
and so $\mathcal{D} \phi$ is a solution.
Lie derivatives with respect to Killing vectors are examples of symmetry operators which commute with the equations of motion.
Further examples of symmetry operators can be created by composing symmetry operators associated with Killing vectors, but these are, in a sense, ``reducible''.
In this section we review two classes of \emph{irreducible} symmetry operators that appear in the Kerr spacetime: those that derive from separation of variables, and those that arise from taking the adjoint of the Teukolsky equation.
Note that, recently, additional symmetry operators have been discussed in the Kerr spacetime~\cite{Aksteiner:2016mol}, which we do not discuss in this paper.
\subsection{Separation of variables} \label{section:separation}
The first class of symmetry operators we consider is associated with the separability of the underlying equations of motion.
To see that there is always a symmetry operator associated with separability, consider as an example the following partial differential equation (in two variables $x, y$):
\begin{equation} \label{eqn:example}
\mathcal{L} \phi \equiv \left[\mathcal{X} (x, \partial_x, \ldots) + \mathcal{Y} (y, \partial_y, \partial_y^2, \ldots)\right] \phi = 0,
\end{equation}
for some differential operators $\mathcal{X}$ and $\mathcal{Y}$.
Since $\mathcal{X}$ only depends upon $x$ and $\mathcal{Y}$ only depends upon $y$, $\mathcal{X}$ and $\mathcal{Y}$ must commute.
Moreover, $\mathcal{L} = \mathcal{X} + \mathcal{Y}$, and so $\mathcal{X}$ and $\mathcal{Y}$ must both commute with $\mathcal{L}$, and so $\mathcal{X}$ and $\mathcal{Y}$ are symmetry operators.
In addition, if there are additional variables $z_1, \ldots, z_n$, and $\mathcal{X}$ and $\mathcal{Y}$ only depend on derivatives with respect to these variables, then this argument still holds.
Irreducible symmetry operators arise in Kerr, similarly, via a separation of variables argument.
As discussed in section~\ref{section:teukolsky}, the Teukolsky equation separates, yielding the two operators $\pb{s} \mathcal{R}$ and $\pb{s} \mathcal{S}$ in equations~\eqref{eqn:R} and~\eqref{eqn:S} (respectively).
These operators are analogous to the operators $\mathcal{X}$ and $\mathcal{Y}$ in equation~\eqref{eqn:example} above, and depend on derivatives with respect to additional variables $t$ and $\phi$.
One combination of $\pb{s} \mathcal{R}$ and $\pb{s} \mathcal{S}$ is particularly interesting, namely
\begin{equation} \label{eqn:D}
\pb{s} \mathcal{D} \equiv \frac{1}{2} \left(\pb{s} \mathcal{R} - \pb{s} \mathcal{S}\right).
\end{equation}
One can show that, for $s = 0$, this is in fact the scalar symmetry operator~\eqref{eqn:carter_operator} discussed by Carter~\cite{PhysRevD.16.3395}.
In the case of linearized gravity, $\pb{s} \mathcal{D}$ is a map from the space of solutions of the homogeneous Teukolsky equation~\eqref{eqn:teukolsky} of spin weight $s$ into itself.
In section~\ref{section:symmetry_separated}, we will review a procedure (a version of Chrzanowski metric reconstruction~\cite{Chrzanowski:1975wv}) which will allow us to construct another operator $\pb{s} \mathcal{D}_{ab}{}^{cd}$ from $\pb{s} \mathcal{D}$ that maps the space of complexified metric perturbations into itself.
The symmetry operator $\pb{s} \mathcal{D}_{ab}{}^{cd}$ will be more useful than $\pb{s} \mathcal{D}$, since the symplectic product for linearized gravity naturally acts on the space of metric perturbations.
\subsection{Adjoint symmetry operators} \label{section:tsi}
In Kerr, for spins higher than 0, there is a second set of irreducible symmetry operators that can be constructed, following an argument due to Wald~\cite{PhysRevLett.41.203}.
This argument holds, as do many of our equations, for all $|s| \leq 2$; however, we will only explicitly use $|s| = 2$ in this paper.
The argument is as follows.
We first define the adjoint of a linear differential operator.
Consider a linear differential operator $\bs{\mathcal L}$ that takes tensor fields of rank $p$ to tensor fields of rank $q$.
We say that an operator which takes tensor fields of rank $q$ to tensor fields of rank $p$ is the adjoint $\bs{\mathcal L}^\dagger$ of $\bs{\mathcal L}$ if, for all tensor fields $\bs{\phi}$ of rank $p$ and tensor fields $\bs{\psi}$ of rank $q$, there exists a vector field $j^a [\bs{\phi}, \bs{\psi}]$ such that
\begin{equation} \label{eqn:adjoint}
\boldsymbol{\psi} \cdot (\bs{\mathcal L} \cdot \boldsymbol{\phi}) - \boldsymbol{\phi} \cdot (\bs{\mathcal L}^\dagger \cdot \boldsymbol{\psi}) = \nabla_a j^a [\boldsymbol{\phi}, \boldsymbol{\psi}].
\end{equation}
Note that this is not the usual definition of adjoint, which has a complex conjugate acting on $\boldsymbol{\psi}$ in the first term and on $(\mathcal{L}^\dagger \boldsymbol{\psi})$ in the second.
Chrzanowski~\cite{Chrzanowski:1975wv} and Gal'tsov~\cite{1982JPhA...15.3737G} use the usual definition, whereas Wald uses the definition~\eqref{eqn:adjoint}.
We now give some examples of adjoints of the operators considered in section~\ref{section:teukolsky}.
First, we note that one can easily show that, for two operators $\bs{\mathcal L}_1$ and $\bs{\mathcal L}_2$,
\begin{equation}
(\bs{\mathcal L}_1 \bs{\mathcal L}_2)^\dagger = \bs{\mathcal L}_2^\dagger \bs{\mathcal L}_1^\dagger.
\end{equation}
Moreover, the adjoints of the various Newman-Penrose operators, using equations~\eqref{eqn:soldering}, \eqref{eqn:spin_coefficients}, and~\eqref{eqn:adjoint}, are given by
\begin{equation} \label{eqn:adjoint_np}
D^\dagger = -D - (\epsilon + \bar{\epsilon}) + \rho + \bar{\rho},
\end{equation}
together with the corresponding expressions obtained via $'$ and $*$ transformations.
Using equations~\eqref{eqn:adjoint} and~\eqref{eqn:einstein_operator}, one finds that $\pb{2} \bs{\mathcal E}$ is self-adjoint:
\begin{equation} \label{eqn:einstein_dagger}
\pb{2} \bs{\mathcal E}^\dagger = \pb{2} \bs{\mathcal E}.
\end{equation}
Similarly, one can show from equations~\eqref{eqn:adjoint_np} and~\eqref{eqn:teukolsky_operator} that
\begin{equation} \label{eqn:teukolsky_dagger}
\pb{s} \Box^\dagger = \pb{-s} \Box,
\end{equation}
as was first noted by Cohen and Kegeles~\cite{PhysRevD.10.1070}.
Finally, the adjoint of the operator $\pb{s} \boldsymbol{\tau}$ [equation~\eqref{eqn:tau}] that enters into the Teukolsky equation~\eqref{eqn:teukolsky}, for $|s| = 2$, is given by
\begin{equation} \label{eqn:tau_dagger}
\pb{s} \tau_{ab}^{\dagger} = \begin{cases}
[m_{(a|} (D + 2\epsilon - \rho) - l_{(a|} (\npdelta + 2\beta - \tau)] [l_{|b)} (\npdelta + 4\beta + 3\tau) - m_{|b)} (D + 4\epsilon + 3\rho)] & s = 2 \\
[\bar{m}_{(a|} (\npDelta - 2\gamma + \mu) - n_{(a|} (\bar{\npdelta} - 2\alpha + \pi)] [n_{|b)} (\bar{\npdelta} - 4\alpha - 3\pi) - \bar{m}_{|b)} (\npDelta - 4\gamma - 3\mu)] \zeta^4 & s = -2 \\
\end{cases}.
\end{equation}
We now take the adjoint of equation~\eqref{eqn:decoupling}, yielding [from equations~\eqref{eqn:teukolsky_dagger} and~\eqref{eqn:einstein_dagger}]
\begin{equation} \label{eqn:adjoint_teukolsky}
\pb{|s|} \boldsymbol{\mathcal{E}} \cdot \pb{s} \boldsymbol{\tau}^\dagger = \pb{s} \boldsymbol{M}^\dagger \pb{-s} \Box.
\end{equation}
Suppose that we have a solution $\pb{-s} \psi$ to the vacuum Teukolsky equation $\pb{-s} \Box \pb{-s} \psi = 0$; note that $\pb{-s} \psi$ is not necessarily the master variable $\pb{-s} \Omega$ associated with $\var g_{ab}$ via equation~\eqref{eqn:M_def}.
Then, from equations~\eqref{eqn:adjoint_teukolsky},
\begin{equation}
\pb{|s|} \boldsymbol{\mathcal{E}} \cdot \pb{s} \boldsymbol{\tau}^\dagger \pb{-s} \psi = 0.
\end{equation}
Thus, $\pb{s} \boldsymbol{\tau}^\dagger \pb{-s} \psi$ is a \emph{complex} metric perturbation that solves the vacuum linearized Einstein equations.
Thus, the operator $\pb{s} \boldsymbol{\tau}^\dagger$ allows the construction of complex vacuum metric perturbations from vacuum solutions to the Teukolsky equation.
From a single solution $\pb{-s} \psi$ to the vacuum Teukolsky equation~\eqref{eqn:teukolsky} of spin weight $-s$, one can therefore apply $\pb{s'} \bs{M}$ (for some other $s'$, where $|s'| = |s|$) to either $\pb{s} \bs{\tau}^\dagger \pb{-s} \psi$ or $\overline{\pb{s} \bs{\tau}^\dagger \pb{-s} \psi}$, both of which yield solutions to the vacuum Teukolsky equation:
\begin{equation} \label{eqn:tsi}
\pb{s'} \Box \pb{s'} \boldsymbol{M} \cdot \pb{s} \boldsymbol{\tau}^\dagger \pb{-s} \psi = 0, \qquad \pb{s'} \Box \pb{s'} \boldsymbol{M} \cdot \overline{\pb{s} \boldsymbol{\tau}^\dagger \pb{-s} \psi} = 0.
\end{equation}
That is, there exist two symmetry operators of the form
\begin{equation} \label{eqn:C}
\pb{s', s} \mathcal{C} \equiv \pb{s'} \boldsymbol{M} \cdot \pb{s} \boldsymbol{\tau}^\dagger, \qquad \pb{s', s} \widetilde{\mathcal C} \equiv \pb{s'} \boldsymbol{M} \cdot \overline{\pb{s} \boldsymbol{\tau}^\dagger}.
\end{equation}
The operator $\pb{s', s} \mathcal{C}$ maps from the space of solutions to the vacuum Teukolsky equation~\eqref{eqn:teukolsky} of spin weight $-s$ to the space of solutions to the vacuum Teukolsky equation of spin weight $s'$.
Similarly, $\pb{s', s} \widetilde{\mathcal C}$ maps from the space of solutions to the complex conjugate of the vacuum Teukolsky equation~\eqref{eqn:teukolsky} of spin weight $-s$ into the space of solutions to the vacuum Teukolsky equation of spin weight $s'$.
As in section~\ref{section:separation}, these operators act on the master variables, rather than metric perturbations.
However, one can also construct the operators (for $|s| = 2$)
\begin{equation} \label{eqn:C_tensor}
\pb{s} \mathcal{C}_{ab}{}^{cd} \equiv \pb{s} \tau_{ab}^\dagger \pb{-s} M^{cd},
\end{equation}
which are symmetry operators for metric perturbations.
That is, they are $\mathbb{R}$-linear maps from the space of complexified solutions to the vacuum linearized Einstein equations into itself.
This follows from the operator identity (derived from equations~\eqref{eqn:adjoint_teukolsky} and~\eqref{eqn:C_tensor})
\begin{equation}
\pb{|s|} \boldsymbol{\mathcal{E}} \cdot \pb{s} \boldsymbol{\mathcal{C}} = \pb{s} \boldsymbol{M}^\dagger \pb{-s} \Box \pb{-s} \boldsymbol{M} = \pb{s} \boldsymbol{M}^\dagger \pb{-s} \boldsymbol{\tau} \cdot \pb{|s|} \boldsymbol{\mathcal{E}},
\end{equation}
where the second equality from equation~\eqref{eqn:decoupling}.
Applying this operator identity to (in general) a complex vacuum metric perturbation, the right-hand side yields zero.
Note that the two cases $s = \pm 2$ in equations~\eqref{eqn:tau_dagger} and~\eqref{eqn:M} differ by a $'$ transformation, along with a factor of $\zeta^4$, and so $\pb{2} \mathcal{C}_{ab}{}^{cd}$ and $\pb{-2} \mathcal{C}_{ab}{}^{cd}$ are related by a $'$ transformation.
Furthermore, the metric perturbations generated by $\pb{\pm 2} \mathcal{C}_{ab}{}^{cd}$ are in a trace-free gauge by construction.
Finally, we note that this argument has been used in a fully tetrad-invariant form, using a spinor form of the Teukolsky equations, to generate symmetry operators for metric perturbations of the sort that we review in this section~\cite{Aksteiner:2016mol}.
For simplicity, we use the Newman-Penrose form of the Teukolsky equations instead.
\subsection{Issues of gauge} \label{section:gauge}
Since the operators $\pb{\pm 2} \tau_{ab}^\dagger$ map into the space of metric perturbations which are solutions to the linearized Einstein equation, the solutions which these operators generate will be in a particular gauge.
This gauge freedom can be understood in the following way: the operators $\pb{\pm 2} \tau_{ab}$ in equation~\eqref{eqn:teukolsky} are only defined up to transformations of the form
\begin{equation} \label{eqn:tau_freedom}
\pb{\pm 2} \tau_{ab} \to \pb{\pm 2} \tau_{ab} + 2 \xi_{(a} \nabla_{b)},
\end{equation}
as they act upon the stress-energy tensor, for which $\nabla_a T^{ab} = 0$.
As such, we find that $\pb{\pm 2} \tau_{ab}^\dagger$ have the corresponding freedom
\begin{equation}
\pb{\pm 2} \tau_{ab}^\dagger \to \pb{\pm 2} \tau_{ab}^\dagger + 2 \nabla_{(a} \xi_{b)}.
\end{equation}
Note here that, in the second term, the covariant derivative acts upon the argument of these operators in addition to acting on $\xi_b$.
The particular choice~\eqref{eqn:tau} of $\pb{\pm 2} \tau_{ab}$ fixes this freedom, and so the metric perturbations generated by $\pb{\pm 2} \mathcal{C}_{ab}{}^{cd}$ are in a particular gauge.
The gauge conditions which they satisfy are~\cite{Chrzanowski:1975wv}
\begin{equation} \label{eqn:radn_gauges}
g^{ab} \pb{\pm 2} \tau_{ab}^\dagger = 0, \qquad l^a \pb{2} \tau_{ab}^\dagger = 0, \qquad n^a \pb{-2} \tau_{ab}^\dagger = 0.
\end{equation}
For $\pb{2} \tau_{ab}^\dagger$, this is the \emph{ingoing radiation gauge condition}, whereas for $\pb{-2} \tau_{ab}^\dagger$, this is the \emph{outgoing radiation gauge condition}.
We now show that the solutions $\pb{2} \boldsymbol{\mathcal{C}} \cdot \var \boldsymbol{g}$ and $\pb{-2} \boldsymbol{\mathcal{C}} \cdot \var \boldsymbol{g}$ do not differ by a gauge transformation, in the case where $\var g_{ab}$ is real.
This is in contrast to the case in electromagnetism~\cite{Grant:2019qyo}, where the analogous solutions do, in fact, differ by a gauge transformation.
While the total solutions $\pb{2} \boldsymbol{\mathcal{C}} \cdot \var \boldsymbol{g}$ and $\pb{-2} \boldsymbol{\mathcal{C}} \cdot \var \boldsymbol{g}$ do not differ by a gauge transformation, we will also show that the imaginary parts of each of these two solutions \emph{are} related by a gauge transformation, and so they represent the same physical solution.
To proceed, we first note the following identities [note a conventional factor of two difference with~\cite{PhysRevD.19.1641}, which comes from the difference between their equation (2.21) and our equation~\eqref{eqn:perturbed_weyl_spinor_metric}]
\begin{subequations} \label{eqn:nonzero_operators}
\begin{align}
\overline{\pb{2} \boldsymbol{M}} \cdot \pb{2} \boldsymbol{\mathcal{C}} &\circeq \frac{1}{2} (D + \epsilon - 3\bar{\epsilon}) (D + 2\epsilon - 2\bar{\epsilon}) (D + 3\epsilon - \bar{\epsilon}) (D + 4\epsilon) \pb{-2} \boldsymbol{M}, \\
\overline{\pb{-2} \boldsymbol{M}} \cdot \pb{2} \boldsymbol{\mathcal{C}} &\circeq \frac{1}{2} \bar{\zeta}^4 (\npdelta + 3\bar{\alpha} + \beta) (\npdelta + 2\bar{\alpha} + 2\beta) (\npdelta + \bar{\alpha} + 3\beta) (\npdelta + 4\beta) \pb{-2} \boldsymbol{M}, \\
\overline{\pb{-2} \boldsymbol{M}} \cdot \overline{\pb{2} \boldsymbol{\mathcal{C}}} &\circeq \frac{3}{2} \overline{\zeta^4 \Psi_2} \left[\bar{\tau} (\npdelta + 4\bar{\alpha}) - \bar{\rho} (\npDelta + 4\bar{\gamma}) - \bar{\mu} (D + 4\bar{\epsilon}) + \bar{\pi} (\bar{\npdelta} + 4\bar{\beta}) + 2\overline{\Psi}_2\right] \overline{\pb{-2} \boldsymbol{M}} \nonumber \\
&= \frac{3}{2} \overline{\zeta^3 \Psi_2} t^a [\nabla_a + 4 (\iota_B \nabla_a o^B)] \overline{\pb{-2} \boldsymbol{M}}, \label{eqn:nonzero_operators_t}
\end{align}
\end{subequations}
where ``$\circeq$'' means ``equality modulo equations of motion''.
Moreover, apart from those that occur in this equation, all other combinations of $\pb{\pm 2} \boldsymbol{M}$ and $\overline{\pb{\pm 2} \boldsymbol{M}}$ acting on $\pb{2} \boldsymbol{\mathcal{C}}$ and $\overline{\pb{2} \boldsymbol{\mathcal{C}}}$ are zero for vacuum solutions.
Here we have used the equation
\begin{equation}
D \rho = (\rho + \epsilon + \bar{\epsilon}) \rho
\end{equation}
(along with its $'$- and $*$-transformed versions) in order to simplify, as well as equation~\eqref{eqn:time}.
One can furthermore use a $'$-transformation to write down versions of equation~\eqref{eqn:nonzero_operators} involving $\pb{-2} \bs{\mathcal C}$, noting that $\Psi_2 \to \Psi_2$ under a $'$-transformation, and $\zeta$ must flip sign (note that $t^a$ keeps the same sign).
To determine whether certain linear combinations of $\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \var g_{cd}$ (and their complex conjugates) differ by gauge transformations, we need the following relation, which only holds for $\var \Psi_4$ and $\var \Psi_0$ coming from the same real vacuum metric perturbations:
\begin{equation} \label{eqn:tsr}
\begin{split}
(D + \epsilon - 3\bar{\epsilon}) (D + 2\epsilon - 2\bar{\epsilon}) (D + 3\epsilon - \bar{\epsilon}) (D + 4\epsilon) \zeta^4 \var \Psi_4 &= (\bar{\npdelta} - \alpha - 3\bar{\beta}) (\bar{\npdelta} - 2\alpha - 2\bar{\beta}) (\bar{\npdelta} - 3\alpha - \bar{\beta}) (\bar{\npdelta} - 4\alpha) \zeta^4 \var \Psi_0 \\
&\hspace{1em}+ 3\overline{\zeta^3 \Psi_2} t^a [\nabla_a - 4 (\iota_B \nabla_a o^B)] \overline{\var \Psi_0};
\end{split}
\end{equation}
we will also need this equation's $'$-transform.
This relation can be derived using the perturbed Bianchi identities and Newman-Penrose equations, as mentioned in~\cite{PhysRevD.14.317}; for a more modern derivation, see for example~\cite{Aksteiner:2016pjt}.
Using equations~\eqref{eqn:nonzero_operators} and~\eqref{eqn:tsr}, along with their $'$-transforms, we find that (applied to a real, vacuum metric perturbation),
\begin{equation} \label{eqn:tsr_operators}
\overline{\pb{2} \bs{M}} \cdot \pb{2} \bs{\mathcal C} \circeq \overline{\pb{2} \bs{M}} \cdot \pb{-2} \bs{\mathcal C} - \overline{\pb{2} \bs{M}} \cdot \overline{\pb{-2} \bs{\mathcal C}}.
\end{equation}
The $'$-transform of this equation merely switches $2 \to -2$.
As remarked below equation~\eqref{eqn:nonzero_operators}, one has that
\begin{equation}
\overline{\pb{2} \bs{M}} \cdot \overline{\pb{2} \bs{\mathcal C}} \circeq 0
\end{equation}
(along with its $'$-transform), and so one therefore has that
\begin{equation}
\overline{\pb{2} \boldsymbol{M}} \cdot \Im\left[\pb{+2} \boldsymbol{\mathcal{C}} - \pb{-2} \boldsymbol{\mathcal{C}}\right] \cdot \var \bs{g} = 0, \qquad \overline{\pb{-2} \boldsymbol{M}} \cdot \Im\left[\pb{+2} \boldsymbol{\mathcal{C}} - \pb{-2} \boldsymbol{\mathcal{C}}\right] \cdot \var \bs{g} = 0. \\
\end{equation}
This equation does not, as it stands, guarantee that $\Im[\pb{2} \mathcal{\bs C} \cdot \var \bs{g}]$ and $\Im[\pb{-2} \mathcal{\bs C} \cdot \var \bs{g}]$ are related by a gauge transformation, just that the master variables associated with these two metric perturbations are equal.
This implies that their difference is a metric perturbation that contributes to $\var M$ and $\var a$; that is, it only has monopole and dipole terms~\cite{1973JMP....14.1453W}.
One would expect that $\Im[\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \var g_{cd}]$, as they are constructed wholly from the radiative Weyl scalars $\var \Psi_0$ and $\var \Psi_4$ (which do not have monopole or dipole pieces), would not have non-radiating pieces.
This statement is in fact correct due to arguments in~\cite{Stewart:1978tm}.
In conclusion, we find that $\Im[\pb{2} \mathcal{\bs C} \cdot \var \bs{g}]$ and $\Im[\pb{-2} \mathcal{\bs C} \cdot \var \bs{g}]$ differ by a gauge transformation:
\begin{equation}
\Im[\pb{2} \mathcal{C}_{ab}{}^{cd} \var g_{cd}] = \Im[\pb{-2} \mathcal{C}_{ab}{}^{cd} \var g_{cd}] + 2 \nabla_{(a} \xi_{b)},
\end{equation}
for some vector field $\xi^a$.
The main theorem of~\cite{Aksteiner:2016pjt} provides an alternative proof of this result, as does the discussion in section 4.3 of~\cite{Aksteiner:2016mol}.
\subsection{Action of symmetry operators on expansions} \label{section:symmetry_separated}
In section~\ref{section:teukolsky}, we showed that the master variables (and their complex conjugates) have convenient expansions [equations~\eqref{eqn:decoupled_expansion} and~\eqref{eqn:decoupled_bar_expansion}] in terms of spin-weighted spheroidal harmonics.
We show in this section that the symmetry operators considered in this paper which act on the master variables are ``diagonal'', in the sense that they act upon each term in these expansions by simply multiplying each term by an overall constant.
We then construct a similar expansion for vacuum metric perturbations, and show that the action of the symmetry operators that we have defined for metric perturbations are also diagonal on this expansion.
First, let us consider the action of the symmetry operator $\pb{s} \mathcal{D}$ defined in equation~\eqref{eqn:D}.
From equations~\eqref{eqn:RS}, \eqref{eqn:DL_raising},~\eqref{eqn:DL_fourier}, and~\eqref{eqn:separated_teukolsky}, it follows that
\begin{equation} \label{eqn:D_action}
\pb{s} \mathcal{D} \pb{s} \Omega = \int_{-\infty}^\infty \sum_{l = |s|}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \pb{|s|} \lambda_{lm\omega} e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta) \pb{s} \widehat{\Omega}_{lm\omega p} (r).
\end{equation}
Later in this section, we will also show that a similar diagonalization occurs for a tensor version of this operator, which we will define in equation~\eqref{eqn:D_tensor}.
Next, we consider the symmetry operators $\pb{s', s} \widetilde{\mathcal C}$ defined in equation~\eqref{eqn:C}.
We begin by noting that these symmetry operators simplify with the choice of Boyer-Lindquist coordinates and the Kinnersley tetrad, yielding the so-called ``spin-inversion'' operators~\cite{Chrzanowski:1975wv, 1982JPhA...15.3737G}:
\begin{subequations} \label{eqn:spinversion}
\begin{align}
\pb{2, 2} \widetilde{\mathcal C} &= \frac{1}{2} \mathscr{D}_0^4, \qquad &\pb{-2, -2} \widetilde{\mathcal C} &= \frac{1}{32} \Delta^2 \left(\mathscr{D}_0^+\right)^4 \Delta^2, \label{eqn:radial_spinversion} \\
\pb{2, -2} \widetilde{\mathcal C} &= \frac{1}{8} \mathscr{L}_{-1}^+ \mathscr{L}_0^+ \mathscr{L}_1^+ \mathscr{L}^+_2, \qquad &\pb{-2, 2} \widetilde{\mathcal C} &= \frac{1}{8} \mathscr{L}_{-1} \mathscr{L}_0 \mathscr{L}_1 \mathscr{L}_2. \label{eqn:angular_spinversion}
\end{align}
\end{subequations}
The constant numerical factors here are consistent with those of Wald~\cite{PhysRevLett.41.203} and Chrzanowski~\cite{Chrzanowski:1975wv}, but disagree with those of other authors (such as~\cite{chandrasekhar1983mathematical, 1982JPhA...15.3737G}) due to normalization conventions.
These operators are referred to as spin-inversion operators for the following reason.
Considering their action on the terms in the expansion~\eqref{eqn:decoupled_bar_expansion} of $\overline{\pb{s} \Omega}$, they are either purely radial [equation~\eqref{eqn:radial_spinversion}] or purely angular [equation~\eqref{eqn:angular_spinversion}].
Due to this fact, along with the expansions in equations~\eqref{eqn:decoupled_expansion} and~\eqref{eqn:decoupled_bar_expansion}, it is apparent that, when acting on the terms in these expansions, the operator $\pb{2, 2} \widetilde{\mathcal C}$ maps from the space of solutions to the radial Teukolsky equation~\eqref{eqn:radial_teukolsky} with $s = -2$ to $s = 2$, and similarly $\pb{-2, -2} \widetilde{\mathcal C}$ maps from solutions with $s = 2$ to $s = -2$.
Similarly, for the angular operators, due to the fact that the expansion for $\overline{\pb{s} \Omega}$ is in terms of $\pb{-s} \Theta_{lm\omega}$, $\pb{2, -2} \widetilde{\mathcal C}$ maps from the space of solutions to angular Teukolsky equation~\eqref{eqn:angular_teukolsky} with $s = 2$ to $s = -2$, and similarly $\pb{-2, 2} \widetilde{\mathcal C}$ maps from $s = -2$ to $s = 2$.
We now show that the spin-inversion operators merely multiply each term in the expansion~\eqref{eqn:decoupled_bar_expansion} by some constant, starting with the angular spin-inversion operators.
The angular Teukolsky equation~\eqref{eqn:angular_teukolsky} is a Sturm-Liouville problem, which only has one solution for a given value of $l$, $m$, and $\omega$ (up to normalization).
If the angular spin-inversion operators, when acting upon individual terms in the expansion~\eqref{eqn:decoupled_bar_expansion}, map between the two spaces of solutions with $s = \pm 2$, then these maps can be entirely characterized by two overall constants, which we denote by $\pb{\pm 2} C_{lm\omega}$:
\begin{equation} \label{eqn:angular_starobinsky}
\mathscr{L}_{-1(\pm m)(\pm \omega)} \mathscr{L}_{0(\pm m)(\pm \omega)} \mathscr{L}_{1(\pm m)(\pm \omega)} \mathscr{L}_{2(\pm m)(\pm \omega)} \pb{\pm 2} \Theta_{lm\omega} \equiv \pb{\pm 2} C_{lm\omega} \pb{\mp 2} \Theta_{lm\omega}.
\end{equation}
This equation is known as the \emph{angular Teukolsky-Starobinsky identity}.
Since these operators are entirely real, this constant $\pb{\pm 2} C_{lm\omega}$ is also real.
Moreover, the normalization condition for $\pb{s} \Theta_{lm\omega}$ implies that~\cite{chandrasekhar1983mathematical}
\begin{equation}
\pb{2} C_{lm\omega} = \pb{-2} C_{lm\omega} \equiv C_{lm\omega},
\end{equation}
where
\begin{equation} \label{eqn:starobinsky}
C_{lm\omega}^2 = \pb{2} \lambda_{lm\omega}^2 (\pb{2} \lambda_{lm\omega} + 2)^2 - 8 \omega^2 \pb{2} \lambda_{lm\omega} [\alpha_{m\omega}^2 (5 \pb{2} \lambda_{lm\omega} + 6) - 12a^2] + 144 \omega^4 \alpha_{m\omega}^4,
\end{equation}
and
\begin{equation}
\alpha_{m\omega}^2 = a^2 - am/\omega.
\end{equation}
We now turn to the case of the radial operators in equation~\eqref{eqn:radial_spinversion}, which are somewhat more complicated.
This is because there are two solutions to the radial equation~\eqref{eqn:radial_teukolsky}, as it is second-order, and not a Sturm-Liouville problem.
However, as noted in section~\ref{section:teukolsky}, the two solutions can be characterized by their eigenvalues under the transformation $(m, \omega) \to (-m, -\omega)$, followed by complex conjugation.
Since the radial spin-inversion operator is also invariant under this transformation, we must therefore have that
\begin{equation} \label{eqn:radial_starobinsky}
\Delta^2 \mathscr{D}_{0(\mp m)(\mp \omega)}^4 \Delta^{(s \pm s)/2} \pb{\pm 2} \widehat{\Omega}_{lm\omega p} \equiv 2^{\pm 2} \pb{\pm 2} C_{lm\omega p} \Delta^{(s \mp s)/2} \pb{\mp 2} \widehat{\Omega}_{lm\omega p}
\end{equation}
(the factor of $2^{\pm 2}$ is purely conventional, and is present only to make our final expressions simpler).
This equation is known as the \emph{radial Teukolsky-Starobinsky identity}.
To determine the values of the constants $\pb{\pm 2} C_{lm\omega p}$, we need to use the fact that $\pb{\pm 2} \Omega$ come from the same real metric perturbation.
The values of these constants given by Teukolsky and Press in their original paper~\cite{1974ApJ...193..443T} only hold for the $p = 1$ case (as pointed out by Bardeen~\cite{bardeen}\footnote{That~\cite{1974ApJ...193..443T} only considers $p = 1$ can be seen from their equation~(3.21), along with the remark below their equation~(3.22) that the quantities $S_2$ and $S_2^\dagger$ that appear in this equation are given by $\pb{2} S_{lm}$ and $\pb{-2} S_{lm}$ (in this chapter, these are denoted $\pb{2} \Theta_{lm\omega}$ and $\pb{-2} \Theta_{lm\omega}$).
These two statements imply that the radial functions $R_s$ discussed in~\cite{1974ApJ...193..443T} obey
\[
\overline{R_s (-m, -\omega)} = R_s (m, \omega).
\]
In this paper, due to differences in notation and the conventions in equation~\eqref{eqn:Theta_flips}, this is equivalent to the statement that $\pb{s} \widehat{\Omega}_{l(-m)(-\omega)} = (-1)^{m + s} \overline{\pb{s} \widehat{\Omega}_{lm\omega}}$, which by equation~\eqref{eqn:eigenvalue_condition} implies that $p = 1$.}).
The values of $\pb{\pm 2} C_{lm\omega p}$ are found using equation~\eqref{eqn:tsr_operators}, since (in terms of $\pb{s} \Omega$) the complex conjugate of this equation (and its $'$-transform) can be written as
\begin{equation} \label{eqn:tsi_scalar}
\pb{-s, -s} \widetilde{\mathcal C}\; \overline{\pb{s} \Omega} = \pb{-s, s} \widetilde{\mathcal C}\; \overline{\pb{-s} \Omega} - \pb{-s, s} \mathcal{C} \pb{-s} \Omega.
\end{equation}
Using equations~\eqref{eqn:spinversion},~\eqref{eqn:angular_starobinsky}, and~\eqref{eqn:radial_starobinsky}, as well as~\eqref{eqn:nonzero_operators_t}, we find that
\begin{subequations} \label{eqn:C_action}
\begin{align}
\pb{-s, s} \widetilde{\mathcal C}\; \overline{\pb{-s} \Omega} &= \frac{1}{8} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} p C_{lm\omega} e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} \pb{-s} \widehat{\Omega}_{lm\omega p}, \\
\pb{-s, -s} \widetilde{\mathcal C}\; \overline{\pb{s} \Omega} &= \frac{1}{8} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} p \pb{s} C_{lm\omega p} e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} \pb{-s} \widehat{\Omega}_{lm\omega p}, \\
\pb{-s, s} \mathcal{C} \pb{-s} \Omega &= \frac{3iM}{2} \sgn(s) \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \omega e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} \pb{-s} \widehat{\Omega}_{lm\omega p},
\end{align}
\end{subequations}
and so equation~\eqref{eqn:tsi_scalar} implies that
\begin{equation} \label{eqn:complex_starobinsky}
\pb{\pm 2} C_{lm\omega p} = C_{lm\omega} \mp 12i pM\omega.
\end{equation}
At this point, we have shown how symmetry operators on the space of master variables act diagonally on the expansions~\eqref{eqn:decoupled_expansion} and~\eqref{eqn:decoupled_bar_expansion}.
We would like a similar diagonalization for the operator $\pb{s} \bs{\mathcal C}$, but (\emph{a priori}) there does not exist an analogous expansion for the metric perturbation.
We now construct such an expansion.
To begin, if a) $\pb{s} \psi$ is a solution to the vacuum Teukolsky equation~\eqref{eqn:teukolsky}, b) it is the master variable associated with some real solution to the linearized Einstein equations, and c)
\begin{equation}
\pb{s} \Omega = \pb{s} M^{ab} \Im[\pb{s} \tau_{ab}^\dagger \pb{-s} \psi],
\end{equation}
then we call $\pb{s} \psi$ a \emph{Debye potential} for $\var g_{ab}$ (for the origin of this terminology, see~\cite{PhysRevD.10.1070}).
The first of these conditions ensures that $\pb{2} \psi$ and $\zeta^{-4} \pb{-2} \psi$ satisfy the same relation as (respectively) $\var \Psi_0$ and $\var \Psi_4$ in equation~\eqref{eqn:tsr}.
The second of these conditions ensures that $\Im[\pb{s} \tau_{ab}^\dagger \pb{-s} \psi]$ and (by the first condition) $\Im[\pb{-s} \tau_{ab}^\dagger \pb{s} \psi]$ are the same as $\var g_{ab}$, up to gauge and $l = 0, 1$ terms.
The easiest way to satisfy these conditions is as follows.
First, note that, by equations~\eqref{eqn:C_tensor} and~\eqref{eqn:C_action},
\begin{equation} \label{eqn:psi_motivation}
\begin{split}
\pb{s} M^{ab} &\Im\left\{\pb{s} \mathcal{C}_{ab}{}^{cd} \Im[\pb{-s} \tau_{cd}^\dagger \pb{s} \Omega]\right\} \\
&= \frac{1}{16} \pb{s} M^{ab} \Re\left[\pb{s} \tau_{ab}^{\dagger} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} p \pb{s} C_{lm\omega p} e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} \pb{-s} \widehat{\Omega}_{lm\omega p}\right] \\
&= \frac{1}{256} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} (C_{lm\omega}^2 + 144M^2 \omega^2) e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} \pb{s} \widehat{\Omega}_{lm\omega p}. \\
\end{split}
\end{equation}
We now define $\pb{s} \psi$, for a given $\pb{s} \Omega$, by
\begin{equation} \label{eqn:psi_def}
\begin{split}
\pb{s} \psi &\equiv 256 \pb{s} M^{ab} \Im\left[\pb{s} \tau_{ab}^\dagger \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \frac{e^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} (\theta) \pb{-s} \widehat{\Omega}_{lm\omega p} (r)}{C_{lm\omega}^2 + 144 M^2 \omega^2}\right] \\
&= 16i \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \frac{pe^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta) \pb{s} \widehat{\Omega}_{lm\omega p} (r)}{\pb{s} C_{lm\omega p}},
\end{split}
\end{equation}
where the second line comes from equation~\eqref{eqn:C_action}, and $\pb{s} \widehat{\Omega}_{lm\omega p}$ is given in terms of $\pb{s} \Omega$ by equations~\eqref{eqn:mode_expansion} and~\eqref{eqn:p_def}.
Since $C_{lm\omega}^2 + 144M^2 \omega^2$ is real, $\pb{s} \psi$ satisfies the first of the above requirements, and by equation~\eqref{eqn:psi_motivation} it also satisfies the second.
Moreover, the second line implies that
\begin{equation}
\pb{s} \widehat{\psi}_{lm\omega(-p)} = \frac{16 ip}{\pb{s} C_{lm\omega p}} \pb{s} \widehat{\Omega}_{lm\omega p}.
\end{equation}
where the expansion coefficients $\pb{s} \widehat{\psi}_{lm\omega p}$ are defined by an expansion analogous to equation~\eqref{eqn:decoupled_expansion}, together with the behavior under complex conjugation given by equation~\eqref{eqn:eigenvalue_condition}.
This condition is satisfied, due to the fact that
\begin{equation}
\overline{\pb{s} C_{l(-m)(-\omega) p}} = \pb{s} C_{lm\omega p},
\end{equation}
by equations~\eqref{eqn:lambda_flips}, \eqref{eqn:starobinsky} and~\eqref{eqn:complex_starobinsky}, as well as by using equation~\eqref{eqn:eigenvalue_condition} for $\pb{s} \widehat{\Omega}_{lm\omega p}$.
While this would also be a perfectly reasonable definition of $\pb{s} \psi$, it is not apparent in this form that $\pb{s} \psi$ is generated by a real metric perturbation, which is crucial, and is explicit in equation~\eqref{eqn:psi_def}.
Finally, note that equations analogous to equation~\eqref{eqn:C_action} also hold for $\pb{s} \psi$ in terms of $\pb{s} \psi_{lm\omega p}$.
We can now define an expansion for the metric perturbation.
First, we define
\begin{equation} \label{eqn:hertz}
\var_\pm g_{ab} \equiv \pb{\pm 2} \tau_{ab}^\dagger \pb{\mp 2} \psi,
\end{equation}
which (as remarked above) satisfy
\begin{equation} \label{eqn:hertz_condition}
\pb{s} M^{ab} \Im[\var_+ g_{ab}] = \pb{s} M^{ab} \Im[\var_- g_{ab}] = \pb{s} \Omega.
\end{equation}
These metric perturbations have convenient expansions of the form
\begin{equation} \label{eqn:hertz_expansion}
\var_\pm g_{ab} = \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} (\var_\pm g_{lm\omega p})_{ab},
\end{equation}
where
\begin{equation} \label{eqn:hertz_coefficients}
(\var_\pm g_{lm\omega p})_{ab} \equiv \pb{\pm 2} \tau_{ab}^\dagger \left[e^{i(m\phi - \omega t)} \pb{\mp 2} \Theta_{lm\omega} (\theta) \pb{\mp 2} \widehat{\psi}_{lm\omega p} (r)\right].
\end{equation}
Note that the relationship between $\var_\pm g_{ab}$ and their coefficients is not $\mathbb{C}$-linear, due to the transformation properties of these coefficients under complex conjugation resulting from equation~\eqref{eqn:eigenvalue_condition}.
This procedure, which allowed us to construct a metric perturbation $\Im[\var_\pm g_{ab}]$ from $\pb{\mp 2} \Omega$ such that the master variables associated with this metric perturbation are $\pb{\pm 2} \Omega$, is similar to the one laid out in~\cite{Chrzanowski:1975wv}, which is referred to in the literature as \emph{Chrzanowski metric reconstruction}.
We now provide an operator form of this procedure: define
\begin{equation} \label{eqn:Pi}
\begin{split}
\pb{s} \Pi_{ab} \pb{s} \Omega &\equiv 256 \pb{s} \mathcal{C}_{ab}{}^{cd} \Im\left[\pb{-s} \tau_{cd}^\dagger \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \frac{e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} \pb{s} \widehat{\Omega}_{lm\omega p}}{C^2_{lm\omega} + 144 M^2 \omega^2}\right] \\
&= 16i \pb{s} \tau_{ab}^\dagger \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \frac{pe^{i(m\phi - \omega t)} \pb{-s} \Theta_{lm\omega} \pb{-s} \widehat{\Omega}_{lm\omega p}}{\pb{-s} C_{lm\omega p}},
\end{split}
\end{equation}
which satisfies
\begin{equation}
\pb{s} M^{ab} \Im[\pb{s} \Pi_{ab} \pb{s} \Omega] = \pb{s} M^{ab} \Im[\pb{-s} \Pi_{ab} \pb{-s} \Omega] = \pb{s} \Omega.
\end{equation}
Note that the operator $\pb{s} \Pi_{ab}$ is non-local, since it requires an expansion in spin-weighted spheroidal harmonics for its definition.
This operator allows us to define a version of the operator $\pb{s} \mathcal{D}$ defined in section~\ref{section:separation} that maps to the space of complexified solutions of the linearized Einstein equations, much like $\pb{s} \mathcal{C}_{ab}{}^{cd}$:
\begin{equation} \label{eqn:D_tensor}
\pb{s} \mathcal{D}_{ab}{}^{cd} \equiv \pb{s} \Pi_{ab} \pb{s} \mathcal{D} \pb{s} M^{cd}.
\end{equation}
We also define a version of this operator \emph{without} the intermediate factor of $\pb{s} \mathcal{D}$:
\begin{equation} \label{eqn:Chi}
\pb{s} \Chi_{ab}{}^{cd} \equiv \pb{s} \Pi_{ab} \pb{s} M^{cd}.
\end{equation}
Now that we have both a definition of an expansion for the metric perturbation, along with a variety of symmetry operators defined which map the space of metric perturbations into itself, we can proceed to show that these symmetry operators act diagonally on these expansions.
Note, again, that there is no convenient notion of an expansion of the form~\eqref{eqn:hertz_expansion} for a general $\var g_{ab}$, and so we only compute the action of our various symmetry operators on $\var_\pm g_{ab}$.
The simplest case is $\pb{s} \mathcal{C}_{ab}{}^{cd}$, which satisfies [by equation~\eqref{eqn:C_action}]\footnote{Note that, as mentioned above below equation~\eqref{eqn:hertz_coefficients}, the relationship between $\var_\pm g_{ab}$ and their coefficients is not $\mathbb{C}$-linear.
This explains the apparent contradiction of the left-hand side of equations~\eqref{eqn:C_tensor_action_angular} and~\eqref{eqn:C_tensor_action_radial} being $\mathbb{C}$-antilinear, but the right-hand sides appearing to be $\mathbb{C}$-linear.}
\begin{subequations} \label{eqn:C_tensor_action}
\begin{align}
\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \overline{\var_\pm g_{cd}} &= \pb{\pm 2} \tau_{ab}^\dagger \pb{\mp 2, \pm 2} \widetilde{\mathcal C}\; \overline{\pb{\mp 2} \psi} \nonumber \\
&= \frac{1}{8} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^{\infty} \sum_{|m| \leq l} \sum_{p = \pm 1} pC_{lm\omega} (\var_\pm g_{lm\omega p})_{ab}, \label{eqn:C_tensor_action_angular} \\
\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \overline{\var_\mp g_{cd}} &= \pb{\pm 2} \tau_{ab}^\dagger \pb{\mp 2, \mp 2} \widetilde{\mathcal C}\; \overline{\pb{\pm 2} \psi} \nonumber \\
&= \frac{1}{8} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^{\infty} \sum_{|m| \leq l} \sum_{p = \pm 1} p \pb{\pm 2} C_{lm\omega p} (\var_\pm g_{lm\omega p})_{ab}, \label{eqn:C_tensor_action_radial} \\
\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \var_\pm g_{cd} &= \pb{\pm 2} \tau_{ab}^\dagger \pb{\mp 2, \pm 2} \mathcal{C} \pb{\mp 2} \psi \nonumber \\
&= \pm \frac{3iM}{2} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^{\infty} \sum_{|m| \leq l} \sum_{p = \pm 1} \omega (\var_\pm g_{lm\omega p})_{ab}.
\end{align}
\end{subequations}
These equations demonstrate that the action on the expansion~\eqref{eqn:hertz_expansion} is diagonal, up to mappings from $\overline{(\var_\pm g_{lm\omega p})_{ab}} \to (\var_\pm g_{lm\omega p})_{ab}$ and $(\var_\mp g_{lm\omega p})_{ab}$, as well as mappings from $(\var_\pm g_{lm\omega p})_{ab} \to (\var_\mp g_{lm\omega p})_{ab}$.
More useful later in this paper will be the action of $\pb{s} \mathcal{C}_{ab}{}^{cd}$ on $\Im[\var_\pm g_{ab}]$:
\begin{equation}
\begin{split}
\pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \Im[\var_+ g_{cd}] &= \pb{\pm 2} \mathcal{C}_{ab}{}^{cd} \Im[\var_- g_{cd}] \\
&= \frac{i}{16} \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^{\infty} \sum_{|m| \leq l} \sum_{p = \pm 1} p \pb{\pm 2} C_{lm\omega p} (\var_\pm g_{lm\omega p})_{ab}.
\end{split}
\end{equation}
Similarly, we will consider the action of $\pb{s} \mathcal{D}_{ab}{}^{cd}$ and $\pb{s} \Chi_{ab}{}^{cd}$ on $\Im[\var_\pm g_{ab}]$.
We have that [by equation~\eqref{eqn:hertz_condition}]
\begin{equation}
\pb{s} \Pi_{ab} \pb{s} \Omega = \pb{s} \Chi_{ab}{}^{cd} \Im[\var_\pm g_{cd}],
\end{equation}
along with [by equations~\eqref{eqn:hertz} and~\eqref{eqn:Pi}]
\begin{equation} \label{eqn:Pi_hertz}
\pb{\pm 2} \Pi_{ab} \pb{\pm 2} \Omega = \var_\pm g_{ab},
\end{equation}
and so we find that
\begin{equation} \label{eqn:Chi_tensor_action}
\pb{\pm 2} \Chi_{ab}{}^{cd} \Im[\var_+ g_{cd}] = \pb{\pm 2} \Chi_{ab}{}^{cd} \Im[\var_- g_{cd}] = \var_\pm g_{ab},
\end{equation}
Similarly, by the $\mathbb{R}$-linearity of equation~\eqref{eqn:Pi_hertz}, we find that [from equation~\eqref{eqn:D_action}]
\begin{equation} \label{eqn:D_tensor_action}
\pb{\pm 2} \mathcal{D}_{ab}{}^{cd} \Im[\var_+ g_{cd}] = \pb{\pm 2} \mathcal{D}_{ab}{}^{cd} \Im[\var_- g_{cd}] = \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^{\infty} \sum_{|m| \leq l} \sum_{p = \pm 1} \pb{2} \lambda_{lm\omega} (\var_\pm g_{lm\omega p})_{ab}.
\end{equation}
\subsection{Projection operators} \label{section:projection}
The final set of symmetry operators that we introduce are projection operators acting on the space of master variables $\pb{s} \Omega$.
Before we introduce these operators, however, it is relevant to discuss the asymptotic properties of the master variables.
First, define the tortoise coordinate $r^*$ by
\begin{equation} \label{eqn:tortoise}
\frac{\ud r^*}{\ud r} \equiv \frac{r^2 + a^2}{\Delta}.
\end{equation}
This coordinate satisfies $r^* \to \infty$ as $r \to \infty$ and $r^* \to -\infty$ as $r \to r_+$, where $r_+$ is the location of the horizon, satisfying $\Delta|_{r = r_+} = 0$.
Now, the vacuum Teukolsky radial equation~\eqref{eqn:radial_teukolsky} is a second-order ordinary differential equation in $r$, and so its solution space is spanned by two solutions (for given values of $s$, $l$, $m$, and $\omega$) that are characterized by their asymptotic behavior at either $r = r_+$ or $r = \infty$.
One can show, from the asymptotic form of the vacuum Teukolsky radial equation~\eqref{eqn:radial_teukolsky}, that one can choose two independent solutions $\pb{s} R_{lm\omega}^{\textrm{in}} (r)$ and $\pb{s} R_{lm\omega}^{\textrm{out}} (r)$ with the following asymptotic forms as $r^* \to -\infty$~\cite{1974ApJ...193..443T}:
\begin{equation} \label{eqn:radial_falloffs_in_out}
\pb{s} R_{lm\omega}^{\textrm{in}} (r) \to e^{-ik_{m\omega} r^*}/\Delta^s, \quad \pb{s} R_{lm\omega}^{\textrm{out}} (r) \to e^{ik_{m\omega} r^*},
\end{equation}
where
\begin{equation}
k_{m\omega} \equiv \omega - am/(2Mr_+).
\end{equation}
Similarly, at $r^* \to \infty$, one can choose two independent solutions $\pb{s} R_{lm\omega}^{\textrm{down}} (r)$ and $\pb{s} R_{lm\omega}^{\textrm{up}} (r)$, which have the following asymptotic forms:
\begin{equation} \label{eqn:radial_falloffs_down_up}
\pb{s} R_{lm\omega}^{\textrm{down}} (r) \to e^{-i\omega r^*}/r, \quad \pb{s} R_{lm\omega}^{\textrm{up}} (r) \to e^{i\omega r^*}/r^{2s + 1}.
\end{equation}
A general solution can therefore be expanded in terms of these solutions as
\begin{equation}
\begin{split}
\pb{s} \widehat{\Omega}_{lm\omega} (r) &= \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{down}} \pb{s} R_{lm\omega}^{\textrm{down}} (r) + \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{up}} \pb{s} R_{lm\omega}^{\textrm{up}} (r) \\
&= \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{in}} \pb{s} R_{lm\omega}^{\textrm{in}} (r) + \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{out}} \pb{s} R_{lm\omega}^{\textrm{out}} (r).
\end{split}
\end{equation}
Moreover, from the asymptotic behavior in equations~\eqref{eqn:radial_falloffs_in_out} and~\eqref{eqn:radial_falloffs_down_up}, we have
\begin{equation}
\overline{\pb{s} R_{l(-m)(-\omega)}^{\textrm{in/out/down/up}} (r)} = \pb{s} R_{lm\omega}^{\textrm{in/out/down/up}} (r),
\end{equation}
and so, from the definition~\eqref{eqn:p_def},
\begin{equation}
\begin{split}
\pb{s} \widehat{\Omega}_{lm\omega p} (r) &= \pb{s} \widehat{\Omega}_{lm\omega p}^{\textrm{down}} \pb{s} R_{lm\omega}^{\textrm{down}} (r) + \pb{s} \widehat{\Omega}_{lm\omega p}^{\textrm{up}} \pb{s} R_{lm\omega}^{\textrm{up}} (r) \\
&= \pb{s} \widehat{\Omega}_{lm\omega p}^{\textrm{in}} \pb{s} R_{lm\omega}^{\textrm{in}} (r) + \pb{s} \widehat{\Omega}_{lm\omega p}^{\textrm{out}} \pb{s} R_{lm\omega}^{\textrm{out}} (r),
\end{split}
\end{equation}
where
\begin{equation}
\pb{s} \widehat{\Omega}_{lm\omega p}^{\textrm{in/out/down/up}} \equiv \frac{1}{2} \left[\pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{in/out/down/up}} + p(-1)^{m + s} \overline{\pb{s} \widehat{\Omega}_{l(-m)(-\omega)}^{\textrm{in/out/down/up}}}\right].
\end{equation}
We now define projection operators associated with this expansion as follows: for example, define $\pb{s} \mathcal{P}^{\textrm{in}}$ by
\begin{equation}
\begin{split}
\pb{s} \mathcal{P}^{\textrm{in}} \pb{s} \Omega &= \pb{s} \mathcal{P}^{\textrm{in}} \int_{-\infty}^\infty \ud \omega \sum_{l = |s|}^\infty \sum_{|m| \leq l} e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta) \left[\pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{in}} \pb{s} R_{lm\omega}^{\textrm{in}} (r) + \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{out}} \pb{s} R_{lm\omega}^{\textrm{out}} (r)\right] \\
&\equiv \int_{-\infty}^\infty \ud \omega \sum_{l = |s|}^\infty \sum_{|m| \leq l} e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} (\theta) \pb{s} \widehat{\Omega}_{lm\omega}^{\textrm{in}} \pb{s} R_{lm\omega}^{\textrm{in}} (r).
\end{split}
\end{equation}
Analogous definitions can be given for $\pb{s} \mathcal{P}^{\textrm{out}}$, $\pb{s} \mathcal{P}^{\textrm{down}}$, and $\pb{s} \mathcal{P}^{\textrm{up}}$.
Since these operators require an expansion in spin-weighted spheroidal harmonics, they are necessarily non-local.
The reason we introduce these projection operators is that, as we show in appendix~\ref{appendix:asymptotics}, whether $\pb{s} \tau_{ab}^\dagger \pb{-s} \Omega$ falls off as $1/r$ (that is, whether it is an asymptotically flat metric perturbation) depends on the values $\pb{-s} \Omega_{lm\omega}^{\textrm{down/out}}$.
This was first remarked by Chrzanowski in~\cite{Chrzanowski:1975wv}.
As such, we define a projected version of $\pb{s} \tau_{ab}^\dagger$, which we call $\pb{s} \mathring{\tau}_{ab}^\dagger$, such that $\pb{s} \mathring{\tau}_{ab}^\dagger \pb{-s} \Omega$ is always well-behaved as $r \to \infty$:
\begin{equation}
\pb{2} \mathring{\tau}_{ab}^\dagger \equiv \pb{2} \tau_{ab}^\dagger \pb{-2} \mathcal{P}^{\textrm{down}}, \qquad \pb{-2} \mathring{\tau}_{ab}^\dagger \equiv \pb{-2} \tau_{ab}^\dagger \pb{2} \mathcal{P}^{\textrm{up}}.
\end{equation}
Using this operator, we can define
\begin{equation} \label{eqn:C_tensor_proj}
\pb{s} \mathring{\mathcal C}_{ab}{}^{cd} \equiv \pb{s} \mathring{\tau}_{ab}^\dagger \pb{-s} M^{cd},
\end{equation}
which allows for the definition of
\begin{equation}
\pb{s} \mathring{\Pi}_{ab}{}^{cd} \pb{s} \Omega \equiv 256 \pb{s} \mathring{\mathcal C}_{ab}{}^{cd} \Im\left[\pb{-s} \tau_{cd}^\dagger \int_{-\infty}^\infty \ud \omega \sum_{l = 2}^\infty \sum_{|m| \leq l} \sum_{p = \pm 1} \frac{e^{i(m\phi - \omega t)} \pb{s} \Theta_{lm\omega} \pb{s} \widehat{\Omega}_{lm\omega p}}{C^2_{lm\omega} + 144 M^2 \omega^2}\right].
\end{equation}
Finally, this last operator allows for the definitions
\begin{equation} \label{eqn:D_Chi_tensor_proj}
\pb{s} \mathring{\mathcal D}_{ab}{}^{cd} \equiv \pb{s} \mathring{\Pi}_{ab} \pb{s} \mathcal{D} \pb{s} M^{cd}, \qquad \pb{s} \ring{\Chi}_{ab}{}^{cd} \equiv \pb{s} \mathring{\Pi}_{ab} \pb{s} M^{cd}.
\end{equation}
\section{Conserved Currents} \label{section:currents}
We next turn to conserved currents that can be constructed using these symmetry operators.
First, we review the general theory of symplectic products, which are bilinear currents constructed from the Lagrangian formulation of a given classical field theory.
We then select a handful of conserved currents that can be constructed using symplectic products and symmetry operators, whose properties we discuss throughout the rest of this paper.
\subsection{Symplectic product} \label{section:symplectic}
Given a theory that possesses a Lagrangian formulation with Lagrangian density $\mathcal{L}$, one method of generating conserved quantities is to use the symplectic product defined in this section.
Following Burnett and Wald~\cite{Burnett57}, we start with a general Lagrangian four-form $\boldsymbol{L} [\boldsymbol{\phi}] \equiv \dual \mathcal{L} [\boldsymbol{\phi}]$ that is locally constructed from dynamical fields $\boldsymbol{\phi}$, where ${}^*$ denotes the Hodge dual.
It then follows that
\begin{equation}
\var \boldsymbol{L} [\boldsymbol{\phi}] \equiv \boldsymbol{E} [\boldsymbol{\phi}] \cdot \var \boldsymbol{\phi} - \ud \boldsymbol{\Theta} [\boldsymbol{\phi}; \var \boldsymbol{\phi}],
\end{equation}
where the three-form $\boldsymbol{\Theta} [\bs \phi; \var \bs \phi]$ is the \emph{symplectic potential}, and $\boldsymbol{E} [\bs \phi]$ is a tensor-valued differential form\footnote{Some of the indices of $\bs E[\bs \phi]$ are contracted with those of $\var \bs \phi$, yielding a four-form $\bs E[\bs \phi] \cdot \var \bs \phi$.} that encodes the equations of motion; that is, $\boldsymbol{E} [\bs \phi] = 0$ on shell.
Thus, on shell, the integral of $\var \boldsymbol{L} [\bs \phi]$ is just a boundary term, which we use to define $\boldsymbol{\Theta} [\boldsymbol{\phi}; \var \boldsymbol{\phi}]$.
We can then define the \emph{symplectic product} by taking a second, independent variation:
\begin{equation}
\boldsymbol{\omega} [\bs \phi; \var_1 \boldsymbol{\phi}, \var_2 \boldsymbol{\phi}] \equiv \var_1 \boldsymbol{\Theta} [\boldsymbol{\phi}; \var_2 \boldsymbol{\phi}] - \var_2 \boldsymbol{\Theta} [\boldsymbol{\phi}; \var_1 \boldsymbol{\phi}].
\end{equation}
Thus, we have that
\begin{equation}
\begin{split}
\ud \boldsymbol{\omega} [\bs \phi; \var_1 \boldsymbol{\phi}, \var_2 \boldsymbol{\phi}] &= \var_1 \boldsymbol{E} [\bs \phi] \cdot \var_2 \boldsymbol{\phi} - \var_2 \boldsymbol{E} [\bs \phi] \cdot \var_1 \boldsymbol{\phi},
\end{split}
\end{equation}
which vanishes if $\var_1 \boldsymbol{\phi}$ and $\var_2 \boldsymbol{\phi}$ are both solutions to the linearized equations of motion.
We define the corresponding vector current by
\begin{equation}
\pb{S} j^a \left[\bs \phi; \var_1 \boldsymbol{\phi}, \var_2 \boldsymbol{\phi}\right] \equiv \left(\dual \boldsymbol{\omega} \left[\bs \phi; \var_1 \boldsymbol{\phi}, \var_2 \boldsymbol{\phi}\right]\right)^a.
\end{equation}
We now turn to two different Lagrangians whose symplectic products are particularly interesting.
First, we consider the symplectic product for the Einstein-Hilbert Lagrangian four-form by $\boldsymbol{L}_{\textrm{EH}} [\bs g] = R \boldsymbol{\epsilon}/(16\pi)$.
For this Lagrangian, we find (following~\cite{Burnett57}, for example; note the difference in sign due to using a different sign convention for $R^a{}_{bcd}$)
\begin{equation}
(\Theta_{\textrm{EH}})_{abc} [\boldsymbol{g}; \var \boldsymbol{g}] = -\frac{1}{8\pi} \epsilon_{abcd} g^{fg} \delta^d{}_{[e} \var C^e{}_{f]g},
\end{equation}
where $\var C^a{}_{bc}$ is the variation of the connection coefficients for $\nabla_a (\lambda)$:
\begin{equation}
\var C^a{}_{bc} = \frac{1}{2} g^{ad} (\nabla_b \var g_{cd} + \nabla_c \var g_{bd} - \nabla_d \var g_{bc}).
\end{equation}
Thus, the symplectic (vector) current is given by
\begin{equation} \label{eqn:symp_form_linearized_gr}
\begin{split}
\pb{S} j_{\textrm{EH}}^a [\var_1 \boldsymbol{g}, \var_2 \boldsymbol{g}] &= \frac{1}{8\pi} \delta^a{}_{[b} \var_1 C^b{}_{c]d} \left[(\var_2 g)^{cd} - \frac{1}{2} (\var_2 g)^e{}_e g^{cd}\right] - \interchange{1}{2} \\
&= \frac{1}{16\pi} \var_1 C^a{}_{bc} (\var_2 g)^{bc} + v^a [\var_1 \bs{g}] (\var_2 g)^b{}_b + w^{ab} [\var_1 \bs{g}] \nabla_b (\var_2 g)^c{}_c - \interchange{1}{2},
\end{split}
\end{equation}
for some tensor fields $v^a [\var \bs{g}]$ and $w^{ab} [\var \bs{g}]$ which are unimportant for the discussion of this paper, as we only consider metric perturbations which are trace-free.
Here, for simplicity, the dependence on the background metric $g_{ab}$ is implicit.
This symplectic product provides a bilinear current on the space of metric perturbations which is conserved for vacuum solutions to the linearized Einstein equations.
Somewhat unexpectedly, one can also define a symplectic product for the master variables themselves.
In order to do so, we need a Lagrangian formulation for the Teukolsky equation.
Such a Lagrangian formulation was recently used to generate Noether currents for the master variables in~\cite{Toth:2018qrx}.
As noted by Bini, Cherubini, Jantzen, and Ruffini~\cite{Bini:2002jx}, the Teukolsky operator can be rewritten as a modified wave operator:
\begin{equation}
\pb{s} \Box = (\nabla^a + s\Gamma^a) (\nabla_a + s\Gamma_a) - 4s^2 \Psi_2,
\end{equation}
where
\begin{equation}
\Gamma^a = -2\left[\gamma l^a + (\epsilon + \rho) n^a - \alpha m^a - (\beta + \tau) \bar{m}^a\right].
\end{equation}
Since the equations of motion are now in the form of a modified wave equation, one can write down a Lagrangian four-form of the form (for $s \geq 0$)
\begin{equation}
\boldsymbol{L}_{\textrm{BCJR}} [\pb{s} \Omega, \pb{-s} \Omega] = \dual (\ud + s\boldsymbol{\Gamma}) \pb{s} \Omega \wedge (\ud - s\boldsymbol{\Gamma}) \pb{-s} \Omega - 96s^2 \Psi_2 \pb{s} \Omega \pb{-s} \Omega \boldsymbol{\epsilon}.
\end{equation}
Note that, in this expression, the metric and $\Gamma^a$ are non-dynamical fields, and therefore do not get varied.
Varying this Lagrangian four-form results in the Teukolsky equations for spins $s$ and $-s$.
One can easily show that
\begin{equation}
\boldsymbol{\Theta}_{\textrm{BCJR}} [\pb{s} \Omega, \pb{-s} \Omega; \var \pb{s} \Omega, \var \pb{-s} \Omega] = \var \pb{s} \Omega \dual (\ud - s\boldsymbol{\Gamma}) \pb{-s} \Omega + \var \pb{-s} \Omega \dual (\ud + s\boldsymbol{\Gamma}) \pb{s} \Omega,
\end{equation}
and so
\begin{equation} \label{eqn:symp_form_BCJR}
\pb{S} j_{\textrm{BCJR}}^a \left[\var_1 \pb{s} \Omega, \var_1 \pb{-s} \Omega; \var_2 \pb{s} \Omega, \var_2 \pb{-s} \Omega\right] = \var_1 \pb{s} \Omega (\nabla^a - s\Gamma^a) \var_2 \pb{-s} \Omega + \var_1 \pb{-s} \Omega (\nabla^a + s\Gamma^a) \var_2 \pb{s} \Omega - \interchange{1}{2}.
\end{equation}
Here, we are dropping any dependence on the background values of $\pb{s} \Omega$ and $\pb{-s} \Omega$, since they do not appear on the right-hand side.
Although this current is bilinear on the space of variations of the master variables, it can be regarded as a bilinear current on the space of master variables themselves, since their equations of motion are linear.
Note further that this symplectic product is not the physical symplectic product for linearized gravity.
\subsection{Currents of interest} \label{section:definitions}
Using the results of sections~\ref{section:symmetry} and~\ref{section:symplectic}, we now define the following currents, for which we will be computing the geometric optics limit and the fluxes at the horizon and null infinity.
The first of these currents is a rescaled version of the symplectic product of $\pb{s} \bs{\mathcal C} \cdot \var \bs{g}$ and its complex conjugate:
\begin{equation} \label{eqn:C_current}
\pb{\pb{s} \mathcal{C}} j^a [\var \bs{g}] \equiv 8i \pb{S} j_{\textrm{EH}}^a \Big[\pb{s} \bs{\mathcal C} \cdot \var \bs{g}, \overline{\pb{s} \bs{\mathcal C} \cdot \var \bs{g}}\Big],
\end{equation}
in terms of the symplectic product~\eqref{eqn:symp_form_linearized_gr} and the symmetry operator~\eqref{eqn:C_tensor}.
The normalization here is chosen to give a nicer limit in geometric optics; similarly, this current is simpler in the limit of geometric optics than other currents that can be constructed from $\pb{s} \bs{\mathcal C}$.
The currents defined in equation~\eqref{eqn:C_current} are entirely local, but they generally diverge at null infinity, as we will show in section~\ref{section:fluxes}.
The divergences can be removed by using $\pb{s} \mathring{\bs{\mathcal C}}$ instead of $\pb{s} \bs{\mathcal C}$.
We therefore define
\begin{equation} \label{eqn:C_current_proj}
\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs{g}] \equiv 8i \sum_{s = \pm 2} \pb{S} j_{\textrm{EH}}^a \bigg[\pb{s} \mathring{\bs{\mathcal C}} \cdot \var \bs{g}, \overline{\pb{s} \mathring{\bs{\mathcal C}} \cdot \var \bs{g}}\bigg],
\end{equation}
where $\pb{2} \mathring{\bs{\mathcal C}}$ is defined in equation~\eqref{eqn:C_tensor_proj}.
The motivation for including the sum over $s$ in this definition is due to the fact that $\pb{2} \mathring{\bs{\mathcal C}}$ and $\pb{-2} \mathring{\bs{\mathcal C}}$ are only nonzero for ingoing and outgoing solutions at null infinity, respectively.
The sum therefore ensures that the total current is nonzero for both types of solutions.
We next define similar currents involving $\pb{s} \bs{\Chi}$ and $\pb{s} \bs{\mathcal D}$:
\begin{align}
\pb{\pb{s} \mathcal{D}} j^a [\var \bs{g}] &\equiv \frac{i}{16} \pb{S} j_{\textrm{EH}}^a \Big[\pb{s} \bs{\Chi} \cdot \var \bs{g}, \overline{\pb{s} \bs{\mathcal D} \cdot \var \bs{g}}\Big], \label{eqn:D_current} \\
\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs{g}] &\equiv \frac{i}{16} \sum_{s = \pm 2} \pb{S} j_{\textrm{EH}}^a \bigg[\pb{s} \mathring{\bs \Chi} \cdot \var \bs{g}, \overline{\pb{s} \mathring{\bs{\mathcal D}} \cdot \var \bs{g}}\bigg]. \label{eqn:D_current_proj}
\end{align}
Unlike the currents~\eqref{eqn:C_current} and~\eqref{eqn:C_current_proj}, both of these currents are nonlocal.
We will see below that the geometric optics limits of these currents are proportional to the Carter constants $K$ of the gravitons, as opposed to $K^4$ for the currents~\eqref{eqn:C_current} and~\eqref{eqn:C_current_proj}.
Finally, we define the currents
\begin{equation} \label{eqn:Omega_current}
\pb{\pb{s} \Omega} j^a [\var \boldsymbol{g}] \equiv \frac{1}{4\pi i} \pb{S} j_{\textrm{BCJR}}^a \left[\pb{s} \Omega, \pb{-s} \Omega; \pb{s, s} \widetilde{\mathcal C}\; \overline{\pb{-s} \Omega}, \pb{-s, s} \widetilde{\mathcal C}\; \overline{\pb{-s} \Omega}\right],
\end{equation}
in terms of the symplectic product for the master variables in equation~\eqref{eqn:symp_form_BCJR} and the symmetry operator~\eqref{eqn:C}.
Note that $\pb{\pm 2} \Omega$ are functions of $\var g_{ab}$, by equation~\eqref{eqn:M_def}.
These currents are very similar to the currents $\pb{\pb{\pm 2} \mathcal{C}} j^a [\var \bs g]$, having the same geometric optics limit, and also being local; however, these currents have the advantage of also having finite fluxes at null infinity.
We now derive various properties of these currents in sections~\ref{section:geometric_optics} and~\ref{section:fluxes}.
For convenience, these properties are summarized at the end of this paper in table~\ref{table:summary}.
\section{Geometric Optics} \label{section:geometric_optics}
Using the symmetry operators in section~\ref{section:symmetry} and the symplectic products in section~\ref{section:symplectic}, one could define a multitude of currents that are conserved for vacuum solutions to the linearized Einstein equations.
In this section, we provide the motivation for the particular currents highlighted in section~\ref{section:definitions}.
This is accomplished by taking the geometric optics limit, in which solutions to the linearized Einstein equations represent null fluids of gravitons.
We express the associated currents in terms of the gravitons' constants of motion.
\subsection{Geometric optics on general backgrounds} \label{section:geometric_optics_gr}
The starting point for geometric optics is a harmonic ansatz for the metric perturbation:
\begin{equation}
\var g_{ab} = \Re\left\{\left[a \varpi_{ab} + O(\epsilon)\right] e^{-i\vartheta/\epsilon}\right\},
\end{equation}
where $a$ and $\vartheta$ are real, $\varpi_{ab}$, the \emph{polarization tensor}, is a complex, symmetric tensor that is normalized to satisfy $\varpi_{ab} \bar{\varpi}^{ab} = 1$, and $\epsilon$ is a dimensionless parameter whose limit is taken to zero.
Inserting this ansatz into the linearized Einstein equations and the Lorenz gauge condition and equating coefficients of powers of $\epsilon$ yields the following results (see, for example, Misner, Thorne, and Wheeler~\cite{mtw}):
\begin{enumerate}
\item[i.] The wavevector $k^a$ defined by
\begin{equation}
k_a \equiv \nabla_a \vartheta
\end{equation}
is tangent to a congruence of null geodesics:
\begin{equation}
k^b \nabla_b k^a = 0, \qquad k_a k^a = 0.
\end{equation}
\item[ii.] The polarization tensor $\varpi_{ab}$ is orthogonal to $k^a$ and parallel-transported along these geodesics:
\begin{equation} \label{eqn:varpi_conditions}
k^a \varpi_{ab} = 0, \qquad k^c \nabla_c \varpi_{ab} = 0.
\end{equation}
\item[iii.] The amplitude $a$ evolves along these geodesics according to
\begin{equation} \label{eqn:etendue}
\nabla_a (a^2 k^a) = 0.
\end{equation}
\end{enumerate}
We now consider this formalism in terms of spinors.
First, as $k^a$ is null, we can write
\begin{equation}
k^{AA'} = \kappa^A \bar{\kappa}^{A'},
\end{equation}
for some spinor $\kappa^A$.
We choose a second spinor $\lambda^A$ such that $(\kappa, \lambda)$ form a spin basis.
The conditions~\eqref{eqn:varpi_conditions} and the normalization of $\varpi_{ab}$ imply that
\begin{equation} \label{eqn:phase_tensor}
\varpi_{ab} = k_{(a} \alpha_{b)} + e_R q_a q_b + e_L \bar{q}_a \bar{q}_b,
\end{equation}
where $q_a \equiv \kappa_A \bar{\lambda}_{A'}$ and $\alpha^a$ is an arbitrary vector satisfying $\alpha^a k_a = 0$.
Because of the gauge freedom $\var g_{ab} \to \var g_{ab} + 2 \nabla_{(a} \xi_{b)}$, the first term can be removed by a gauge transformation (which preserves the Lorenz gauge condition), and so we can safely set $\alpha^a = 0$.
The last two terms in equation~\eqref{eqn:phase_tensor} are physically measurable.
The complex coefficients $e_R$ and $e_L$ correspond to right and left circular polarization.
By the normalization of $\varpi_{ab}$, we have that $|e_R|^2 + |e_L|^2 = 1$.
Moreover, these factors of $e_R$ and $e_L$ appear in the expansion for $(\var \Psi)_{ABCD}$:
\begin{equation} \label{eqn:Psi_go}
(\var \Psi)_{ABCD} = -\frac{1}{\epsilon^2} a \kappa_A \kappa_B \kappa_C \kappa_D \left(e_R e^{-i\vartheta/\epsilon} + \bar{e}_L e^{i\vartheta/\epsilon}\right) + O(1/\epsilon).
\end{equation}
\subsection{Conserved currents} \label{section:geometric_optics_currents}
When considering nonlinear quantities in geometric optics, such as conserved currents, we will discard rapidly oscillating terms.
This effectively takes a spacetime average of these quantities over a scale that is large compared to $\epsilon$, but small compared to the radius of curvature of the background spacetime (see, for example,~\cite{Isaacson:1968zza}, or~\cite{Burnett:1989gp} for rigorous treatments of this averaging procedure via weak limits).
Such an average we will denote by $\langle \cdot\rangle$.
We start with a few simple results.
First, if a conserved current reduces in the limit of geometric optics to
\begin{equation} \label{eqn:conserved_quantity}
\langle j^a\rangle = \frac{1}{\epsilon^n} [a^2 Q k^a + O(\epsilon)],
\end{equation}
for some quantity $Q$ and integer $n$, then $Q$ is a conserved quantity along the integral curves of $k^a$.
To see this, note that the leading order term in the conservation equation $\nabla_a \langle j^a\rangle = 0$ yields
\begin{equation} \label{eqn:optical_conservation}
0 = a^2 k^a \nabla_a Q + Q \nabla_a (a^2 k^a) = a^2 k^a \nabla_a Q,
\end{equation}
from equation~\eqref{eqn:etendue}.
All currents that we consider in this paper will be of the form~\eqref{eqn:conserved_quantity} in the geometric optics limit..
The second result is that, under the assumption~\eqref{eqn:conserved_quantity}, the conserved charge associated with the current $j^a$ reduces to a sum over all gravitons of the conserved quantity $Q$ for each graviton.
This result means that equation~\eqref{eqn:conserved_quantity} is a physically appealing assumption.
The proof proceeds as follows~\cite{mtw}: first, we note that the effective stress-energy tensor appropriate to gravitational radiation in the geometric optics regime is given by~\cite{Isaacson:1968zza}
\begin{equation}
\langle T_{ab}^{\textrm{eff}}\rangle = \frac{1}{32\pi} \left\langle (\nabla_a \var g_{cd}) [\nabla_b (\var g)^{cd}]\right\rangle + O(1/\epsilon) = \frac{a^2}{32\pi \epsilon^2} \left[k_a k_b + O(\epsilon)\right].
\end{equation}
On the other hand, the stress-energy tensor for a collection of gravitons with number-flux $\mathcal N_a$ and momentum $p_a = \hbar k_a/\epsilon$ is given by~\cite{mtw}
\begin{equation}
T_{ab}^{\textrm{eff}} = p_{(a} \mathcal N_{b)},
\end{equation}
and so we find that
\begin{equation}
a^2 k_a = 32 \pi \hbar \epsilon \mathcal N_a [1 + O(\epsilon)].
\end{equation}
Upon integrating a current $j^a$ given by equation~\eqref{eqn:conserved_quantity} over a hypersurface $\Sigma$, one finds the charge
\begin{equation} \label{eqn:conserved_charge}
\int_\Sigma \langle j^a\rangle \ud^3 \Sigma_a = \frac{32\pi \hbar}{\epsilon^{n - 1}} \sum_\alpha Q_\alpha [1 + O(\epsilon)],
\end{equation}
where $\alpha$ labels the gravitons passing through the hypersurface.
That is, the charge is proportional to the sum of the conserved quantities over all of the gravitons passing through the surface.
\subsection{Computations} \label{section:geometric_optics_examples}
We now turn to computations of geometric optics limits for the conserved currents discussed in this paper.
For these calculations, we first define the quantities $\kappa_0$, $\kappa_1$, $r_a$, and $s_a$:
\begin{equation}
\kappa_0 \equiv o_A \kappa^A, \qquad \kappa_1 \equiv \iota_A \kappa^A, \qquad r^a \equiv \sigma^a{}_{AA'} o^A \bar{\kappa}^{A'}, \qquad s^a \equiv \sigma^a{}_{AA'} \iota^A \bar{\kappa}^{A'}.
\end{equation}
These quantities are constructed from the spinor $\kappa_A$ (which is related to the wavevector $k^a$) and the principal spin basis $(o, \iota)$.
They satisfy
\begin{equation}
\begin{gathered}
|\zeta \kappa_0 \kappa_1|^2 = \frac{\epsilon^2}{2 \hbar^2} K, \qquad r_a r^a = s_a s^a = r_a k^a = s_a k^a = 0, \\
r_a \bar{r}^a = |\kappa_0|^2, \qquad s_a \bar{s}^a = |\kappa_1|^2, \qquad r_a \bar{s}^a = -\kappa_0 \bar{\kappa}_1,
\end{gathered}
\end{equation}
where $K = \hbar^2 K_{ab} k^a k^b/\epsilon^2$ is the Carter constant for the gravitons.
The factors of $\hbar$ arise in this classical computation as part of converting from the wavevectors of the gravitons to their momenta, and hence their conserved quantities.
We now begin calculating the conserved currents defined in section~\ref{section:definitions}.
Since, to leading order in geometric optics, the differential operators present in this paper become c-numbers, a straightforward calculation starting from equations~\eqref{eqn:M} and~\eqref{eqn:tau_dagger} shows that
\begin{subequations}
\begin{align}
\pb{s} \tau_{ab}^\dagger &= \frac{1}{\epsilon^2} \begin{cases}
\kappa_0^2 r_a r_b + O(\epsilon) & s = 2 \\
\zeta^4 \kappa_1^2 s_a s_b + O(\epsilon) & s = -2
\end{cases}, \label{eqn:tau_dagger_go} \\
\pb{s} M^{ab} &= \frac{1}{2\epsilon^2} \begin{cases}
\kappa_0^2 r^a r^b + O(\epsilon) & s = 2 \\
\zeta^4 \kappa_1^2 s^a s^b + O(\epsilon) & s = -2
\end{cases}, \label{eqn:M_go}
\end{align}
\end{subequations}
and [starting from equation~\eqref{eqn:Psi_go}] that
\begin{equation} \label{eqn:Omega_go}
\pb{s} \Omega = -\frac{a}{\epsilon^2} (e_R e^{-i\vartheta/\epsilon} + \bar{e}_L e^{i\vartheta/\epsilon})\begin{cases}
\kappa_0^4 + O(\epsilon) & s = 2 \\
(\zeta \kappa_1)^4 + O(\epsilon) & s = -2
\end{cases}.
\end{equation}
As such, we find that
\begin{equation} \label{eqn:C_geometric_optics}
\pb{s} \mathcal{C}_{ab}{}^{cd} \var g_{cd} = -\frac{a}{\epsilon^4} \zeta^4 (\kappa_1 \kappa_0)^2 (e_R e^{-i\vartheta/\epsilon} + \bar{e}_L e^{i\vartheta/\epsilon}) \begin{cases}
r_a r_b \kappa_1^2 + O(\epsilon) & s = 2 \\
s_a s_b \kappa_0^2 + O(\epsilon) & s = -2
\end{cases}.
\end{equation}
This implies that
\begin{equation}
\left\langle(\pb{s} \mathcal{C}_{bc}{}^{de} \var g_{de}) \nabla^a \overline{\pb{s} \mathcal{C}^{bc}{}_{de} \var g^{de}}\right\rangle = -\frac{2\pi i}{\hbar^7} K^4 (|e_R|^2 - |e_L|^2) \mathcal N^a [1 + O(\epsilon)].
\end{equation}
Thus, we find that the current $\pb{\pb{s} \mathcal{C}} j^a [\var \bs g]$ is given in this limit by
\begin{equation} \label{eqn:C_current_go}
\begin{split}
\left\langle\pb{\pb{s} \mathcal{C}} j^a [\var \boldsymbol{g}]\right\rangle &= \frac{1}{2\pi} \left\langle\Im\left[(\pb{s} \mathcal{C}_{bc}{}^{de} \var g_{de}) \nabla^a \overline{\pb{s} \mathcal{C}^{bc}{}_{de} \var g^{de}}\right]\right\rangle [1 + O(\epsilon)] \\
&= \frac{1}{\hbar^7} K^4 \left(|e_R|^2 - |e_L|^2\right) \mathcal N^a [1 + O(\epsilon)].
\end{split}
\end{equation}
As such, these currents are a generalization of the Carter constant for point particles to linearized gravity in the Kerr spacetime, at least in the limit of geometric optics.
We now turn to the current $\pb{\pb{s} \mathcal{D}} j^a [\var \bs g]$.
First, note that, from equations~\eqref{eqn:D} and~\eqref{eqn:RS},
\begin{equation}
\pb{s} \mathcal{D} \pb{s} \Omega = \frac{1}{\epsilon^2} |\zeta \kappa_0 \kappa_1|^2 \pb{s} \Omega [1 + O(\epsilon)],
\end{equation}
and so
\begin{equation}
\pb{s} \mathcal{D}_{ab}{}^{cd} \var g_{cd} = \frac{K}{2 \hbar^2} \pb{s} \Chi_{ab}{}^{cd} \var g_{cd} [1 + O(\epsilon)].
\end{equation}
Now, note that $\pb{s} \Chi_{ab}{}^{cd} \var g_{cd}$, by equations~\eqref{eqn:Chi} and~\eqref{eqn:Pi}, can be written (in the limit of geometric optics, where differential operators commute to leading order) as a product of the form
\begin{equation}
\pb{s} \Chi_{ab}{}^{cd} \var g_{cd} = 4 \left(\pb{s, s} \tilde{\mathcal C} \overline{\pb{-s, -s} \tilde{\mathcal C}}\right)^{-1} \pb{s} \mathcal{C}_{ab}{}^{cd} \overline{\pb{-s} \mathcal{C}_{cd}{}^{ef}} \var g_{ef} [1 + O(\epsilon)],
\end{equation}
where the operator $\left(\pb{s, s} \tilde{\mathcal C} \overline{\pb{-s, -s} \tilde{\mathcal C}}\right)^{-1}$ is a nonlocal operator having the effect of multiplying each coefficient of the expansion~\eqref{eqn:decoupled_expansion} by $64/(C_{lm\omega}^2 + 144M^2 \omega^2)$.
This operator is a nonlocal inverse to $\pb{s, s} \tilde{\mathcal C} \overline{\pb{-s, -s} \tilde{\mathcal C}}$, by equation~\eqref{eqn:C_action}.
For its geometric optics limit, note that
\begin{subequations} \label{eqn:C_go}
\begin{align}
\pb{2, -2} \widetilde{\mathcal C}\; \overline{\pb{2} \Omega} &= \frac{1}{2\epsilon^4} (\bar{\zeta} \kappa_0 \bar{\kappa}_1)^4 \overline{\pb{2} \Omega} [1 + O(\epsilon)], \quad &\pb{-2, 2} \widetilde{\mathcal C}\; \overline{\pb{-2} \Omega} &= \frac{1}{2\epsilon^4} (\zeta \bar{\kappa}_0 \kappa_1)^4 \overline{\pb{-2} \Omega} [1 + O(\epsilon)], \\
\pb{2, 2} \widetilde{\mathcal C}\; \overline{\pb{-2} \Omega} &= \frac{1}{2\epsilon^4} |\kappa_0|^8 \overline{\pb{-2} \Omega} [1 + O(\epsilon)], \quad &\pb{-2, -2} \widetilde{\mathcal C}\; \overline{\pb{2} \Omega} &= \frac{1}{2\epsilon^4} |\zeta \kappa_1|^8 \overline{\pb{2} \Omega} [1 + O(\epsilon)],
\end{align}
\end{subequations}
and so
\begin{equation}
\left(\pb{s, s} \tilde{\mathcal C} \overline{\pb{-s, -s} \tilde{\mathcal C}}\right)^{-1} \pb{s} \Omega = \frac{4 \epsilon^8}{|\zeta \kappa_0 \kappa_1|^8} \pb{s} \Omega[1 + O(\epsilon)].
\end{equation}
Moreover, we have that [from equations~\eqref{eqn:tau_dagger_go} and~\eqref{eqn:M_go}]
\begin{equation}
\pb{s} \mathcal{C}_{ab}{}^{cd} \overline{\pb{-s} \mathcal{C}_{cd}{}^{ef}} \var g_{ef} = -\frac{a}{4 \epsilon^8} |\zeta \kappa_0 \kappa_1|^8 (\bar{e}_R e^{i\vartheta/\epsilon} + e_L e^{-i\vartheta/\epsilon}) \begin{cases}
r_a r_b/\kappa_0^2 + O(\epsilon) & s = 2 \\
s_a s_b/\kappa_1^2 + O(\epsilon) & s = -2
\end{cases},
\end{equation}
from which it follows that
\begin{equation}
\pb{s} \Chi_{ab}{}^{cd} \var g_{cd} = -4a (\bar{e}_R e^{i\vartheta/\epsilon} + e_L e^{-i\vartheta/\epsilon}) \begin{cases}
r_a r_b/\kappa_0^2 + O(\epsilon) & s = 2 \\
s_a s_b/\kappa_1^2 + O(\epsilon) & s = -2
\end{cases}.
\end{equation}
The current in question is then given by
\begin{equation} \label{eqn:D_current_go}
\langle\pb{\pb{s} \mathcal{D}} j^a [\var \boldsymbol{g}]\rangle = \frac{1}{\hbar} K \left(|e_R|^2 - |e_L|^2\right) \mathcal N^a [1 + O(\epsilon)].
\end{equation}
This therefore provides another, entirely \emph{non-local} notion of the Carter constant for linearized gravity in the Kerr spacetime.
There are, of course, other currents whose charges reduce to the Carter constant in the geometric optics limit.
Another class of currents come from the symplectic product for the master variables, instead of the metric perturbation.
One current of interest from this class is given by equation~\eqref{eqn:Omega_current}, which has a limit in geometric optics given by [from equations~\eqref{eqn:symp_form_BCJR},~\eqref{eqn:C_go}, and~\eqref{eqn:Omega_go}]
\begin{equation} \label{eqn:Omega_current_go}
\langle\pb{\pb{s} \Omega} j^a [\var \bs{g}]\rangle = \frac{1}{\hbar^7} K^4 (|e_R|^2 - |e_L|^2) \mathcal N^a [1 + O(\epsilon)].
\end{equation}
The results of this section [equations~\eqref{eqn:C_current_go},~\eqref{eqn:D_current_go}, and~\eqref{eqn:Omega_current_go}] give the geometric optics limits for the currents that do not involve projection operators.
We now consider the two remaining currents, $\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs g]$ and $\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$.
For simplicity, we first consider $\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs g]$ (the exact same argument holds for $\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$).
This current is the sum of two terms, the first of which is equal to $\pb{\pb{-2} \mathcal C} j^a [\var \bs g]$, except that it contains a projection which eliminates the ingoing modes at null infinity.
Similarly, the second term is equal to $\pb{\pb{2} \mathcal C} j^a [\var \bs g]$, except it eliminates all outgoing modes.
Consider the case where $\var g_{ab}$ represents a null fluid of gravitons where the gravitons are purely outgoing at future null infinity; that is, $k^a$ is tangent to an outgoing null congruence.
The geometric optics limit in this case would be the same as that of $\pb{\pb{-2} \mathcal C} j^a [\var \bs g]$.
Similarly, if $k^a$ is an ingoing null congruence, the geometric optics limit would be the same as that of $\pb{\pb{2} \mathcal C} j^a [\var \bs g]$.
Since these geometric optics limits are equal by equation~\eqref{eqn:C_current_go}, we recover the following result:
\begin{equation}
\left\langle\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \boldsymbol{g}]\right\rangle = \frac{1}{\hbar^7} K^4 \left(|e_R|^2 - |e_L|^2\right) \mathcal N^a [1 + O(\epsilon)],
\end{equation}
when $\var g_{ab}$ represents an ingoing or outgoing null fluid of gravitons.
A similar argument gives a similar result for $\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$.
However, the geometric optics limits for $\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs g]$ and $\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$ are only given by simple expressions when $k^a$ is either tangent to an ingoing or outgoing null congruence, but not for general geometric optics solutions $\var g_{ab}$.
We conclude this discussion with a brief review of a classification scheme for conserved currents in geometric optics that we used in~\cite{Grant:2019qyo}.
In the limit of geometric optics, one often finds that conserved currents depend on the quantities $e_R$ and $e_L$ in one of the following four ways; depending on this dependence, we call such currents \emph{energy}, \emph{zilch}, \emph{chiral}, and \emph{antichiral currents}:
\begin{equation}
\langle j^a\rangle = Q \mathcal N^a \begin{cases}
1 + O(\epsilon) & \textrm{energy currents} \\
(|e_R|^2 - |e_L|^2) + O(\epsilon) & \textrm{zilch currents} \\
e_R \bar{e}_L + O(\epsilon) & \textrm{chiral currents} \\
\bar{e}_R e_L + O(\epsilon) & \textrm{antichiral currents}
\end{cases}.
\end{equation}
This classification scheme is a specialization of that of~\cite{Anco:2002xn}.
For conserved currents that are $\mathbb{R}$-bilinear functionals of $(\var \Psi)_{ABCD}$ (a property which is satisfied by all currents considered in this paper), there is a relationship between $Q$ and the type of current in this classification: for energy and zilch currents,
\begin{equation}
Q = Q_{a_1 \cdots a_n} p^{a_1} \cdots p^{a_n},
\end{equation}
where $Q_{a_1 \cdots a_n}$ is a rank $n$ Killing tensor and $n$ is odd for energy currents and even for zilch currents.
Moreover, for chiral and antichiral currents, $Q$ cannot be written in the above form.
Since we wanted to construct conserved currents which were related to the Carter constant, which is a conserved quantity arising from a rank two Killing tensor, it is unsurprising that all currents which we considered were zilch currents.
Another interesting result of this classification scheme is an odd result for the symplectic product for the master variables.
The symplectic product for linearized gravity, when applied to $\var g_{ab}$ and $\lie_\xi \var g_{ab}$, gives an energy current in geometric optics, and the associated conserved quantity is proportional to $\xi^a p_a$ (which would be proportional to the energy in the case $\xi^a = t^a$).
This current is known as the \emph{canonical energy current}.
However, using the symplectic product for the master variables, one finds that a similar current, obtained by using $\pb{\pm s} \Omega$ and $\lie_\xi \pb{\pm s} \Omega$, gives a chiral current.
In this sense, the symplectic product for the master variables cannot be used to construct a current whose geometric optics limit behaves like energy.
\section{Fluxes at null infinity and the horizon} \label{section:fluxes}
Another desirable property for a conserved current is that its flux through the horizon ($H$) and through null infinity ($\scri$) be finite.
In this section, we provide formulae for these fluxes, using results for the asymptotic falloffs in appendix~\ref{appendix:asymptotics}.
More details on the definitions of these fluxes are given in appendix~\ref{appendix:fluxes_integration}.
We begin with some notation: first, the Boyer-Lindquist coordinate system is not well suited to working at the horizon or null infinity.
Instead, one uses the ingoing and outgoing coordinate systems $(v, r, \theta, \psi)$ and $(u, r, \theta, \chi)$, defined in terms of Boyer-Lindquist coordinates and the tortoise coordinate~\eqref{eqn:tortoise} by
\begin{subequations}
\begin{align}
v &= t + r^*, \qquad &\psi &= \phi + \int \frac{a \ud r}{\Delta}, \\
u &= t - r^*, \qquad &\chi &= \phi - \int \frac{a \ud r}{\Delta}.
\end{align}
\end{subequations}
The ingoing coordinate system is relevant near the future horizon ($H^+$) and past null infinity ($\scri^-$), while the outgoing coordinate system is relevant near the past horizon ($H^-$) and future null infinity ($\scri^+$).
When dealing with a generic surface $S$, we will write $w$ and $\alpha$ instead of either $v$ and $\psi$ or $u$ and $\chi$:
\begin{equation}
w = \begin{cases}
v & \textrm{ at $H^+$, $\scri^-$} \\
u & \textrm{ at $H^-$, $\scri^+$}
\end{cases}, \qquad \alpha = \begin{cases}
\psi & \textrm{ at $H^+$, $\scri^-$} \\
\chi & \textrm{ at $H^-$, $\scri^+$}
\end{cases}.
\end{equation}
This greatly simplifies definitions.
For example, we will write the flux of a current $\pb{\ldots} j^a$ through a surface $S$ as $\ud^2 \pb{\ldots} Q/\ud w \ud \Omega|_S$, which we will define more explicitly in equation~\eqref{eqn:differential_fluxes}, where the differential solid angle is defined by
\begin{equation}
\ud \Omega \equiv \sin \theta \ud \theta \ud \alpha.
\end{equation}
We next remark that, in this paper, we compute fluxes of the conserved currents~\eqref{eqn:C_current}, \eqref{eqn:C_current_proj}, \eqref{eqn:D_current}, and~\eqref{eqn:D_current_proj} \emph{only} when acting upon the metric perturbations $\Im[\var_\pm g_{ab}]$.
We are free to do so, as these metric perturbations are related by a gauge transformation to any $l \geq 2$ metric perturbation $\var g_{ab}$.
Moreover, this specialization allows us to use equations~\eqref{eqn:C_tensor_action} and~\eqref{eqn:D_tensor_action} in order to write the fluxes in terms of the fluxes of the currents
\begin{equation} \label{eqn:mode_currents}
\pb{\pm 2} j^a_{ll'm\omega pp'} \equiv \pb{S} j_{\textrm{EH}}^a \Big[(\var_\pm \bs{g})_{lm\omega p}, \overline{(\var_\pm \bs{g})_{l'm\omega p'}}\Big],
\end{equation}
assuming that we average over $w$ and $\alpha$.
These currents are functions of the Debye potentials $\pb{\pm 2} \psi$, instead of the metric perturbation.
In particular, they are functions of the coefficients $\pb{s} \widehat{\psi}_{lm\omega p}^{\; \textrm{in/out/down/up}}$.
In terms of the fluxes of the currents~\eqref{eqn:mode_currents}, we have that (averaging over $w$ and $\alpha$)
\begin{subequations}
\begin{align}
\left\langle\frac{\ud^2 \pb{\pb{s} \mathcal{C}} Q}{\ud w \ud \Omega}\right\rangle_{w, \alpha} &= \frac{i}{32} \int_{-\infty}^\infty \ud \omega \sum_{l, l' = 2}^\infty \sum_{|m| \leq \min(l, l')} \sum_{p, p' = \pm 1} pp' \pb{s} C_{lm\omega p} \overline{\pb{s} C_{l'm\omega p'}} \frac{\ud^2 \pb{s} Q_{ll'm\omega pp'}}{\ud w \ud \Omega}, \\
\left\langle\frac{\ud^2 \pb{\pb{s} \mathcal{D}} Q}{\ud w \ud \Omega}\right\rangle_{w, \alpha} &= \frac{i}{16} \int_{-\infty}^\infty \ud \omega \sum_{l, l' = 2}^\infty \sum_{|m| \leq \min(l, l')} \sum_{p, p' = \pm 1} \pb{2} \lambda_{l'm\omega} \frac{\ud^2 \pb{s} Q_{ll'm\omega pp'}}{\ud w \ud \Omega}.
\end{align}
\end{subequations}
As these quantities are all $\mathbb{R}$-bilinear, it is convenient to define
\begin{equation}
\pb{s} \Upsilon^{\textrm{in}/\textrm{out}/\textrm{down}/\textrm{up}}_{ll'm\omega pp'} \equiv \pb{s} \widehat{\psi}^{\textrm{in}/\textrm{out}/\textrm{down}/\textrm{up}}_{lm\omega p} \overline{\pb{s} \widehat{\psi}^{\textrm{in}/\textrm{out}/\textrm{down}/\textrm{up}}_{l'm\omega p'}}.
\end{equation}
Moreover, the fluxes will each have a nontrivial angular dependence.
To determine this, we define, for some quantity $q[\pb{s} \psi]$, with coefficients $q_{lm\omega p} [\pb{s} \psi]$ in an expansion, the angular dependences $\pb{q} S^{\textrm{in}/\textrm{out}/\textrm{down}/\textrm{up}}_{lm\omega p} (\theta)$ by
\begin{equation} \label{eqn:S_def}
q_{lm\omega p} (t, r, \theta, \phi) \equiv \begin{cases}
\pb{s} \widehat{\psi}^{\textrm{in}}_{lm\omega p} e^{i(m\psi - \omega v)} \pb{q} S^{\textrm{in}}_{lm\omega p} (\theta) \Delta^{n_q^{\textrm{in}}} + \pb{s} \widehat{\psi}^{\textrm{out}}_{lm\omega p} e^{i(m\chi - \omega u)} \pb{q} S^{\textrm{out}}_{lm\omega p} (\theta) \Delta^{n_q^{\textrm{out}}} & r \to r_+ \\
\pb{s} \widehat{\psi}^{\textrm{down}}_{lm\omega p} e^{i(m\psi - \omega v)} \pb{q} S^{\textrm{down}}_{lm\omega p} (\theta) r^{n_q^{\textrm{down}}} + \pb{s} \widehat{\psi}^{\textrm{up}}_{lm\omega p} e^{i(m\chi - \omega u)} \pb{q} S^{\textrm{up}}_{lm\omega p} (\theta) r^{n_q^{\textrm{up}}} & r \to \infty
\end{cases},
\end{equation}
for some integers $n_q^{\textrm{in}/\textrm{out}/\textrm{down}/\textrm{up}}$.
Assuming appropriate smoothness conditions, equation~\eqref{eqn:S_def} simplifies further if we specialize to the various surfaces at which we are computing these quantities:
\begin{equation}
\left.q_{lm\omega p} (t, r, \theta, \phi)\right|_S \sim \begin{cases}
\pb{s} \widehat{\psi}^{\textrm{in}}_{lm\omega p} e^{i(m\psi - \omega v)} \pb{q} S^{\textrm{in}}_{lm\omega p} (\theta) \Delta^{n_q^{\textrm{in}}} & S = H^+ \\
\pb{s} \widehat{\psi}^{\textrm{out}}_{lm\omega p} e^{i(m\chi - \omega u)} \pb{q} S^{\textrm{out}}_{lm\omega p} (\theta) \Delta^{n_q^{\textrm{out}}} & S = H^- \\
\pb{s} \widehat{\psi}^{\textrm{down}}_{lm\omega p} e^{i(m\psi - \omega v)} \pb{q} S^{\textrm{down}}_{lm\omega p} (\theta) r^{n_q^{\textrm{down}}} & S = \scri^- \\
\pb{s} \widehat{\psi}^{\textrm{up}}_{lm\omega p} e^{i(m\chi - \omega u)} \pb{q} S^{\textrm{up}}_{lm\omega p} (\theta) r^{n_q^{\textrm{up}}} & S = \scri^+
\end{cases}.
\end{equation}
In other words, only ``in'' modes contribute at $H^+$, ``out'' modes at $H^-$, etc.
The various quantities $q$ which we will be considering will be components of metric perturbations and perturbed connection coefficients.
The relevant integers $n_q^{\textrm{in/out/down/up}}$ are (effectively) given in table~\ref{table:gr_asymp}.
Moreover, the various angular dependences are given by equations~\eqref{eqn:perturbed_metric_asymp} and~\eqref{eqn:perturbed_chris_asymp}, and computed in appendix~\ref{appendix:asymptotics}.
Using table~\ref{table:gr_asymp} and equations~\eqref{eqn:symplectic_flux_l} and~\eqref{eqn:symplectic_flux_n}, we find that
\begin{subequations} \label{eqn:base_fluxes_down}
\begin{align}
\left.\frac{\ud^2 \pb{+2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{\scri^+} &= 0, \\
\left.\frac{\ud^2 \pb{+2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{\scri^-} &= -\frac{i}{64\pi} \pb{-2} \Upsilon^{\textrm{down}}_{ll'm\omega pp'} \pb{\var_+ C_{l\bar{m}\bar{m}}} S^{\textrm{down}}_{lm\omega p} \overline{\pb{\var_+ g_{\bar{m}\bar{m}}} S^{\textrm{down}}_{l'm\omega p'}} + \overline{\interchange{l, p}{l', p'}}, \\
\left.\frac{\ud^2 \pb{+2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{H^+} &= -\frac{i}{64\pi} \pb{-2} \Upsilon^{\textrm{in}}_{ll'm\omega pp'} \pb{\var_+ C_{l\bar{m}\bar{m}}} S^{\textrm{in}}_{lm\omega p} \overline{\pb{(\var_+ g)_{\bar{m}\bar{m}}} S^{\textrm{in}}_{l'm\omega p'}} + \overline{\interchange{l, p}{l', p'}}, \\
\left.\frac{\ud^2 \pb{+2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{H^-} &= \frac{i\Sigma_+}{32\pi} \pb{-2} \Upsilon^{\textrm{out}}_{ll' m\omega pp'} \left(\pb{\var_+ C_{n\bar{m}\bar{m}}} S^{\textrm{out}}_{lm\omega p} \overline{\pb{\var_+ g_{\bar{m}\bar{m}}} S^{\textrm{out}}_{l'm\omega p'}} - \pb{\var_+ C_{n(l\bar{m})}} S^{\textrm{out}}_{lm\omega p} \overline{\pb{\var_+ g_{n\bar{m}}} S^{\textrm{out}}_{l'm\omega p'}}\right) \nonumber \\
&\hspace{1em}+ \overline{\interchange{l, p}{l', p'}},
\end{align}
\end{subequations}
where the superscript ``down'' indicates that we have performed a projection such that $\pb{s} \widehat{\psi}^{\textrm{up}}_{lm\omega p} = 0$, and
\begin{subequations} \label{eqn:base_fluxes_up}
\begin{align}
\left.\frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{\scri^+} &= \frac{i}{32\pi} \pb{2} \Upsilon^{\textrm{up}}_{ll'm\omega pp'} \pb{\var_- C_{nmm}} S^{\textrm{up}}_{lm\omega p} \overline{\pb{\var_- g_{mm}} S^{\textrm{up}}_{l'm\omega p'}} + \overline{\interchange{l, p}{l', p'}}, \\
\left.\frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{\scri^-} &= 0, \\
\left.\frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{H^+} &= -\frac{i}{64\pi} \pb{2} \Upsilon^{\textrm{in}}_{ll'm\omega pp'} \left(\pb{\var_- C_{lmm}} S^{\textrm{in}}_{lm\omega p} \overline{\pb{\var_- g_{mm}} S^{\textrm{in}}_{l'm\omega p'}} - \pb{\var_- C_{l(nm)}} S^{\textrm{in}}_{lm\omega p} \overline{\pb{\var_- g_{lm}} S^{\textrm{in}}_{l'm\omega p'}}\right) \nonumber \\
&\hspace{1em}+ \overline{\interchange{l, p}{l', p'}}, \\
\left.\frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{H^-} &= \frac{i\Sigma_+}{32\pi} \pb{2} \Upsilon^{\textrm{out}}_{ll'm\omega pp'} \pb{\var_- C_{nmm}} S^{\textrm{out}}_{lm\omega p} \overline{\pb{\var_- g_{mm}} S^{\textrm{out}}_{l'm\omega p'}} + \overline{\interchange{l, p}{l', p'}},
\end{align}
\end{subequations}
and the superscript ``up'' denotes the fact that we have performed a projection to set $\pb{s} \widehat{\psi}^{\textrm{down}}_{lm\omega p} = 0$.
If these projections are not performed, then the respective fluxes \emph{diverge}, as is evident from table~\ref{table:gr_asymp} and equation~\eqref{eqn:symplectic_components}.
Since the fluxes of $\pb{\pb{s} \mathcal{C}} j^a$ and $\pb{\pb{s} \mathcal{D}} j^a$ can be written in terms of those of $\pb{s} j_{ll'm\omega pp'}^a$, there are issues with these currents as well.
These divergences motivated the introduction of the projection operators in section~\ref{section:projection}.
With these projection operators, we have sacrificed locality (which we had already sacrificed in $\pb{\pb{s} \mathcal{D}} j^a$) in order to obtain finite fluxes.
As mentioned at the end of section~\ref{section:geometric_optics_examples}, the geometric optics limits are similar to those of the currents $\pb{\pb{s} \mathcal{C}} j^a$ and $\pb{\pb{s} \mathcal{D}} j^a$.
We also have that
\begin{subequations} \label{eqn:projection_fluxes}
\begin{align}
\left\langle\frac{\ud^2 \pb{\pb{2} \mathring{\mathcal C}} Q}{\ud w \ud \Omega}\right\rangle_{w, \alpha} &= \frac{i}{32} \int_{-\infty}^\infty \ud \omega \sum_{l, l' = 2}^\infty \sum_{|m| \leq \min(l, l')} \sum_{p, p' = \pm 1} \nonumber \\
&\hspace{3.5em}\times pp' \left\{\pb{2} C_{lm\omega p} \overline{\pb{2} C_{l'm\omega p'}} \frac{\ud^2 \pb{2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud w \ud \Omega} + \pb{-2} C_{lm\omega p} \overline{\pb{-2} C_{l'm\omega p'}} \frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud w \ud \Omega}\right\}, \\
\left\langle\frac{\ud^2 \pb{\pb{2} \mathring{\mathcal D}} Q}{\ud w \ud \Omega}\right\rangle_{w, \alpha} &= \frac{i}{16} \int_{-\infty}^\infty \ud \omega \sum_{l, l' = 2}^\infty \sum_{|m| \leq \min(l, l')} \sum_{p, p' = \pm 1} \pb{2} \lambda_{l'm\omega} \left\{\frac{\ud^2 \pb{2} Q^{\textrm{down}}_{ll'm\omega pp'}}{\ud w \ud \Omega} + \frac{\ud^2 \pb{-2} Q^{\textrm{up}}_{ll'm\omega pp'}}{\ud w \ud \Omega}\right\}.
\end{align}
\end{subequations}
Using equations~\eqref{eqn:base_fluxes_down},~\eqref{eqn:base_fluxes_up}, and~\eqref{eqn:projection_fluxes}, we have completely determined the fluxes of the charges $\pb{\pb{2} \mathring{\mathcal C}} j^a$ and $\pb{\pb{2} \mathring{\mathcal D}} j^a$.
Using the symplectic product for linearized gravity, we have not been able to construct a \emph{local} current with finite fluxes which reduces to the Carter constant in geometric optics.
However, we can do so using the symplectic product we defined in equation~\eqref{eqn:symp_form_BCJR} for the master variables.
We find that the fluxes for $\pb{\pb{s} \Omega} j^a$, averaged over $w$ and $\alpha$, are given by an expansion of the form
\begin{equation} \label{eqn:Omega_flux}
\left\langle\frac{\ud^2 \pb{\pb{s} \Omega} Q}{\ud w \ud \Omega}\right\rangle_{w, \alpha} \equiv \int_{-\infty}^\infty \ud \omega\; \sum_{l, l' = 2}^\infty\; \sum_{|m| < l, l'} \sum_{p, p' = \pm 1} \frac{\ud^2 \pb{\pb{s} \Omega} Q_{ll'm\omega pp'}}{\ud w \ud \Omega},
\end{equation}
where
\begin{subequations}
\begin{align}
&\begin{aligned}
\left.\frac{\ud^2 \pb{\pb{s} \Omega} Q_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{\scri^+} = \frac{\omega}{32\pi} \bigg\{&C_{l'm\omega} \pb{s} \Theta_{lm\omega} \pb{s} \Theta_{l'm\omega} \Big[\pb{s} \widehat{\psi}^{\textrm{up}}_{lm\omega p} \overline{\pb{-s} \widehat{\psi}^{\textrm{up}}_{l'm\omega p'}} + \overline{\interchange{l, p, s}{l', p', -s}}\Big] \\
&+ \pb{s} C_{l'm\omega p'} \pb{-s} \Theta_{lm\omega} \pb{-s} \Theta_{l'm\omega} \Big[\pb{-s} \widehat{\psi}^{\textrm{up}}_{lm\omega p} \overline{\pb{s} \widehat{\psi}^{\textrm{up}}_{l'm\omega p'}} + \overline{\interchange{l, p, s}{l', p', -s}}\Big]\bigg\},
\end{aligned} \\
&\begin{aligned}
\left.\frac{\ud^2 \pb{\pb{s} \Omega} Q_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{\scri^-} = -\frac{\omega}{32\pi} \bigg\{&C_{l'm\omega} \pb{s} \Theta_{lm\omega} \pb{s} \Theta_{l'm\omega} \Big[\pb{s} \widehat{\psi}^{\textrm{down}}_{lm\omega p} \overline{\pb{-s} \widehat{\psi}^{\textrm{down}}_{l'm\omega p'}} + \overline{\interchange{l, p, s}{l', p', -s}}\Big] \\
&+ \pb{s} C_{l'm\omega p'} \pb{-s} \Theta_{lm\omega} \pb{-s} \Theta_{l'm\omega} \Big[\pb{-s} \widehat{\psi}^{\textrm{down}}_{lm\omega p} \overline{\pb{s} \widehat{\psi}^{\textrm{down}}_{l'm\omega p'}} + \overline{\interchange{l, p, s}{l', p', -s}}\Big]\bigg\},
\end{aligned}
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
&\begin{aligned}
\left.\frac{\ud^2 \pb{\pb{s} \Omega} Q_{ll'm\omega pp'}}{\ud v \ud \Omega}\right|_{H^+} = -\frac{Mr_+ k_{m\omega}}{16\pi} \bigg\{&C_{l'm\omega} \pb{s} \Theta_{lm\omega} \pb{s} \Theta_{l'm\omega} \Big[\pb{s} \kappa_{m\omega} \pb{s} \widehat{\psi}^{\textrm{in}}_{lm\omega p} \overline{\pb{-s} \widehat{\psi}^{\textrm{in}}_{l'm\omega p'}} \\
&\hspace{10.5em}+ \overline{\interchange{l, p, s}{l', p', -s}}\Big] \\
&+ \pb{s} C_{l'm\omega p'} \pb{-s} \Theta_{lm\omega} \pb{-s} \Theta_{l'm\omega} \Big[\pb{-s} \kappa_{m\omega} \pb{-s} \widehat{\psi}^{\textrm{in}}_{lm\omega p} \overline{\pb{s} \widehat{\psi}^{\textrm{in}}_{l'm\omega p'}} \\
&\hspace{12.5em}+ \overline{\interchange{l, p, s}{l', p', -s}}\Big]\bigg\},
\end{aligned} \\
&\begin{aligned}
\left.\frac{\ud^2 \pb{\pb{s} \Omega} Q_{ll'm\omega pp'}}{\ud u \ud \Omega}\right|_{H^-} = \frac{Mr_+ k_{m\omega}}{16\pi} \bigg\{&C_{l'm\omega} \pb{s} \Theta_{lm\omega} \pb{s} \Theta_{l'm\omega} \Big[\pb{s} \kappa_{m\omega} \pb{s} \widehat{\psi}^{\textrm{out}}_{lm\omega p} \overline{\pb{-s} \widehat{\psi}^{\textrm{out}}_{l'm\omega p'}} \\
&\hspace{10.5em}+ \overline{\interchange{l, p, s}{l', p', -s}}\Big] \\
&+ \pb{s} C_{l'm\omega p'} \pb{-s} \Theta_{lm\omega} \pb{-s} \Theta_{l'm\omega} \Big[\pb{-s} \kappa_{m\omega} \pb{-s} \widehat{\psi}^{\textrm{out}}_{lm\omega p} \overline{\pb{s} \widehat{\psi}^{\textrm{out}}_{l'm\omega p'}} \\
&\hspace{12.5em}+ \overline{\interchange{l, p, s}{l', p', -s}}\Big]\bigg\},
\end{aligned}
\end{align}
\end{subequations}
where
\begin{equation}
\pb{s} \kappa_{m\omega} = 1 - \frac{is(r_+ - M)}{2Mr_+ k_{m\omega}}.
\end{equation}
\section{Discussion} \label{section:discussion}
\begin{table}[t!]
\centering
\caption{\label{table:summary} Summary of the properties of the conserved currents considered in this paper.
For convenience, we give the equation numbers (within section~\ref{section:definitions}) in which these currents are defined.
We then give the limit of the corresponding charges in geometric optics, where $K$ is the Carter constant of a graviton (see section~\ref{section:geometric_optics} for the definitions of the polarization coefficients $e_R$ and $e_L$, as well as the justification of the factors of $\hbar$).
The next column indicates whether the fluxes of these currents through future and past null infinity ($\scri^\pm$) and the future and past horizons ($H^\pm$) are finite.
We finally indicate which of these currents are local functionals of the metric perturbation.}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{|*{8}{c|}} \hline
& Definition & Geometric optics limit & \multicolumn{4}{c|}{Finite fluxes?} & \\
Current & (equation) & of charge (per graviton) & $\scri^+$ & $\scri^-$ & $H^+$ & $H^-$ & Local? \\\hline
$\pb{\pb{2} \mathcal{C}} j^a [\var \bs g]$ & \multirow{2}{*}{\eqref{eqn:C_current}} & \multirow{2}{*}{$K^4 (|e_R|^2 - |e_L|^2)/\hbar^7$} & $\times$ & \checkmark & \checkmark & \checkmark & \checkmark \\
$\pb{\pb{-2} \mathcal{C}} j^a [\var \bs g]$ & & & \checkmark & $\times$ & \checkmark & \checkmark & \checkmark \\\hline
$\pb{\pb{2} \mathring{\mathcal C}} j^a [\var \bs g]$ & \eqref{eqn:C_current_proj} & $K^4 (|e_R|^2 - |e_L|^2)/\hbar^7\;$~\footnote{\label{foot:tab} \!\!This result only holds, if the null fluid of gravitons is either completely ingoing or outgoing at null infinity; see the discussion near the end of section~\ref{section:geometric_optics_examples} for more details.} & \checkmark & \checkmark & \checkmark & \checkmark & $\times$ \\\hline
$\pb{\pb{2} \mathcal{D}} j^a [\var \bs g]$ & \multirow{2}{*}{\eqref{eqn:D_current}} & \multirow{2}{*}{$K (|e_R|^2 - |e_L|^2)/\hbar$} & $\times$ & \checkmark & \checkmark & \checkmark & $\times$ \\
$\pb{\pb{-2} \mathcal{D}} j^a [\var \bs g]$ & & & \checkmark & $\times$ & \checkmark & \checkmark & $\times$ \\\hline
$\pb{\pb{2} \mathring{\mathcal D}} j^a [\var \bs g]$ & \eqref{eqn:D_current_proj} & $K (|e_R|^2 - |e_L|^2)/\hbar\;$~\textsuperscript{\ref{foot:tab}} & \checkmark & \checkmark & \checkmark & \checkmark & $\times$\\\hline
\begin{tabular}{c}
$\pb{\pb{2} \Omega} j^a [\var \bs g]$ \\
$\pb{\pb{-2} \Omega} j^a [\var \bs g]$ \\
\end{tabular} & \eqref{eqn:Omega_current} & $K^4 (|e_R|^2 - |e_L|^2)/\hbar^7$ & \checkmark & \checkmark & \checkmark & \checkmark & \checkmark \\\hline
\end{tabular}
\end{table}
In this paper, we have constructed a class of conserved currents for linearized gravity whose conserved charges reduce to the sum of the Carter constants (to some positive power) for a null fluid of gravitons in the geometric optics limit.
These conserved currents are constructed from symplectic products of two solutions constructed via the method of symmetry operators.
Moreover, some of these currents yield finite fluxes at the horizon and null infinity, although most that are finite at null infinity are not local.
A full summary of their properties is given in table~\ref{table:summary}.
Note that only the currents $\pb{\pb{s} \Omega} j^a$ are both local and possess finite fluxes.
That some of these currents possess diverging fluxes at null infinity is not ideal.
It may be possible to find a symmetry operator, differing from those that appear in this paper by a gauge transformation, that is both local and maps to a solution with a non-divergent symplectic product.
In the absence of a clear example of such a symmetry operator, we have instead decided to consider nonlocal symmetry operators which are easier to define.
We have also shown that there exists a symplectic product for the master variables (instead of the metric perturbation) which yields finite fluxes.
This symplectic product can also be used to construct a current which gives (positive powers of) the Carter constant in the limit of geometric optics.
However, note that this is not the physical symplectic product for linearized gravity.
One motivation for seeking conserved currents is the hope to derive, for the dynamical system of a point particle coupled to linearized gravity in the Kerr spacetime, a ``unified conservation law'' that would generalize the conservation of the Carter constant for a point particle by itself.
The local currents considered in this paper could be relevant for such a conservation law, but the potential relevance of the nonlocal currents is less obvious.
We plan to further explore these currents, particularly their applications, in future work.
\section*{Acknowledgments}
We thank Lars Andersson and Kartik Prabhu for helpful conversations.
We acknowledge the support of NSF Grants PHY-1404105 and PHY-1707800 to Cornell University.
\appendix
\section{Integration along the horizon and null infinity} \label{appendix:fluxes_integration}
The flux of a current $\pb{\ldots} j^a$ through a surface $S$ of constant $r$ (such as the horizon or null infinity) is defined by
\begin{equation} \label{eqn:differential_fluxes}
\left.\frac{\ud^2 \pb{\ldots} Q}{\ud w \ud \Omega}\right|_S \equiv \lim_{\to S} (r^2 + a^2) \pb{\ldots} j^a N_a,
\end{equation}
where $N_a$ is the surface normal, and the factor of $r^2 + a^2$ comes from the fact that the determinant of the induced metric on surfaces of constant $r$ is $(r^2 + a^2) \sin \theta$.
The surface normals are proportional to $(\ud r)_a$,
\begin{equation}
N_a \propto (\ud r)_a = n_a - \frac{\Delta}{2\Sigma} l_a,
\end{equation}
and the usual scaling freedom is fixed by requiring\footnote{Note that, if one were integrating these currents on a finite portion of these surfaces, the normalization of $N_a$ would not matter.
However, for equation~\eqref{eqn:differential_fluxes} to hold---that is, when integrating over an infinitesimal portion $\ud w$, for $w = u$ or $v$, we must normalize $N_a$ appropriately.} that either $N^a \nabla_a u = 1$ (for $H^-$ and $\scri^+$) or $N^a \nabla_a v = 1$ (for $H^+$ and $\scri^-$).
It turns out, however, that these requirements are the same, and fix the normalization such that
\begin{equation}
N_a = \frac{1}{r^2 + a^2} \left(\Sigma n_a - \frac{\Delta}{2} l_a\right).
\end{equation}
As such, we find that
\begin{subequations}
\begin{align}
\left.\frac{\ud^2 Q}{\ud v \ud \Omega}\right|_{H^+} &= \lim_{r \to r_+, v \textrm{ fixed}} \Sigma \left(j_n - \frac{\Delta}{2\Sigma} j_l\right), \\
\left.\frac{\ud^2 Q}{\ud u \ud \Omega}\right|_{H^-} &= \lim_{r \to r_+, u \textrm{ fixed}} \Sigma \left(j_n - \frac{\Delta}{2\Sigma} j_l\right), \\
\left.\frac{\ud^2 Q}{\ud v \ud \Omega}\right|_{\scri^-} &= \lim_{r \to \infty, v \textrm{ fixed}} r^2 \left(j_n - \frac{1}{2} j_l\right), \\
\left.\frac{\ud^2 Q}{\ud u \ud \Omega}\right|_{\scri^+} &= \lim_{r \to \infty, v \textrm{ fixed}} r^2 \left(j_n - \frac{1}{2} j_l\right).
\end{align}
\end{subequations}
From this discussion, for the calculations in section~\ref{section:fluxes}, we need the components of symplectic products along $l_a$ and $n_a$:
\begin{subequations} \label{eqn:symplectic_components}
\begin{align}
\pb{S} j^{\textrm{EH}}_l \left[\var_+ \boldsymbol{g}, \overline{\var_+ \boldsymbol{g}}\right] &= -\frac{1}{16\pi} \Im\left[(\var_+ C)_{l\bar{m}\bar{m}} \overline{(\var_+ g)^{\bar{m}\bar{m}}}\right], \label{eqn:symplectic_flux_l} \\
\pb{S} j^{\textrm{EH}}_n \left[\var_+ \boldsymbol{g}, \overline{\var_+ \boldsymbol{g}}\right] &= -\frac{1}{16\pi} \Im\left[(\var_+ C)_{n\bar{m}\bar{m}} \overline{(\var_+ g)^{\bar{m}\bar{m}}} - (\var_+ C)_{n(l\bar{m})} \overline{(\var_+ g)^{(n\bar{m})}}\right], \label{eqn:symplectic_flux_n}
\end{align}
\end{subequations}
where $l$, $n$, $m$, and $\bar{m}$ subscripts denote contraction on an index with the corresponding null tetrad vector, and where the non-zero perturbed connection coefficients are
\begin{subequations}
\begin{align}
(\var_+ C)_{l\bar{m}\bar{m}} &= -\frac{1}{2} [D + 2(\epsilon - \bar{\epsilon}) - \rho] (\var_+ g)_{\bar{m}\bar{m}}, \\
(\var_+ C)_{n(l\bar{m})} &= -\frac{1}{4} (D + 2\epsilon + \rho) (\var_+ g)_{(n\bar{m})} - \frac{1}{2} \tau (\var_+ g)_{\bar{m}\bar{m}}, \\
(\var_+ C)_{n\bar{m}\bar{m}} &= -\frac{1}{4} (\npdelta + 2\bar{\alpha}) (\var_+ g)_{(n\bar{m})} - \frac{1}{2} [\npDelta + 2(\gamma - \bar{\gamma}) - 2\mu] (\var_+ g)_{\bar{m}\bar{m}}.
\end{align}
\end{subequations}
One can obtain the analogous expressions for $\var_-$ by performing a $'$ transformation.
For the symplectic product defined using the master variables, we find that
\begin{subequations}
\begin{align}
\pb{S} j^{\textrm{BCJR}}_l \left[\var_1 \pb{s} \Omega, \var_1 \pb{-s} \Omega; \var_2 \pb{s} \Omega, \var_2 \pb{-s} \Omega\right] &= \var_1 \pb{s} \Omega (D - s\Gamma_l) \var_2 \pb{-s} \Omega + \var_1 \pb{-s} \Omega (D + s\Gamma_l) \var_2 \pb{s} \Omega - \interchange{1}{2}, \\
\pb{S} j^{\textrm{BCJR}}_n \left[\var_1 \pb{s} \Omega, \var_1 \pb{-s} \Omega; \var_2 \pb{s} \Omega, \var_2 \pb{-s} \Omega\right] &= \var_1 \pb{s} \Omega (\npDelta - s\Gamma_n) \var_2 \pb{-s} \Omega + \var_1 \pb{-s} \Omega (\npDelta + s\Gamma_n) \var_2 \pb{s} \Omega - \interchange{1}{2}.
\end{align}
\end{subequations}
\section{Asymptotic behavior} \label{appendix:asymptotics}
In order to determine fluxes at null infinity and the horizon, we also need to know the asymptotic behavior of the quantities that appear in equation~\eqref{eqn:symplectic_components} and its $'$ transform.
These are given in table~\ref{table:gr_asymp}.
To determine these falloff rates, we write the quantities that appear in~\eqref{eqn:symplectic_components} and its $'$ transform in terms of differential operators acting upon the Debye potential, using the operators defined in equation~\eqref{eqn:D_L_operators}: the perturbed metric satisfies
\begin{subequations} \label{eqn:var_g_exp}
\begin{align}
(\var_+ g)_{(n\bar{m})} &= -\frac{1}{\sqrt{2} \bar{\zeta}} \left[\left(\mathscr{D}_0 + \frac{1}{\zeta} - \frac{2}{\bar{\zeta}}\right) \left(\mathscr{L}_2^+ - \frac{3ia \sin \theta}{\zeta}\right) + \left(\mathscr{L}_2^+ + \frac{ia \sin \theta}{\zeta} + \frac{2ia \sin \theta}{\bar{\zeta}}\right) \left(\mathscr{D}_0 - \frac{3}{\zeta}\right)\right] \pb{-2} \psi, \\
(\var_+ g)_{\bar{m}\bar{m}} &= -\left(\mathscr{D}_0 + \frac{1}{\zeta}\right) \left(\mathscr{D}_0 - \frac{3}{\zeta}\right) \pb{-2} \psi, \\
(\var_- g)_{(lm)} &= \frac{\zeta^2}{2 \sqrt{2} \bar{\zeta} \Delta} \left[\left(\mathscr{L}_2 + \frac{ia \sin \theta}{\zeta} + \frac{2ia \sin \theta}{\bar{\zeta}}\right) \left(\mathscr{D}_0^+ - \frac{3}{\zeta}\right) + \left(\mathscr{D}_0^+ + \frac{1}{\zeta} - \frac{2}{\bar{\zeta}}\right) \left(\mathscr{L}_2 - \frac{3ia \sin \theta}{\zeta}\right)\right] \Delta^2 \pb{2} \psi, \\
(\var_- g)_{mm} &= \frac{\zeta^2}{4\bar{\zeta}^2} \left(\mathscr{D}_0^+ + \frac{1}{\zeta}\right) \left(\mathscr{D}_0^+ - \frac{3}{\zeta}\right) \Delta^2 \pb{2} \psi,
\end{align}
\end{subequations}
whereas the relevant perturbed connection coefficients are given by
\begin{subequations} \label{eqn:var_gamma_exp}
\begin{align}
(\var_+ C)_{l\bar{m}\bar{m}} &= -\frac{1}{2} \left(\mathscr{D}_0 + \frac{1}{\zeta}\right) (\var_+ g)_{\bar{m}\bar{m}}, \\
(\var_+ C)_{n(l\bar{m})} &= -\frac{1}{4} \left(\mathscr{D}_0 - \frac{1}{\zeta}\right) (\var_+ g)_{(n\bar{m})} + \frac{ia \sin \theta}{2\sqrt{2} \Sigma} (\delta_+ g)_{\bar{m}\bar{m}}, \\
(\var_+ C)_{n\bar{m}\bar{m}} &= -\frac{1}{4\sqrt{2} \bar{\zeta}} \left(\mathscr{L}_{-1}^+ - \frac{2ia \sin \theta}{\bar{\zeta}}\right) (\var_+ g)_{(n\bar{m})} + \frac{\Delta}{4 \Sigma} \left(\mathscr{D}_0^+ - \frac{2}{\zeta} - \frac{2}{\bar{\zeta}}\right) (\var_+ g)_{\bar{m}\bar{m}}, \\
(\var_- C)_{nmm} &= \frac{\Delta}{4 \Sigma} \left(\mathscr{D}_0^+ - \frac{1}{\zeta} + \frac{2}{\bar{\zeta}}\right) (\var_- g)_{mm}, \\
(\var_- C)_{l(nm)} &= \frac{1}{8 \Sigma} \left(\mathscr{D}_0^+ - \frac{3}{\zeta}\right) \Delta (\var_- g)_{(lm)} + \frac{ia \sin \theta}{2\sqrt{2} \zeta^2} (\var_- g)_{mm}, \\
(\var_- C)_{lmm} &= -\frac{1}{4\sqrt{2} \zeta} \mathscr{L}_{-1} (\var_- g)_{(lm)} - \frac{1}{2} \left(\mathscr{D}_0 - \frac{2}{\zeta}\right) (\var_- g)_{mm}.
\end{align}
\end{subequations}
In order to compute the asymptotic behavior of these quantities, one needs to determine the asymptotic behavior of derivatives of the master variables.
However, applying the na\"ive approach, which uses the asymptotic expansions given by equations~\eqref{eqn:radial_falloffs_in_out} and~\eqref{eqn:radial_falloffs_down_up}, along with
\begin{equation} \label{eqn:naive}
\begin{split}
&\left.\begin{aligned}
\mathscr{D}_{0(\pm m)(\pm \omega)} f(r) e^{\pm i\omega r^*} &= \frac{\ud f}{\ud r} e^{\pm i\omega r^*} \\
\mathscr{D}_{0(\pm m)(\pm \omega)} f(r) e^{\mp i\omega r^*} &= \left[\frac{\ud f}{\ud r} \mp 2i\omega f(r)\right] e^{\mp i\omega r^*} \\
\end{aligned}\right\} r^* \to \infty, \\
&\left.\begin{aligned}
\mathscr{D}_{0(\pm m)(\pm \omega)} f(r) e^{\pm ik_{m\omega} r^*} &= \frac{\ud f}{\ud r} e^{\pm i k_{m\omega} r^*} \\
\mathscr{D}_{0(\pm m)(\pm \omega)} f(r) e^{\mp ik_{m\omega} r^*} &= \left[\frac{\ud f}{\ud r} \mp \frac{4Mr^+}{\Delta} ik_{m\omega} f(r)\right] e^{\mp ik_{m\omega} r^*} \\
\end{aligned}\right\} r^* \to -\infty,
\end{split}
\end{equation}
results in cancellations in the leading-order behavior.
Instead, we use the radial Teukolsky-Starobinsky identity~\eqref{eqn:radial_starobinsky}, which provides a differential equation that is independent of the radial Teukolsky equation~\eqref{eqn:radial_teukolsky}.
Using the radial Teukolsky equation, one can reduce the radial Teukolsky-Starobinsky identity to the following expression for derivatives of $\pb{s} \widehat{\Omega}_{lm\omega p} (r)$~\cite{chandrasekhar1983mathematical}:
\begin{equation} \label{eqn:deriv_relation}
\mathscr{D}_{0(\mp m)(\mp \omega)} \Delta^{(2 \pm 2)/2} \pb{\pm 2} \widehat{\Omega}_{lm\omega p} \equiv \pb{\pm 2} \Xi_{lm\omega p} \Delta^{(2 \pm 2)/2} \pb{\pm 2} \widehat{\Omega}_{lm\omega p} + \pb{\pm 2} \Pi_{lm\omega p} \Delta^{(2 \mp 2)/2} \pb{\mp 2} \widehat{\Omega}_{lm\omega p},
\end{equation}
where this equation defines the coefficients $\pb{\pm s} \Xi_{lm\omega p}$ and $\pb{\pm s} \Pi_{lm\omega p}$.
These equations also clearly hold for $\pb{s} \widehat{\psi}_{lm\omega p} (r)$.
\begin{table}[!t]
\centering
\caption{\label{table:gr_asymp} Asymptotic behavior of the solutions for linearized gravity.}
\begin{tabular}{|l|c c|c c|} \hline
& \multicolumn{2}{c|}{Ingoing [$e^{i(m\psi - \omega v)} \times$]} & \multicolumn{2}{c|}{Outgoing [$e^{i(m\chi - \omega u)} \times$]} \\
& $r \to r_+$ & $r \to \infty$ & $r \to r_+$ & $r \to \infty$ \\\hline
$(\var_+ g_{lm\omega p})_{n\bar{m}}$ & $\Delta$ & $1/r^2$ & $1$ & $r$ \\
$(\var_+ g_{lm\omega p})_{\bar{m}\bar{m}}$ & $1$ & $1/r$ & $1$ & $1$ \\
$(\var_- g_{lm\omega p})_{lm}$ & $1/\Delta$ & $r$ & $1$ & $1/r^2$ \\
$(\var_- g_{lm\omega p})_{mm}$ & $1$ & $1$ & $1$ & $1/r$ \\
$(\var_+ C_{lm\omega p})_{l\bar{m}\bar{m}}$ & $1/\Delta$ & $1/r$ & $1$ & $1/r$ \\
$(\var_+ C_{lm\omega p})_{n(l\bar{m})}$ & $1$ & $1/r^2$ & $1$ & $1/r^2$ \\
$(\var_+ C_{lm\omega p})_{n\bar{m}\bar{m}}$ & $\Delta$ & $1/r^2$ & $1$ & $1$ \\
$(\var_- C_{lm\omega p})_{nmm}$ & $\Delta$ & $1/r$ & $1$ & $1/r$ \\
$(\var_- C_{lm\omega p})_{l(nm)}$ & $1$ & $1/r^2$ & $1$ & $1/r^2$ \\
$(\var_- C_{lm\omega p})_{lmm}$ & $1/\Delta$ & $1$ & $1$ & $1/r^2$ \\\hline
\end{tabular}
\end{table}
Plugging equation~\eqref{eqn:deriv_relation} [for $\pb{s} \psi_{lm\omega p} (r)$] into equations~\eqref{eqn:var_g_exp} and~\eqref{eqn:var_gamma_exp}, and then taking the limits $r \to \infty$ and $r \to r_+$, yields the asymptotic forms given in table~\ref{table:gr_asymp}.
Using this same calculation, we can determine the angular dependences of the quantities in~\eqref{eqn:var_g_exp} and~\eqref{eqn:var_gamma_exp}.
Defining, for $s \geq 0$,
\begin{equation}
\pb{\pm s} \eta^+_{lm\omega} = \pm 2i(2s - 1) \omega r_+ - \pb{2} \lambda_{lm\omega}, \qquad \pb{\pm s} \eta^\infty_{lm\omega} = \pm 2(2s - 1) \omega a \cos \theta + \pb{2} \lambda_{lm\omega},
\end{equation}
they are given by
\begin{subequations} \label{eqn:perturbed_metric_asymp}
\begin{align}
&\pb{\var_+ g_{n\bar{m}}} S_{lm\omega p}^{\textrm{in}} = \frac{4ik_{m\omega} \sqrt{Mr_+} \pb{-2} \kappa_{m\omega}}{\zeta_+} \mathscr{L}_{2(-m)(-\omega)} \pb{-2} \Theta_{lm\omega}, \\
&\mathrlap{\pb{\var_+ g_{n\bar{m}}} S_{lm\omega p}^{\textrm{out}} = \frac{\pb{-2} \eta^+_{lm\omega} \zeta_+ + 8 Mr_+ ik_{m\omega} \pb{-1} \kappa_{m\omega}}{4 (Mr_+)^{3/2} ik_{m\omega} \pb{-1} \kappa_{m\omega} \zeta_+^2} \mathscr{L}_{2(-m)(-\omega)} \pb{-2} \Theta_{lm\omega},} \\
&\pb{\var_+ g_{n\bar{m}}} S_{lm\omega p}^{\textrm{down}} = 2\sqrt{2} i\omega \mathscr{L}_{2(-m)(-\omega)} \pb{-2} \Theta_{lm\omega}, &\pb{\var_+ g_{n\bar{m}}} S_{lm\omega p}^{\textrm{up}} = -\sqrt{2} \mathscr{L}_{2(-m)(-\omega)} \pb{-2} \Theta_{lm\omega}, \\
&\pb{\var_+ g_{\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{in}} = 4 (2Mr_+)^{3/2} k_{m\omega}^2 \pb{-2} \kappa_{m\omega} \pb{-1} \kappa_{m\omega} \pb{-2} \Theta_{lm\omega}, \\
&\mathrlap{\pb{\var_+ g_{\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{out}} = -\frac{24 Mr_+ i\omega k_{m\omega} \pb{-1} \kappa_{m\omega} \zeta_+ + [i\zeta_+ (2 - \pb{-1} \eta^+_{lm\omega}) + 8Mr_+ k_{m\omega}] \pb{-2} \eta^+_{lm\omega}}{4 ik_{m\omega}^2 (2Mr_+)^{5/2} \pb{-1} \kappa_{m\omega} \zeta_+} \pb{-2} \Theta_{lm\omega},} \\
&\pb{\var_+ g_{\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{down}} = 4 \omega^2 \pb{-2} \Theta_{lm\omega}, &\pb{\var_+ g_{\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{up}} = \frac{i \pb{2} \eta^\infty_{lm\omega}}{\omega} \pb{-2} \Theta_{lm\omega}, \\
&\pb{\var_- g_{lm}} S_{lm\omega p}^{\textrm{in}} = \frac{\pb{2} \eta^+_{lm\omega} \zeta_+ - 8Mr_+ ik_{m\omega} \pb{1} \kappa_{m\omega}}{8 (Mr_+)^{3/2} ik_{m\omega} \pb{1} \kappa_{m\omega}} \mathscr{L}_{2m\omega} \pb{2} \Theta_{lm\omega}, \\
&\pb{\var_- g_{lm}} S_{lm\omega p}^{\textrm{out}} = 2 \sqrt{Mr_+} ik_{m\omega} \pb{2} \kappa_{m\omega} \zeta_+ \mathscr{L}_{2m\omega} \pb{2} \Theta_{lm\omega}, \\
&\pb{\var_- g_{lm}} S_{lm\omega p}^{\textrm{down}} = \frac{\mathscr{L}_{2m\omega}}{\sqrt{2}} \pb{2} \Theta_{lm\omega}, &\pb{\var_- g_{lm}} S_{lm\omega p}^{\textrm{up}} = \sqrt{2} i \omega \mathscr{L}_{2m\omega} \pb{2} \Theta_{lm\omega}, \\
&\mathrlap{\pb{\var_- g_{mm}} S_{lm\omega p}^{\textrm{in}} = \frac{24 Mr_+ i\omega k_{m\omega} \pb{1} \kappa_{m\omega} \zeta_+ + [i\zeta_+ (2 - \pb{1} \eta^+_{lm\omega}) - 8Mr_+ k_{m\omega}] \pb{2} \eta^+_{lm\omega}}{16 ik_{m\omega}^2 (2Mr_+)^{5/2} \pb{1} \kappa_{m\omega} \zeta_+} \pb{2} \Theta_{lm\omega},} \\
&\pb{\var_- g_{mm}} S_{lm\omega p}^{\textrm{out}} = -(2Mr_+)^{3/2} k_{m\omega}^2 \pb{2} \kappa_{m\omega} \pb{1} \kappa_{m\omega} \pb{2} \Theta_{lm\omega}, \\
&\pb{\var_- g_{mm}} S_{lm\omega p}^{\textrm{down}} = \frac{i \pb{-2} \eta^\infty_{lm\omega}}{4 \omega} \pb{2} \Theta_{lm\omega}, &\pb{\var_- g_{mm}} S_{lm\omega p}^{\textrm{up}} = -\omega^2 \pb{2} \Theta_{lm\omega},
\end{align}
\end{subequations}
and
\begin{subequations} \label{eqn:perturbed_chris_asymp}
\begin{align}
&\mathrlap{\pb{\var_+ C_{l\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{in}} = 4(2Mr_+)^{5/2} ik_{m\omega}^3 \pb{-2} \kappa_{m\omega} \pb{-1} \kappa_{m\omega} \pb{-2} \Theta_{lm\omega},} \\
&\mathrlap{\begin{aligned}
\pb{\var_+ C_{l\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{out}} &= \Big\{4Mr_+ ik_{m\omega} \pb{1} \kappa_{m\omega} [24 Mr_+ i\omega k_{m\omega} \pb{-1} \kappa_{m\omega} + i(2 - \pb{-1} \eta^+_{lm\omega}) \pb{-2} \eta^+_{lm\omega}] \\
&\hspace{2em}- \zeta_+ \{8Mr_+ i\omega k_{m\omega} [3 \pb{-1} \kappa_{m\omega} (2 - \pb{1} \eta^+_{lm\omega}) - 4 \pb{-2} \eta^+_{lm\omega}] \\
&\hspace{5em}+ i \pb{-2} \eta^+_{lm\omega} [|\pb{-1} \eta^+_{lm\omega}|^2 + 4(\pb{2} \lambda_{lm\omega} + 1)]\}\Big\} \frac{\pb{-2} \Theta_{lm\omega}}{16 k_{m\omega}^3 (2Mr_+)^{7/2} |\pb{-1} \kappa_{m\omega}|^2 \zeta_+},
\end{aligned}} \\
&\pb{\var_+ C_{l\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{down}} = 4 i \omega^3 \pb{-2} \Theta_{lm\omega}, &\pb{\var_+ C_{l\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{up}} = -\frac{i \pb{2} \eta^\infty_{lm\omega}}{2 \omega} \pb{-2} \Theta_{lm\omega}, \\
&\mathrlap{\pb{\var_+ C_{n(l\bar{m})}} S_{lm\omega p}^{\textrm{in}} = -4 (Mr_+)^{3/2} k_{m\omega}^2 \pb{-2} \kappa_{m\omega} \pb{-1} \kappa_{m\omega} \zeta_+^{-2} (\zeta_+ \mathscr{L}_{2(-m)(-\omega)} - ia \sin \theta) \pb{-2} \Theta_{lm\omega},} \\
&\mathrlap{\begin{aligned}
\pb{\var_+ C_{n(l\bar{m})}} S_{lm\omega p}^{\textrm{out}} &= \zeta_+^{-3} \Big\{[24Mr_+ i\omega k_{m\omega} \pb{-1} \kappa_{m\omega} + i(2 - \pb{-1} \eta^+_{lm\omega}) \pb{-2} \eta^+_{lm\omega}] \zeta_+ (\zeta_+ \mathscr{L}_{2(-m)(-\omega)} - ia \sin \theta) \\
&\hspace{4em}+ 8Mr_+ k_{m\omega} \pb{-2} \eta^+_{lm\omega} (2\zeta_+ \mathscr{L}_{2(-m)(-\omega)} - ia \sin \theta) \\
&\hspace{4em}+ 6 (4Mr_+)^2 ik_{m\omega}^2 \pb{-1} \kappa_{m\omega} \mathscr{L}_{2(-m)(-\omega)}\Big\} \frac{\pb{-2} \Theta_{lm\omega}}{64 (Mr_+)^{5/2} ik_{m\omega}^2 \pb{-1} \kappa_{m\omega}},
\end{aligned}} \\
&\mathrlap{\pb{\var_+ C_{n(l\bar{m})}} S_{lm\omega p}^{\textrm{down}} = -\sqrt{2} \omega^2 \mathscr{L}_{2(-m)(-\omega)} \pb{-2} \Theta_{lm\omega},} \\
&\mathrlap{\begin{aligned}
\pb{\var_+ C_{n(l\bar{m})}} S_{lm\omega p}^{\textrm{up}} &= -\Big\{[4 \omega^2 a^2 (2 \cos^2 \theta - 3) - 12 i \omega (M + iam) + \pb{2} \lambda_{lm\omega} (\pb{2} \lambda_{lm\omega} + 2)] \mathscr{L}_{2(-m)(-\omega)} \\
&\hspace{3em}+ 4a\omega \sin \theta (12 a\omega \cos \theta + \pb{2} \eta^\infty_{lm\omega})\Big\} \frac{\pb{-2} \Theta_{lm\omega}}{8\sqrt{2} \omega^2},
\end{aligned}} \\
&\mathrlap{\begin{aligned}
\pb{\var_+ C_{n\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{in}} &= \sqrt{Mr_+} ik_{m\omega} \pb{-2} \kappa_{m\omega} \zeta_+^{-4} \Big\{\zeta_+^2 [(2 - \pb{-1} \eta^+_{lm\omega}) - \mathscr{L}_{(-1)(-m)(-\omega)} \mathscr{L}_{2(-m)(-\omega)}] \\
&\hspace{12em}+ 16Mr_+ ik_{m\omega} \pb{-3/2} \kappa_{m\omega} \zeta_+ + 2a^2 \sin^2 \theta \\
&\hspace{12em}+ ia \sin \theta \zeta_+ \mathscr{L}_{2(-m)(-\omega)}\Big\} \frac{\pb{-2} \Theta_{lm\omega}}{\sqrt{2}},
\end{aligned}} \\
&\mathrlap{\begin{aligned}
\pb{\var_+ C_{n\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{out}} &= \zeta_+^{-4} \Big\{\pb{-2} \eta^+_{lm\omega} [\zeta_+ (ia \sin \theta - \zeta_+ \mathscr{L}_{(-1)(-m)(-\omega)}) \mathscr{L}_{2(-m)(-\omega)} - 8Mr_+ ik_{m\omega} \zeta_+ + 2a^2 \sin^2 \theta] \\
&\hspace{3.5em}- 8Mr_+ ik_{m\omega} \pb{-1} \kappa_{m\omega} (\zeta_+ \mathscr{L}_{(-1)(-m)(-\omega)} - ia \sin \theta) \mathscr{L}_{2(-m)(-\omega)} \\
&\hspace{3.5em}+ [24Mr_+ \omega k_{m\omega} \pb{-1} \kappa_{m\omega} + (2 - \pb{-1} \eta^+_{lm\omega}) \pb{-2} \eta^+_{lm\omega}] \zeta_+^2\Big\} \frac{\pb{-2} \Theta_{lm\omega}}{(8Mr_+)^{3/2} ik_{m\omega} \pb{-1} \kappa_{m\omega}}, \\
\end{aligned}} \\
&\pb{\var_+ C_{n\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{down}} = -5 \omega^2 \pb{-2} \Theta_{lm\omega}, &\pb{\var_+ C_{n\bar{m}\bar{m}}} S_{lm\omega p}^{\textrm{up}} = -\frac{2 \pb{2} \eta^\infty_{lm\omega} - \mathscr{L}_{(-1)(-m)(-\omega)} \mathscr{L}_{2(-m)(-\omega)}}{4} \pb{-2} \Theta_{lm\omega}, \\
&\mathrlap{\begin{aligned}
\pb{\var_- C_{nmm}} S_{lm\omega p}^{\textrm{in}} &= \Big\{4Mr_+ ik_{m\omega} \pb{-1} \kappa_{m\omega} [24Mr_+ i\omega k_{m\omega} \pb{1} \kappa_{m\omega} + i(2 - \pb{1} \eta^+_{lm\omega}) \pb{2} \eta^+_{lm\omega}] \\
&\hspace{2em}+ \zeta_+ \{8Mr_+ i\omega k_{m\omega} [3 \pb{1} \kappa_{m\omega} (2 - \pb{-1} \eta^+_{lm\omega}) - 4 \pb{2} \eta^+_{lm\omega}] \\
&\hspace{5em}+ i \pb{2} \eta^+_{lm\omega} [|\pb{1} \eta^+_{lm\omega}|^2 + 4(\pb{2} \lambda_{lm\omega} + 1)]\}\Big\} \frac{\pb{2} \Theta_{lm\omega}}{k_{m\omega}^3 (8Mr_+)^{7/2} |\pb{1} \kappa_{m\omega}|^2 \zeta_+^3},
\end{aligned}} \\
&\mathrlap{\pb{\var_- C_{nmm}} S_{lm\omega p}^{\textrm{out}} = -\frac{(2Mr_+)^{5/2} ik_{m\omega}^3 \pb{2} \kappa_{m\omega} \pb{1} \kappa_{m\omega}}{2\zeta_+^2} \pb{2} \Theta_{lm\omega},} \\
&\pb{\var_- C_{nmm}} S_{lm\omega p}^{\textrm{down}} = \frac{i \pb{-2} \eta^\infty_{lm\omega}}{16 \omega} \pb{2} \Theta_{lm\omega}, &\pb{\var_- C_{nmm}} S_{lm\omega p}^{\textrm{up}} = -\frac{i \omega^3}{2} \pb{2} \Theta_{lm\omega}, \\
&\mathrlap{\begin{aligned}
\pb{\var_- C_{l(nm)}} S_{lm\omega p}^{\textrm{in}} &= \zeta_+^{-3} \Big\{[24Mr_+ i\omega k_{m\omega} \pb{1} \kappa_{m\omega} + i \pb{2} \eta^+_{lm\omega} (2 - \pb{1} \eta^+_{lm\omega})] \zeta_+ (\zeta_+ \mathscr{L}_{2m\omega} + ia \sin \theta) \\
&\hspace{4em}- 8Mr_+ k_{m\omega} \pb{2} \eta^+_{lm\omega} (2 \zeta_+ \mathscr{L}_{2m\omega} + ia \sin \theta) \\
&\hspace{4em}+ 6(4Mr_+)^2 ik_{m\omega}^2 \pb{1} \kappa_{m\omega} \mathscr{L}_{2m\omega}\Big\} \frac{\pb{2} \Theta_{lm\omega}}{256 (Mr_+)^{5/2} ik_{m\omega}^2 \pb{1} \kappa_{m\omega}},
\end{aligned}} \\
&\mathrlap{\pb{\var_- C_{l(nm)}} S_{lm\omega p}^{\textrm{out}} = -(Mr_+)^{3/2} k_{m\omega}^2 \pb{2} \kappa_{m\omega} \pb{1} \kappa_{m\omega} \zeta_+^{-2} (\zeta_+ \mathscr{L}_{2m\omega} + ia \sin \theta) \pb{2} \Theta_{lm\omega},} \\
&\mathrlap{\begin{aligned}
\pb{\var_- C_{l(nm)}} S_{lm\omega p}^{\textrm{down}} &= -\Big\{[4 \omega^2 a^2 (2\cos^2 \theta - 3) + 12 i\omega (M - iam) + \pb{2} \lambda_{lm\omega} (\pb{2} \lambda_{lm\omega} + 2)] \mathscr{L}_{2m\omega} \\
&\hspace{3em}+ 4 a\omega \sin \theta (12a\omega \cos \theta + \pb{-2} \eta^\infty_{lm\omega})\Big\} \frac{\pb{2} \Theta_{lm\omega}}{32\sqrt{2} \omega^2},
\end{aligned}} \\
&\pb{\var_- C_{l(nm)}} S_{lm\omega p}^{\textrm{up}} = -\frac{\omega^2 \mathscr{L}_{2m\omega}}{2\sqrt{2}} \pb{2} \Theta_{lm\omega}, \\
&\mathrlap{\begin{aligned}
\pb{\var_- C_{lmm}} S_{lm\omega p}^{\textrm{in}} &= \zeta^{-2} \Big\{\pb{2} \eta^+_{lm\omega} [2a^2 \sin^2 \theta - \zeta_+ (\zeta_+ \mathscr{L}_{(-1)m\omega} + 3ia \sin \theta) \mathscr{L}_{2m\omega} - 8Mr_+ ik_{m\omega} \zeta_+] \\
&\hspace{3.5em}+ 8Mr_+ ik_{m\omega} \pb{1} \kappa_{m\omega} (\zeta_+ \mathscr{L}_{(-1)m\omega} + 3ia \sin \theta) \mathscr{L}_{2m\omega} \\
&\hspace{3.5em}- [24Mr_+ \omega k_{m\omega} \pb{1} \kappa_{m\omega} + \pb{2} \eta^+_{lm\omega} (2 - \pb{1} \eta^+_{lm\omega})] \zeta_+^2\Big\} \frac{\pb{2} \Theta_{lm\omega}}{2 (8Mr_+)^{3/2} ik_{m\omega} \pb{1} \kappa_{m\omega}},
\end{aligned}} \\
&\mathrlap{\begin{aligned}
\pb{\var_- C_{lmm}} S_{lm\omega p}^{\textrm{out}} &= \sqrt{Mr_+} ik_{m\omega} \pb{2} \kappa_{m\omega} \zeta_+^{-2} \Big\{\zeta_+ (8Mr_+ ik_{m\omega} \pb{2} \kappa_{m\omega} - 3ia \sin \theta \mathscr{L}_{2m\omega}) + 2a^2 \sin^2 \theta \\
&\hspace{11.5em}-\zeta_+^2 [\mathscr{L}_{(-1)m\omega} \mathscr{L}_{2m\omega} + (2 - \pb{1} \eta^+_{lm\omega})]\Big\} \frac{\pb{2} \Theta_{lm\omega}}{2\sqrt{2}},
\end{aligned}} \\
&\mathrlap{\pb{\var_- C_{lmm}} S_{lm\omega p}^{\textrm{down}} = -\frac{2 \pb{-2} \eta^\infty_{lm\omega} + \mathscr{L}_{(-1)m\omega} \mathscr{L}_{2m\omega}}{8} \pb{2} \Theta_{lm\omega},} &\pb{\var_- C_{lmm}} S_{lm\omega p}^{\textrm{up}} = -\frac{3 \omega^2}{2} \pb{2} \Theta_{lm\omega}.
\end{align}
\end{subequations}
\bibliographystyle{unsrt}
\bibliography{adwg}
\end{document}
| 98,446
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TITLE: Infinite convex combinations in a Banach space
QUESTION [19 upvotes]: Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds:
For any sequence $(x_k)_{k\ge0}$ in $C$, and for
any sequence of non-negative real numbers $(\lambda_k)_ {k\ge0}$ with
$\sum_{k=0}^\infty \lambda_k=1$ the
series $\sum_{k=0}^\infty \lambda_k x_k$ converges to an element of $C$.
(the term $\sigma$-convex seems quite natural for this property, and indeed it is used e.g. in this 1976 paper; though I'm not certain that this is the standard current terminology).
Clearly, any $\sigma$-convex set is convex and bounded; a bounded convex set need not be $\sigma$-convex (e.g. the convex hull $\Delta$ of the orthonormal basis of $\ell^2$). A closed bounded convex set is $\sigma$-convex; and an open bounded convex set is $\sigma$-convex, too. Also, the intersection of $\sigma$-convex sets is $\sigma$-convex, and the image of a $\sigma$-convex set via a bounded linear operator is $\sigma$-convex.
Question: is there a topological
characterization of those bounded
convex subsets of a Banach space which
are $\sigma$-convex?
Given the above mentioned facts, a reasonable conjecture could be, that a bounded convex set is $\sigma$-convex if and only in it is a Baire space.
[edit] a simple counterexample in $X:=\mathbb{R}\times \ell^2 $ is $C:=(0,1]\times B\, \cup\, \{0\}\times\Delta$ where $B$ is the open unit ball of $\ell^2$, and $\Delta$ is the non-$\sigma$-convex set described above. This set is bounded, convex, and Baire, though it's not $\sigma$-convex.
$$*$$
[edit] As far as I see, the interesting feature of $\sigma$-convex sets is the following "iteration lemma" (it's a piece of the Open Mapping Theorem, that in my opinion is worth to be a lemma in itself, also because its proof is repeated in several theorems).
Lemma. Let $X$ be a Banach space; $C\subset X$ $\sigma$-convex; $B\subset X$ a
bounded subset, $0 < t < 1$ be such
that $$B\subset C + tB \, . $$ Then
$$(1-t)B\subset C \, .$$
(proof: as in the OMT: start from $b_0\in B$, represent it as $b_0=c_0+tb_1$, and iterate; one gets $(1-t)b_0$ as sum of an infinite convex combination in $C$). Curiously, this is also a characterization, in that any bounded set $C$ for which the above property holds for any bounded set $B$ (and even for just a fixed $0 < t < 1$) is indeed $\sigma$-convex.
REPLY [1 votes]: Possibly I misunderstood your question, but I do not see a hope for a suitable characterization because of the following example (inspired by the comment of Bill Johnson): Consider a set $A$ in $\ell_2^2$ ($2$-dimensional Euclidean space) consisting of the union of the open unit ball and some subset of the unit sphere. The set $A$ is always $\sigma$-convex. On the other hand $A$ is (this can be shown directly) homeomorphic to a subset $O$ consisting of the union of the open unit ball of $\ell_1^2$ and a suitable subset of the sphere. The set $O$ is $\sigma$-convex (and even convex) in some exceptional cases only. Unless the set $A$ contains at most one point on the boundary, the homeomorphism can be modified in such a way that the (modified) $O$ is not $\sigma$-convex.
| 175,400
|
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Progressives Are Starting to Notice the Weird Cultural Climate
The cultural climate out there is weird. Just about every American adult feels it, although we try to go about life as normal, pushing our unease aside.
But what about the kids? Does the cultural unrest affect them as well?
Writer George Packer suggests it does. “When the Culture War Comes for the Kids” is Packer’s lengthy contribution to the October issue of The Atlantic, chronicling the journey of his children through the American school system.
Packer’s progressive ideology makes his family’s educational experiences intriguing. As Packer explains, he and his wife are doing their best to give their children a good education – standing in line at ungodly hours to get into a desirable school, moving their child to a more progressive-minded form of education, and so on.
But over time things began to change in school and the culture at large, Packer says. The “new mood was progressive but not hopeful” and arose “toward the end of the Obama years.”
At the heart of the new progressivism was indignation, sometimes rage, about ongoing injustice against groups of Americans who had always been relegated to the outskirts of power and dignity. … Over time the new mood took on the substance and hard edges of a radically egalitarian ideology.
Eventually, Packer notes, this mood went beyond policy reforms and invaded “private spaces where we think and imagine and talk and write, and the public spaces where institutions shape the contours of our culture and guard its perimeter.” Many began to view political correctness as “a problem,” a point with which Packer seems to agree.
One of the difficulties, Packer explains, is with identity politics:.
Packer saw such identity politics play out in the school system. They reared their head when the family swam against the tide and chose not to opt out of standardized tests. They appeared again when the school adopted gender neutral bathrooms, causing students great discomfort. These policies were enacted as means of helping an oppressed minority; instead, they ended up oppressing the majority.
Such a scenario is not new. In the mid-19th century, Alexis de Tocqueville explained that the quest for ultra-equality often results in negative attitudes which set individuals at odds with one another:.
De Tocqueville continues:.
De Tocqueville’s words are an apt description of our day. As adults we have grudgingly accepted this as reality – life is no longer as we once knew it. The trouble is, as Packer so aptly describes, this reality is not limited to us, for it is being passed on to our children.
Is it possible that our quest for equality, instead of being liberating, may end up enslaving the next generation?
—
[Image Credit: Max Pixel]
| 108,207
|
Mcbreen
Dr. Brian Mcbreen, MD is a Doctor primarily located in Washington, DC. He has 22 years of experience. His specialties include Internal Medicine. Dr. Mcbreen is affiliated with George Washington University Hospital, Georgetown University Hospital and Georgetown University. Dr. Mcbreen has received 3 awards. He speaks English.
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2
NetBeans 3.6, released last month by the NetBeans.org open-source community, offers Java developers an attractive and capable programming environment that features a native look and feel on a variety of popular platforms. With so many tool sets for the supposedly platform-neutral Java paying scant heed to non-x86 developers, we commend NetBeans.org for also providing convenient download options and instructions for Windows, Solaris (both for x86 and SPARC), Mac OS X, Linux and the Java Archive format.
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| 175,892
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TITLE: assessing linear relationships as logarithms
QUESTION [4 upvotes]: I am teaching myself maths. I am not sure how to approach this problem. It is assessing linear relationships of the form $y=mx+c$ as logarithms.
Here I have gotten as far as taking the gradient ($\log e$) of $\log \frac{s}{t} = -0.6363...$ so $e = 0.23$ but I do not know how to derive or separate $-n$ or what it represents. Could someone tell me how to proceed? Thank you.
Alan
REPLY [1 votes]: If you take logarithms of both sides of the formula mentioned in the problem you get:
$$\ln(s) = \ln(k)-n t$$
So in the terminology of your post $y=\ln(s), c=\ln(k), m=-n, x=t$.
To create the graph, augment the table with one more row and populate it with $\ln(s)$ taking logaritms of the second row. Then use the $t$-values of the first row for the horizontal coordinate the values of the third row as the vertical coordinate. You will have 5 points in your graph and if they are close to a straight line you have your graphical verification. Next calculate (graphicaly, an approximation) the slope of that line and you get $m$. Finaly the point that it intersects the vertical axis gives you $c$.
| 72,079
|
TITLE: Galois group of $x^{15}-1$
QUESTION [2 upvotes]: Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity.
a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$.
b) Show that $G$ is isomorphic to a direct product of two cyclic groups. Construct this isomorphism.
The group $|G|=|G(\mathbb Q (\zeta)/ \mathbb Q)|= [\mathbb Q (\zeta):\mathbb Q)]=U_{15}$
Now what is the generator of $\mathbb F_{15}^*$? We have to find the primitive root modulo 15. I am confused.
Please help. Thanks in advance.
REPLY [2 votes]: When constructing an automorphism of $\mathbb{Q}(\zeta)$, the question is where $\zeta$ goes to. But because roots from $x^{15}-1$ should be interchanged, there are only $15$ possibilities. These are the possibilities:
$$f(\zeta)\in\{\zeta,\zeta^2,\zeta^4,\zeta^7,\zeta^8,\zeta^{11},\zeta^{13},\zeta^{14}\}$$
There are no other possibilities, because if for example $f(\zeta)=\zeta^3$, then $f(\zeta^5)=\zeta^{15}=1=f(1)$, but elements from $\mathbb{Q}$ should be fixed and the automorphism $f$ is invertible.
Now we want to construct an ismorphism between this group and $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$. This is a natural choice, because there are no other possibilities to write a group with eight elements as a product of two cyclic groups.
A possible isomorphism from $G$ to $\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}$ is the following:
\begin{align}
G&\rightarrow\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/4\mathbb{Z}\\
\zeta&\mapsto(0,0)\\
\zeta^2&\mapsto(0,1)\\
\zeta^4&\mapsto(0,2)\\
\zeta^7&\mapsto(1,3)\\
\zeta^8&\mapsto(0,3)\\
\zeta^{11}&\mapsto(1,2)\\
\zeta^{13}&\mapsto(1,1)\\
\zeta^{14}&\mapsto(1,0)
\end{align}
Because this group with eight elements is not cyclic (there does not exist an element with order eight), $(\mathbb{Z}/15\mathbb{Z})^\star$ does not have a primitive root.
| 143,949
|
TITLE: Set of extreme points in $\ell^1(2) \oplus_2 \mathbb{R}.$
QUESTION [0 upvotes]: Let $\ell^\infty(2)$ be a subset of $\mathbb{R}^2$ with maximum norm, that is, $\|(x,y)\|_\infty=\max\{|x|,|y| \}.$
Given two Banach spaces $X$ and $Y,$ if we have $X\oplus_2Y,$ we mean that $(x,y)\in X \oplus_2 Y$ with norm $\|(x,y\|_2= \sqrt{\|x\|_X^2 + \|y\|_Y^2}.$
We say that $x^*$ is an extreme point of unit ball $B_{X^*}$ of $X^*$ if whenever $x^*=\lambda y^* +(1-\lambda)z^*$ for some $\lambda\in[0,1]$ and $y^*,z^*\in B_{X^*},$ then $x^* = y^*$ or $x^*=z^*.$
Geometrically speaking, extreme points are 'corners' of unit ball.
Given a Banach space $X$, its set of extreme points $Ext(X)$ is defined to be a subset of closed unit ball of its dual $X^*.$
Also, if $X$ and $Y$ are isometrically isomorphic, we denote it as $X\cong Y.$
Question: Let $E = \ell^\infty(2)\oplus_2 \mathbb{R}$ be given. What is its $Ext(E)?$
First, I guess that $E^* \cong E$ (not very sure) due to $(\ell^2)^* \cong \ell^2$ and $F^* \cong F$ for any finite dimensional space $F$. So $Ext(E) \subseteq E^* \cong E.$
However, I have no idea how $E^*$ look like, thus cannot see its corner points. Any hint would be appreciated.
UPDATED: Based on Daniel Fischer's comment, dual space of $E$ is $\ell^1(2) \oplus_2 \mathbb{R}.$
So my question is reduced to finding extreme points of closed unit ball in $\ell^1(2)
\oplus_2 \mathbb{R}.$
REPLY [1 votes]: Some points to make first:
Extreme points are not preserved by isomorphisms, so thinking of what the given space is isomorphic to is not helpful. They are preserved by isometric isomorphisms.
A space is strictly convex if $\|a+b\|<\|a\|+\|b\|$ whenever $a,b$ are non-collinear vectors. In a strictly convex space, every boundary point of the unit ball is its extreme point.
If $X$ and $Y$ are strictly convex, then $X\oplus_2Y$ is also strictly convex. Indeed, to have equality
$$\|(x_1,y_1) + (x_2,y_2)\| = \|(x_1,y_1)\| + \|(x_2,y_2)\|$$
one needs $x_1,x_2$ to be collinear, $y_1,y_2$ to be collinear, and also the Minkowski inequality
$$
\sqrt{(\|x_1\|+\|x_2\|)^2 + (\|y_1\|+\|y_2\|)^2}
\le \sqrt{\|x_1\|^2 + \|y_1\|^2} + \sqrt{\|x_2\|^2 + \|y_2\|^2}
$$
must turn into equality. All this together implies that the vectors $(x_1,y_1)$ and $(x_2,y_2)$ are collinear.
From 1-3 the solution of the question about $\ell^\infty(2)\oplus_2 \mathbb{R}$ follows.
Given a Banach space $X$, its set of extreme points $\operatorname{Ext}(X)$ is defined to be a subset of closed unit ball of its dual $X^∗$
It's not defined like that, as pointed out in comments.
| 52,085
|
TITLE: If $A^2=A,\,B^2=B,\,C^2=C$ and $A+B+C=I,$ prove $AB=BC=AC=\mathbb{O}$.
QUESTION [1 upvotes]: Let $A,\,B,\,C$ be square matrices such that $A^2=A,\,B^2=B,\,C^2=C$ and $A+B+C=I.$
Prove that $AB=BC=CA=\mathbb{O}.$
Attempt. We have $A(B+C)=A(I-A)=A-A^2=\mathbb{O}$ and similarly we get
$B(A+C)=C(A+B)=\mathbb{O}$, but this is where i get so far.
Thanks in advance for the help.
REPLY [6 votes]: Roland's answer gives a great and simple linear algebra argument.
Here is a purely algebraic development:
\begin{align*}
(A+B+C)^2 & = A^2 + B^2 + C^2 + AB + AC + BC + BA + CA + CB \\
& = I + A(B+C) + BC + (B+C)A + CB \\
& = I + A(I - A) + (I-A)A + BC + CB \\
& = I + A - A^2 + A - A^2 + BC + CB \\
& = I + BC + CB \\
I & = I + BC + CB \\
0 & = BC + CB \\
\end{align*}
By permuting the roles of $A,B,C$, we get $BC + CB = AC + CA = AB + BA = 0$.
This would allow us to conclude if we knew that $AB = BA$. It is not one of the hypotheses, but at this point we can deduce it:
\begin{align*}
AB + BA & = 0 \\
AB & = - BA \\
A^2B & = - ABA \\
AB & = - ABA \\
AB & = BA^2 \\
AB & = BA \\
\end{align*}
| 36,933
|
\begin{document}
\newtheorem{Thm}{Theorem}[section]
\newtheorem{Def}[Thm]{Definition}
\newtheorem{Lem}[Thm]{Lemma}
\newtheorem{Rem}[Thm]{Remark}
\newtheorem{Cor}[Thm]{Corollary}
\newtheorem{Prop}[Thm]{Proposition}
\newtheorem{Example}[Thm]{Example}
\newcommand{\g}[0]{\textmd{g}}
\newcommand{\pr}[0]{\partial_r}
\newcommand{\dif}{\mathrm{d}}
\newcommand{\bg}{\bar{\gamma}}
\newcommand{\md}{\rm{md}}
\newcommand{\cn}{\rm{cn}}
\newcommand{\sn}{\rm{sn}}
\newcommand{\seg}{\mathrm{seg}}
\newcommand{\Ric}{\mbox{Ric}}
\newcommand{\Iso}{\mbox{Iso}}
\newcommand{\ra}{\rightarrow}
\newcommand{\Hess}{\mathrm{Hess}}
\newcommand{\RCD}{\mathsf{RCD}}
\title{Almost volume cone implies almost metric cone for annuluses centered at a compact set in $\RCD(K, N)$-spaces}
\author{Lina Chen}
\address[Lina Chen]{Department of mathematics, Nanjing University, Nanjing China}
\email{chenlina\_mail@163.com}
\thanks{Supported partially by NSFC Grant 12001268 and a research fund from Nanjing University.}
\maketitle
\pagestyle{fancy}\lhead{Almost volume cone implies almost metric cone}\rhead{Lina Chen}
\begin{abstract}
\setlength{\parindent}{10pt} \setlength{\parskip}{1.5ex plus 0.5ex
minus 0.2ex}
In \cite{CC1}, Cheeger-Colding considered manifolds with lower Ricci curvature bound and gave some almost rigidity results about warped products including almost metric cone rigidity and quantitative splitting theorem. As a generalization of manifolds with lower Ricci curvature bound, for metric measure spaces in $\RCD(K, N)$, $1<N<\infty$, splitting theorem \cite{Gi13} and ``volume cone implies metric cone" rigidity for balls and annuluses of a point \cite{PG} have been proved. In this paper we will generalize Cheeger-Colding's \cite{CC1} result about ``almost volume cone implies almost metric cone for annuluses of a compact subset " to $\RCD(K, N)$-spaces. More precisely, consider a $\RCD(K, N)$-space $(X, d, \mathfrak m)$ and a Borel subset $\Omega\subset X$. If the closed subset $S=\partial \Omega$ has finite outer curvature, the diameter $\op{diam}(S)\leq D$ and the mean curvature of $S$ satisfies
$$m(x)\leq m, \, \forall x\in S,$$
and
$$\mathfrak m(A_{a, b}(S))\geq (1-\epsilon)\int_a^b \left(\op{sn}'_H(r)+ \frac{m}{n-1}\op{sn}_H(r)\right)^{n-1}dr \mathfrak m_S(S)$$
then $A_{a', b'}(S)$ is measured Gromov-Hausdorff close to a warped product $(a', b')\times_{\op{sn}'_H(r)+ \frac{m}{n-1}\op{sn}_H(r)}Y$, $A_{a, b}(S)=\{x\in X\setminus \Omega, \, a<d(x, S)<b\}$, $a<a'<b'<b$, $Y$ is a metric space with finite components with each component a $\RCD(0, N-1)$-space when $m=0, K=0$ or a $\RCD(N-2, N-1)$-space for other cases and $H=\frac{K}{N-1}$. Note that when $m=0, K=0$, our result is a kind of quantitative splitting theorem and in other cases it is an almost metric cone rigidity.
To prove this result, different from \cite{Gi13, PG}, we will use \cite{GiT}'s second order differentiation formula and a method similar as \cite{CC1}.
\end{abstract}
\section{Introduction}
In \cite{CC1}, Cheeger-Colding gave the following ``almost volume cone implies almost metric cone" rigidity.
\begin{Thm}[\cite{CC1}] \label{ann-rig}
If a complete $N$-manifold $M$ with $\op{Ric}_M\geq (N-1)H$ and a compact subset $\Omega\subset M$ satisfies that the mean curvature
\begin{equation} m(x)\leq m, \forall \, x\in S=\partial \Omega, \label{mean-com}\end{equation}
$$\op{diam}(S)\leq D,$$
and
\begin{equation} \volume(A_{a, b}(S))\geq (1-\epsilon) \int_a^b \left(\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)\right)^{n-1}dr \volume(S),\label{alm-max}\end{equation}
then
$$d_{GH}(A_{a+\alpha, b-\alpha}(S), (a+\alpha, b-\alpha)\times_{\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)} Y)\leq \Psi(\epsilon | N, H, m, a, b, \alpha, D).$$
where $A_{a, b}(S)=\{x\in X\setminus \Omega, \, a<d(x, S)<b\}$, $Y$ is a length metric space with at most $C(N, H, a, b, D)$ components $Y_i$ such that $\op{diam}(Y_i)\leq c(N, H, m, a, b, \alpha, D)$ and
$$\op{sn}_H(r)=\begin{cases} \frac{\sin\sqrt{H}r}{\sqrt H}, & H>0;\\ r, & H=0;\\ \frac{\sinh \sqrt{-H}r}{\sqrt{-H}}, & H<0. \end{cases}$$
\end{Thm}
If $S$ is a hypersurface of $M$, by Heintze-Karcher \cite{HK}, \eqref{mean-com} implies that
\begin{equation} \volume(A_{a, b}(S))\leq \int_a^b \left(\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)\right)^{N-1}dr \volume(S).\label{loc-vol}\end{equation}
In particular, if $M$ is compact with $\tilde D=\op{diam}(X)$,
\begin{equation}
\volume(M)\leq \int_{[-\tilde D, \tilde D]} \left(\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)\right)^{N-1}dr \volume(S). \label{gvol-com}\end{equation}
And when $H>0$, the equality holds in \eqref{gvol-com} iff $M$ and $N$ have constant curvature (see \cite{HK}).
In \cite{Ket2}, Ketterer extended Heintze-Karcher \cite{HK}'s results about the volume comparison \eqref{loc-vol} and the rigidity result for $K>0$ in $\RCD(K, N)$-spaces with $S=\partial \Omega$, $\Omega$ is Borel and $H=\frac{K}{N-1}$.
In this note, we will generalize Theorem~\ref{ann-rig} to $\RCD(K, N)$-spaces which can also be treated as a quantitative version and a generalization of Ketterer \cite{Ket2}'s work (see Theorem~\ref{HK-general}) to arbitrary $K$.
In the following, we will use the same definitions of mean curvature, finite outer curvature and measure on $S$, $\mathfrak m_S$ as in \cite{Ket2} (see Section 2.7 for these definitions).
\begin{Thm} \label{main}
If a metric measure space $(X, d, \mathfrak m)\in \op{RCD}(K, N)$, $3\leq N<\infty$, $\op{supp}(\mathfrak m)=X$ and a Borel subset $\Omega\subset X$ satisfies that $S=\partial \Omega$ is closed, $\op{diam}(S)\leq D$, $\mathfrak m(S)=0$, $S$ has finite outer curvature, the mean curvature
\begin{equation}m(x)\leq m, \, \forall \, x\in S, \label{mean-bound}\end{equation}
and
\begin{equation} \mathfrak m(A_{a, b}(S))\geq (1-\epsilon)\int_a^b \left(\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)\right)^{n-1}dr \mathfrak m_S(S), \label{lvol-com}\end{equation}
then
$$d_{mGH}(A_{a', b'}(S), (a', b')\times_{\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)} Y)\leq \Psi(\epsilon | N, K, m, a, b, D),$$
where $H=\frac{K}{N-1}$, $a'=a+(b-a)/3, b'=b-(b-a)/3$ and $(Y, d_Y, \mathfrak m_Y)$ has at most $C(N, K, a, b, D)$ components $Y_i$ with each $Y_i\in \RCD(0, N-1)$ for $m=0, K=0$, $Y_i \in \RCD(N-2, N-1)$ for $m\neq 0$ or $K\neq 0$.
\end{Thm}
\begin{Rem}
(i) When $K=0, m=0$, \eqref{lvol-com} becomes
\begin{equation}\mathfrak m(A_{a, b}(S))\geq (1-\epsilon)(b-a)\mathfrak m_S(S)\label{lvol-com-imp0}\end{equation}
and Theorem~\ref{main} is a kind of quantitative splitting theorem.
In \cite{Hu}, Huang gave a quantitative splitting rigidity under \eqref{lvol-com-imp0} and a measure-decreasing-along-distance-function (MDADF) condition. The assumption that $S$ has finite outer measure and mean curvature upper bound \eqref{mean-bound} implies MDADF condition (see Lemma~\ref{vol-ele-com} and the definition of finite outer curvature). A better result we have is that $Y$ can be chosen as each component in $\RCD(0, N-1)$.
(ii) When $K\neq 0$ or $m\neq 0$, assume
\begin{equation}m=(N-1)\frac{\op{sn}'_H(r_0)}{\op{sn}_H(r_0)},\label{mean-imp}\end{equation}
then \eqref{lvol-com} becomes
\begin{equation} \frac{\mathfrak m(A_{a, b}(S))}{\mathfrak m_S(S)}\geq (1-\epsilon)\frac{\svolann{H}{a+r_0, b+r_0}}{\svolsp{H}{r_0}}. \label{lvol-com-imp1}\end{equation}
In \cite{PG}, Philippis-Gigli pointed out that ``volume cone implies metric cone" holds for annulus centered at a point where they assume
$$\frac{\mathfrak m(A_{a, b}(x))}{\mathfrak m(\partial B_a(x))}\geq (1-\epsilon)\frac{\svolann{H}{a, b}}{\svolsp{H}{a}},$$
and derived a quantitative rigidity as in Theorem~\ref{main}.
Here $$\mathfrak m(\partial B_a(x))=\limsup_{\delta\to 0}\frac{\mathfrak m(A_{a, a+\delta}(x))}{\delta}.$$
Our result is a generalization of \cite{PG} in some sense.
(iii) When the equality holds in \eqref{lvol-com}, we have that $A_{a', b'}(S)$ has a warped product structure and $\partial B_{a'}(S)$ has constant mean curvature.
\end{Rem}
Now we give a sketch of the proof of Theorem~\ref{main}. Consider a sequence of $\RCD(K, N)$-spaces, $(X_i, d_i, \mathfrak m_i)$ which is measured Gromov-Hausdorff convergent to a $\RCD(K, N)$-space $(X, d, \mathfrak m)$. Assume $S_i=\partial \Omega_i, \Omega_i\subset X_i$ is Borel with $\mathfrak m_i(S_i)=0$ and $\op{diam}(S_i)\leq D$.
Define a signed distance function associated to $\Omega_i$.
$$d_{s, i}(x)=\begin{cases} d_i(x, S_i), & x\in X_i\setminus \Omega_i;\\ -d_i(x, S_i), & x\in \Omega_i. \end{cases}$$
Obviously, $d_{s, i}$ is $1$-Lipschitz.
Then by \cite[Proposition 2.70]{Vi} or \cite[Proposition 2.12]{MN}, there is a $1$-Lipschitz function $d_s: X\to \Bbb R$ such that
$d_{s, i}$ converges uniformly to $d_s$ on any compact set. Let $S=\{x\in X, \, d_s(x)=0\}$, $\Omega=\{x\in X, \, d_s(x))\leq 0\}$. Then as in \cite[Lemma 3.25]{Hu}, we know that $d_s$ is a signed distance function associated to $\Omega$.
Assume $S_i$ has finite outer curvature and the mean curvature
$$m(x)\leq m, \forall\, x\in S_i,$$
\begin{equation} \mathfrak m(A_{a, b}(S_i))\geq (1-\epsilon_i)\int_a^b \left(\op{sn}'_H(r)+ \frac{m}{n-1}\op{sn}_H(r)\right)^{n-1}dr \mathfrak m_{S_i}(S_i), \, \epsilon_i\to 0. \label{lqvol-com}\end{equation}
To prove Theorem~\ref{main}, we only need to show that $A_{a', b'}(S)$ is isometric to $(a', b')\times_{\op{sn}'_H(r)+ \frac{m}{n-1}\op{sn}_H(r)} Y$ where $(Y, d_Y, \mathfrak m_Y)\in \RCD(0, N-1)$ for $m=0, K=0$, $(Y, d_Y, \mathfrak m_Y)\in \RCD(N-2, N-1)$ for the other cases.
To obtain this result, first we have that:
($\ast$) With intrinsic metric $A_{a', b'}(S)$ is isometric to a warped product $(a', b')\times_{\op{sn}'_H(r)+ \frac{m}{n-1}\op{sn}_H(r)} Y$ (for the definition see Section 2.6).
We will follow the process as in \cite{CC1}. The assumption $S$ has finite outer curvature and $m(X)\leq m$, together with the laplacian formula derived by \cite{CMo}, we will derive the laplacian comparison of $d_s$ and relative volume comparison (see Lemma~\ref{glap-com} and Lemma~\ref{rel-vol}). Then by the volume condition \eqref{lqvol-com}, we will get a laplacian estimate of $d_s$ in Theorem~\ref{lap-main}. These laplacian estimates and improved Bochner's inequality in $\RCD$-spaces (\cite{RS, Han18}, see also Theorem~\ref{Boc-ine}) give the Hessian estimates Theorem~\ref{hess-est}. Then using the second order differentiation in $\RCD(K, N)$-space \cite{GiT}, we can show that the metric in the path connected component of $A_{a', b'}(S)$ satisfies the Pythagoras theorem when $m=0$, $K=0$ and Cosine law for the other cases. This gives the warped product structure of $A_{a', b'}(S)$. And the relative volume comparison gives that $Y$ has at most $C(N, K, D, b, a)$ components.
Assume $Y$ has one component. Then ($\ast$) and that $A_{a', b'}(S)\subset (X, d, \mathfrak m)\in \RCD(K, N)$ enable us to derive that:
($\ast\ast$) $(Y, d_Y, \mathfrak m_Y)\in \RCD(0, N-1)$ for $m=0, K=0$ and $(Y, d_Y, \mathfrak m_Y)\in \RCD (N-2, N-1)$ for the other cases (see Section 6).
Endow $Y$ with an admissible metric and an admissible measure from the warped product structure ($\ast$). A similar argument as in \cite{CDNPSW} shows that $(Y, d_Y, \mathfrak m_Y)$ is infinitesimally Hilbertian and satisfies Sobolev to Lipschitz property (see Theorem~\ref{measure-pro}). Now by local to global property we can see that $\Bbb R\times Y$ (when $m=0, K=0$) and the Euclidean cone $C(Y)$ (when $m\neq 0, K=0$) are $\RCD(0, N)$-spaces. Then \cite{Gi13} and \cite{Ket2} gives ($\ast\ast$) for $K=0$. For $K\geq 0$, we will follow the argument in the proof of \cite[Theorem 1.2]{Ket2} where Ketterer showed that if the $(K, N)$-cone $(C(Y), d_K, \mathfrak m_N)$ is a $\RCD(K, N)$-space, then $Y$ is $\RCD(N-2, N-1)$-space.
The paper is organized as follows. In Section 2, we will present some basic definitions and facts we need in the poof of Theorem~\ref{main}. Then by studying the relative volume comparison for annuluses centered at a compact subset, we give the Laplacian estimates of the distance function from the compact subset in Section 3. Then in Section 4, we will give the corresponding Hessian estimates in $A_{a', b'}(S)$. In Section 5, by the second differential formula in $\RCD(K, N)$-spaces\cite{GiT}, using the Hessian estimates and a methods as in \cite{CC1} we will derive that with the intrinsic metric the annulus $A_{a', b'}(S)$ satisfies Pythagoras theorem or Cosine law. In Section 6, we will give the warped product structure of $A_{a', b'}(S)$ and by studying the properties of the section $Y$ of the warped product, we will prove that $Y$ is a $\RCD$-space.
The author would like to thank Professor Xian-tao Huang's advice about that $Y$ may contain more than one components in the main results.
\section{Preliminary}
In this section, we recall some basic definitions and properties that we need in the proof of Theorem~\ref{main}. Let $(X, d, \mathfrak m)$ be a metric measure space satisfying that $(X, d)$ is a complete, separable and locally compact geodesic metric space endowed with a nonnegative Radon measure $\mathfrak m$ which is supported on $X$ and is finite on any bounded sets. We refer readers to the survey \cite{Am} for an overview of the topic and bibliography about curvature-dimension bounds in metric measure spaces.
\subsection{Calculus in metric measure spaces} For the details of this subsection one can confer \cite{Gi13}.
Consider a metric measure space $(X, d, \mathfrak m)$ as above.
Let $C([0, 1], X)$ be the space of continuous curves with weak convergence topology and let $\mathcal{P}(C([0,1], X))$ be the space of Borel probability measures of $C([0,1], X)$. A measure $\pi\in \mathcal{P}(C([0,1], X))$ is called a \textbf{test plan} if for some $c>0$,
$$(e_t)_{\sharp}(\pi)\leq c \mathfrak m, \forall \, t\in [0, 1], \quad \int\int_0^1 |\dot\gamma(t)|dt d\pi(\gamma)<\infty,$$
where $|\dot\gamma(t)|=\lim_{h\to 0}d(\gamma(t+h), \gamma(t))/|h|$ and $e_t: C([0, 1], X) \to X$, $e_t(\gamma)=\gamma(t)$ is the evaluation map.
Sobolev class $S^2(X, d, \mathfrak m)$ is defined as the space of $f: X\to \Bbb R$, such that there exists $G\in L^2(X, \mathfrak m)$,
$$\int |f(\gamma(1))-f(\gamma(0))|d\pi(\gamma)\leq \int\int_0^1G(\gamma(t))|\dot\gamma(t)|dt d\pi(\gamma), \, \forall \, \text{test plan } \pi, $$
where $G$ is called a weak upper gradient of $f$. Let $|\nabla f|_w$ be the minimal (in $\mathfrak m$-a.e. sense) weak upper gradient of $f$.
The space $W^{1,2}(X, d, \mathfrak m)=L^2(X, \mathfrak m)\cap S^2(X, d, \mathfrak m)$ endowed with the norm
$$\|f\|^2_{W^{1,2}}=\|f\|_{L^2}^2+ \||\nabla f|_w\|^2_{L^2},$$
is a Banach space.
Define the \textbf{Cheeger energy} as $\op{Ch}: L^2(X, \mathfrak m)\to [0, \infty]$
$$\op{Ch}(f)=\begin{cases} \frac12\int |\nabla f|_w^2d\mathfrak m, & f\in W^{1,2}(X, d, \mathfrak m)\\
+\infty, & \text{ otherwise}. \end{cases}$$
We say $(X, d, \mathfrak m)$ is \textbf{infinitesimally Hilbertian} if $W^{1,2}(X, d, \mathfrak m)$ is a Hilbert space, i.e., the Cheeger energy is a quadratic form.
In the following of this section we always assume that $(X, d, \mathfrak m)$ is infinitesimally Hilbertian.
For an open subset $\Omega\subset X$, let $W^{1,2}_{\op{loc}}(\Omega)$ be the space of function $f: \Omega\to \Bbb R$ that locally equal to some function in $W^{1, 2}(X, d, \mathfrak m)$.
For $f, g\in W^{1,2}_{\op{loc}}(\Omega)$, define
$$\Gamma(f, g)=\left<\nabla f, \nabla g\right>=\liminf_{\epsilon\downarrow 0}\frac{|\nabla (g+\epsilon f)|_w^2-|\nabla g|_w^2}{2\epsilon}.$$
In fact $\left<\nabla f, \nabla g\right>$ can be achieved by taking limit directly $\mathfrak m$-a.e(cf. \cite{Gi14}). By \cite{Gi14}, the map $\Gamma: W^{1,2}_{\op{loc}}(\Omega)\times W^{1,2}_{\op{loc}}(\Omega)\to L^1_{\op{loc}}(\Omega)$ is symmetric, bilinear and $\Gamma(f, f)=|\nabla f|_w^2$.
\begin{Def}
For $f\in W^{1,2}_{\op{loc}}(\Omega)$, if there exists a Radon measure $\mu$ on $\Omega$ such that
$$-\int\left<\nabla f, \nabla g\right>=\int gd\mu$$
holds for any Lipschitz function $g: \Omega\to \Bbb R$, $\op{supp}g\subset\subset \Omega$, then $\mu$ is called the \textbf{distributional Laplacian} or \textbf{measure valued Laplacian} of $f$ and denote it by $\left.\Delta f\right|_{\Omega}$.
\end{Def}
Let $D(\Delta, \Omega)$ be the space of $f$ which has a distribution Laplacian. By the property of $\Gamma$, we know that $D(\Delta, \Omega)$ is a vector space and the Laplacian is linear. For $f\in W^{1,2}(X, d, \mathfrak m)\cap D(\Delta, X)$, if $\Delta f= h\mathfrak m$, $h\in L^2(X, \mathfrak m)$, we denote $\Delta f=h$.
\subsection{Tangent and cotangent modules} The details of this subsection can be found in \cite{Gi18}.
Consider a measured space $(X, \mathcal A, \mathfrak m)$ where $\mathcal A$ is its $\sigma$-algebra. Let $\mathcal B(X)=\mathcal A/\sim$, where $A, B\in \mathcal A, A\sim B$ iff $\mathfrak m((A\setminus B)\cup (B\setminus A))=0$.
A Banach space $(\mathcal M, \|\cdot\|)$ is called a \textbf{$L^{\infty}(X, \mathfrak m)$-premodule} if there is a bilinear map
$$L^{\infty}(X, \mathfrak m)\times \mathcal M\to \mathcal M, \quad (f, v)\mapsto f\cdot v,$$
such that for each $v\in \mathcal M$, $f, g\in L^{\infty}(X, \mathfrak m)$,
$$(fg)\cdot v=f\cdot (g\cdot v), \, 1\cdot v=v, \, \|f\cdot v\|\leq \|f\|_{L^{\infty}(X,\mathfrak m)}\|v\|.$$
An $L^{\infty}(X,\mathfrak m)$-premodule $(\mathcal M, \|\cdot\|)$ is called a \textbf{$L^{\infty}(X, \mathfrak m)$-module} if
(1) Locality: for each $x\in \mathcal M$, $A_n\in \mathcal B(X)$,
$$\forall n, \, \chi_{A_n}\cdot v=0 \Rightarrow \chi_{\cup_n A_n}\cdot v=0;$$
(2) Gluing: for every sequence $\{v_n\}\subset \mathcal M$, $\{A_n\}\subset \mathcal B(X)$, if
$$\chi_{A_i\cap A_j}\cdot v_i=\chi_{A_i\cap A_j}\cdot v_j, \forall i,j, \quad \limsup_{n\to \infty}\|\sum_{i=1}^n\chi_{A_i}\cdot v_i\|<\infty,$$
then there is $v\in \mathcal M$,
$$\chi_{A_i}\cdot v=\chi_{A_i}\cdot v_i, \forall i, \quad \|v\|\leq \liminf_{n\to \infty}\|\sum_{i=1}^n\chi_{A_i}\cdot v_i\|.$$
A \textbf{module morphism} is a map $T: \mathcal M_1\to \mathcal M_2$ which is bounded and linear by viewing $\mathcal M_1$ and $\mathcal M_2$ as Banach spaces and satisfies the locality condition
$$T(f\cdot v)=f\cdot T(v), \, \forall v\in \mathcal M_1, f\in L^{\infty}(X, \mathfrak m).$$
Denote all module morphism from $\mathcal M_1$ to $\mathcal M_2$ by $\op{Hom}(\mathcal M_1, \mathcal M_2)$. The \textbf{dual module} $\mathcal M^*=\op{Hom}(\mathcal M, L^1(X,\mathfrak m))$.
If there is a non-negative map $|\cdot| : \mathcal M\to L^p(X, \mathfrak m)$, $p\in [0, \infty]$ satisfying that
$$\||v|\|_{L^p(X, \mathfrak m)}=\|v\|, \quad |f\cdot v|=|f||v|, \mathfrak m-a.e., \forall \, v\in \mathcal M, f\in L^{\infty}(X,\mathfrak m),$$
then $\mathcal M$ is called a \textbf{$L^p(X,\mathfrak m)$-normed $L^{\infty}(X,\mathfrak m)$-premodule (resp. module)} when $\mathcal M$ is a $L^{\infty}(X, \mathfrak m)$-premodule (resp. module). $|\cdot|$ is called the pointwise $L^{p}(X,\mathfrak m)$-norm.
And locally, for $A\in \mathcal B(X)$, we can define $\left.\mathcal M\right|_A=\{v\in \mathcal M, \, |v|=0\,\, \mathfrak m-a.e. \text{ on }A^c\}$ which is a $L^p(X,\mathfrak m)$-normed $L^{\infty}(X,\mathfrak m)$-module.
A $L^2(X,\mathfrak m)$-normed $L^{\infty}(X,\mathfrak m)$-module which is a Hilbert space under $\|\cdot\|$ is called a \textbf{Hilbert module}.
Consider a $L^2(X,\mathfrak m)$-normed $L^{\infty}(X,\mathfrak m)$-module $\mathcal M$, $V\subset \mathcal M$. Let $\op{Span}(V)$ be the collection of $v\in \mathcal M$ such that there is a Borel decomposition $\{X_n\}$ of $X$, and for each $n$, there are $v_{1, n}, \cdots, v_{k_n, n}\in V$, $f_{1, n}, \cdots, f_{k_n, n}\in L^{\infty}(X,\mathfrak m)$,
$$\chi_{X_n}v=\sum_1^{k_n}f_{i, n}v_{i, n}, \forall n.$$
And we say $V$ generates $\mathcal M$ if $\overline{\op{Span}(V)}=\mathcal M$.
\begin{Def}[\cite{Gi18}]
There is a unique, up to isomorphism, Hilbert module $L^2(T^*X)$ endowed with a linear map $d: W^{1, 2}(X, d, \mathfrak m)\to L^2(T^*X)$ satisfying
$$|df|=|\nabla f|_w, \mathfrak m-a.e., \forall f\in W^{1,2}(X, d, \mathfrak m); \quad d(W^{1,2}(X, d, \mathfrak m)) \text{ generates }L^2(T^*X).$$
We call $L^2(T^*X)$ the cotangent module of $(X, d, \mathfrak m)$. The dual of $L^2(T^*X)$ is called the tangent module of $(X, d, \mathfrak m)$ and denoted by $L^2(TX)$. Elements of $L^2(TX)$ is called vector fields. And denote by $\nabla f$ the dual of $df$.
\end{Def}
Let $D(\op{div})\subset L^2(TX)$ be the space of vector fields $v$ satisfying that there is $f\in L^2(X, \mathfrak m)$ such that for any $g\in W^{1,2}(X, d, \mathfrak m)$,
$$\int fg d\mathfrak m=-\int dg(v) d\mathfrak m.$$
$f$ is called the \textbf{divergence} of $v$ and denoted by $\op{div}(v)$. If $f\in D(\Delta)$, then $\nabla f\in D(\op{div})$ and $\op{div}(\nabla f)=\Delta f$ (see \cite[Proposition 2.3.14]{Gi18}).
For two Hilbert module $\mathcal H_1, \mathcal H_2$, we can define the tensor product $\mathcal H_1\otimes \mathcal H_2$ and the exterior product $\mathcal H_1\wedge \mathcal H_2$ (see Section 1.5 in \cite{Gi18}). And denote the pointwise $L^2(X, \mathfrak m)$-normal of the tensor product $L^2((T^*)^{\otimes 2}X)$ by $|\cdot|_{\op{HS}}$.
\subsection{$\op{CD}(K, N)$-spaces and $\RCD(K, N)$-spaces}
In this subsection, we recall the definitions of $\op{CD}(K, N)$-spaces and $\RCD(K, N)$-spaces. The notion of curvature dimension condition $\op{CD}(K, N)$ was introduced by Lott-Villani (\cite{LV}) and Strum (\cite{St1, St2}) independently. The Riemannian curvature dimension condition $\RCD(K, N)$ was introduced by a series of works \cite{AGS, Gi15, EKS, CMi, AMS19}.
Let $(X, d, \mathfrak m)$ be as in the beginning of this section. Let $\mathcal{P}_2(X)$ be the space of Borel probability measures $\mu$ on $(X, d)$ satisfying $\int_X d(x_0, x)^2d\mu(x)<\infty $ for some $x_0\in X$. For $\mu, \nu\in \mathcal{P}_2(X)$, define
$$W_2(\mu, \nu)=\left(\inf\int\int_0^1|\dot\gamma(t)|^2dt d\pi(\gamma)\right)^{\frac12},$$
where the infimum is taken among all $\pi\in \mathcal{P}(C([0,1], X))$ with $(e_0)_*(\pi)=\mu$, $(e_1)_*(\pi)=\nu$. In fact, the minimal can always be achieved. We call the plan $\pi$ which achieves the minimal an \textbf{optimal transportation} and denote the set of optimal transportations by $\op{OpGeo}(\mu, \nu)$.
For $N\geq 1, K$, let $\sigma_{K, N}: [0, 1]\times \Bbb R^{+}\to \Bbb R$ be as
$$\sigma_{K, N}^{t}(\theta)=\left\{\begin{array}{cc}
+\infty, & K\theta^2\geq N\pi^2,\\
\frac{\sin(t\theta\sqrt{K/N})}{\sin(\theta\sqrt{K/N})}, & 0<K\theta^2<N\pi^2,\\
t, & K\theta^2=0,\\
\frac{\sinh(t\theta\sqrt{-K/N})}{\sinh(\theta\sqrt{-K/N})},& K\theta^2<0.\end{array}\right.$$
and let
$$\tau_{K, N}^{t}(\theta)=t^{\frac1N}\sigma_{K, N-1}^t(\theta)^{\frac{N-1}{N}}.$$
\begin{Def}[\cite{LV, St1, St2}]
Given $K\in \Bbb R, N\geq 1$, we say a metric measure space $(X, d, \mathfrak m)$ is a $\op{CD}(K, N)$-space if for any two measures $\mu_0, \mu_1\in \mathcal{P}_2(X)$ with bounded support which contains in $\mathfrak m$'s support, there exists $\pi\in \op{OpGeo}(\mu_0, \mu_1)$ such that for each $t\in [0, 1]$
$$-\int\rho_t^{1-\frac{1}{N}}dm\leq -\int \tau_{K, N}^{1-t}(d(\gamma(0),\gamma(1)))\rho_0^{-\frac{1}{N}}(\gamma(0))+\tau_{K, N}^t(d(\gamma(0),\gamma(1)))\rho_1^{-\frac{1}{N}}(\gamma(1))d\pi(\gamma),$$
where $(e_t)_{\sharp}\pi=\rho_t\mathfrak m + \mu_t, \mu_t\bot \mathfrak m$. We call $(X, d, \mathfrak m)$ is a $\op{CD}^*(K, N)$-space if the above inequality holds for $\sigma^t_{K, N}$ instead of $\tau^t_{K, N}$.
\end{Def}
\begin{Def}[\cite{AGS, Gi15}]
A metric measure space $(X, d, \mathfrak m)$ is a $\RCD(K, N)$-space (resp. $\RCD^*(K, N)$-space) if it is an infinitesimally Hilbertian $\op{CD}(K, N)$-space (resp. $\op{CD}^*(K, N)$-space).
\end{Def}
And we say a metric measure space $(X, d, \mathfrak m)$ is a $\op{CD}_{\op{loc}}(K, N)$-space if for a cover $\{A_i\}$ of $X$, $A_i\subset X$, $\cup_iA_i=X$, $\op{CD}(K, N)$ holds in each $A_i$. At the local level $\bigcap_{K'<K}\op{CD}_{\op{loc}}^*(K', N)$ coincide with $\bigcap_{K'<K}\op{CD}_{\op{loc}}(K', N)$.
We call $(X, d, \mathfrak m)$ is \textbf{essentially non-branching} if for any $\mu, \nu\in \mathcal{P}_2(X)$ with bounded support, each $\pi\in\op{OpGeo}(\mu, \nu)$ is concentrated on a Borel set of non-branching geodesics. It was proved in \cite{CMi}, an essentially non-branching metric measure space $(X, d, \mathfrak m)$ is $\op{CD}(K, N)$ if and only if it is $\op{CD}^*(K, N)$ if and only if it is $\op{CD}^*_{\op{loc}}(K, N)$. By \cite{AGS, AGMR, RS}, a $\RCD(K, \infty)$-space is essentially non-branching.
\begin{Thm}[\cite{EKS}] \label{equivalent}
Assume a metric measure space $(X, d, \mathfrak m)$ with $\op{supp}(\mathfrak m)=X$ satisfies the infinitesimally Hilbertian and Sobolev to Lipschitz property, i.e. any $f\in W^{1,2}(X, d, \mathfrak m)$ with $|\nabla f|_w\leq 1$ $\mathfrak m$-a.e. admits a $1$-Lipschitz representative. Then the followings are equivalence:
(i) $(X, d, \mathfrak m)\in \op{CD}^*(K, N)$;
(ii) The Bakry-Ledoux pointwise gradient estimate $\op{BL}(K, N)$ holds: for $f$ of finite Cheeger energy,
$$|\nabla H_t(f)|^2_w+\frac{4Kt^2}{N(e^{2Kt}-1)}|\Delta H_t f|^2\leq e^{-2Kt} H_t(|\nabla f|^2_w), \mathfrak m-a.e.$$
where
$$\frac{d}{dt}H_t(f)=\Delta H_t(f), \quad H_0(f)=f.$$
(ii) The Bochner/Bakry-\'Emery inequality $\op{BK}(K, N)$ holds: for $f\in D(\Delta), \Delta f\in W^{1,2}(X, d, \mathfrak m)$, $g\in D(\Delta), g\geq 0$, $\Delta g\in L^{\infty}(X, \mathfrak m)$,
$$\frac12\int \Delta g|\nabla f|^2_w d\mathfrak m-\int g\left<\nabla(\Delta f), \nabla f\right>d\mathfrak m\geq K\int g|\nabla f|^2_wd\mathfrak m +\frac1{N}\int g(\Delta f)^2 d\mathfrak m.$$
\end{Thm}
Last, let's recall the following existence of good cut-off functions.
\begin{Lem}[\cite{MN}]\label{cut-off}
Let $(X, d, \mathfrak m)$ be an $\RCD(K, N)$-space for $N<\infty$ and let compact subset $S=\partial \Omega$ with $\Omega$ Borel. For each $R>0$, $0<10r_1<r_2<R$, there exists a Lipschitz function $\phi: X\to [0, 1]$ such that
(i) $\phi=1$ on $A_{3r_1, \frac{r_2}{3}}(S)$, $\phi=0$ on $X\setminus A_{2r_1, \frac{r_2}{2}}(S)$;
(ii) $r_1^2|\Delta \phi|+r_1|\nabla \phi|\leq C(K, N, R)$ a.e. on $A_{2r_1, 3r_1}(S)$;
(ii) $r_2^2|\Delta \phi|+r_2|\nabla \phi|\leq C(K, N, R)$ a.e. on $A_{\frac{r_2}3, \frac{r_2}2}(S)$.
\end{Lem}
\subsection{Regular Lagrangian flow}
In this subsection, we recall the definition and some facts about Regular Lagrangian flow (see \cite{AT14, GR18}).
\begin{Def}
Consider a metric measure space $(X, d, \mathfrak m)\in \RCD(K, N)$ and a time-dependent vector field $V_t\in L^2([0, 1], L^2_{\op{loc}}(TX))$. We say that
$$F: [0, 1]\times X\to X$$
is a Regular Lagrangian flow (RLF for brief) of $V_t$ if
(i) $\left(F_s\right)_{\sharp}\mathfrak m\leq C\mathfrak m$ for some $C>0$;
(ii) For $\mathfrak m$-a.e. $x\in X$, the curve $s\mapsto F_s(x), s\in [0, 1]$ is continuous and $F_0(x)=x$;
(iii) For each $f\in W^{1,2}(X, d, \mathfrak m)$, for $\mathfrak m$-a.e. $x\in X$, the function $s\mapsto f(F_s(x))$ belongs to $W^{1,1}(0, 1)$ and
$$\frac{d}{ds} f(F_s(x))=df (V_s)(F_s(x)), \mathfrak m\times \mathcal L^1-a.e. (x, s).$$
\end{Def}
\begin{Thm}[\cite{AT14}] \label{exist-RLF}
For a time-dependent vector $V_t\in L^1([0, T], L^2(TX))$ with $V_t\in D(\op{div})$ for a.e. t, if
\begin{equation*}
\op{div}(V_t)\in L^1([0, T], L^2(X, \mathfrak m)), \max\{-\op{div}(V_t), 0\}\in L^1([0, T], L^{\infty}(X,\mathfrak m)), \nabla V_t\in L^1([0, T], L^2(T^{\otimes2}X)),
\end{equation*}
then there exists a unique, up to $\mathfrak m$-a.e. equality, RLF $(F_t)_{t\in [0, T]}$ for $V_t$ and
$$(F_t)_{\sharp}\mathfrak m\leq \exp\left(\int_0^t \|\max\{-\op{div}(V_s), 0\}\|_{L^{\infty}}ds\right)d\mathfrak m.$$
\end{Thm}
The RLFs are closely related with the continuity equation. A $W_2$-continuous curve $(\mu_t)_{t\in [0,T]}\subset \mathcal P(X)$ with $\mu_t\leq C\mathfrak m$ and a vector $V_t\in L^2([0, T], L^2(TX))$ is said a solution of the \textbf{continuity equation}
\begin{equation}\frac{d}{dt}\mu_t+\op{div}(V_t\mu_t)=0 \label{continuity-equ}\end{equation}
iff for each $f\in W^{1, 2}(X, d, \mathfrak m)$, the map $t\mapsto \int f d\mu_t$ is absolutely continuous and
$$\frac{d}{dt}\int f d\mu_t=\int df (V_t)d\mu_t, a.e. t\in [0, T].$$
For $V_t$ as in Theorem~\ref{exist-RLF} and $F_t$ the unique RLF of $V_t$, let $\mu_t=(F_t)_{\sharp}\mu_0$, $\mu_0\in \mathcal P_2(X)$.
By \cite{AT14}, $(\mu_t, V_t)$ is a solution of the continuity equation.
\subsection{The differential formula in $\RCD(K, N)$-spaces}
In this subsection, we alway assume $(X, d, \mathfrak m)$ is a $\RCD(K, N)$-space.
Define the class of test functions as
$$\op{Test}(X)=\{f\in D(\Delta)\cap L^{\infty}(X, \mathfrak m), \, |\nabla f|_w\in L^{\infty}(X, \mathfrak m), \Delta f\in W^{1,2}(X, d, \mathfrak m)\}.$$
It was shown in \cite{Sav14, Han18} that if $f\in \op{Test}(X)$, then $|\nabla f|^2\in D(\Delta)$ and one may define
$$\Gamma_2(f)=\frac12\Delta|\nabla f|^2-\left<\nabla f, \nabla \Delta f\right>.$$
\begin{Def}[\cite{Gi18}]
Let $W^{2,2}(X, d, \mathfrak m)$ be the set of $f\in W^{1,2}(X, d, \mathfrak m)$ satisfying that there is $A\in L^2((T^*)^{\otimes 2}(X))$ such that for any $g_1, g_2, h\in \op{Test}(X)$
$$2\int hA(\nabla g_1, \nabla g_2)d\mathfrak m=-\int \left<\nabla f, \nabla g_1\right>\op{div}(hg_2)+\left<\nabla f, \nabla g_2\right>\op{div}(hg_1)+h\left<\nabla f, \nabla\left<\nabla g_1, \nabla g_2\right>\right>d\mathfrak m.$$
$A$ is called the Hessian of $f$, denoted by $\op{Hess}(f)$.
\end{Def}
It was proved in \cite{Gi18} that $D(\Delta)\subset W^{2,2}(X, d, \mathfrak m)$ and let $H^{2,2}(X)$ be the closure of $D(\Delta)$ in $W^{2,2}(X, d, \mathfrak m)$.
\begin{Thm}[Improved Bocher inequality, \cite{Sav14, Han18}] \label{Boc-ine}
For $(X, d, \mathfrak m)\in \op{RCD}(K, N)$, $K\in \Bbb R, N\in [1, \infty)$ and $f\in \op{Test}(X)$, we have that $f\in W^{2,2}(X, d, \mathfrak m)$ and
$$\Gamma_2(f)\geq \left(K|\nabla f|^2+|\op{Hess}(f)|_{HS}^2\right)\mathfrak m.$$
\end{Thm}
Given a function $\phi: X\to \Bbb R\cup \{-\infty\}$ not identically $-\infty$, its \textbf{$c$-transform} $\phi^c: X\to \Bbb R\cup \{-\infty\}$ is defined as
$$\phi^c(x)=\inf_{y\in X}\frac{d^2(x, y)}{2}-\phi(y).$$
We call $\phi$ is \textbf{$c$-concave} if $\phi^{cc}=\phi$. Let $\partial^c\phi\subset X^2$ be the set of $(x, y)\in X^2$ such that
$$\phi(z)-\phi(x)\leq \frac{d^2(z, y)}2-\frac{d^2(x, y)}2, \, \forall\, z\in X.$$
A test plan $\pi\in \op{OpGeo}(\mu, \nu)$ if and only if there is a $c$-concave function $\phi$ such that $\op{supp}((e_0, e_1)_{\sharp}\pi)\subset \partial ^c\phi$ and such $\phi$ is called \textbf{Kantorovich potential} from $\mu$ to $\nu$. And for any $t\in (0, 1)$, $t\phi$ is a Kantorovich potential from $\mu$ to $(e_t)_{\sharp}(\pi)$.
\begin{Thm}[\cite{Gi13}, see also \cite{GiT}]\label{first-diff1}
Consider a $\RCD(K, N)$-space $(X, d, \mathfrak m)$. Let $\pi$ be an optimal geodesic test plan with bounded support
and consider $f\in W^{1,2}(X, d, \mathfrak m)$. We have that the map $t\mapsto f\circ e_t$ is $C^1([0, 1])$ and for $t\in [0, 1]$
\begin{equation}
\frac{d}{dt}f\circ e_t =\left<\nabla f, \nabla \phi_t\right>\circ e_t,
\end{equation}
where $\phi_t$ is any function such that for some $s\neq t$, $s\in [0, 1]$, $-(s-t)\phi_t$ is a Kantorovich potential from $\mu_t=(e_t)_{\sharp}(\pi)$ to $\mu_s=(e_t)_{\sharp}(\pi)$.
\end{Thm}
\begin{Thm}[\cite{GiT}]
Let the assumption be as in Theorem~\ref{first-diff1} and let $f\in H^{2,2}(X)$. Then the map $t\mapsto f\circ e_t$ is in $C^2([0,1])$ and for $t\in [0, 1]$
\begin{equation}
\frac{d^2}{dt^2}f\circ e_t=\op{Hess}(f)(\nabla \phi_t, \nabla \phi_t)\circ e_t.
\end{equation}
\end{Thm}
A corollary of Theorem~\ref{first-diff1} is the following first order differential estimates along geodesics (see \cite[Corollary 3.14]{Deng}):
\begin{Cor}[\cite{Deng}] \label{first-diff2}
Let $(X, d, \mathfrak m)$ be a $\RCD(K, N)$-space and let $p\in X$, $f\in W^{1, 2}(X, d, \mathfrak m)$. For $\mathfrak m$-a.e. $x\in X$, the map $t\mapsto f(\gamma_{x, p}(t))$ is in $W^{1,1}_{\op{loc}}([0, d(p, x)))$ and
\begin{equation}
\frac{d}{dt} f(\gamma_{x, p}(t))= -df(\nabla d_p)(\gamma_{x, p}(t)), \text{ for } a.e. t\in [0, d(p, x)),
\end{equation}
where $\gamma_{x, p}$ is a unit speed geodesic from $x$ to $p$.
\end{Cor}
For a fixed point $p\in X$ and a measure $\mu\in \mathcal P_2(X)$ with $\mu\leq C\mathfrak m$, let $\gamma_{x, p}(t), t\in [0, 1]$ be a constant speed geodesic from $x\in \op{supp}(\mu)$ to $p$, let $D=\sup _{x\in \op{supp}(\mu)}d(x, p)$ and let $\mu_t=(\gamma_{\cdot, p}(t))_{\sharp} (\mu)$. By \cite[Theorem 3.15]{Deng}, we have that $(\mu_t)_{t\in [0, 1-\delta]}\leq C(K, N, D, \delta)\mathfrak m$ is a $W_2$-geodesic for any $1>\delta>0$, and $(\mu_t)_{t\in [0, 1-\delta]}$ solves the continuity equation
$$\frac{d}{dt} \mu_t+\op{div}(-\nabla d_p \mu_t)=0.$$
And we have the second order differential formula, as \cite[Proposition 3.21]{Deng},
\begin{Cor} \label{second-diff2}
Let the assumption be as in Corollary~\ref{first-diff2}, let $\Pi\leq C(\mathfrak m\times \mathfrak m)$ be a nonnegative, compactly supported measure on $X\times X$ and let $f\in H^{2,2}(X)$. Then for each $t, s\in (0, 1]$,
\begin{equation}
\int \left<\nabla f, \nabla d_x\right>(\tilde{\gamma}_{x, y}(t))-\left<\nabla f, \nabla d_x\right>(\tilde{\gamma}_{x, y}(s))d\Pi(x, y)=\int_s^t\int d(x, y)\op{Hess}(f)(\nabla d_x, \nabla d_x)(\tilde{\gamma}_{x, y}(\tau))d\Pi(x, y)d\tau,
\end{equation}
where $\tilde{\gamma}_{x, y}:[0, 1]\to X$ is a constant speed geodesic from $x$ to $y$.
\end{Cor}
\subsection{Warped product and $(K, N)$-cone over metric measure spaces}
Consider two metric measure spaces $(Z, d_Z, \mathfrak m_Z)$ and $(Y, d_Y, \mathfrak m_Y)$ and two continuous maps $w_d, w_m: Z\to [0, \infty)$ with $\{w_d=0\}\subset \{w_m=0\}$. A \textbf{warped product} $Z\times_w Y$ is the space $Z\times Y$ admitting the metric
$$d_w(p, q)=\inf\{l_w(\gamma),\, \gamma \text{ is a absolutely continuous curve between } p, q \},$$
where $\gamma=(\gamma^Z, \gamma^Y)$,
$$l_w(\gamma)=\int_0^1\sqrt{|\dot \gamma_t^Z|^2+w_d^2(\gamma_t^Z)|\dot \gamma_t^Y|^2}dt.$$
And the measure $\mathfrak m_w$ on $Z\times_w Y$ is defined as
$$d\mathfrak m_w= w_md\mathfrak m_Z\otimes d\mathfrak m_Y.$$
\begin{Def}
The $(K, N)$-cone $(C(Y), d_{K}, \mathfrak m_N)$ over a metric measure space $(Y, d, \mathfrak m_Y)$ is defined as follows: let $H=K/(N-1)$ and $(t, y_1), (t, y_2)\in C(Y)$,
$$C(Y)=\begin{cases} \left[0, \pi/\sqrt H\right]/ \{\{0, \pi/\sqrt H\}\times Y\}, & K>0;\\ [0, \infty)\times Y/\{0\times Y\}, & K\leq 0; \end{cases}$$
$$d_K((t, y_1), (s, y_2))=\begin{cases} \sqrt{t^2+s^2-2st\cos\min\{\pi, d(y_1, y_2)\}} & K=0;\\
(\op{sn}'_H)^{-1}\left(\op{sn}'_H(t)\op{sn}'_H(s)+H\op{sn}_H(s)\op{sn}_H(t)\cos \min\{\pi, d(y_1, y_2)\}\right), & K\neq 0; \end{cases}$$
$$\mathfrak m_N=\op{sn}^{N-1}_H(t)dt\otimes \mathfrak m_Y.$$
\end{Def}
It is obvious that if $\op{diam}(Y)\leq \pi$, then $C(Y)= I_K\times_{\op{sn}_H}^{N-1}Y$, where $I_K=[0, \frac{\pi}{\sqrt H}]$ for $K>0$ and $I_K=[0, \infty)$ for $K\leq 0$ and $w_m=\op{sn}_H^{N-1}$, $w_d=\op{sn}_H$. And in \cite{Ket1}, Ketterer showed that for $N\geq 2$, $K\geq 0$, a $(K, N)$-cone over $(Y, d, \mathfrak m)$ is a $\RCD(K, N)$-space if and only if $(Y, d, \mathfrak m)$ is a $\RCD(N-2, N-1)$-space. More precisely,
\begin{Thm}[\cite{Ket1}] \label{cone-rcd}
(i) Assume a metric measure space $(Y, d, \mathfrak m)\in \RCD(N-2, N-1)$, $N\geq 2$, $K\geq 0$ and $\op{diam}(Y)\leq \pi$, Then the $(K, N)$-cone over $(Y, d, \mathfrak m)$, $(C(Y), d_K, \mathfrak m_N)\in \RCD(K, N)$;
(ii) Assume the $(K, N)$-cone over a metric measure space $(Y, d, \mathfrak m)$, $(C(Y), d_K, \mathfrak m_N)\in \RCD(K, N)$ with $N\geq 2$, then $(Y, d, \mathfrak m)\in \RCD(N-2, N-1)$ with $\op{diam}(Y)\leq \pi$.
\end{Thm}
\subsection{Measure decomposition and Heintze-Karcher inequality in $\RCD(K, N)$-spaces}
In this subsection, we briefly recall the following measure decomposition in $\RCD(K, N)$-spaces by \cite{CM17, CMo} and the Heintze-Karcher inequality in $\RCD(K, N)$-spaces given by \cite{Ket2}. Note that in \cite{CM17}, to derive the measure decomposition result they also assume the condition that $\mathfrak m(X)<\infty$. In \cite{CMo}, they showed that this assumption is unnecessary.
Consider a measurable space $(R, \mathcal R, \mathfrak m)$ and a map $\mathfrak Q: R\to Q$. One can equip $Q$ with a $\sigma$-algebra $\mathcal Q$: $B\in \mathcal Q$ iff $\mathfrak Q^{-1}(B)\in \mathcal R$. Let $\mathfrak Q_{\sharp} \mathfrak m=\mathfrak q$ which is a probability measure on $Q$.
A \textbf{disintegration} of $\mathfrak m$ which is consistent with $\mathfrak Q$ is a map $\mathcal R\times Q\to [0, 1]$, $(A, \alpha)\mapsto \mathfrak m_{\alpha}(A)$ satisfying that:
(i) $\mathfrak m_{\alpha}$ is a probability measure on $(R, \mathcal R)$ for each $\alpha\in Q$;
(ii) $\alpha\mapsto \mathfrak m_{\alpha}(A)$ is $\mathfrak q$-measurable for each $A\in \mathcal R$;
(iii) For $A\in \mathcal R, B\in \mathcal Q$,
$$\mathfrak m(A\cap \mathfrak Q^{-1}(B))=\int_B \mathfrak m_{\alpha}(A)d\mathfrak q(\alpha).$$
We call $\{\mathfrak m_{\alpha}\}_{\alpha\in Q}$ a disintegration of $\mathfrak m$ and call $\mathfrak m_{\alpha}$ the conditional probability measures.
A disintegration $\{\mathfrak m_{\alpha}\}_{\alpha\in Q}$ is strongly consistent with $\mathfrak Q$ if for $\mathfrak q$-a.e. $\alpha$, $\mathfrak m_{\alpha}(\mathfrak Q^{-1}(\alpha))=1$.
Let $(X, d, \mathfrak m)$ be as in the beginning of this section and be essentially non-branching. Let $u: X\to \Bbb R$ be a $1$-Lipschitz map.
A set $A\subset X\times X$ is $d$-cyclically monotone if for any finite set of points $(x_1, y_1), \cdots, (x_n, y_n)\in A$
$$\sum_{i=1}^nd(x_i, y_i)\leq \sum_{i=1}^n d(x_i, y_{i+1}), \, y_{n+1}=y_1.$$
A $d$-cyclically monotone set associated with $u$ is defined as
$$\Gamma=\{(x, y)\in X\times X, \, u(x)-u(y)=d(x, y)\}.$$
It is obvious that $(x, y)\in \Gamma$ implies that for any minimizing geodesic $\gamma$ from $x$ to $y$, $(\gamma_t, \gamma_s)\in \Gamma$ for $0\leq s\leq t\leq 1$.
Let the transport rays $R=\Gamma\cup \Gamma^{-1}$, where $\Gamma^{-1}=\{(x, y)\in X\times X, \, (y, x)\in \Gamma\}$. Let $\mathcal T=P_1(R\setminus \{(x, y), x=y\in X\})\subset X$ where $P_1(x, y)=x$. And the sets
$$A_{+}=\{x\in \mathcal T, \, \exists z, w\in \Gamma(x), (z, w)\notin R\},$$
$$A_{-}=\{x\in \mathcal T, \, \exists z, w\in \Gamma^{-1}(x), (z, w)\notin R \},$$
are called forward and backward branching points respectively,
where $\Gamma(x)=\{y\in X, (x, y)\in \Gamma\}$, and similar define $\Gamma^{-1}(x), R(x)$.
The initial and final points are
$$\mathfrak a=\{x, \, \Gamma^{-1}(x)=\{x\} \}, \,\, \mathfrak b=\{x, \, \Gamma(x)=\{x\}\}.$$
Let $\mathcal T_u=\mathcal T\setminus (A_{+}\cup A_{-})$. It was proved that
$$x\sim y \Leftrightarrow (x, y)\in R$$ is an equivalence relation on $\mathcal T_u$ (\cite{CM17, CMo}). Denote this relation by $R_u$ and the equivalence classes by $\{X_{\alpha}\}_{\alpha\in Q}$.
Each $X_{\alpha}$ is isometric to an interval $I_{\alpha}\subset \Bbb R$ via an isometry $\gamma_{\alpha}: I_{\alpha}\to X_{\alpha}$ such that $d(\gamma_{\alpha}(t), \gamma_{\alpha}(s))=|t-s|$ for $t, s\in I_{\alpha}$ .
The map $\gamma_{\alpha}$ extends to a geodesic in $X$ which is also denoted by $\gamma_{\alpha}$. Denote the closure $\bar I_{\alpha}$ of $I_{\alpha}$ by $[a(X_{\alpha}), b(X_{\alpha})]$.
\begin{Thm}[\cite{CM17, CMo}]\label{dis-com}
Let $(X, d, \mathfrak m)$ be an essentially non-branching $\op{CD}(K, N)$-space with $\op{supp}(\mathfrak m)=X$. Let $u: X\to \Bbb R$ be a $1$-Lipschitz function. Then
(i) there exists a disintegration $\{\mathfrak m_{\alpha}\}_{\alpha\in Q}$ of $ \mathfrak m\llcorner_{\mathcal T_u}$ that is strongly consistent;
(ii) there is $Q'\subset Q$ such that $\mathfrak{q}(Q\setminus Q')=0$ and for each $\alpha\in Q'$, $\mathfrak m_{\alpha}$ is a Radon measure with $\mathfrak m_{\alpha}=h_{\alpha}\mathcal{H}^1\llcorner_{X_{\alpha}}$ and $(X_{\alpha}, d, \mathfrak m_{\alpha})$ verifies the condition $\op{CD}(K, N)$. More precisely, for $\alpha\in Q'$, it holds that
\begin{equation}h_{\alpha}(\gamma(t))^{\frac1{N-1}}\geq \sigma_{K, N-1}^{1-t}(|\gamma'|)h_{\alpha}(\gamma(0))^{\frac1{N-1}}+\sigma_{K, N-1}^{t}(|\gamma'|)h_{\alpha}(\gamma(1))^{\frac1{N-1}},\label{ele-com}\end{equation}
for each constant speed geodesic $\gamma: [0, 1]\to (a(X_{\alpha}), b(X_{\alpha}))$.
\end{Thm}
And by \cite[Corollary 4.3]{Ket2}, we have that
\begin{Lem}[\cite{Ket2}] \label{ele-com0}
Let the assumption be as in Theorem~\ref{dis-com}. Then for each $0\leq a<b\leq 1$,
$$\frac{h_{\alpha}(b)}{h_{\alpha}(a)}\leq \left(\op{sn}'_H(b-a)+\frac{(\ln h_{\alpha})_+'(a)}{N-1}\op{sn}_H(b-a)\right)_+^{N+1},$$
where
$$(\ln h_{\alpha})_+'(a)=\lim_{h\downarrow 0}\frac{\ln h_{\alpha}(a+h)-\ln h_{\alpha}(a)}{h},$$
$h_{\alpha}(t)=h_{\alpha}(\gamma(t))$ and $f_+=\max\{f, 0\}$.
\end{Lem}
Consider $(X, d, \mathfrak m)$ be as above lemma. Let $\Omega\subset X$ be a Borel subset and let $S=\partial \Omega$ with $\mathfrak m(S)=0$. Let
$$d_s=\begin{cases} d(x, S), & x\in X\setminus \Omega;\\ -d(x, S), & x\in \Omega. \end{cases}$$
Then $d_s$ is $1$-Lipschitz. By Theorem~\ref{dis-com}, there is a disintegration of $d_s$, $\{\mathfrak m_{\alpha}\}_{\alpha\in Q}$ and a partition $\{X_{\alpha}\}_{\alpha\in Q}$ of $X$ up a measure zero set (By \cite[Lemma 3.4]{CMo}, $\mathfrak m(\mathcal T\setminus \mathcal T_{d_s})=0$. And $\mathcal T\supset X\setminus S$. Thus $\mathfrak m(S)=0$ implies $\mathfrak m(X\setminus \mathcal T_{d_s})=0$).
Let
$A= \mathfrak{Q}^{-1}(\mathfrak{Q}(S\cap \mathcal T_{d_s}))$. Then for each $\alpha\in \mathfrak Q(A)$, there is a unique $t_{\alpha}\in (a(X_{\alpha}), b(X_{\alpha}))$ such that $X_{\alpha}\cap S=\{\gamma(t_{\alpha})\}\neq \emptyset$. Identify the measurable set $\mathfrak Q(A)\subset Q$ with $A\cap S$ and one can assume $t_{\alpha}=0$.
Let $\hat Q=\{\alpha\in Q, \, \overline{X_{\alpha}}\setminus \mathcal T_{u}\subset \mathfrak a\cup \mathfrak b\}$. By \cite[Theorem 7.10]{CM17}, $\mathfrak q(Q\setminus \hat Q)=0$. Let $\mathcal T^*_{d_s}=\mathfrak Q^{-1}(\hat Q\cap Q')$. The sets $B_{\op{in}}=\Omega^{\circ}\cap \mathcal T^*_{d_s}\setminus (A\cap \mathcal T^*_{d_s})$ and
$B_{\op{out}}=\Omega^{c}\cap \mathcal T^*_{d_s}\setminus (A\cap \mathcal T^*_{d_s})$
are measurable (see \cite[Remark 5.1]{Ket2}).
We say that $S$ has \textbf{finite outer curvature} if $\mathfrak m(B_{\op{out}})=0$, $S$ has finite inner curvature if $\mathfrak m(B_{\op{in}})=0$, and $S$ has \textbf{finite curvature} if $\mathfrak m(B_{\op{in}}\cup B_{\op{out}})=0$.
If $S$ has finite outer curvature, we can define its \textbf{outer mean curvature} as
$$p\in S\mapsto H^{+}(p)=\left\{\begin{array}{cc}
\frac{d^{+}}{dr}\ln h_{\alpha}(\gamma_{\alpha}(0)), & p=\gamma_{\alpha}(0)\in S\cap A\cap \mathcal T^{*}_{d_s}\\
-\infty, & p\in R_{d_s}(B_{\op{in}})\cap S,\\
c (\text{for some } c\in \Bbb R), & \text{ otherwise}.\end{array}\right.$$
Switch the roles of $\Omega$ and $\overline{\Omega^c}$ and assume $S$ has finite inner curvature, we call the corresponding outer mean curvature the \textbf{inner mean curvature} and write as $H^{-}$.
If $S$ has finite curvature, the \textbf{mean curvature} defined as $\max\{H^+, -H^-\}=m$.
The \textbf{surface measure} of $S$ is defined as
$$\int_S \phi(x)d\mathfrak m_S(x)=\int_{\mathfrak Q(A\cap \mathcal T^*_{d_s})}\phi(\gamma_{\alpha}(0))h_{\alpha}(0)d\mathfrak q(\alpha),$$
for any continuous function $\phi: X\to \Bbb R$.
In \cite{Ket2}, Ketterer generalized the Heintze-Karcher inequality to $\op{CD}(K, N)$-spaces.
\begin{Thm} \label{HK-general}
Assume $(X, d, \mathfrak m)\in \op{CD}(K, N)$ is an essentially non-branching metric measure space, $N>1$. Let $\Omega\subset X$ be a closed Borel subset and let $S=\partial \Omega$ such that $\mathfrak m(S)=0$ and $S$ has finite outer curvature. Then
$$\mathfrak m(B_t(\Omega)\setminus \Omega)\leq \int_S\int_0^t J_{K, H^+(p), N}(r)dr d\mathfrak m_S(p), \forall \, t\in (0, \tilde D],$$
where $\tilde D=\op{diam}(X)$, $J_{K, H, N}(r)=\left(\op{sn}'_{K/N-1}(r)+\frac{H}{N-1}\op{sn}_{K/N-1}(r)\right)_+^{N-1}$.
If $S$ has finite curvature, then
\begin{equation}\mathfrak m(X)\leq \int_S\int_{-\tilde D}^{\tilde D} J_{K, m(p), N}(r)dr d\mathfrak m_S(p).\label{vol-1}\end{equation}
In particular, if $(X, d, \mathfrak m)$ is a $\op{RCD}(K, N)$-space, $K>0$, then the equality holds in \eqref{vol-1} iff there is a $\op{RCD}(N-2, N-1)$-space $(Y, d_Y, \mathfrak m_Y)$ such that $(X, d, \mathfrak m)=(C(Y), d_K, \mathfrak m_N)$
and $S$ is a constant mean curvature surface in $X$.
\end{Thm}
\section{Laplacian estimates}
Let $(X_i, d_i, \mathfrak m_i)$ be a sequence of $\RCD(K, N)$-spaces and let $\Omega_i\subset X_i$ be a Borel subset with $S_i=\partial \Omega_i$ closed and $\op{diam}(S_i)\leq D$. Define a signed distance function associated to $\Omega_i$.
$$d_{s, i}(x)=\begin{cases} d_i(x, S_i), & x\in X_i\setminus \Omega_i;\\ -d_i(x, S_i), & x\in \Omega_i. \end{cases}$$
Obviously, $d_{s, i}$ is $1$-Lipschitz.
Assume $(X_i, d_i, \mathfrak m_i)$ is measured Gromov-Hausdorff convergent to a metric measure space $(X, d, \mathfrak m)\in \RCD(K, N)$.
Then by \cite[Proposition 2.70]{Vi} or \cite[Proposition 2.12]{MN}, there is a $1$-Lipschitz function $d_s: X\to \Bbb R$ such that
$d_{s, i}$ converges uniformly to $d_s$ on any compact set. Let $S=\{x\in X, \, d_s(x)=0\}$, $\Omega=\{x\in X, \, d_s(x))\leq 0\}$. Then as in \cite[Lemma 3.25]{Hu}, we know that $d_s$ is a signed distance function associated to $\Omega$. Let $H=\frac{K}{N-1}$ and for $m\neq 0$ or $K\neq 0$, let
$$m=(N-1)\frac{\op{sn}'_H(r_0)}{\op{sn}_H(r_0)}.$$
In this section, we will show that
\begin{Thm}[Laplacian estimates] \label{lap-main}
Let $(X, d, \mathfrak m)$, $d_s$ be as above. Assume $S_i$ has finite outer curvature, $\op{diam}(S_i)\leq D$ and the mean curvature $m_i(x_i)\leq m$ for each $i$ and any $x_i\in S_i$.
Then if \eqref{lqvol-com}
$$\frac{\mathfrak m(A^+_{a, b}(S_i))}{\mathfrak m(S_i)}\geq (1-\epsilon_i) \int_a^b \left(\op{sn}'_H(r)+ \frac{m}{N-1}\op{sn}_H(r)\right)^{n-1}dr,$$
holds with $\epsilon_i\to 0$, $d_s\in \Delta(A_{a, b}(S))$. In particular, for $x\in A_{a, b}(S)$,
(i) For $m=0$ and $K=0$, we have that $\Delta d_s=0$;
(ii) For $m\neq 0$ or $K\neq 0$, $$\Delta d_s=(N-1)\frac{\op{sn}'_H(d_s+r_0)}{\op{sn}_H(d_s+r_0)}.$$
\end{Thm}
To prove the above Laplacian estimates, we first recall the following Laplacian formula in \cite[Corollary 4.16]{CMo}.
\begin{Thm}[\cite{CMo}]
\label{lap-com}
Let $(X, d, \mathfrak m)$ be a $\RCD(K, N)$-space. Consider the signed distance function $d_s$ associated with a Borel subset $\Omega\subset X$ and a compact boundary $S=\partial \Omega$ with $\mathfrak m(S)=0$. And assume the associated disintegration of $d_s$, $\mathfrak m=\int_Q\int_{X_{\alpha}} h_{\alpha}(r)dr d\mathfrak q(\alpha)$. Then $d_s\in D(\Delta, X\setminus S)$ and
$$\Delta d_s\llcorner_{X\setminus S}=\left(\ln h_{\alpha}\right)' \mathfrak m\llcorner_{X\setminus S}-\int_Q h_{\alpha}\left(\delta_{a(X_{\alpha})\cap (X\setminus \Omega)}+\delta_{b(X_{\alpha})\cap \Omega}\right)d\mathfrak q(\alpha),$$
where $\left(\ln h_{\alpha}\right)'$ is roughly the directional derivative of $\ln h_{\alpha}$ in the direction of $\nabla d_s$.
In particular
$$[\Delta d_s\llcorner_{X\setminus S}]_{\op{reg}}=\left(\ln h_{\alpha}\right)' \mathfrak m\llcorner_{X\setminus S}$$ and
$$[\Delta d_s\llcorner_{X\setminus S}]_{\op{sing}}=-\int_Q h_{\alpha}\left(\delta_{a(X_{\alpha})\cap (X\setminus \Omega)}+\delta_{b(X_{\alpha})\cap \Omega}\right)d\mathfrak q(\alpha).$$
\end{Thm}
Note that in \cite[Corollary 4.16]{CMo}, there is a negative sign in front of the derivative $(\ln h)'_{\alpha}$, where they defined $h'_{\alpha}$ as
$$h'_{\alpha}(x)=\lim_{t\to 0}\frac{h_{\alpha}(g_t(x))-h_{\alpha}(x)}{t},$$
and $g_t(x)=y$ such that $d_s(x)-d_s(y)=t$. Roughly speaking the derivative $h'_{\alpha}$ there is the directional derivative in the direction of $-\nabla d_s$. Compared the one in Lemma~\ref{ele-com0} in this paper, we always denote
$$h'_{\alpha}(x)=\lim_{t\to 0}\frac{h_{\alpha}(\gamma(t_0+t))-h_{\alpha}(\gamma(t_0))}{t},$$
where $\gamma(t_0)=x$ and $\gamma$ is a unit speed geodesic in $X_{\alpha}$ such that $d_s(\gamma(l))-d_s(\gamma(t))=l-t$.
Now using \eqref{ele-com}, as the discussion of \cite[Lemma 4.1]{Ket2} and the proof of Laplacian comparison in manifolds with lower Ricci curvature bound, we have the following Laplacian comparison.
\begin{Lem} \label{glap-com}
Let the assumption be as in Theorem~\ref{lap-com} and assume $S$ has finite outer curvature. Assume that for each $x\in S$, the mean curvature $$m(x)\leq m.$$
(i) If $m=0$ and $K= 0$, we have
$$[\Delta d_s\llcorner_{X\setminus (\Omega\cup S)}]_{\op{reg}}(x)\leq 0;$$
(ii) If $m\neq 0$ or $K\neq 0$, then
$$[\Delta d_s\llcorner_{X\setminus (\Omega\cup S)}]_{\op{reg}}(x)\leq (N-1)\frac{\op{sn}'_{H}(d_s(x)+r_0)}{\op{sn}_{H}(d_s(x)+r_0)}.$$
\end{Lem}
\begin{proof}
Let $u(t)=h^{\frac1{N-1}}_{\alpha}(\gamma(t))$ where $h_{\alpha}$ satisfies \eqref{ele-com} as in Theorem~\ref{dis-com}. Then $u$ is semi-concave and satisfies
$$u''+Hu\leq 0$$
in the distributional sense. And the limits
$$u'_+(r)=\lim_{\delta\downarrow 0}\frac{u(r+\delta)-u(r)}{\delta}, \quad u'_-(r)=\lim_{\delta\downarrow 0}\frac{u(r-\delta)-u(r)}{-\delta}$$
exits.
Take $\phi\in C_0^{\infty}((-1, 1))$, $\int_{-1}^1\phi=1, \phi_{\epsilon}(t)=\frac1{\epsilon}\phi(\frac{t}{\epsilon})$ and let
$$\tilde u(s)=\int_{-\epsilon}^{\epsilon}\phi_{\epsilon}(-r)u(s-r)dr.$$
Then
$$\tilde u''(s)\leq -H\tilde u,$$
and
$$\left(\frac{\tilde u'}{\tilde u}\right)'=\frac{\tilde u''}{\tilde u}-\left(\frac{\tilde u'}{\tilde u}\right)^2\leq -H-\left(\frac{\tilde u'}{\tilde u}\right)^2.$$
Let $f=\frac{\tilde u'}{\tilde u}$ and let $f_H=\frac{\op{sn}'_H}{\op{sn}_H}$. Then $f'_H=-H-f^2_H$ and
\begin{eqnarray*}
\left(\op{sn}_H^2(s)(f(s)-f_H(s))\right)' & =& 2\op{sn}_H(s)\op{sn}'_H(s)(f(s)-f_H(s))+\op{sn}_H^2(s)(f'(s)-f'_H(s))\\
&\leq & 2\op{sn}^2_H(s) f_H(s)(f(s)-f_H(s))-\op{sn}^2_H(s)(f^2(s)-f_H^2(s))\\
&=& -\op{sn}_H^2(s)(f(s)-f_H^2(s))^2\leq 0.
\end{eqnarray*}
Thus for $s\geq 0$,
$$\op{sn}_H^2(s+r_0)(f(s+r_0)-f_H(s+r_0))-\op{sn}_H^2(r_0)(f(r_0)-f_H(r_0))\leq 0.$$
If $f(r_0)\leq f_H(r_0)$, then
$$f(s+r_0)\leq f_H(s+r_0).$$
Now note that as $\epsilon\to 0$, $\tilde u\to u$ and $\tilde u'_+\to u'_+$ and $\frac{u'}{u}=\frac1{N-1}\left(\ln h_{\alpha}\right)'$. And by the Laplacian formula Theorem~\ref{lap-com}, we have (ii).
For (i), as above we have that $f'\leq 0$. Thus for $s\geq 0$,
$$f(s+r_0)\leq f(r_0)\leq 0.$$
\end{proof}
\begin{Rem} The above laplacian comparison can also be seen in \cite{BKMW}.
\end{Rem}
Now by Lemma~\ref{glap-com} and Lemma~\ref{ele-com0}, we have the following volume element comparison.
\begin{Lem}[Volume element comparison] \label{vol-ele-com}
Let the assumption be as in Lemma~\ref{glap-com}. Then for any $r_1, r_2\in (0, b(X_{\alpha}))$, $r_1\leq r_2$,
(i) for $m=0$ and $K=0$,
$$h_{\alpha}(r_2)\leq h_{\alpha}(r_1);$$
(ii) for $m\neq 0$ or $K\neq 0$,
$$\frac{h_{\alpha}(r_2)}{\op{sn}_H^{N-1}(r_2+r_0)}\leq \frac{h_{\alpha}(r_1)}{\op{sn}_H^{N-1}(r_1+r_0)}.$$
\end{Lem}
\begin{proof}
By Lemma~\ref{ele-com0}, for $r_1\leq r_2$ as above,
\begin{eqnarray*}
h_{\alpha}(r_2)&\leq & J_{K, H^+(\gamma_{\alpha}(r_1)), N}(r_2-r_1)h_{\alpha}(r_1)\\
&= & \left(\op{sn}'_{H}(r_2-r_1)+\frac{\left(\ln h_{\alpha}\right)'_+(r_1)}{N-1}\op{sn}_{H}(r_2-r_1)\right)_+^{N-1} h_{\alpha}(r_1).
\end{eqnarray*}
Thus for $m=0$ and $K=0$, by Lemma~\ref{glap-com}, $(\ln h_{\alpha})'\leq 0$, we have
$$h_{\alpha}(r_2)\leq h_{\alpha}(r_1);$$
For $m\neq 0$ or $K\neq 0$, by Lemma~\ref{glap-com},
$$h_{\alpha}(r_2)\leq \frac{\op{sn}_H^{N-1}(r_2+r_0)}{\op{sn}_H^{N-1}(r_1+r_0)}h_{\alpha}(r_1).$$
\end{proof}
In the following, let $h_{\alpha}(r)=0$ when $r$ increases and $h_{\alpha}(r)$ becomes undefined. If $S$ has finite outer curvature, then for almost all $a\geq 0$,
$$\mathfrak m(\partial B_a(S))=\limsup_{\delta\to 0}\frac{\mathfrak m(A_{a, a+\delta}(S))}{\delta}=\int_{\mathfrak Q(\partial B_a(S)\cap \mathcal T^*_{d_s})} h_{\alpha}(a)d\mathfrak q(\alpha).$$
Using above lemma, we have the following relative volume comparison.
\begin{Lem}[Relative volume comparison]\label{rel-vol}
Let the assumption be as in Lemma~\ref{glap-com}. Then for $0\leq a<b$,
(i) if $m=0$ and $K=0$, then
\begin{equation}\mathfrak m(\partial B_b(S)\leq \mathfrak m(\partial B_a(S));\label{sph-com0}\end{equation}
\begin{equation}\mathfrak m(A_{a, b}(S))\leq (b-a)\mathfrak m(\partial B_a(S));\label{ann-sph0}\end{equation}
(ii) if $m\neq 0$ or $K\neq 0$,
\begin{equation}
\mathfrak m(\partial B_b(S))\leq \mathfrak m(\partial B_a(S))\frac{\svolsp{H}{b+r_0}}{\svolsp{H}{a+r_0}}, \label{sph-com}
\end{equation}
\begin{equation}
\mathfrak m(A_{a, b}(S))\leq \mathfrak m(\partial B_a(S))\frac{\svolann{H}{a+r_0, b+r_0}}{\svolsp{H}{a+r_0}}. \label{ann-sph}
\end{equation}
\end{Lem}
\begin{proof}
The proof is similar as the the one of \cite[Theorem 1.1]{Ket2},
\begin{eqnarray*}
\mathfrak m(A_{a, b}(S))&=& \mathfrak m(A_{a, b}(S)\cap \mathcal T^*_{d_s}\setminus B_{\op{out}})=\int_{\mathfrak Q(A_{a, b}(S)\cap \mathcal T^*_{d_s})}\int_{A_{a, b}(S)\cap X_{\alpha}}h_{\alpha}(r)dr d\mathfrak q(\alpha)
\end{eqnarray*}
By Lemma~\ref{vol-ele-com}, for $m\neq 0$ or $K\neq 0$,
\begin{eqnarray*}
\mathfrak m(A_{a, b}(S))& \leq & \int_{\mathfrak Q(A_{a, b}(S)\cap \mathcal T^*_{d_s})}\int_a^b \left(\frac{\op{sn}_{H}(r+r_0)}{\op{sn}_{H}(a+r_0)}\right)^{N-1}drh_{\alpha}(a) d\mathfrak q(\alpha)\\
& \leq & \mathfrak m(\partial B_a(S))\int_a^b \left(\frac{\op{sn}_{H}(r+r_0)}{\op{sn}_{H}(a+r_0)}\right)^{N-1}dr\\
&=& \mathfrak m(\partial B_a(S))\frac{\svolann{H}{a+r_0, b+r_0}}{\svolsp{H}{a+r_0}};
\end{eqnarray*}
For $m=0$ and $K=0$,
$$\mathfrak m(A_{a, b}(S))\leq (b-a) \mathfrak m(\partial B_a(S)).$$
\end{proof}
By above relative volume comparison, as in \cite{CC1}, we have that
\begin{Lem} \label{qua-com}
Let the assumption be as in Lemma~\ref{rel-vol} and assume \eqref{lvol-com}. Then for each $d\in (a, b)$,
(i) if $m=0$ and $K=0$,
\begin{equation}
\frac{\mathfrak m(\partial B_d(S))}{\mathfrak m(\partial B_a(S))}\geq 1-\epsilon\frac{b-a}{b-d};\label{csph-com0}
\end{equation}
(ii) if $m\neq 0$ or $K\neq 0$,
\begin{equation}
\frac{\mathfrak m(\partial B_d(S))}{\mathfrak m(\partial B_a(S))}\geq \left(1-\epsilon\frac{\svolann{H}{a+r_0,b+r_0}}{\svolann{H}{d+r_0,b+r_0}}\right)\frac{\svolsp{H}{d+r_0}}{\svolsp{H}{a+r_0}}. \label{csph-com1}
\end{equation}
\end{Lem}
\begin{proof}
By $$\frac{\mathfrak m (A_{a,b}(S))}{\mathfrak m (S)}\geq (1-\epsilon)\cdot \frac{\svolann{H}{a+r_0,b+r_0}}{\svolsp{H}{r_0}},$$
we have that,
$$\mathfrak m(A_{a,b}(S))\frac{\svolsp{H}{r_0}}{\mathfrak m(S)} - \svolann{H}{a+r_0,b+r_0} \geq - \epsilon \svolann{H}{a+r_0,b+r_0}.$$
Thus for all $a< d< b$,
\begin{eqnarray*}
1 - \epsilon\frac{\svolann{H}{a+r_0,b+r_0}}{\svolann{H}{d+r_0,b+r_0}} &\leq &1 + \frac{\mathfrak m(A_{a,b}(S))}{\svolann{H}{d+r_0,b+r_0}}\frac{\svolsp{H}{r_0}}{\mathfrak m(S)} - \frac{\svolann{H}{a+r_0,b+r_0}}{\svolann{H}{d+r_0,b+r_0}}\\
&= & \frac{\mathfrak m(A_{a,b}(S))}{\svolann{H}{d+r_0,b+r_0}}\frac{\svolball{H}{r_0}}{\mathfrak m(S)} - \frac{\svolann{H}{a+r_0,d+r_0}}{\svolann{H}{d+r_0,b+r_0}}.
\end{eqnarray*}
By relative volume comparison Lemma~\ref{rel-vol},
$$\svolann{H}{a+r_0,d+r_0} \geq \mathfrak m(A_{a,d}(S))\cdot \frac{\svolball{H}{r_0}}{\mathfrak m(S)}.$$
So by \eqref{ann-sph},
\begin{eqnarray*}
1 - \epsilon\frac{\svolann{H}{a+r_0,b+r_0}}{\svolann{H}{d+r_0,b+r_0}} &\leq & \frac{\mathfrak m(A_{a,b}(S))}{\svolann{H}{d+r_0,b+r_0}}\frac{\svolball{H}{r_0}}{\mathfrak m(S)} - \frac{\mathfrak m(A_{a,d}(S))}{\svolann{H}{d+r_0,b+r_0}} \frac{\svolball{H}{r_0}}{\mathfrak m(S)}\\
&= & \frac{\mathfrak m(A_{d,b}(S))}{\svolann{H}{d+r_0,b+r_0}}\frac{\svolsp{H}{r_0}}{\mathfrak m(S)}\\
&\leq & \frac{\mathfrak m(\partial B_d(S))}{\svolsp{H}{d+r_0}}\frac{\svolsp{H}{r_0}}{\mathfrak m(S)}.
\end{eqnarray*}
Note that by \eqref{sph-com0} and \eqref{sph-com}, \eqref{lvol-com} implies that for $0\leq a<b$,
\begin{equation*}
\mathfrak m(A_{a, b}(S))\geq (1-\epsilon)(b-a)\mathfrak m(\partial B_a(S)), \text{ for } m=0 \text{ and } K=0;
\end{equation*}
\begin{equation*}
\mathfrak m(A_{a, b}(S))\geq (1-\epsilon)\frac{\svolann{H}{a+r_0, b+r_0}}{\svolsp{H}{a+r_0}}\mathfrak m(\partial B_a(S)), \text{ for } m\neq 0 \text{ or } K\neq 0.
\end{equation*}
Then same argument as above gives the results.
\end{proof}
Now we will use above volume estimates Lemma~\ref{rel-vol}, Lemma~\ref{qua-com} and Laplacian comparison Lemma~\ref{glap-com} to prove the Laplacian estimates Theorem~\ref{lap-main}.
\begin{proof}[Proof of Theorem~\ref{lap-main}]
By Lemma~\ref{glap-com}, we have that
for $m=0, K=0$, $$\Delta d_{s, i}\leq 0;$$
for $m\neq 0$ or $K\neq 0$,
$$\Delta d_{s, i} \leq (N-1)\frac{\op{sn}'_H(r+r_0)}{\op{sn}_H(r+r_0)}.$$
And for $a<d<b$ (see also \cite[Proposition 3.6]{CDNPSW})
\begin{eqnarray*}
-\kern-1em\int_{A_{a, d}(S_i)}\Delta d_{s, i} &=&\frac1{\mathfrak m_i(A_{a, d}(S_i))}\int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s_i}})}\int_{A_{a, d}(S_i)\cap X_{\alpha}}\Delta d_{s, i}h_{\alpha}(r)dr d\mathfrak q(\alpha)\\
&=&\frac1{\mathfrak m_i(A_{a, d}(S_i))} \int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s_i}})}\int_a^d\frac{h'_{\alpha}(r)}{h_{\alpha}(r)}h_{\alpha}(r)dr d\mathfrak q(\alpha)\\
&=&\frac1{\mathfrak m_i(A_{a, d}(S_i))} \int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s_i}})}h_{\alpha}(d)-h_{\alpha}(a) d\mathfrak q(\alpha)\\
& =& \frac{\mathfrak m_i(\partial B_d(S_i))-\mathfrak m_i(\partial B_a(S_i))}{\mathfrak m_i(A_{a, d}(S_i))}.
\end{eqnarray*}
Thus for $m=0$ and $K=0$, by \eqref{csph-com0} and \eqref{sph-com0}
$$-\kern-1em\int_{A_{a, d}(S_i)}\Delta d_{s, i} \geq
\frac{-\epsilon_i(b-a)}{b-d-\epsilon_i(b-a)}\frac{\mathfrak m_i(\partial B_d(S_i))}{\mathfrak m_i(A_{a, d}(S_i))}\geq \frac{-\epsilon_i(b-a)}{b-d-\epsilon_i(b-a)}\frac1{d-a};$$
And for $m\neq 0$ or $K\neq 0$, if $\svolsp{H}{d+r_0}>\svolsp{H}{a+r_0}$ which is always holds for $K\leq 0$, by \eqref{ann-sph} and \eqref{csph-com1}
\begin{eqnarray*}-\kern-1em\int_{A_{a, d}(S_i)}\Delta d_{s, i} &\geq & \frac{1}{\svolsp{H}{a+r_0}}\left((1-\epsilon_i C(N, H, a, b, d, r_0))\svolsp{H}{d+r_0}-\svolsp{H}{a+r_0}\right)\frac{\mathfrak m(\partial B_a(S))}{\mathfrak m(A_{a, d}(S))}\\
&\geq & \frac{(1-\epsilon_i C(N, H, a, b, d, r_0))\svolsp{H}{d+r_0}-\svolsp{H}{a+r_0}}{\svolann{H}{a+r_0, d+r_0}};
\end{eqnarray*}
If $\svolsp{H}{b+r_0}\leq \svolsp{H}{a+r_0}$ for $K>0$, as the $m=0$, $K=0$ case,
\begin{eqnarray*}-\kern-1em\int_{A_{a, d}(S_i)}\Delta d_{s, i} &\geq & \frac{1}{\svolsp{H}{a+r_0}}\left(\svolsp{H}{d+r_0}-(1+\epsilon_i C(N, H, a, b, d, r_0))\svolsp{H}{a+r_0}\right)\frac{\mathfrak m(\partial B_d(S))}{\mathfrak m(A_{a, d}(S))}\\
&\geq & \frac{\svolsp{H}{d+r_0}-(1+\epsilon_i C(N, H, a, b, d, r_0))\svolsp{H}{a+r_0}}{\svolann{H}{a+r_0, d+r_0}};
\end{eqnarray*}
And by Lemma~\ref{ele-com0},
\begin{eqnarray*}
& & -\kern-1em\int_{A_{a, d}(S_i)} (N-1)\frac{\op{sn}'_H(r+r_0)}{\op{sn}_H(r+r_0)}\\
& =& \frac{1}{\mathfrak m(A_{a, d}(S_i))}\int_{A_{a, d}(S_i)\cap \mathcal T_{d_{s, i}}\setminus B_{\op{out}}}(N-1)\frac{\op{sn}'_{H}(r+r_0)}{\op{sn}_H(r+r_0)}d\mathfrak m\\
& \leq & \frac{1}{\mathfrak m(A_{a, b}(S_i))}\int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s, i}})}\int_a^d(N-1)\frac{\op{sn}'_{H}(r+r_0)}{\op{sn}_H(r+r_0)} \frac{h_{\alpha}(r)}{h_{\alpha}(a)}h_{\alpha}(a)dr d\mathfrak q(\alpha)\\
& \leq & \frac{1}{\mathfrak m(A_{a, b}(S_i))}\int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s, i}})}\int_a^d(N-1)\frac{\op{sn}'_{H}(r+r_0)}{\op{sn}_H(r+r_0)} \frac{\op{sn}^{N-1}_H(r+r_0)}{\op{sn}^{N-1}_H(a+r_0)}h_{\alpha}(a)dr d\mathfrak q(\alpha)\\
& = & \frac{1}{\mathfrak m(A_{a, b}(S_i))\op{sn}_H^{N-1}(a+r_0)}\int_{\mathfrak Q(A_{a, d}(S_i)\cap \mathcal T^*_{d_{s, i}})}\int_a^d d\op{sn}_H^{N-1}(r+r_0) h_{\alpha}(a) d\mathfrak q(\alpha)\\
&=& \frac{\mathfrak m(\partial B_a(S_i))\left(\op{sn}_H^{N-1}(d+r_0)-\op{sn}_H^{N-1}(a+r_0)\right)}{\mathfrak m(A_{a, b}(S_i))\op{sn}_H^{N-1}(a+r_0)}\\
&\leq & \begin{cases}(1+\epsilon_i)\frac{\svolsp{H}{a+r_0}}{\svolann{H}{a+r_0, d+r_0}} \frac{\op{sn}_H^{N-1}(d+r_0)-\op{sn}_H^{N-1}(a+r_0)}{\op{sn}_H^{N-1}(a+r_0)}, & \op{sn}_H^{N-1}(d+r_0)>\op{sn}_H^{N-1}(a+r_0)\\
\frac{\svolsp{H}{a+r_0}}{\svolann{H}{a+r_0, d+r_0}} \frac{\op{sn}_H^{N-1}(d+r_0)-\op{sn}_H^{N-1}(a+r_0)}{\op{sn}_H^{N-1}(a+r_0)}, & \op{sn}_H^{N-1}(d+r_0)\leq \op{sn}_H^{N-1}(a+r_0)\end{cases}\\
&\leq & (1+\epsilon_i) \frac{\svolsp{H}{d+r_0}-\svolsp{H}{a+r_0}}{\svolann{H}{a+r_0, d+r_0}}
\end{eqnarray*}
We have that
\begin{equation}
-\kern-1em\int_{A_{a, d}(S_i)}\Delta d_{s, i}\geq (1-\Psi(\epsilon_i | N, K, a, d, b, m))-\kern-1em\int_{A_{a, d}(S_i)} (N-1)\frac{\op{sn}'_H(r+r_0)}{\op{sn}_H(r+r_0)}.
\end{equation}
Now as the proof of Claim 1 in \cite[Proposition 4.3]{Ch}, passing to the limit, we have that on $A_{a, d}(S)$
(i) for $m=0$ and $K=0$,
$$\Delta d_s=0;$$
(ii) for $m\neq 0$ or $K\neq 0$,
$$\Delta d_s=(N-1)\frac{\op{sn}'_H(d_s+r_0)}{\op{sn}_H(d_s+r_0)}.$$
\end{proof}
\begin{Cor} Let the assumption be as in Theorem~\ref{lap-main}. For $m\neq 0$ or $K\neq 0$, let
$$f_H(x)=f_H(d_s(x)), \quad f_H(r)=\int \op{sn}_H(r+r_0)dr.$$
Then in $A_{a, b}(S)$
$$\nabla f_H=\op{sn}_{H}(d_s+r_0)\nabla d_s,$$
and
$$\Delta f_H= N\op{sn}'_{H}(d_s+r_0).$$
\end{Cor}
\section{Hessian estimates}
In this section we will use the Laplacian estimates Theorem~\ref{lap-main} to give an estimate of $\op{Hess}(d_s)$ for $m=0$ and $K=0$ and estimates of $\op{Hess}(f_H)$ for $m\neq 0$ or $K\neq 0$.
\begin{Thm}[Hessian estimates] \label{hess-est}
Let the assumption be as in Theorem~\ref{lap-main} and let $a<d<b$.
(i) For $m=0$, $K=0$, in $A_{a+\frac{d-a}4, b-\frac{d-a}4}(S)$ $\mathfrak m$-a.e.
\begin{equation}
\op{Hess}(d_s)=0.
\label{hess-0}
\end{equation}
(ii) For $m\neq 0$ or $K\neq 0$, in $A_{a+\frac{d-a}4, b-\frac{d-a}4}(S)$ $\mathfrak m$-a.e.
\begin{equation}
\op{Hess}(f_H)=\op{sn}'_H(d_s+r_0).\label{hess-1}
\end{equation}
\end{Thm}
\begin{proof}
Take a cut-off function $\phi: X\to [0, 1]$ as in Lemma~\ref{cut-off} such that
(1) $\phi=1$ on $A_{a+\frac{d-a}4, b-\frac{d-a}4}(S)$, $\phi=0$ on $X\setminus A_{a+\frac{d-a}5, b-\frac{d-a}5}(S)$;
(2) $|\Delta \phi|+|\nabla \phi|\leq C(K, N, a, b)$ a.e. on $A_{a+\frac{d-a}5, b-\frac{d-a}5}(S)$.
For $m=0, K=0$, by the improved Bochner inequality Theorem~\ref{Boc-ine} and Theorem~\ref{lap-main},
\begin{eqnarray*}
0&=&-\frac12\int_X \Gamma(\phi, |\nabla d_s|^2)=\int_X\phi \frac12 \Delta|\nabla d_s|^2\\
&\geq & \int_X\phi\left( |\op{Hess}(d_s)|^2_{\op{HS}}+ \Gamma(d_s, \Delta d_s)\right)=\int_X\phi |\op{Hess}(d_s)|^2_{\op{HS}}.
\end{eqnarray*}
And thus in $A_{a+\frac{d-a}4, b-\frac{d-a}4}(S)$ $\mathfrak m$-a.e.
\begin{equation*}\op{Hess}(d_s)=0. \end{equation*}
For $m\neq 0, K=0$, $f_0=\frac12 (r+r_0)^2$,
\begin{eqnarray*}
0&=&-\int_X \Gamma(\phi, \frac12|\nabla f_0|^2-f_0)=\int_X\phi \left(\frac12 \Delta|\nabla f_0|^2-\Delta f_0\right)\\
&\geq & \int_X\phi\left( |\op{Hess}(f_0)|^2_{\op{HS}}+ \Gamma(f_0, \Delta f_0)-N\right)\geq \int_X\phi |\op{Hess}(f_0)-1|^2_{\op{HS}}.
\end{eqnarray*}
For $K=N-1$, $f_1=-\cos (r+r_0)$,
\begin{eqnarray*}
0&=&-\frac12\int_X \Gamma(\phi, |\nabla f_1|^2+f_1^2-1)=\frac12\int_X\phi \left(\Delta|\nabla f_1|^2+2|\nabla f_1|^2+2f_1\Delta f_1\right)\\
&\geq & \int_X\phi\left( |\op{Hess}(f_1)|^2_{\op{HS}}+(N-1)|\nabla f_1|^2+ \Gamma(f_1, \Delta f_1)+|\nabla f_1|^2+f_1\Delta f_1\right)\\
&=& \int_X\phi\left( |\op{Hess}(f_1)|^2_{\op{HS}}+ (N-1)\sin^2(r+r_0)-N\sin^2(r+r_0) +\sin^2(r+r_0)-N\cos^2(r+r_0)\right)\\
&\geq& \int_X\phi |\op{Hess}(f_1)-\cos(r+r_0)|^2_{\op{HS}}.
\end{eqnarray*}
For $K=-(N-1)$, $f_{-1}=\cosh (r+r_0)$,
\begin{eqnarray*}
0&=&-\frac12\int_X \Gamma(\phi, |\nabla f_{-1}|^2-f_{-1}^2+1)=\frac12\int_X\phi \left(\Delta|\nabla f_{-1}|^2-2|\nabla f_{-1}|^2-2f_{-1}\Delta f_{-1}\right)\\
&\geq & \int_X\phi\left( |\op{Hess}(f_{-1})|^2_{\op{HS}}-(N-1)|\nabla f_{-1}|^2+ \Gamma(f_{-1}, \Delta f_{-1})-|\nabla f_{-1}|^2-f_{-1}\Delta f_{-1}\right)\\
&=& \int_X\phi\left( |\op{Hess}(f_{-1})|^2_{\op{HS}}- (N-1)\sinh^2(r+r_0)+N\sinh^2(r+r_0) -\sinh^2(r+r_0)-N\cosh^2(r+r_0)\right)\\
&\geq& \int_X\phi |\op{Hess}(f_{-1})-\cosh(r+r_0)|^2_{\op{HS}}.
\end{eqnarray*}
\end{proof}
\section{Pythagoras theorem and Cosine law}
Let $(X, d, \mathfrak m)$ be as in Theorem~\ref{lap-main}. In this section, we will use a method as in \cite{CC1} to show that the metric in $A_{a+\delta, b-\delta}(S)$ (some $\delta>0$) satisfies Pythagoras theorem for $m=0$ and $K=0$ or Cosine law for $m\neq 0$ or $K\neq 0$.
First note that for $t\in [-(b-a)/3, (b-a)/3]$, $td_s$ is $c$-concave, thus we can apply the differential formula in Section 2.5 to $d_s$.
\begin{Lem} \label{c-concave}
Let the assumption be as in Theorem~\ref{lap-main}. Then $td_s$ is $c$-concave for $t\in [-\frac{b-a}3, \frac{b-a}3]$, and
$$(td_s)^c(y)=-td_s-\frac{t^2}2.$$
\end{Lem}
\begin{proof}
The proof is the same as in \cite{CC1}. First take small $\tau>0$ and consider
$$\mathcal T^i_{a, b}=\{\mathcal T_{d_{s, i}}\cap A_{a, b-\tau}(S_i), \, \mathcal T_{d_{s, i}}\cap A_{b-\tau, b}(S_i)\neq \emptyset \}.$$
Then for any $x\in \mathcal T^i_{a, b}$, there is $y\in A_{b-\tau, b}(S_i)$, such that
$$d_{s, i}(y)-d_{s, i}(x)=d_i(x, y).$$
Since $\mathfrak m_i(X_i\setminus \mathcal T_{d_{s, i}})=0$,
$$\mathfrak m_i(\mathcal T_{d_{s, i}}\cap A_{b-\tau, b}(S_i))=\mathfrak m_i(A_{b-\tau, b}(S_i)).$$
And by volume element comparison Lemma~\ref{vol-ele-com} and \eqref{lqvol-com},
\begin{eqnarray*}
\frac{\mathfrak m(\mathcal T^i_{a, b})}{\mathfrak m_i(A_{a, b}(S_i))}&=& \frac{\mathfrak m_i(\mathcal T^i_{a, b})}{\svolann{H}{a+r_0, b-\tau+r_0}}\frac{\svolann{H}{a+r_0, b+r_0}}{\mathfrak m_i(A_{a, b}(S_i))}\frac{\svolann{H}{a+r_0, b-\tau+r_0}}{\svolann{H}{a+r_0, b+r_0}}\\
&\geq & \frac{A_{b-\tau, b}(S_i)}{\svolann{H}{b-\tau, b}}\frac{\svolann{H}{a+r_0, b+r_0}}{\mathfrak m_i(A_{a, b}(S_i))}\frac{\svolann{H}{a+r_0, b-\tau+r_0}}{\svolann{H}{a+r_0, b+r_0}}\\
&\geq & (1-\epsilon_i)\frac{\mathfrak m_i(\partial B_a(S_i))}{\svolsp{H}{a+r_0}}\frac{\svolann{H}{a+r_0, b+r_0}}{\mathfrak m_i(A_{a, b}(S_i))}\frac{\svolann{H}{a+r_0, b-\tau+r_0}}{\svolann{H}{a+r_0, b+r_0}}\\
&\geq & (1-\epsilon_i)\frac{\svolann{H}{a+r_0, b-\tau+r_0}}{\svolann{H}{a+r_0, b+r_0}}.
\end{eqnarray*}
Since $\tau$ can be taken arbitrary small, without loss of generality, we may let $\tau\to 0$ and derive that
for each $x\in A_{a, b}(S_i)$, there is $y\in \partial B_b(S_i)$ such that
\begin{equation}d_i(x, y)\geq b-d_{s, i}(x)\geq (1-\epsilon_i)d_i(x, y). \label{geo-con}\end{equation}
Since $d_s$ is $1$-Lipschitz, as in \cite{Gi13}, for $x, y\in X$,
$$td_s(x)-td_s(y)\leq |t|d(x, y)\leq \frac{t^2}{2}+\frac{d^2(x, y)}{2}.$$
And by the definition of $c$-transform,
$$(td_s)^c(y)=\inf_{x\in X}\frac{d^2(x,y)}{2}-td_s(x)\geq -td_s(y)-\frac{t^2}{2}.$$
For the opposite inequality, note that for $y\in A_{a, b}(S)$, there is $y_i\in A_{a, b}(S_i)$ and $y_i^-\in \partial B_a(S_i)$, $y_i^+\in \partial B_b(S_i)$ such that $y_i\to y$
$$d_i(y_i, y_i^-)=d_{s, i}(y_i)-a, \, d_{s, i}(y_i^+)-d_{s, i}(y_i)-d_i(y_i^+, y_i)\geq -\epsilon_i.$$
Then
$$b-a\leq d_i(y_i^+, y_i^-)\leq d_i(y_i, y_i^-)+d_i(y_i, y_i^+)\leq b-a+\epsilon_i.$$
Let $\gamma_i^{-}$ be a unit speed minimal geodesic from $y_i^-$ to $y$ and let $\gamma_i^+$ be a unit speed minimal geodesic from $y$ to $y_i^+$, $\gamma_i\to \gamma$. Assume $\gamma_i^-\cup\gamma_i^+\to \gamma\in X$.
For $y\in A_{\frac{b+2a}3, \frac{2b+a}3}(S)$, for each $t\in (-\frac{b-a}3, \frac{b-a}3)$, we can take $\gamma_t=\gamma(d_s(y)+t)$ such that
$$d_s(\gamma_t)-d_s(y)=t=\op{sign}(t)d(y, \gamma_t).$$
Thus
$$\left(td_s\right)^c(y)\leq \frac{d^2(\gamma_t, y)}2-td_s(\gamma_t)=-td_s(y)-\frac{t^2}2.$$
\end{proof}
Take a cut-off function $\phi: X\to [0, 1]$ as in the proof of Theorem~\ref{hess-est}. Consider the vector field $\phi\nabla d_s$. Then as the discussion in section 3.4 of \cite{CDNPSW}, by \cite{AT14} (see Theorem~\ref{exist-RLF}), we know that the Regular Lagrangian flow $F_t$ for $\phi\nabla d_s$ exists and is unique.
Using Hessian estimates Theorem~\ref{hess-est}, Lemma~\ref{c-concave} and the differential formula Theorem~\ref{first-diff1}, Corollary~\ref{first-diff2} and Corollary~\ref{second-diff2}, we will derive Pythagoras theorem for $m=0$, $K=0$, and Cosine law for $m\neq 0$ or $K\neq 0$.
Note that by Corollary~\ref{first-diff2}, we have that for a Lipschitz map $f$,
\begin{equation}f(p)= f(x)-\lim_{\delta\to 0}\int_{\delta}^{1} d(x, p)\left<\nabla f, \nabla d_p\right>(\gamma_{p, x}(t)) dt, \label{diff-form-1}\end{equation}
where $\gamma_{p, x}$ is a constant geodesic from $p$ to $x$ and $d_p$ is the distance function from $p$. In the following for simplicity we will write \eqref{diff-form-1} just as
$$f(x)-f(p)=\int_0^1d(x, p)\left<\nabla f, \nabla d_p\right>(\gamma_{p, x}(t)) dt.$$
\begin{Thm}[Pythagoras theorem and Cosine law] \label{cos-law}
Let the assumption be as in Theorem~\ref{lap-main}. Let $F_t$ be the regular Lagrangian flow of $\phi\nabla d_s$. Denote $A=A_{a+\frac{b-a}3, b-\frac{b-a}3}(S)$. Let $B=B_r(x)\subset A$ be a geodesic ball in $A$.
(i) For $m=0$ and $K=0$,
\begin{equation}
\int_{B\times B}\left|d^2(x, y)-(d_s(x)-d_s(y))^2-d^2\left(x, F_{-d_s(y)+d_s(x)}(y)\right)\right| d\mathfrak m(x) \mathfrak m(y)=0;
\end{equation}
(ii) For $m\neq 0$ or $K\neq 0$,
(ii-1) if $K=0$, we have that
\begin{equation}
\int_{B\times B}\left|\frac{(d_s(x)+r_0)^2+(d_s(y)+r_0)^2-d^2(x, y)}{2(d_s(x)+r_0)(d_s(y)+r_0)}-\frac{2\left(r_0+d_s(x)\right)^2-d^2\left(x, F_{-d_s(y)+d_s(x)}(y)\right)}{2\left(r_0+d_s(x)\right)^2}\right| d\mathfrak m(x) \mathfrak m(y)=0;
\end{equation}
(ii-2) if $K=\pm(N-1)$, and thus $H=\pm 1$,
\begin{equation}
\int_{B\times B}\left|\frac{\op{sn}'_H(d(x,y))-\op{sn}'_H(d_s(x)+r_0)\op{sn}'_H(d_s(y)+r_0)}{\op{sn}_H(d_s(x)+r_0)\op{sn}_H(d_s(y)+r_0)}-\frac{\op{sn}'_H\left(d\left(x, F_{-d_s(y)+d_s(x)}(y)\right)\right)-{\op{sn}'}^2_H\left(r_0+d_s(x)\right)}{\op{sn}^2_H\left(r_0+d_s(x)\right)}\right| =0;
\end{equation}
\end{Thm}
\begin{proof}
Assume for $x, y\in B$, $d_s(x)=t_0$, $d_s(y)=t$. Let $\gamma_{\tau}(l): [0, 1]\to B$ be a constant speed geodesic from $x$ to $F_{-\tau}(y)$, $\tau\in[0, t-t_0]$.
For $m=0$, $K=0$,
\begin{eqnarray*}
& &\int_{B\times B}\frac12\left|d^2(x, y)-(t-t_0)^2-d^2\left(x, F_{t_0-t}(y)\right)\right| d\mathfrak m(x) d\mathfrak m(y)\\
&=& \int_{B\times B}\frac12\left|\left.\left(d^2(x, F_{-\tau}(y))-(t-t_0-\tau)^2\right)\right|_{t-t_0}^0\right| d\mathfrak m(x) d\mathfrak m(y)\\
&\leq & \int_{B\times B} \int_0^{|t-t_0|}\left|d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla d_s\right>(F_{-\tau}(y))-(t-t_0-\tau) \right|d\tau d\mathfrak m(x)d\mathfrak m(y) \\
&=& \int_{B\times B} \int_0^{|t-t_0|}\left|d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla d_s\right>(F_{-\tau}(y))-\left.d_s(\gamma_{\tau}(l))\right|^{1}_0 \right|d\tau d\mathfrak m(x)d\mathfrak m(y) \\
& \leq & \int_{B\times B} \int_0^{|t-t_0|}\int_0^{1}\left|d(x, F_{-\tau}(y))\left(\left<\nabla d_x, \nabla d_s\right>(F_{-\tau}(y))-\left<\nabla d_x, \nabla d_s\right>(\gamma_{\tau}(l))\right)\right| dl d\tau d\mathfrak m(x)d\mathfrak m(y) \\
&\leq & \int_0^1 \int_{l}^{1}\int_{B\times B}\int_0^{|t-t_0|}d^2(x, F_{-\tau}(y))\left|\op{Hess}_{d_s}(\gamma'_{\tau}(\xi), \gamma'_{\tau}(\xi))\right| d\tau d\mathfrak m(x)d\mathfrak m(y) d\xi dl\\
&\leq & \int_0^{\frac{b-a}3}\int_0^1 \int_{l}^{1}\int_{B\times B}d^2(x, F_{-\tau}(y))\left|\op{Hess}_{d_s}(\gamma'_{\tau}(\xi), \gamma'_{\tau}(\xi))\right| d\mathfrak m(x)d\mathfrak m(y) d\xi dld\tau\\
& =& 0.
\end{eqnarray*}
Now prove the cosine law for $K=0$ and $m\neq 0$ where $f_0(x)=\frac12 (d_s+r_0)^2$.
\begin{eqnarray*}
& & \int_{B\times B}\left|\frac{(t_0+r_0)^2+(t+r_0)^2-d^2(x, y)}{2(t_0+r_0)(t+r_0)}-\frac{2\left(r_0+t_0\right)^2-d^2\left(x, F_{t_0-t}(y)\right)}{2\left(r_0+t_0\right)^2}\right| d\mathfrak m(x) \mathfrak m(y)\\
&=& \int_{B\times B}\left|\left.\frac{(t_0+r_0)^2+(t+r_0-\tau)^2-d^2(x, F_{-\tau}(y))}{2(t_0+r_0)(t+r_0-\tau)}\right|_{t-t_0}^0\right| d\mathfrak m(x) \mathfrak m(y)\\
&\leq & \frac{1}{2(a+r_0)^3}\int_{B\times B}\int_0^{|t-t_0|} \left|(r_0+t-\tau)^2-(t_0+r_0)^2 + d^2(x, F_{-\tau}(y))\right.\\
& & \left.-d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla (d_s+r_0)^2\right>(F_{-\tau}(y))\right| d\tau\\
& = & \frac{1}{(a+r_0)^3}\int_{B\times B}\int_0^{|t-t_0|} \left|\left.f_0(\gamma_{\tau}(l)\right|_0^1 + \frac12d^2(x, F_{-\tau}(y))-d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla f_0\right>(F_{-\tau}(y))\right| d\tau\\
& = & \frac{1}{(a+r_0)^3}\int_{B\times B}\int_0^{|t-t_0|} \left|\left.\left(f_0(\gamma_{\tau}(l)) - \frac12\left(d(x, F_{-\tau}(y))-d(x, \gamma_{\tau}(l))\right)^2\right)\right|_{0}^1\right.\\
& & \left.-d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla f_0\right>(F_{-\tau}(y))\right| d\tau\\
&\leq & \frac{1}{(a+r_0)^3}\int_{B\times B} \int_0^{|t-t_0|}\int_{0}^1d(x, F_{-\tau}(y))\left|\left<\nabla f_0, \nabla d_x\right>(\gamma_{\tau}(l)) -\left<\nabla d_x, \nabla f_0\right>(F_{-\tau}(y))\right.\\
& &\left.+ d(x, F_{-\tau}(y))-d(x, \gamma_{\tau}(l))\right| dld\tau \\
& \leq & \frac{1}{(a+r_0)^3}\int_0^1 \int_{l}^{1}\int_{B\times B}\int_0^{|t-t_0|} d^2(x, F_t(y))\left|\op{Hess}_f(\nabla d_x, \nabla d_x)(\gamma_{\tau}(\xi))-1\right| d\tau d\mathfrak m(x)d\mathfrak m(y)d\xi dl\\
& =& 0
\end{eqnarray*}
For $K=(N-1)$, consider $f_1=-\cos(d_s+r_0)$, we can see that
\begin{eqnarray*}
& & \int_{B\times B}\left|\frac{\cos(d(x, y))-\cos(t_0+r_0)\cos(t+r_0)}{\sin(t_0+r_0)\sin(t+r_0)}-\frac{\cos\left(d\left(x, F_{t_0-t}(y)\right)\right)-\cos^2\left(r_0+t_0\right)}{\sin^2\left(r_0+t_0\right)}\right|d\mathfrak m(x) d\mathfrak m(y) \\
&=& \int_{B\times B}\left|\left.\frac{\cos d(x, F_{-\tau}(y))-\cos(t_0+r_0)\cos(t+r_0-\tau)}{\sin(t_0+r_0)\sin(t+r_0-\tau)}\right|_{0}^{t-t_0}\right|d\mathfrak m(x) d\mathfrak m(y) \\
&\leq & c \int_{B\times B}\int_0^{|t-t_0|} \left|\sin d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla f_1\right>(F_{-\tau}(y))- \cos (t_0+r_0)+\cos d(x, F_{-\tau}(y))\cos(t+r_0-\tau)\right| d\tau \\
&= & c \int_{B\times B}\int_0^{|t-t_0|} \left|\sin d(x, F_{-\tau}(y))\left<\nabla d_x, \nabla f_1\right>(F_{-\tau}(y))+ \cos d(x, \gamma_{\tau}(0))f_1(\gamma_{\tau}(0))-\cos d(x, \gamma_{\tau}(1))f_1(\gamma_{\tau}(1))\right| d\tau \\
&= & c \int_{B\times B}\int_0^{|t-t_0|} \left|\left.\left(\sin d(x, \gamma_{\tau}(l))\left<\nabla d_x, \nabla f_1\right>(F_{-\tau(y)})- \cos d(x, \gamma_{\tau}(l))f_1(\gamma_{\tau}(l))\right)\right|_0^1\right| d\tau \\
&\leq & c \int_0^{\frac{b-a}{3}}\int_{B\times B}\left|\int_0^1d(x, F_{-\tau}(y))\cos \left(d(x, \gamma_{\tau}(l)) \left<\nabla f_1, \nabla d_x\right>(F_{-\tau}(y))\right.\right.\\
& & \left.\left.-\cos d(x, \gamma_{\tau}(l)) \left<\nabla f_1, \nabla d_x\right>(\gamma_{\tau}(l))+\sin d(x, \gamma_{\tau}(l)) f_1(\gamma_{\tau}(l))\right)dl\right| d\mathfrak m(x)d\mathfrak m(y)d\tau\\
&= & c \int_0^{\frac{b-a}{3}}\int_{B\times B}\left|\int_0^1d(x, F_{-\tau}(y))\left(\cos d(x, \gamma_{\tau}(l))\left.\left<\nabla f, \nabla d_x\right>(\gamma_{\tau}(\xi))\right|_l^1+\left.\sin d(x, \gamma_{\tau}(\xi))\right|_0^l f_1(\gamma_{\tau}(l))\right)dl\right| \\
&= & c \int_0^{\frac{b-a}{3}}\int_{B\times B}\left|\int_0^1d(x, F_{-\tau}(y))\cos d(x, \gamma_{\tau}(l))\left.\left<\nabla f, \nabla d_x\right>(\gamma_{\tau}(\xi))\right|_l^1 dl\right.\\
& & +\left.\int_0^1\int_0^l \cos d(x, \gamma_{\tau}(\xi)) d^2(x, F_{-\tau}(y))f_1(\gamma_{\tau}(l))d\xi dl \right|\\
&= & c \int_0^{\frac{b-a}{3}}\int_{B\times B}\left|\int_0^1d^2(x, F_{-\tau}(y))\cos d(x, \gamma_{\tau}(l))\int_l^1\op{Hess}_{f_1}(\gamma'_{\tau}(\xi), \gamma'_{\tau}(\xi))d\xi dl\right.\\
& & +\left.\int_0^1\int_l^1 \cos d(x, \gamma_{\tau}(l)) d^2(x, F_{-\tau}(y))f_1(\gamma_{\tau}(\xi))d\xi dl \right|\\
&= & c \int_0^{\frac{b-a}{3}}\int_0^1 \int_{l}^{1}\int_{B\times B} d^2(x, F_{-\tau}(y))\left|\cos d(x, \gamma_{\tau}(l))\right| \left|\op{Hess}_{f_1}(\nabla d_x, \nabla d_x)(\gamma_{\tau}(\xi))+ f_1(\gamma_{\tau}(\xi))\right| \\
& = & 0.
\end{eqnarray*}
For $K=-(N-1)$, the same argument as $K=(N-1)$ gives the result.
\end{proof}
By the Pythagoras theorem and Cosine law above, we have that
\begin{Thm} \label{flow-con}
The regular Lagrangian flow $F: (-\frac{b-a}3, \frac{b-a}3)\times A\to A_{a, b}(S)$ admits a continuous representation with respect to the measure $\mathcal L^1\times \mathfrak m$ which we still denote by $F$. In particular the Pythagoras theorem or Cosine law in Theorem~\ref{cos-law} holds pointwise for $F$.
\end{Thm}
\begin{proof}
By the definition of regular Lagrangian flow, for $\mathfrak m$-a.e. $x$, $t\mapsto F_t(x)$ is continuous. And since
$$d(F_t(x), F_l(y))= d(F_t(x), F_l(y))-d(F_t(x), F_l(x))+d(F_t(x), F_l(x))\leq d(F_l(y), F_l(x))+d(F_t(x), F_l(x)),$$
it sufficient to show that for $\mathfrak m\times \mathfrak m$-a.e. $x, y\in A$ with $d_s(x)=d_s(y)=l$ and $d(x, y)<\delta<1$, for $t\in (-\frac{b-a}3, \frac{b-a}3)$,
\begin{equation} \label{con-equ}
d(F_t(x), F_t(y))\leq \Psi(\delta | K, a, b, r_0).
\end{equation}
If $m=0$, $K=0$, by Pythagoras theorem, we see that for $\mathfrak m\times \mathfrak m$-a.e. $x, y\in A$ with $d_s(x)=d_s(y)=l$, for $t\in (-\frac{b-a}3, \frac{b-a}3)$,
$$d(x, y)=d(F_t(x), F_t(y));$$
If $K=0$, $m\neq 0$, by Cosine law,
$$\frac{2(l+r_0)^2-d^2(x, y)}{2(l+r_0)^2}=\frac{2(l+r_0+t)^2-d^2(F_t(x), F_t(y))}{2(l+r_0+t)^2},$$
that is
$$d^2(F_t(x), F_t(y))=\left(\frac{l+t+r_0}{l+r_0}\right)^2d^2(x, y)\leq \frac{(b+r_0)^2}{(a+r_0)^2}d^2(x,y).$$
If $K=(N-1)$, by Cosine law,
$$\frac{\cos d(x, y)-\cos^2(l+r_0)}{\sin^2(l+r_0)}=\frac{\cos d(F_t(x), F_t(y))-\cos^2(l+t+r_0)}{\sin^2(l+t+r_0)},$$
that is
$$\frac{\cos d(F_t(x), F_t(y))-1}{\sin^2(b+r_0)}\leq \frac{\cos d(F_t(x), F_t(y))-1}{\sin^2(l+t+r_0)}=\frac{\cos d(x, y)-1}{\sin^2(l+r_0)}\leq \Psi(\delta | a, b, r_0).$$
Thus \eqref{con-equ} holds.
For $K=-(N-1)$, it is similar as the $K=(N-1)$ case.
\end{proof}
\section{Warped product structure} In this section, we will finish the proof of Theorem~\ref{main}. First by the Pythagoras theorem or Cosine law, we derive a warped product structure $(a', b')\times_{\op{sn}'_H(r)+\frac{m}{N-1}\op{sn}_H(r)}Y$ of $A_{a', b'}(S)$ and see that $Y$ has finite components. Then using this warped product and the $\RCD$-condition of $A_{a', b'}(S)$ we prove that each component of $Y$ is infinitesimally Hilbertian and satisfies the Sobolev to Lipschitz property (see Theorem~\ref{measure-pro}). And last, by a methods as in \cite{Ket1}, we show that each component of $Y$ is a $\RCD$-space (see Theorem~\ref{Y-rcd}).
\subsection{Warped product structure}
Let $a'=a+(b-a)/3, b'=b-(b-a)/3$ and let $A=A_{a', b'}(S)$ be as in Theorem~\ref{cos-law}. Let $F: [-\frac{b-a}3, \frac{b-a}3]\times A\to A_{a, b}(S)$ be a continuous map as in Theorem~\ref{flow-con}.
Let $Y=\partial B_{a'}(S)$ and let $\iota: Y\to A_{a, b}(S)$ be the inclusion map. Assume $Y$ has one component.
For $y_1, y_2\in Y$, define $d_Y(y_1, y_2)$ and the measure $\mathfrak m_Y$ as follows:
(i) for $m=0$, $K=0$,
$$ d_Y(y_1, y_2)= d(y_1, y_2);$$
And for $\bar E\subset Y$,
\begin{equation}\mathfrak m_Y (\bar E)=\frac{\mathfrak m(\{x\in A, F_{-d_s(x)+a'}(x)\in \bar E\})}{b'-a'};\label{measure-1}\end{equation}
(ii) for $m\neq 0$ or $K\neq 0$,
$$d_Y(y_1, y_2)=\frac1{\op{sn}_H(r_0+a')}\inf\{L(\bar \gamma),\, \bar \gamma\in \iota(Y), \bar\gamma_0=y_1, \bar \gamma_1=y_2\},$$
where
$$L(\bar\gamma)=\limsup_{\delta\to 0}\left\{\sum d(\bar\gamma(t_i), \bar\gamma(t_{i+1})),\, d(\bar\gamma(t_i), \bar\gamma(t_{i+1}))\leq \delta, 0\leq t_1\leq \cdots \leq t_n\leq b \text{ is a divison of }\bar\gamma\right\}$$
For $\bar E\in Y$, define the measure $\mathfrak m_Y(\bar E)$ as
\begin{equation}\mathfrak m_Y(\bar E)=\frac{\mathfrak m(\{x\in A, F_{-d_s(x)+a'}(x)\in \bar E\})}{\int_{a'}^{b'}\op{sn}^{N-1}_H(t+r_0)dt}.\label{measure-2}\end{equation}
Define a map from $A$ to a warped product space as the following:
For $m=0$ and $K=0$, define
$$\Phi: A\to (a', b')\times Y, \, x\mapsto \Phi(x)=(d_s(x), F_{-d_s(x)+a'}(x)).$$
Then by Pythagoras theorem, for $x, x'\in A$,
$$d^2(x, x')=(d_s(x)-d_s(x'))^2+ d^2(F_{-d_s(x)+a'}(x), F_{-d_s(x')+a'}(y))=d^2(\Phi(x), \Phi(x')).$$
For $m\neq 0$ or $K\neq 0$, define
$$\Phi: A\to (a'+r_0, b'+r_0)\times_{\op{sn}_H(r)} Y, \, x\mapsto \Phi(x)=(d_s(x)+r_0, F_{-d_s(x)+a'}(x)).$$
And by Cosine law and the definition fo $d_Y$, we have that for $x, x'\in A$
$$d(x, x')=d_w(\Phi(x), \Phi(x')).$$
In fact, assume $\gamma$ is a minimal geodesic from $x$ to $x'$ with $d(x, x')$ sufficient small such that $\gamma\subset A$. Let $F_{-d_s(\gamma(t))+a'} (\gamma(t))=\bar \gamma(t)$. Dived $\gamma$ with
$0=t_0\leq t_1\leq \cdots \leq t_n=1$ such that $d(\gamma(t_i), \gamma(t_{i+1}))\leq \delta$. Then
\begin{eqnarray*}
d(x, x') &= &\lim_{\delta\to 0}\sum d(\gamma(t_i), \gamma(t_{i+1}))\\
&=& \lim_{\delta\to 0}\sum \frac{\op{sn}_H(d_s(\gamma(\xi_i))+r_0)}{\op{sn}_H(a'+r_0)}d(\bar \gamma(t_i), \bar \gamma(t_{i+1}))\\
&=& \lim_{\delta\to 0}\sum \op{sn}_H(d_s(\gamma(\xi_i))+r_0)d_Y(\bar \gamma(t_i), \bar \gamma(t_{i+1}))\\
&=& d_w(\Phi(x), \Phi(x')).
\end{eqnarray*}
And more precisely, we have that for $d_Y(y_1, y_2)\leq \delta$, some $\delta>0$,
(ii-1) if $K=0$ and $m\neq 0$,
$$ d_Y(y_1, y_2)=\arccos\left(1-\frac{d^2(y_1, y_2)}{2(r_0+a+(b-a)/3)^2}\right);$$
(ii-2) if $K=N-1$,
$$d_Y(y_1, y_2)=\arccos\left(\frac{\cos d(y_1, y_2)-1}{\sin^2(r_0+a+(b-a)/3)^2}+1\right);$$
(ii-3) if $K=-(N-1)$,
$$d_Y(y_1, y_2)=\arccos\left(\frac{1-\cosh d(y_1, y_2)}{\sinh^2(r_0+a+(b-a)/3)^2}+1\right).$$
If $Y$ has more than one components, by above discussion, restricted to each component of $A$, $\Phi$ is an isometry. Now as the discussion in \cite[Claim 5.8]{Hu}, by relative volume comparison, we know that the number of $Y$'s components $\leq C(N, H, D, b, a)$. In fact, for each component $Y_k$, there is $y_k\in Y_k$ such that $x_k=F_{(b'-a')/2}(y_k)\in A$ and thus $B_{(b'-a')/3}(x_k)\subset A$. And for $l\neq k$, $B_{(b'-a')/3}(x_k)\cap B_{(b'-a')/3}(x_l)=\emptyset$. By relative volume comparison and that $A\subset B_{D+b}(p)$ for any $p\in S$, we know that $A$ contains at most $C(N, H, D, b, a)$ points which are $(b'-a')/3$ separated.
In the following, we will alway assume $Y$ has one component.
By the uniqueness of solutions of the continuity equation \eqref{continuity-equ} (more precisely the local uniqueness, see \cite[Lemma 3.14]{CDNPSW}), we have that
for any $E\subset A$, $t\in [-(b-a)/3, (b-a)/3]$,
(i) for $m=0$, $K=0$,
\begin{equation}(F_t)_{\sharp}\mathfrak m(E)=\mathfrak m(E);\label{flow-vol1}\end{equation}
(ii) for $m\neq 0$ or $K\neq 0$, for a.e. $x\in A$, $\mu_t=(F_t)_{\sharp} \mathfrak m$ satisfies
\begin{equation}\frac{d}{dt} d\mu_t(x)+\Delta d_s(F_{-t}(x)) d\mu_t(x)=0\label{flow-vol2}\end{equation}
which implies that
\begin{equation}d\mu_t(x)=\frac{\op{sn}_H^{N-1}(d_s(x)-t+r_0)}{\op{sn}_H^{N-1}(d_s(x)+r_0)}d\mu_0(x).\label{flow-measure}\end{equation}
By the definition of $\mathfrak m_Y$, \eqref{measure-1}, \eqref{measure-2}, and the property \eqref{flow-vol1}, \eqref{flow-vol2} and the Laplacian estimates Theorem~\ref{lap-main}, as the proof of \cite[Proposition 5.28]{Gi13} (see also \cite[Lemma 5.11]{CDNPSW}), we have that for $a'\leq c<d\leq b'$ and a Borel subset $\bar E\subset Y$,
\begin{equation}\mathfrak m(E_c^d)=\begin{cases} \mathfrak m_Y(\bar E)(d-c), & m=0 \text{ and } K=0;\\
\mathfrak m_Y(\bar E)\int_{c}^{d} \op{sn}_H^{N-1}(t+r_0)dt, & m\neq 0 \text{ or } K\neq 0,\end{cases}\label{flow-measure-2}\end{equation}
where
$$E_c^d=\{x\in A, \, c\leq d_s(x)\leq d, F_{-d_s(x)+a'}\in \bar E\}.$$
In fact, for $m\neq 0$ or $K\neq 0$, \eqref{flow-measure-2} can also be derived
by \eqref{flow-measure} and the fact that
$$\int_{a'-d}^{a'-c}\mu_t(\bar E) dt=\mathfrak m(E_c^d).$$
Since $\Phi$ is a isometry, we know that $A$ has a warped product structure. And thus $(a'+r_0, b'+r_0)\times_{\op{sn}_H} Y$ is a $\op{CD}_{loc}(K, N)$-space for $m\neq 0$ or $K\neq 0$, $(a', b')\times Y$ is a $\op{CD}_{\op{loc}}(0, N)$-space for $m=0$ and $K=0$.
In the following, we denote
$$(Y_w, d_w, \mathfrak m_w)=\begin{cases} (\Bbb R\times Y, d, \mathcal L^1\otimes \mathfrak m_Y), & m=0, K=0;\\
(C(Y), d_K, \mathfrak m_N), & m\neq 0 \text{ or } K\neq 0.\end{cases} $$
\subsection{Properties of the metric measure space $(Y, d_Y, \mathfrak m_Y)$}
In this subsection, we consider the metric measure space $(Y, d_Y, \mathfrak m_Y)$ as above and as in \cite{CDNPSW} we will show that
\begin{Thm} \label{measure-pro}
Consider the metric measure space $(Y, d_Y, \mathfrak m_Y)$ defined as in above subsection. We have that $(Y, d_Y, \mathfrak m_Y)$ is infinitesimally Hilbertian, satisfies the almost everywhere locally doubling property, supports a local Poincar\'e inequality and is a measured-length space.
\end{Thm}
For any $[a'', b'']\subset [a', b']$, let
$$T: [a'', b'']\times Y\to A, \, (t, y)\mapsto T(t, y)=F_{t}(y),$$
$$\hat T: C([0, 1], Y)\times [a'', b'']\to C([0, 1], X), \, (\bar \gamma_s, t)\mapsto \hat T(\bar\gamma, t)_s=F_{t}(\bar\gamma_s).$$
And let
$$P: A_{a, b}(S)\to Y, \, x\mapsto P(x)=F_{-d_s(x)+a'}(x),$$
$$\hat P: C([0, 1], A_{a, b}(S))\to C([0, 1], Y), \, \gamma_s\mapsto \hat P(\gamma)_s=F_{-d_s(\gamma_s)+a'}(\gamma_s).$$
Consider the inclusion map $\iota: Y\to A_{a, b}(S)$. By the definition of $d_Y$,
for $m=0, K=0$, $\iota$ is an isometric embedding; for $m\neq 0$ or $K\neq 0$, $\iota$ is a Lipschitz map.
Since Poincar\'e inequality is invariant under a bi-Lipschitz map (cf. \cite[Section 4.3]{BB}, more precisely the proof of \cite[Proposition 4.16]{BB}) and $\iota: Y\to A_{a', b'}(S)$ is Lipschitz, the local Poincar\'e inequality in $A_{a', b'}(S)$ implies the local Poincar\'e inequality in $(Y, d_Y, \mathfrak m_Y)$. And since $A_{a', b'}(S)\subset X$ which is a $\RCD$-space, by \cite{Ra}, $A_{a', b'}(S)$ supports a local Poincar\'e inequality.
To see $(Y, d_Y, \mathfrak m_Y)$ is infinitesimally Hilbertian we only need to show that
\begin{Lem} \label{inf-hil}
Given a nonnegative function $h\in \op{Lip}(\Bbb R)$ with $h(t)=0$ for $t\in \Bbb R\setminus [a+\frac{b-a}4, b-\frac{b-a}4]$ and $h(t)=1$ for $t\in [a', b']$, for each $g\in L^2(Y, \mathfrak m_Y)$, define $f(x)=g(P(x))h(d_s(x))$. Then
$g\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$ if and only if $f\in W^{1,2}(X, d, \mathfrak m)$ and for $x\in A$,
(i) for $m=0$, $K=0$,
$$|\nabla f|(x)= |\nabla g|(F_{-d_s(x)+a'}(x))=|\nabla^Y g|(F_{-d_s(x)+a'}(x));$$
(ii) for $m\neq 0$ or $K\neq 0$,
$$|\nabla f|(x)=\frac{\op{sn}_H(a'+r_0)}{\op{sn}_H(d_s(x)+r_0)}|\nabla g|(F_{-d_s(x)+a'}(x))=\frac{1}{\op{sn}_H(d_s(x)+r_0)}|\nabla^Y g|(F_{-d_s(x)+a'}(x)).$$
\end{Lem}
\begin{proof}
The proof is similar as in \cite[Proposition 5.12, Theorem 5.13]{CDNPSW}.
Consider test plans $\bar \Pi$ on $Y$. Let $\Pi=\hat T_{\sharp} \left(\bar \Pi\times (b''-a'')^{-1}\mathcal L^1_{[a'', b'']}\right)$, where $[a'', b'']\subset [a', b']$ as above.
Claim 1: $\Pi$ is a test plan of $X$.
First note that
$$\int_{C([0,1], X)}\int_0^1|\dot{\gamma}_s|^2ds d\Pi(\gamma)=\int_{C([0, 1], Y)}(b''-a'')^{-1}\int_{a''}^{b''}\int_0^1|\hat T(\bar \gamma, t)'_s|^2ds dt d\bar\Pi(\bar \gamma),$$
and
\begin{equation*}
|\hat T(\bar \gamma, t)'_s| = \lim_{h\to 0}\frac{d(\hat T(\bar \gamma, t)_{s+h}, \hat T(\bar \gamma, t)_s)}{|h|}
= \lim_{h\to 0}\frac{d(F_t(\bar \gamma_{s+h}), F_t(\bar \gamma_s))}{|h|}.
\end{equation*}
By Theorem~\ref{flow-con},
for $m=0$, $K=0$,
$$|\hat T(\bar \gamma, t)'_s| =|\dot {\bar\gamma}_s|,$$
and thus
$$\int_{C([0,1], X)}\int_0^1|\dot{\gamma}_s|^2ds d\Pi(\gamma)=\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_s|^2ds d\bar\Pi(\bar \gamma)<\infty;$$
For $m\neq 0, K=0$,
$$|\hat T(\bar \gamma, t)'_s| =\lim_{h\to 0}\frac{r_0+a'+t}{r_0+a'}\frac{d(\bar \gamma_{s+h}, \bar \gamma_s)}{|h|}=\frac{t+r_0+a'}{r_0+a'}|\dot{\bar \gamma}_s|$$
and thus
$$\int_{C([0,1], X)}\int_0^1|\dot{\gamma}_s|^2ds d\Pi(\gamma)=\frac{(r_0+a'+b')^2}{(r_0+a')^2} \int_{C([0, 1], Y)}\int_0^1|\bar\gamma'_s|^2ds d\bar\Pi(\bar \gamma)<\infty.$$
For $K=(N-1)$,
$$|\hat T(\bar \gamma, t)'_s| =\lim_{h\to 0}\frac{|\sin(r_0+a'+t)|}{|\sin(r_0+a')|}\frac{d(\bar \gamma_{s+h}, \bar \gamma_s)}{|h|}=\frac{|\sin(r_0+a'+t)|}{|\sin(r_0+a')|} |\dot{\bar \gamma}_s|$$
and thus
$$\int_{C([0,1], X)}\int_0^1|\dot{\gamma}_s|^2ds d\Pi(\gamma)\leq \frac1{|\sin(r_0+a')|^2}\int_{C([0, 1], Y)}\int_0^1|\bar\gamma'_s|^2ds d\bar\Pi(\bar \gamma)<\infty.$$
For $K=-(N-1)$, it is similar as the case $K=N-1$.
And for the set $E_c^d=\{x\in A_{a, b}(S), \, a'\leq c\leq d_s(x)\leq d\leq b', x\in E\}$, where $E$ is a Borel set in $Y$,
\begin{eqnarray*}
\left(e_t\right)_{\sharp}\Pi(E_c^d) &=& \bar \Pi\times (b''-a'')^{-1}\left.\mathcal L^1\right|_{[a'', b'']}((e_t\circ \hat T)^{-1} E_c^d)\\
&=& \bar \Pi(e_t^{-1}E)(b''-a'')^{-1}\left.\mathcal L^1\right|_{[a'', b'']}([c, d])\\
&\leq& C\mathfrak m_Y(E)\leq C' \mathfrak m(E_c^d).
\end{eqnarray*}
Claim 2: If $f\in W^{1,2}(X, d, \mathfrak m)$, then $g\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$.
Assume $f\in W^{1,2}(X, d, \mathfrak m)$, then
\begin{eqnarray*}
\int_{C([0,1], Y)}g(\bar \gamma_1)-g(\bar\gamma_0)d\bar \Pi(\bar \gamma)&=& \int_{C([0,1], Y)}(b''-a'')^{-1}\int_{a''}^{b''}g(\bar \gamma_1)h(t)-g(\bar\gamma_0)h(t)dtd\bar\Pi(\bar \gamma)\\
& =& \int_{C([0, 1], X)} f(\gamma_1)-f(\gamma_0) d\Pi(\gamma)\\
&=& \int_{C([0,1], X)}\int_0^1 |\nabla f||\dot \gamma_s| ds d\Pi(\gamma)\\
&=& \int_{C([0,1], Y)}(b''-a'')^{-1}\int_{a''}^{b''} \int_0^1 |\nabla f|(\hat T(\bar \gamma_s, t))|\hat T(\bar \gamma, t)'_s| ds dt d\bar \Pi(\bar\gamma),
\end{eqnarray*}
and thus $g\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$ and
$$|\nabla g|(y)\leq (b''-a'')^{-1}\int_{a''}^{b''}|\nabla f|(T(y, t))\op{Lip}(F_t)dt.$$
Let $a''\to t_0, b''\to t_0$.
For $m=0, K=0$,
$$|\nabla g|(y)\leq |\nabla f|(F_{t_0}(y)).$$
For $m\neq 0$, $K=0$,
$$|\nabla g|(y)\leq \frac{t_0+r_0+a'}{r_0+a'} |\nabla f|(F_{t_0}(y)).$$
For $K\neq 0$,
$$|\nabla g|(y)\leq \frac{\op{sn}_H(t_0+r_0+a')}{\op{sn}_H(r_0+a')} |\nabla f|(F_{t_0}(y)).$$
Claim 3: If $g\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$, then $f\in W^{1,2}(X, d, \mathfrak m)$.
By the definition of $f$, we only need to consider text plan $\Pi$ which is supported in $C([0, 1], A_{a, b}(S))$.
In fact, for $\gamma\in C([0, 1], X)$ with finite length, without loss of generality we may assume $\gamma(0)$, $\gamma(1)\in A_{a+(b-a)/4, b-(b-a)/4}(S)$, we can divide $\gamma$ into finite pieces $\gamma^1, \gamma^2, \cdots, \gamma^{n+1}$ by taking cut points
$$0<t_1<t_2<\cdots <t_n<1,$$
such that for $A_1=\partial B_a(S)$, $A_2=\partial B_b(S)$, $A_3=\partial B_{a+(b-a)/4}(S)$, $A_4=\partial B_{b-(b-a)/4}(S)$,
(1) $\gamma(t_i)\in \cup_1^4 A_j$;
(2) if $\gamma(t_i)\in A_j$ then $\gamma(t_{i+1})\notin A_j$;
(3) $\left.\gamma\right|_{[t_i, t_{i+1}]}\subset A_{a, b}(S)$ or $\left.\gamma\right|_{[t_i, t_{i+1}]}\subset X\setminus A_{a+(b-a)/4, b-(b-a)/4}(S)$.
Assume the pieces $\gamma^{i1}, \cdots, \gamma^{ik}\subset A_{a, b}(S)$. Note that $f(\gamma(t_i))=0$ for each $i$, then
$$|f(\gamma_1)-f(\gamma_0)|\leq |f(\gamma^{i1}_1)-f(\gamma^{i1}_0)|+\cdots + |f(\gamma^{ik}_1)-f(\gamma^{ik}_0)|.$$
If for any $\gamma\in C([0, 1], A_{a, b}(S))$, we have
$$|f(\gamma_1)-f(\gamma_0)|\leq \int_0^1 G(\gamma_t)|\dot\gamma_t| dt,$$
then we can take
$$\tilde G(x)=\begin{cases} G(x), & x\in A_{a, b}(S),\\
0, & x\in X\setminus A_{a, b}(S),
\end{cases}$$
such that
$$|f(\gamma_1)-f(\gamma_0)|\leq \int_0^1\tilde G(\gamma_t)|\dot \gamma_t|dt.$$
Let $\bar \Pi=\hat P_{\sharp}(\Pi)$ where $\Pi$ is supported in $C([0, 1], A_{a, b}(S))$.
As claim 1, we have that $\bar \Pi$ is a test plan of $Y$. To see this as above
$$\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_t|^2 dt d\bar \Pi=\int_{C([0, 1], X)}\int_0^1 \left|\frac{d}{dt}F_{-d_s(\gamma_t)+a'}(\gamma_t)\right|^2dtd\Pi.$$
For $m=0$, $K=0$,
$$d^2(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t}))=d^2(\gamma_{t+h}, \gamma_t)-|d_s(\gamma_{t+h})-d_s(\gamma_t)|^2\leq d^2(\gamma_{t+h}, \gamma_t),$$
Thus
$$\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_t|^2 dt d\bar \Pi\leq \int_{C([0, 1], X)}\int_0^1 \left|\dot\gamma_t\right|^2dtd\Pi<\infty.$$
For $m\neq 0$, $K=0$, by
$$\frac{d^2(\gamma_{t+h}, \gamma_t)-(d_s(\gamma_{t+h})+r_0)^2-(d_s(\gamma_{t})+r_0)^2}{2(d_s(\gamma_{t+h})+r_0)(d_s(\gamma_{t})+r_0)}=\frac{d^2(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t}))-2(a'+r_0)^2}{2(a'+r_0)^2},$$
we have
$$d^2(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t}))\leq \frac{(a'+r_0)^2}{(d_s(\gamma_{t+h})+r_0)(d_s(\gamma_{t})+r_0)}d^2(\gamma_{t+h}, \gamma_t).$$
Thus
$$\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_t|^2 dt d\bar \Pi\leq \int_{C([0, 1], X)}\int_0^1 \left(\frac{a'+r_0}{d_s(\gamma_t)+r_0}\right)^2\left|\dot\gamma_t\right|^2dtd\Pi<\infty.$$
For $K=(N-1)$, by
$$\frac{\cos d(\gamma_{t+h}, \gamma_t)-\cos(d_s(\gamma_{t+h})+r_0)\cos(d_s(\gamma_{t})+r_0)}{\sin(d_s(\gamma_{t+h})+r_0)\sin(d_s(\gamma_{t})+r_0)}=\frac{\cos d(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t})) -\cos^2(a'+r_0)}{\sin^2(a'+r_0)}$$
i.e.
$$\frac{\cos d(\gamma_{t+h}, \gamma_t)-\cos(d_s(\gamma_{t+h})-d_s(\gamma_t))}{\sin(d_s(\gamma_{t+h})+r_0)\sin(d_s(\gamma_{t})+r_0)}=\frac{\cos d(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t})) -1}{\sin^2(a'+r_0)},$$
$$\cos d(\gamma_{t+h}, \gamma_t)-1\leq \frac{\sin(d_s(\gamma_{t+h})+r_0)\sin(d_s(\gamma_{t})+r_0)}{\sin^2(a'+r_0)}\left(\cos d(F_{-d_s(\gamma_{t+h})+a'}(\gamma_{t+h}), F_{-d_s(\gamma_{t})+a'}(\gamma_{t})) -1\right).$$
thus
$$\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_t|^2 dt d\bar \Pi\leq \int_{C([0, 1], X)}\int_0^1 \left(\frac{\sin(a'+r_0)}{\sin (d_s(\gamma_t)+r_0)}\right)^2\left|\dot\gamma_t\right|^2dtd\Pi<\infty.$$
For $K=-(N-1)$, similarly as the $K=(N-1)$ case we have
$$\int_{C([0, 1], Y)}\int_0^1|\bar \gamma'_t|^2 dt d\bar \Pi\leq \int_{C([0, 1], X)}\int_0^1 \left(\frac{\sinh(a'+r_0)}{\sinh (d_s(\gamma_t)+r_0)}\right)^2\left|\dot\gamma_t\right|^2dtd\Pi<\infty.$$
And for $E\subset Y$,
\begin{eqnarray*}
(e_t)_{\sharp} \bar \Pi(E) &=& \Pi ((e_t\circ \hat P)^{-1}(E))\\
&=& \Pi (e_t^{-1}(E_a^b))=(e_t)_{\sharp}\Pi(E_a^b)\\
&\leq & c\mathfrak m(E_a^b)\leq c' \mathfrak m_Y(E).
\end{eqnarray*}
Now, since $g\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$ and
\begin{eqnarray*}
& & \int_{C([0,1], X)} f(\gamma_1)-f(\gamma_0) d\Pi(\gamma)\\
& =& \int_{C([0,1], X)}g(F_{-d_s(\gamma_1)+a'}(\gamma_1))h(d_s(\gamma_1))-g(F_{-d_s(\gamma_0)+a'}(\gamma_0))h(d_s(\gamma_0)) d\Pi(\gamma)\\
&\leq & \int_{C([0,1], Y)}g(\bar \gamma_1)-g(\bar \gamma_0) d\bar \Pi(\bar \gamma)+ \int_{C([0,1], X)}g(F_{-d_s(\gamma_0)+a'}(\gamma_0))\int_0^1h'(d_s(\gamma_t))|\dot \gamma_t| dtd\Pi(\gamma)\\
&\leq & \int_{C([0,1], Y)}\int_0^1|\nabla g|(\bar \gamma_t) |\bar \gamma'_t|dtd\bar \Pi(\bar \gamma)+ \int_{C([0,1], X)}g(F_{-d_s(\gamma_0)+a'}(\gamma_0))\int_0^1h'(d_s(\gamma_t))|\dot \gamma_t| dtd\Pi(\gamma)\\
&\leq & \int_{C([0,1], X)}\int_0^1|\nabla g|(F_{-d_s(\gamma_t)+a'}(\gamma_t))|\hat P(\gamma)'_t|+g(F_{-d_s(\gamma_0)+a'}(\gamma_0))h'(d_s(\gamma_t))|\dot \gamma_t| dtd\Pi(\gamma)
\end{eqnarray*}
Thus for $x\in A$,
$$|\nabla f|(x)\leq |\nabla g|(F_{-d_s(x)+a'}(x))\op{Lip}(\hat P).$$
For $m=0$, $K=0$,
$$|\nabla f|(x)\leq |\nabla g|(F_{-d_s(x)+a'}(x));$$
For $m\neq 0$, $K=0$,
$$|\nabla f|(x)\leq \frac{a'+r_0}{d_s(x)+r_0}|\nabla g|(F_{-d_s(x)+a'}(x));$$
For $K\neq 0$,
$$|\nabla f|(x)\leq \frac{\op{sn}_H(a'+r_0)}{\op{sn}_H(d_s(x)+r_0)}|\nabla g|(F_{-d_s(x)+a'}(x)).$$
\end{proof}
Recall the definitions of almost everywhere locally doubling property and measured-length property.
\begin{Def}
A metric measure space $(X, d, \mathfrak m)$ is almost everywhere locally doubling if there is a full measure Borel subset $\hat X\subset X$ satisfying that for each $x\in \hat X$, there exists an open set $U\ni x$, constants $C, R>0$, such that for $r\in (0, R)$, $y\in U$,
$$\mathfrak m(B_{2r}(y))\leq C\mathfrak m(B_{r}(y)).$$
\end{Def}
\begin{Def}
A metric measure space $(X, d, \mathfrak m)$ is measured-length if there is a full measure subset $\hat X\subset X$ satisfying the following: For $x_0, x_1\in \hat X$, there exist $\epsilon>0$, a map
$$(0, \epsilon]^2\to \mathcal P(C([0, 1], X)), \, (t_0, t_1)\mapsto \Pi^{t_0, t_1},$$
such that
(i) For $\phi\in C_{b}(C([0, 1], X))$, the map
$$(0, \epsilon]^2\to \Bbb R, \, (t_0, t_1)\mapsto \int \phi d\Pi^{t_0, t_1}$$
is Borel;
(ii) For $i=0, 1$,
$$(e_i)_{\sharp}\Pi^{t_0, t_1}=\frac{1_{B_{t_i}(x_i)}}{\mathfrak m(B_{t_i}(x_i))}\mathfrak m;$$
(iii)
$$\limsup_{t_0, t_1\downarrow 0}\int \int_0^1 |\dot \gamma_t|^2 dt d\Pi^{t_0, t_1}(\gamma)\leq d^2(x_0, x_1).$$
\end{Def}
Now we prove $(Y, d_Y, \mathfrak m_Y)$ is almost everywhere locally doubling and measured-length.
\begin{Lem}
The metric measure space $(Y, d_Y, \mathfrak m_Y)$ is almost everywhere locally doubling.
\end{Lem}
\begin{proof}
For $E\subset Y$, let $E_r=\{x\in A, \, 0\leq d_s(x)-a'\leq r, F_{-d_s(x)+a'}\in E\}$.
For each $x\in Y$, there is open set $Y\supset U\ni x$, $R>0$, such that for each $ y\in U$, $r<R$,
(i) for $m=0$, $K=0$,
\begin{eqnarray*}
\mathfrak m_Y(B_{2r}( y))& =&\frac{\mathfrak m(B_{2r}(y)_{2r})}{2r}\leq \frac{\mathfrak m(B_{4r}(F_r(\iota(y))))}{2r}\\
&\leq& \frac{\svolball{H}{4r}}{\svolball{H}{r}}\frac{\mathfrak m(B_{r}(F_r(\iota(y))))}{2r}\leq \frac{\svolball{H}{4r}}{\svolball{H}{r}}\frac{\mathfrak m(B_{r}(\iota(y))_{2r})}{2r}\\
& \leq & \frac{\svolball{H}{4r}}{\svolball{H}{r}}\mathfrak m_Y(B_{r}( y).
\end{eqnarray*}
(ii) for $m\neq 0$ or $K\neq 0$, note that for any $r<R$, there are $0<c(K, R)< C(K, R)$ such that
$$B_{c(K, R)r}(F_r(\iota(y)))\subset B_r(y)_{2r}\subset B_{2r}(y)_{2r}\subset B_{C(K, R)r}(F_r(\iota(y))).$$
Then by relative volume comparison, a similar argument as the $m=0$, $K=0$ case gives the almost everywhere locally doubling property.
\end{proof}
\begin{Lem}
The metric measure space $(Y, d_Y, \mathfrak m_Y)$ is a measured-length space.
\end{Lem}
The proof of this lemma is the same as the one \cite[Proposition 5.14]{CDNPSW}. Here we omit it. And by the following theorem, we can derive a series of properties about $(Y_w, d_w, \mathfrak m_w)$.
\begin{Thm}[\cite{GH, CDNPSW}] \label{glob-hil}
Consider a warped product space $Y_w=I\times_w Y$ where $I$ is a bounded interval in $\Bbb R$ and $w_d, w_m: I\to [0, \infty)$ with $w_m>0$ for points in the interior of $I$. Assume $(Y, d_Y, \mathfrak m_Y)$ is a.e. locally doubling, measured length, infinitesimally Hilbertian, then $(Y_w, d_w, \mathfrak m_w)$ is a.e. locally doubling, measured length, infinitesimally Hilbertian and it has the Sobolev to Lipschitz property.
\end{Thm}
\subsection{$(Y, d_Y, \mathfrak m_Y)$ is a $\RCD$-space}
For $m=0$, $K=0$, we will show that $(\Bbb R\times Y, d, \mathcal L^1\otimes \mathfrak m_Y)$ satisfies the $\op{CD}_{\op{loc}}(0, N)$ condition. Then by the local-to-global property \cite[Theorem 3.14]{EKS} and \cite{RS}, we know that $\Bbb R\times Y$ is essentially non-branching and is a $\op{CD}(0, N)$-space. Finally an argument as in \cite{Gi13} gives that $(Y, d_Y, \mathfrak m_Y)\in \RCD(0, N-1)$.
For $K\neq 0$ or $m\neq 0$, Ketterer \cite[Theorem 1.2]{Ket1} (see Theorem~\ref{cone-rcd}) proved that if $(C(Y), d_K, \mathfrak m_N)\in \RCD^*(K, N)$, then $(Y, d_Y, \mathfrak m_Y)\in \RCD^*(N-2, N-1)$ and $\op{diam}(Y)\leq \pi$. We have known that:
$(a'+r_0, b'+r_0)\times_{\op{sn}_H(r)} Y$ is isometric to $A_{a', b'}(S)\subset X$ which is a $\op{RCD}(K, N)$-space.
We will show that Ketterer's result (ii) of Theorem~\ref{cone-rcd} holds under this weaker condition.
First recall some basic definitions about Dirichlet forms. See \cite{AGS} or \cite{Ket1} for more details.
Let $(X, d, \mathfrak m)$ be a locally compact, separable Hausdorff metric measure space. A symmetric Dirichlet form $\mathcal E^X$ defined in $D(\mathcal E^X)\subset L^2(X, \mathfrak m)$ is a $L^2(X, \mathfrak m)$-lower semi-continuous, quadratic form that satisfies the Markov property. The domain $D(\mathcal E^X)$ is a Hilbert space with respect to
$$(u, u)_{D(\mathcal E^X)}=(u, u)_{L^2(X, \mathfrak m)}+\mathcal E^X(u, u).$$
There is a self-adjoint, negative-definite operator $(L^X, D_2(L^X))$ on $L^2(X, \mathfrak m_X)$ where
$$D_2(L^X)=\{u\in D(\mathcal E^X),\, \exists v\in L^2(X, \mathfrak m), -(v, w)_{L^2(X, \mathfrak m)}=\mathcal E^X(u, w), \forall w\in D(\mathcal E^X)\}.$$
Let $v=L^X u$.
Denote $D^{\infty}(\mathcal E^X)=D(\mathcal E^X)\cap L^{\infty}(X, \mathfrak m)$. For $u, \phi\in D^{\infty}(\mathcal E^X)$, define
$$\Gamma^X(u; \phi)=\mathcal E^X(u, u\phi)-\frac12\mathcal E^X(u^2, \phi),$$
which can be extended by continuity to $u\in D(\mathcal E^X)$.
For $u, v\in D(\mathcal E^X), \phi\in D^{\infty}(\mathcal E^X)$, define
$$\Gamma^X(u, v; \phi)=\frac12\left(\Gamma^X(u;\phi)+\Gamma^X(v;\phi)-\Gamma^X(u-v; \phi)\right).$$
And let
$$2\Gamma_2^X(u, v;\phi)=\Gamma^X(u, v; L^X\phi)-2\Gamma^X(u, L^Xv; \phi), \quad \Gamma^X_2(u;\phi)=\Gamma^X_2(u, u;\phi).$$
where
$u, v\in D(\Gamma^X_2)=\{u\in D_2(L^X), \, L^Xu\in D(\mathcal E^X)\}$, test function $\phi\in D_+^{b, 2}(L^X)=\{\phi\in D_2(L^X),\, \phi, L^X\phi\in L^{\infty}(X, \mathfrak m), \phi>0\}$.
Let $D'$ be the set of $u$ such that the map $\phi\mapsto \Gamma^X(u;\phi)$ is an absolutely continuous measure w.r.t. $\mathfrak m$ which is denoted by $\Gamma^X(u)\mathfrak m$. If $D'=D(\mathcal E^X)$, we call $\mathcal E^X$ admits a ``carr\'e du champ" operator.
If $\mathcal E^X$ is strongly local and admits a ``carr\'e du champ" operator (see \cite[Section 2.1]{Ket1}), one can define $D_{\op{loc}}(\mathcal E^X)\subset L^2_{\op{loc}}(X, \mathfrak m)$ and thus there is a intrinsic distance of $\mathcal E^X$,
$$d_{\mathcal E^X}(x, y)=\sup\{u(x)-u(y), \, u\in D_{\op{loc}}(\mathcal E^X)\cap C(X), \Gamma^X(u)\leq 1, \mathfrak m-a.e.\}.$$
Note that if $(X, d, \mathfrak m)$ is infinitesimally Hilbertian, the Cheeger energy of $X$, $\op{Ch}^X$ is a symmetric Dirichlet form. And the corresponding $L^Xu=\Delta_Xu$, the Laplacian of $u$, $\Gamma^X$ is the same as the one defined in Subsection 2.1. Compared with $\Gamma_2$ defined in $\RCD(K, N)$ (Subsection 2.5), $\Gamma_2^X$ here is in a weak sense.
\begin{Thm} \label{Y-rcd}
Let $(Y, d_Y, \mathfrak m_Y)$ be as the one in the beginning of this section.
(i) For $m=0$, $K=0$, $(\Bbb R\times Y, d, \mathcal L^1\otimes \mathfrak m_Y)$ is a $\RCD(0, N)$-space and thus $(Y, d_Y, \mathfrak m_Y)$ is a $\RCD(0, N-1)$-space;
(ii) For $m\neq 0$, $K=0$, $(C(Y), d_0, \mathfrak m_N)$ is a $\RCD(0, N)$-space and $(Y, d_Y, \mathfrak m_Y)$ is a $\RCD(N-2, N-1)$-space;
(iii) For $K\neq 0$, $(Y, d_Y, \mathfrak m_Y)$ is a $\RCD(N-2, N-1)$-space. Especially for $K>0$, $(C(Y), d_K, \mathfrak m_N)$ is a $\RCD(K, N)$-space.
\end{Thm}
\begin{proof}
For $m=0$ and $K=0$, note that for $\delta>0$, $(a'+\delta, b'-\delta)\times Y$ is locally a $\op{CD}(0, N)$-space. And for any $t\in \Bbb R$, $(t, t+b'-a'-2\delta)\times Y$ is isometric to $(a'+\delta, b'-\delta)\times Y$ and thus is locally a $\op{CD}(0, N)$-space. Since $\Bbb R\times Y$ can be covered by $(t, t+b'-a'-2\delta)\times Y$, $t\in \Bbb R$, we know $(\Bbb R\times Y, d, \mathcal L^1\otimes \mathfrak m_Y)$ is in $\op{CD}_{\op{loc}}(0, N)$.
By the discussion above Theorem 3.14 in \cite{EKS} and the infinitesimaly Hilbertian property of $\Bbb R\times Y$ (Theorem~\ref{measure-pro} and Theorem~\ref{glob-hil}), we know that $\Bbb R\times Y$ is essentially non-branching (see \cite{RS}). And then by local to global property \cite{CMi}, $(\Bbb R\times Y, d, \mathcal L^1\otimes \mathfrak m_Y)$ is $\RCD(0, N)$-space.
Now
$(Y, d_Y, \mathfrak m_Y)$ is a $\RCD(0, N-1)$-space by the argument in \cite{Gi13}.
For $m\neq 0$ or $K\neq 0$, first endow $Y$ with a new metric $d'_Y$ such that $\op{diam}(Y)\leq \pi$:
$$d'_Y(y_1, y_2)=\begin{cases} d_Y(y_1, y_2), & \text{ if } d_Y(y_1, y_2)\leq \pi;\\
\pi, & \text{ if } d_Y(y_1, y_2)>\pi.\end{cases}
$$
Then $(Y, d_Y)$ is locally isometric to $(Y, d'_Y)$ and the $(K, N)$-cone $C(Y, d_Y)=C(Y, d'_Y)$. In the following we will assume $Y$ endowed with the metric $d'_Y$.
For $m\neq 0$, $K=0$, note that for any $r>0$, $(C(Y), r^{-1} d_0, c_r \mathfrak m_N)$ is isometric to $(C(Y), d_0, \mathfrak m_N)$, where
$$c_r=\left(\int_{B_r(x)} 1-\frac{d_0(x, y)}{r} d\mathfrak m_N(y)\right)^{-1}.$$
Then for any $t>0$, $(a'+r_0+\delta, b'+r_0-\delta)\times_r Y\subset (C(Y), (a'+r_0+\delta)/t d_0, c_{t/(a'+r_0+\delta)}\mathfrak m_N)$ is isometric to the one in $(C(Y), d_0, \mathfrak m_N)$ which is locally a $\op{CD}(0, N)$-space. Rescaling back, we can see that
$$\left(t, \frac{b'+r_0-\delta}{a+r_0+\delta}t\right)\times_r Y\in (C(Y), d_0, \mathfrak m_N)$$
is in $\op{CD}_{\op{loc}}(0, N)$ which implies $(C(Y), d_0, \mathfrak m_N)\in \op{CD}_{\op{loc}}(0, N)$. As the above discussion we know that $(C(Y), d_0, \mathfrak m_N)$ is a $\RCD(0, N)$-space. Then (ii) of Theorem~\ref{cone-rcd} implies that $(Y, d'_Y, \mathfrak m_Y)$ is a $\RCD(0, N-1)$-space. And so is $(Y, d_Y, \mathfrak m_Y)$.
For (iii), we follow the argument in the proof of \cite[Theorem 1.2]{Ket1}.
By Theorem~\ref{measure-pro}, we know that $(Y, d_Y, \mathfrak m_Y)$ is infinitesimally Hilbertian and satisfies the Sobolev to Lipschitze property. Thus by \cite{EKS} to prove $(Y, d_Y, \mathfrak m_Y)$ is a $\RCD(N-2, N-1)$-space, one only need to show it satisfying the $(N-2, N-1)$ Bakry-Ledoux estimate:
for any $f\in W^{1,2}(Y, d_Y, \mathfrak m_Y)$, $t>0$,
\begin{equation}|\nabla (H_t(f))|^2+\frac{4(N-2)t^2}{(N-1)(e^{2(N-2)t}-1)}|\Delta H_t(f)|^2\leq e^{-2(N-2)t}H_t(|\nabla f|^2), \mathfrak m_Y-a.e. \label{B-L}\end{equation}
As in the proof of \cite[Theorem 1.2]{Ket1}, we can derive \eqref{B-L} by the following kind of Bakry-Emery inequality: for any $u\in D(\Gamma^Y_2)$, \begin{eqnarray}
& & \frac12\int_Y L^Y\phi \Gamma^Y(u) d\mathfrak m_Y-\int_Y \Gamma^Y(u, L^Y u)\phi d\mathfrak m_Y \nonumber\\
& \geq & (N-1)\int_Y \Gamma^Y(u)\phi d\mathfrak m_Y+\frac1{N}\int_Y (L^Y u)^2 \phi d\mathfrak m_Y -\frac1{(N+1)N}\int_F(L^Y u+N u)^2\phi d\mathfrak m_Y. \label{B-E}
\end{eqnarray}
Namely the methods and calculations from \eqref{B-E} to \eqref{B-L} are similar as the proof of the equivalence of Bakry-Emery inequality and Bakry-Ledoux estimate in \cite{EKS} (see \cite[Proposition 4.7, 4.9]{EKS}, see also the proof of \cite[Theorem 1.2]{Ket1}). Here we omit it and only point out that in the proof one needs the regularity of $\op{Ch}^Y$: The intrinsic distance of $\op{Ch}^Y$, $d_{\op{Ch}^Y}=d_Y$ and $\op{Ch}^Y$ is strongly regular \cite[Lemma 5.14]{Ket1}.
In \cite[Lemma 5.14]{Ket1}, Ketterer assumed that $(C(Y), d_K, \mathfrak m_N)$ is a $\RCD^*$-space. By examining the proof there, we can see that it only needs the Sobolev to Lipschitz property of $(C(Y), d_K, \mathfrak m_N)$ which can be seen from Theorem~\ref{glob-hil}.
In the following, we will see how to derive \eqref{B-E} from the relations between the Cheeger energies $\op{Ch}^Y$, $\op{Ch}^{C(Y)}$ and the warped product $\mathcal E^C$, where $\mathcal E^C$ is a symmetric form on $L^2(C(Y), \op{sn}_H^{N-1}(t)dt\otimes \mathfrak m_Y)$ defined as
$$\mathcal E^C(u)=\int_Y \int_{I_K} |u'_y(t)|^2dt d\mathfrak m_Y+\int_{I_K} \op{Ch}^Y(u_p)\op{sn}_H^{N-3}(t) dt$$
where $u\in C_0^{\infty}(I_K)\otimes D(\op{Ch}^Y)$ and $u_p=u(p, \cdot)$, $u_y=u(\cdot, y)$.
First note that:
Claim 1: As in \cite[Lemma 5.11]{Ket1}, we have that the intrinsic distance $d_{\mathcal E^C}$ of $\mathcal E^C$ satisfies that, for $x, y\in A_{a', b'}(S)$,
$$d_{\mathcal E^C}(x, y)=d_K(x, y).$$
Claim 2: As \cite[Corollary 5.12]{Ket1}, for $u\in D_{\op{loc}}(\mathcal E^C)\cap L^{\infty}(A_{a, b}(S))$, we have that
$$\mathcal E^C(u)=\op{Ch}^{C(Y)}(u).$$
By the definition of $\mathcal E^C$, for $u_1\otimes u\in C_0^{\infty}((a', b'))\otimes D(\Gamma^Y_2), 1\otimes \phi\in 1\otimes D_{+}^{b, 2}(L^Y)$, a careful calculation as in \cite{Ket1} gives that (see also (33) in \cite{Ket1}):
\begin{eqnarray}
& & \Gamma_2^C(u_1\otimes u; 1\otimes \phi) = \int_{C(Y)} \Gamma_2^{I_K, \op{sn}_H^{N-1}}(u_1)u^2\phi+\frac{u_1^2}{\op{sn}_H^2}\Gamma_2^Y(u)\phi+\frac12\Gamma^{I_K}(u_1)\frac{1}{\op{sn}_H^2}L^Y(u)\phi \nonumber\\
& & +\int_{C(Y)}\left(\frac12L^{I_K, \op{sn}_H^{N-1}}\left(\frac{u_1^2}{\op{sn}_H^2}\right)\Gamma^Y(u)-\frac{u_1}{\op{sn}_H^2}L^{I_K, \op{sn}_H^{N-1}}(u_1)\Gamma^Y(u)-\Gamma^{I_K}\left(u_1, \frac{u_1}{\op{sn}_H^2}\right)uL^Y(u)\right)\phi, \label{BE-weak}
\end{eqnarray}
where
$L^{I_K}=d^2/d t^2$ and
$$\mathcal E^{I_K, \op{sn}_H^{N-1}}(u)=\int_{I_K}(u')^2\op{sn}_H^{N-1} dt,$$
thus
$$L^{I_K, \op{sn}_H^{N-1}}(u)=u''+\frac{N-1}{\op{sn}_H}\op{sn}'_H u'.$$
By Claim 2, $\mathcal E^C=\op{Ch}^{C(Y)}$. And the Bakry-\'Emery inequality holds in $A_{a', b'}(S)$:
$$\Gamma_2^C(u_1\otimes u; 1\otimes \phi) \geq K\Gamma^C(u_1\otimes u; \phi)+\frac1{N}\int (L^C u_1\otimes u)^2\phi.$$
Then as in \cite[Theorem 3.10]{Ket1}, by taking $u_1=\sin t$, for $K=N-1$, and $u_1=\sinh t$, for $K=-(N-1)$, $t\in (a', b')$ in \eqref{BE-weak} and using the above Bakry-\'Emery inequality, we have \eqref{B-E}.
To see Claim 1, note that in the proof of \cite[Lemma 5.11]{Ket1}, the only place where Ketterer used the $\RCD^*$-condition of $C(Y)$ is to derive the inequality (see (47) in \cite{Ket1})
\begin{equation}
d_{\mathcal E^C}\leq d_K. \label{dist-comp}
\end{equation}
Here we want to show \eqref{dist-comp} holds locally in $A_{a', b'}(S)$. This can be seen by restricting all Ketterer's estimates in $A_{a', b'}(S)$ and then the curvature-dimension condition can be applied similarly.
In the proof of \cite[Corollary 5.12]{Ket2}, one only needs the local doubling property and local Poincar\'e inequality of $Y$ and $C(Y)$, and Claim 1. Thus Claim 2 is derived.
\end{proof}
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Airbus Parent EADS Earnings Dimmed by Cash Flow Issues
The European Aeronautic Defence and Space Co. (EADS) reported first-quarter earnings before markets opened this morning. Airbus, the company's largest division, competes with U.S. aircraft maker Boeing Co. (NYSE: BA) for both commercial and military plane sales.
In the first quarter, Boeing posted revenues of $18.9 billion, compared with sales totaling $16.23 billion at EADS. Boeing's net earnings totaled $1.1 billion, compared with about $315.8 million at EADS. Boeing's earnings per share (EPS) totaled $1.44 in the first quarter, compared with EPS of $0.38 at EADS.
And both companies have their problems. Boeing's problems are well-known: long delays in the delivery of the company's 787 Dreamliner and an electrical system problem that kept the all 787s grounded for three months. At EADS, similar delays stalled delivery of the Airbus A350 XWB, which is just about to enter flight testing after the first plane got a nifty paint job.
The big issue for EADS - and Airbus - is that it is bleeding cash. The company's free cash flow in the first quarter was negative - about $4.15 billion negative. The company is burning through cash as it prepares to manufacture the A350, a military transport, and its A380 jumbo jet. The company expects to get its free cash flow back at break-even for the year.
Filed under: 24/7 Wall St. Wire, Aerospace, Earnings Tagged: BA
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Upgrade Facebook Application with Monetization Features
Bajet $250-750 USD
Enhance WhispR application to support points. A point based system will allow users to accumulate points whenn an offer is completed via the OfferPal Network, friends are Invited, app is installed, etc. Points will be used for every message they read or use certain features of the application. OfferPal Network provides a simple to use API detailed below.
We are also implementing a landing page. The objective of this page is to inform our users of tips, promote advertisements, display other messages other users are sending, and provide an at-a-glance of view of Whispr activity (ie: number of new messages). We are also adding an “Add Points” page where the user can link to actions that can add points to their account.
More notable additions include expansion of our send-message module which will allow users to message friends of your friend’s social network and message users who are not on Facebook via email. Usage of these new features will be premium features and will deduct points from the user’s balance. The original send-message feature will remain free to use.
| 213,981
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How To Fill Your Heart (Without Diamonds)
By Rabbi Justus Baird
Excerpted from a High Holy Day sermon (full text here)
Coco Chanel once quipped, “The best things in life are free. The second best things are very, very expensive.” One of those expensive second-best things that many of us desire is a diamond. But as recently as the 1940’s, Americans weren’t familiar with engagement rings and did not associate diamonds with romance or marriage. Today, more than 75% of American women wear a diamond ring. So, how did our attitudes toward diamonds shift so radically?
The answer: brilliant marketing.
Two extremely talented women developed a two-prong approach to stimulate demand for diamonds. The first was developed by Frances Gerety who was working late on the DeBeers presentation that was due the next morning. As she headed to bed, she realized that she forgot to create a slogan for the ad campaign. So Gerety scribbled “A diamond is forever.” As a tagline, “a diamond is forever” became so successful and so popular that Advertising Age named it as the slogan of the century.
The second approach was product placement, the specialty of Dorothy Dignam, who practically invented the practice. Dignam’s theory was “the big ones sell the little ones.” She got movie studios to include the word ‘diamond’ in their titles and to include scenes about diamonds in films.
These two women are responsible for the ingrained emotional attachment that all of us have to diamond rings.
When I first encountered this story I began to question what it is exactly that I desire and how I came to desire it. Do I desire the right things? Is the culture around me shaping my desires too much?
One of the great spiritual challenges of our time is cultivating satisfaction, gratitude and joy in the face of a tsunami of messages that we do not have enough. Experts suggest that at a minimum we are confronted by a few hundred marketing messages every single day. We see and hear them on big screens, small screens and phone screens. Experts tell us that social media is becoming one massive envy-trigger.
If the Torah had an advertising slogan for dealing with our instinct to want what someone else has, it would be the tenth commandment. The Hebrew is lo tachmod, which we often translate as do not covet, do not lust after or crave. According to the tenth commandment, “we shall not covet our neighbor’s wife, house, field, slave, animal, or anything else that is our neighbor’s.”
Perhaps we could freshen up the tenth commandment with a new marketing campaign. Today we shall not covet our neighbor’s new kitchen, new car, hand-held device, vacation plans, their smart children, and certainly not their jewelry. Jewish tradition teaches us to love our neighbors, not to love our neighbor’s things.
In birkat hamazon [blessing after meals],?
We need spiritual tools to survive the modern onslaught of constant messages to buy more, eat more, desire more. So I offer a practice that has been helpful for me. see our friend with a bigger this or nicer that. We desire more. We enlarge our heart-vessel, which no longer fills up with gratefulness. If it keeps expanding, it will never overflow, and we never experience joy or thanksgiving. All we can feel is that our heart is not yet full. We see how much more we could have instead of relishing, savoring, adoring, and delighting in, everything we already do have.
If you purchase that thing you think you really want, consider investing.
Rabbi Justus Baird serves as Dean at Auburn Seminary.
| 286,369
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HOUSE CLEANING 20 yrs experience, references. Polish woman, English speaking. Call Jola At (908) 872- 7146
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Well, than anything I am use to.
Let me be real here – I was totally disappointed to get this phone… I had been drooling over the HTC DNA since the announcement was made about it. Why was I disappointed when I found out I was getting this phone? Well, the Microsoft App Store is so new that, though they have over 125,000 apps, they don’t have some of the key ones that I turn to every single day.
However, they are working on that! So, while I am disappointed that I can’t do instagram on it, there are many things that I am seriously loving about this new phone. In fact, as I was in bed last night trying to figure out parts of it so I could set my alarm clock, I hit the OneNote button and voice recorded some of my thoughts. Cool, right?
This post is just one of the many that I will be writing over the course of learning this phone and it is just my initial thoughts and such. Do not let it sway you either direction… I have only had it a couple days and have LOTS of stuff to learn!
Initial Pros:
- It is gorgeous! No really, it is a really pretty phone. So, call me weird, but that matters to me.
- It is Microsoft - I use Microsoft Office in every aspect of my daily working life. I LOVE being able to sync my phone with my computer!
- It is not afraid of Apple! Though Microsoft is decidedly PC, this phone integrates (with a simple app download) seamlessly with your Mac computer. It also will play (and sync) with your iTunes library letting you set up your music right there on your phone!
- Verizon Wireless 4G LTE! Phones just do not get any faster than this! I said it before and I will say it again… it is just as fast as pen and paper! Maybe faster because I don’t have to search for my phone as I would have to for pen and paper. LOL!
- Live Tiles! Live Tiles are certainly a Windows Phone thing and I already love them. I can customize my home screen in so many ways and not be overwhelmed. I can’t wait to learn more about them and the things that they can do.
- The UH-MAY-ZING camera! The front camera is just as good as the back camera! A Smartphone first for me!
- Missing Key Apps – I REALLY need the pinterest and instagram apps on my phone! As a blogger (well, and a Mom), I use both many times through the day to show off my new items or discuss issues or just to show off my darling Children!
- Way less Free Apps - I love to play games on my phone (and so do my kids) but I am finding that some of the ones that we play are either not there or are not free within the Microsoft App Store as they are within the Android store
- The Learning Curve – There seem to be LOTS of cool things about this phone, but you have to learn them. They aren’t covered in the “Start Here” manual – so the deeper you dig, the cooler it gets. But, the initial thing is kind of frustrating.
Now, let’s see the Initial Cons:
I have tried the Nokia Lumia 900 which is a Windows phone. I will agree that it definitely takes some getting used to since I used Androids prior to this phone. I love the color (blue) and that it has Windows. However, I didn’t care for the boxiness of the main screen. I did find a game that is unique to Window phones though that I enjoy. It’s called Wordament. You might check it out. Also, I was deeply disappointed with the lack of certain apps for it (especially Instagram).
Lindsay @ Laughing Lindsay´s last blog post ..Unilever Spreads Good to Know Holiday Cookie Baking Giveaway
it sounds good but the learning curve thing could be a deterrent.
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I just went from Android to iPhone and it was a seemless transition. I looked into the Windows phone but it just didn’t seem like it was going to meet my needs.
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I’m still playing around with my Windows phone. As of now I’m not happy with small amount of apps, especially the ones I use so often. My daughter calls it a gaming phone.
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I’m an iPhone (and AT&T) gal but this phone sounds really neat.
Mama Bee Does´s last blog post ..Shadora jewelry {review & 50.00 give away}
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They can’t require the government to open them, they have a right over process. That said who knows what a court would decide.
Via Daily Caller:
New.
“Otero County will also consider litigation in regards to the State of New Mexico failing to follow its constitutional duties towards the people of Otero County,” Griffin added.
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TITLE: poker hand probability question
QUESTION [2 upvotes]: 8 pokers hands are dealt from a shuffled deck without replacement.
a. Find the probability that at least one of the 8 hands is a heart flush(all five cards are hearts).
Pr(at least one of 8 hands is heart flush) $= 1 -$ Pr(none of eight hands is heart flush) $=$
$$ \left( 8\frac{\binom{13}{5}}{\binom{52}{5}}-28\frac{\binom{13}{5}}{\binom{52}{5}}\frac{\binom{8}{5}}{\binom{47}{5}}\right)$$
Is this answer correct?
b. Find the expected value and variance of the total number of eight hands which are heart flushes.
Expected value $= 8\dfrac{13 \choose 5}{52 \choose 5}$
Variance $= 8\dfrac{13 \choose 5}{52 \choose 5}\left(1- \dfrac{13 \choose 5}{52 \choose 5}\right)$
Are these answers correct?
REPLY [3 votes]: Let $A$ be the probability the first player has a heart flush. Then $A=\frac{13 \choose 5}{52 \choose 5}.$
Let $B$ be the probability the first and the second player both have heart flushes. Then $B=\frac{13 \choose 10}{52 \choose 10}.$
The probability at least one player has a heart flush is then ${8 \choose 1}A-{8 \choose 2}B=8A-28B$ which is in effect what you have written for question (a).
The probability of zero heart flushes is $1-8A+28B$, of one $8A-56B$ and of two $28B$.
Your expected value of $8A$ in (b) is correct. You can see this directly, or as $0\times(1-8A+28B) +1\times(8A-56B)+2\times 28B$.
Your variance of $8A(1-A)$ or $8A(1-8A)$ is slightly wrong. You could work out the variance to be $0^2\times(1-8A+28B) +1^2\times(8A-56B)+2^2\times 28B - (8A)^2 = 8A(1-8A)+56B$.
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Dr. Michael Faktor D.M.D has a huge heart when it comes to donating his time and talents to others in need. He has made trips to provide free dentistry in places like Cambodia and Indonesia. In October of 2009, Dr. Faktor went to Cambodia with his assistant and a local high school student. While in Cambodia, the three of them traveled in a boat along the Mekong River to floating villages, providing a wide array of care from extractions to fluoride treatments for children. In 2010, Dr. Faktor is hoping to make it to Africa to do more of the same!
Emergencies and Patients of all ages are Welcome
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Friday, January 15, 2021
HITLER: Poor Trump comparison
| September 13, 2020 1:00 AM
In reference to the opinion page on Aug. 23, there was a statement by a Red Stolley: “Aren’t we lucky? Germany had Hitler, we have Trump.”
That is the most idiotic statement I have heard in a long time. I just can’t get past it. I am surprised no one has challenged it.
Millions of Jews were killed. There were numerous concentration camps where unspeakable activities were conducted on innocent women and children, as well as men. Gas chambers and open pits where the bodies were bulldozed into. There were just too many horrible activities carried out under Hitler’s rule.
I hope the world does not see another person like that. If there is one, sounds like Red Stolley would be a candidate to reside with him.
DUANE SEVERSON
Coeur d’Alene
| 409,918
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+91 (40) 2377 1763 info@vmmodular.com
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We offer a wide range of shutters for modular kitchens, wardrobes and other furn ...[Read More...] iture. Laminate shutters are the most popular among them. In addition to it, there are other choices - acrylic shutters, glass shutters and PU (painted) shutters. They are available in a very rich colour palette. Browse through each category to find your favourite shade.
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Simple implications of these obscure semicolon tattoos
Pictures of tattoos is a simple punctuation, and last month we introduced the dot tattoos. Mini tattoo style, but the semicolon behind the tattoos significance.
Such of design is not from boring people or obsession punctuation of nerd of hand, everyone are know has. Semicolon is between between period and comma of pause of punctuation, actually sentences originally is can terminated of, but author is select continues to down, temporarily not designated Shang period. Because he also has words to said, so non-profit organization decided with this symbol to as symbol.
Semicolon tattoos, small, inconspicuous, it represents the opposite of hope and confidence in the continuation, deeply engraved on the wrist, like a mark, or a mantra. Remind anyone of us has the power to continue and ability.
Plans launched has has over two years, joined which of just psychiatric patients and kindness people, for in life in the suffered had hit of friends also are is a very good of encourages, you does not necessarily to Thorn green to said support, but please let side people are know this plans of exists’s, good stay this crowd, they are efforts struggled, so please don’t put they as Monster view.
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TITLE: Prove that the Michael line is not metrizable
QUESTION [0 upvotes]: Exercise Show that the Michael Line is not metrizable.
The Michael line is given by $\{U \cup A: U$ is open in $\mathbb{R}$ usual topology and $A \subset \mathbb{R} \setminus \mathbb{Q}\}$
Note that in a metric space, closed sets are $G_\delta$ sets, i.e. the countable intersection of open sets.
Proof
To see that the Michael line $\mathbb{M}$ is not metrizable, to derive a contradiction, suppose that $\mathbb{M}$ is metrizable. We may treat $\mathbb{M}$ as a metric space. Let $E$ be closed in $\mathbb{M}$. Since $E$ is closed, then $E$ is the countable intersection of $\mathbb{M}$-open sets, i.e.
$$E = \displaystyle\bigcap_{i \in \mathbb{N}} (U_i \cup A_i)$$
It follows then that the complelement $\mathbb{R} \setminus E$ is given by
$$\mathbb{R} \setminus E = \mathbb{R} \setminus \displaystyle\bigcap_{i \in \mathbb{N}} (U_i \cup A_i)$$
Since $E$ is $G_\delta$, then $\mathbb{R} \setminus E$ is $F_\sigma$, a countable union of closed sets.
$$\mathbb{R} \setminus \displaystyle\bigcap_{i \in \mathbb{N}} (U_i \cup A_i) = \displaystyle\bigcup_{i \in \mathbb{N}} \mathbb{R} \setminus (U_i \cup A_i)$$
I seem to have run into a wall at this point. Where can I derive a contradiction? Any advice would be of great value.
REPLY [1 votes]: In the Michael line, $\mathbb{Q}$ is a closed set but not a $G_\delta$ set.
Proof:
$\mathbb{R}\setminus\mathbb{Q} $ is an open set, so $\mathbb{Q}$ is closed.
Suppose you have a pair of sequences $U_n$ and $B_n$,
such that $U_n$ is an open set in the usual Euclidean topology on $\mathbb{R}$ for all $n$
and $B_n\subset\mathbb{R}\setminus\mathbb{Q}$ for all $n$.
Suppose furthermore that $\mathbb{Q}\subset \bigcap_{n}(U_n\cup B_n)$.
Now, $\mathbb{Q}\subset \bigcap_{n}U_n$,
since each $B_n$ does not contain any rational number.
This means $\mathbb{Q}\subset U_n$ for all $n$,
which in turn means $U_n=\mathbb{R}$ for all $n$ because $\mathbb{Q}$ is dense in the usual Euclidean topology on $\mathbb{R}$.
The conclusion is $\bigcap_{n}(U_n\cup B_n)=\mathbb{R}$,
so that $\bigcap_{n}(U_n\cup B_n)$ is not equal to $\mathbb{Q}$.
| 207,158
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Level: Elementary
Water For Birds - an experiment to determine the effectiveness of using moving water to attract birds
Level: Various
On February 14, The Moon will be directly opposite the Earth from the Sun and will be fully illuminated as seen from Earth (i.e. a full moon). This phase occurs at 11:53 p.m. local time. This full moon was known by early Native American tribes as the Full Snow Moon because the heaviest snows usually fell during this time of the year. Since hunting is difficult, this moon has also been known by some tribes as the Full Hunger Moon.
Source:
Previous Issues
A_2<< a different appearance for each sex (just like humans) and seasonal dimorphism mean a different appearance for each season (breeding vs. non-breeding)..
This week the terrible news came out that many of us have been dreading. The number of Monarch Butterflies (Danaus plexippus) that made it safely to sanctuaries in Mexico for the winter was the lowest number on record. Please click here for an excellent discussion for the numerous reasons for the precipitous decline of one of our favorite butterflies. But all hope is not lost. There are many things that you can do to help, both at school and at home. Plant a garden with nectar plants, host plants and especially milkweed. Milkweed is the only thing that Monarch caterpillars can eat. If you come to Fernbank's plant sale on March 29th, we will help you find the best plants for your existing or new garden. If you have any questions, please email me a t.neal@fernbank.edu.
AT THE
Toothpick Bridge Building Event
Fernbank Science Center
Saturday, February 22, 2014
1:00 PM to 3:00 PM
Sponsored by the American Society of Civil Engineers Georgia Section, Society of Manufacturing Engineers Atlanta Chapter 61, Fernbank Science Center and the Structural Engineers Association of Georgia. For more information please visit:
Science Article of the Month
Robots with insect-like brains: Robot can learn to navigate through its environment guided by external stimuli - Scientists.:Dark-eyed Junco Archive
Tree of the Month
Can you I.D. this tree?
Get the answer in the next issue.
Hint: Often used for baseball bats
Previous Issue: Southern Red Oak
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Molecule of the Month
Do you know this molecule?
Get the answer in the next issue.
Hint: Hot!
Previous Issue: peppermint
Archive
| 291,803
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This course is designed to present practitioners with tools and techniques modeling, analyzing and improving processes, with ample opportunity to apply through case study and exercises. This course will present:
- tools and instructions for documenting business processes – obtaining process information, formatting and depicting processes in the form of process maps and other supporting documents
- tools and instructions for creating some high-level systems views of an organization
- description and explanation of the variables that influence process performance
- tools and instructions for analyzing a process and its performance
- tools and instructions for selecting and applying a variety of improvement techniques
Workshop Length:
2 and 3 Days
Who Should Attend:
Process Improvement Practitioners, Business Analysts, and anyone tasked with participating in process improvement efforts.
| 2,639
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$ 83.00 $ 166.00 You save: 50% ($ 83.00)
FINAL SALE - no returns or exchanges
The Crosby sweatpant is made of soft brushed fleece. Elastic waist with drawstring. Side pockets. Top & bottom sold separately. We've paired it with the matching Chase sweatshirt
Inseam size Small is approx. 26.4"
Color: Oat | Fabric: 50% Cotton 50% Polyester
Model is wearing size S. She is 5'8"| 33" bust | 25" waist | 35" hipsReturn Policy
| 333,082
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Phone: (619) 291 4605
phil.bergman@junglemusic.net
Nursery Hours
& Location:
Monday -Saturday
9AM-4PM
450 Ocean View Ave., Encinitas, CA
92024
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POTTING
SOIL
For Palm Trees
by
Phil Bergman
This articles describes the soil that we use at our nursery.
We use it on all of the palms we grow. We amend it for certain
species but use the formulae below for 90% of the palms. Note, this
article is about a potting soil, i.e. a soil to used in pots. It's
not about "garden soil" or soil you'd import to put in garden areas.
Introduction
Over my more than thirty-seven years as a nurseryman, I have surveyed many palm growers to obtain their formulas for potting soil.
And, I inquired about their results with their soil. I would like to recap a few of the conclusions drawn at that time, most of which I still agree with today.
Is There Just One Perfect Palm Potting Soil?
First, there was no single formulae that everyone used. And yet, most growers were growing successfully. Secondly, it depended on what area you were doing your growing. It also depended on whether it was for greenhouse or outdoor culture. Finally, it depended upon whether you were growing seedlings or large plants. The one thing that almost everyone had in common was the use of sand. Only one grower used the classical peat
moss/perlite combination. I can say after talking to so many others
that there is not perfect, "must contain these things" type of potting
soil. Rather, many soils will work and you have to experiment a bit
and see what works for you and your area.
One must also remember that not all palms like the same sort of mix. It would be ludicrous to assume that
Brahea
armata and Nypa
fruticans would like the same soil. Yet, most growers try to use a universal mix for the majority of their palms. This is certainly a practical approach, but not the most scientific one.
What Should a Potting Soil Do?
A potting soil should offer the plant a substratum for both stability and obtaining its needed water, nutrients and aeration. If a soil is too porous, it will dry out too rapidly and the plant either has to be watered daily or suffer the consequences of desiccation. If it is too dense, it may become waterlogged and lead to rot. Thus, there is an ideal soil that offers drainage but adequate water retention and root support. I like to guarantee good drainage by the usage of
perlite, sand, and pumice in my mix. One will find the needed frequency of watering with experience with their soil.
What Should a Good Potting Soil Contain?
As mentioned above every grower has his own formulae for potting soil.
Potential components might include:TopsoilBark or wood chips
Shredded organic materialPerlitePeat mossPumice or lava stone
SandFertilizer (slow release)Dolomite to balance out excess
acidityMicroelements such as iron, magnesium, manganese, etc.
Below I will describe which of these components above we utilize. As
examples, a mix of only topsoil and sand may stay wet for some time and
not work for you. A classical a cheap soil would be half peat moss
and half perlite. This is probably ok for greenhouse seedlings but
doesn't do well over the long run. Also, the classical Hawaiian mix of lava stone and peat will tend to dry out very quickly and succumb to our dry hot winds
in Southern California without frequent watering. As a consumer, a palm
grower must try his soil(s), be willing to alter or change them and then
come up with what works the best for him. Also, he must know to adjust his watering according
to how fast his soil dries out.
Soil Acidity and Alkalinity
With the usage of peat moss, humus and bark, soils
can tend toward being acidic, i.e. a pH lower than 7. These three
components result in be on the acidic side. We use dolomite powder
to increase the pH. Remember to crush up the rocks you find in the
bag. The soil I describe below has a pH of about 6.3 to 6.5. This seems to be acceptable to most palm species, especially the more tropical ones. It is rare indeed that I have to adjust the pH on the soil formulae below to accommodate a particular palm. However, I do use some dolomite in my mix during preparation to offset the very acidic organic material.
If one has alkaline water in their municipality, salts can build up leading to a higher pH. A soil too alkaline can lead to various nutritional problems and microelement deficiencies. Thus, adjust the amount of dolomite
to your soil's pH. You may find, on the contrary, that your soil is
too alkaline if which case you may need to add sulfur and certainly not
use dolomite.
Availability of the Components to Make Potting Soil
Additionally, a palm grower has to use what is available to him. Every locality has different supplies for making the soil. This gets back to the idea that there are many good potential soils, whether you are using ground up macadamia shells, rice husks, or coconut fiber. You just have to experiment with what's available and talk to experienced growers.
And, you must use what you can acquire in your area. It's super
expensive to have materials shipped to you from out of state.
Jungle Music's Potting Soil for Palms
The potting soil I have been using for
many years is below. But, believe me that I've altered it and
changed a few things many times.
10% amended topsoil (has about 1/3 high quality topsoil and 2/3 humus)
15% pumice #2
15% 0 - 1/8 inch pine bark
15% nitrolized redwood shavings
20% perlite #2
10% coarse washed sand, #12
15% coarse peat moss
To this I add per cubic yard: one lb.
Dolomite, one lb. Osmocote (14-14-14), and lb. of a microelement mix called
Micromax Plus. The entire mixture is turned many times before using. I use about 100 yards of this mix a year and have it professionally mixed in batches of ten to twenty yards each time.
Adjustments to a Potting Soil
For brand new seedlings or very young plants, I may amend this mixture with more redwood shavings and perlite to open it up..
In the
greenhouse, I have found that most mixes seem to age quicker, getting more
dense and retain more water with age. The same thing happens outdoors but not quite as quickly. The above mix seems to be adequate in most environments for about 2 years. After that point, repotting into a larger pot with fresh mix around the existing rootball is needed.
Remember that new mixes tend to repel water and adequate care must be taken to ensure that the soil is thoroughly and adequately watered during the first watering. Penetrating agents can be used but are not necessary if one waters multiple times on the new mix. My rule is to water at least three times, each time bringing the water to the top of the pot.
The small amount of fertilizer in the mix above is to theoretically offset any nitrogen loss from the large amount of organic material in the mix. It is not meant to replace a regular fertilization program. The microelements I feel are important for overall plant health as very little topsoil is being used. If plants yellow in time in the above mix, I will top-dress with blood meal and this seems to green them up.
CONCLUSION
Remember, there is no single perfect potting soil. Our formulae
above is for potted plants and is not meant to be used as garden soil.
For potting soil, that magical and scientifically formulated soil produced
by big companies may or may not be best. If one works for you, then it works for you.
Be willing to experiment a bit to find the best working soil in your area.
Also important is that most palm trees like good drainage and some organic
material. You are welcome to try out mix above. We do sell in
in bags at the nursery but not in bulk. And, feel free to email me
if you have any questions.
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Description
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Old Chinese woven red coral rope necklace. This necklace is made of tiny 2mm rich tomato coral beads woven into a peyote stitch rope a tiny bit short of 16 inches. It has a brass spring clasp which could easily be substituted with a gold clasp. The necklace is in excellent condition. The threading is completely intact with no missing beads.
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How These Two TV Shows Are Changing the Rules for Adapting Books
Hackers have targeted about 19,000 French websites since a rampage by Islamic extremists left 20 dead last week, France's cyber-defence-defence for the French military, said about 19,000 French websites had faced cyber-attacks in recent days, some carried out by well-known Islamic hacker groups.
The attacks, mostly relatively minor denial-of-service attacks, hit sites as varied as military regiments to pizza shops but none appeared to have caused serious damage, he said.
"What's new, what's important, is that this is 19,000 sites - that's never been seen before," Coustilliere said. "This is the first time that a country has been faced with such a large wave of cyber-contestation."
Coustilliere called the attacks a response to the massive demonstrations against terrorism that drew 3.7 million people into the streets Sunday across France. He pointed to "more or less structured groups" that used tactics like posting symbols of jihadist groups on companies' Web sites.
Two of the Paris terror attackers claimed allegiances to al-Qaida in Yemen, and another - who targeted a kosher supermarket - to the Islamic State group.
Military authorities have launched round-the-clock surveillance to protect the government sites still coming under attack.
According to Arbor Networks, a private company that monitors threats to the Internet, in the past 24 hours alone, France has been the target of 1,070 denial of service attacks. That's about a quarter as many as the United States, but the U.S. hosts 30 times as many websites.
The terror attacks in Paris occurred in an atmosphere of rising anti-Semitism in France, and have prompted scattered violence against Muslims and Muslim sites around France in an apparent backlash. They have also put many French Muslims on the defensive..
U.S. Secretary of State John Kerry, who., which again had Muhammad on its cover. Even though it has a special increased print run of 5 million copies, it sold out before dawn Thursday in Paris kiosks for a second straight day.
Some Muslims, who believe their faith forbids depictions of the prophet, reacted with dismay or anger to the new cover. Pakistani lawmakers passed a resolution and marched outside parliament Thursday to protest the publication.
A leader of Yemen's al-Qaida branch officially claimed responsibility for the attacks at Charlie Hebdo, saying in a video online that the slayings were in "vengeance for the prophet."
But U.S. and French intelligence officials are leaning toward an assessment that the Paris terror attacks were inspired by al-Qaida but not directly supervised by the group. That view would put the violence in the category of homegrown terror attacks, which are extremely difficult to detect and thwart.
For the latest tech news and reviews, follow Gadgets 360 on Twitter, Facebook, and subscribe to our YouTube channel.
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\begin{document}
\title{Lee monoid $L_4^1$ is non-finitely based}
\author{ Inna A. Mikhailova\footnote{email: inna.mikhaylova@gmail.com, Ural Federal University, Ekaterinburg, Russia The first author was supported by the Russian Foundation for Basic Research, project
no.\ 17-01-00551, the Ministry of Education and Science
of the Russian Federation, project no.\ 1.3253.2017, and
the Competitiveness Program of Ural Federal University.
}\quad and Olga B. Sapir\footnote{email: olga.sapir@gmail.com}\\}
\date{}
\maketitle
\begin{abstract} We establish a new sufficient condition under which a monoid is non-finitely based and apply this condition to show that
the 9-element monoid $L_4^1$ is non-finitely based. The monoid $L_4^1$ was the only unsolved case in the finite basis problem for
Lee monoids $L_\ell^1$, obtained by adjoining an identity element to the semigroup $L_\ell$ generated by two idempotents $a$ and $b$ subjected to the relation $0=abab \cdots$ (length $\ell$). We also prove a syntactic sufficient condition which is equivalent to the sufficient condition of Lee under which a semigroup is non-finitely based. This gives a new proof to the results of Zhang-Luo and Lee that the semigroup $L_\ell$ is non-finitely based for each $\ell \ge 3$.
\vskip 0.1in
\noindent{\bf 2010 Mathematics subject classification}: 20M07, 08B05
\noindent{\bf Keywords and phrases}: Lee monoids, identity, finite basis problem, non-finitely based, variety, isoterm
\end{abstract}
\section{Introduction}
An algebra is said to be {\em finitely based} (FB) if there is a finite subset of its identities from which all of its identities may be deduced.
Otherwise, an algebra is said to be {\em non-finitely based} (NFB). By the celebrated McKenzie's result \cite{RM} the classes of FB and NFB finite algebras are recursively inseparable. It is still unknown whether the set of FB finite semigroups is recursive although a very large volume of work is devoted to this problem (see
the survey \cite{MV}).
Recently, Lee suggested to investigate the finite basis property of the semigroups
\[
L_\ell = \langle a,b \mid aa=a, bb=b, \underbrace{ababab\cdots}_{\text{length }\ell} = 0 \rangle, \quad \ell \geq 2
\]
and the monoids $L_\ell^1$ obtained by adjoining an identity element to $L_\ell$.
The 4-element semigroup $L_2 =A_0$ is long known to be finitely based \cite{1980}.
Zhang and Luo proved \cite{WTZ1} that the 6-element semigroup $L_3$ is NFB
and Lee generalized this
result into a sufficient condition \cite{EL} which implies that for all $\ell \geq 3$, the semigroup $L_\ell$ is NFB \cite{EL1}.
As for the monoids $L_\ell^1$, the 5-element monoid $L_2^1$ was also proved to be FB by Edmunds \cite{1977}, while the 7-element monoid $L_3^1$ is recently shown to be NFB by Zhang \cite{WTZ}. It is proved in \cite{OS} that for each $\ell \ge 5$ the monoid $L_\ell^1$ is NFB. The goal of this article is to prove that $L_4^1$ is NFB.
To this aim we establish a new sufficient condition under which a monoid is NFB.
Throughout this article, elements of a countably infinite alphabet $\mathfrak A$ are called {\em variables} and elements of the free monoid $\mathfrak A^*$ and free semigroup $\mathfrak A^+$ are called {\em words}.
We say that a word $\bf u$ has {\em the same type} as $\bf v$ if $\bf u$ can be obtained from $\bf v$ by changing the individual exponents of variables. For example, the words $x^2yxzx^5y^2xzx^3$ and
$xy^2x^3zxyx^2zx$ are of the same type.
An {\em island} formed by a variable $x$ in a word {\bf u} is a maximal subword of $\bf u$ which is a power of $x$. For example, the word $xyyx^5yx^3$ has three islands formed by $x$ and two islands formed by $y$.
We use $x^+$ to denote $x^n$ when $n$ is a positive integer and its exact value is unimportant.
If $\bf u$ is a word over a two-letter alphabet then the {\em height} of $\bf u$ is the number of islands in $\bf u$. For example,
the word $x^+$ has height 1, $x^+y^+$ has height 2, $x^+y^+x^+$ has height 3, and so on.
For each $\ell \ge 2$ consider the following property of a semigroup $S$.
$\bullet$ (C$_\ell$) If the height of ${\bf u} \in \{x,y\}^+$ is at most $\ell$, then
$\bf u$ can form an identity of $S$ only with a word of the same type.
We use $\var S$ to refer to the variety of semigroups generated by $S$.
The following result from \cite{OS} gives us a connection between Lee semigroups, Lee monoids and Properties (C$_\ell$).
\begin{fact} \label{final} \cite[Corollary 7.2]{OS} Let $\ell \ge 2$ and $S$ be a semigroup (resp. monoid).
Then $S$ satisfies Property (C$_\ell$) if and only if $\var S$ contains $L_{\ell}$ (resp. $L_{\ell}^1$).
\end{fact}
In view of Fact \ref{final}, the sufficient condition of Lee \cite{EL} (see Fact \ref{EL61} below) is equivalent to the following sufficient condition.
\begin{sufcon} \label{EL} (cf. Theorem \ref{EL62})
Let $S$ be a semigroup that satisfies Property (C$_3$) and $k \ge 2$.
If for each $n \ge 2$, $S$ satisfies the identity
\begin{equation} \label{eL30}
x^ky^k_1y^k_2 \dots y^k_nx^k \approx x^ky^k_ny^k_{n-1} \dots y^k_1x^k\end{equation}
then $S$ is NFB.
\end{sufcon}
Note that every monoid that satisfies \eqref{eL30} violates Property (C$_2$). Therefore, Sufficient Condition \ref{EL} cannot be used to establish the non-finite basis property of any monoid. Theorem 2.7 in \cite{OS1} implies the result of Zhang \cite{WTZ} that $L_3^1$ is NFB and can be reformulated as follows.
\begin{sufcon} \label{L3} \cite{OS1}
Let $M$ be a monoid that satisfies Property (C$_3$).
If for each $n>0$, $M$ satisfies the identity
\begin{equation} \label{eL3}
(x_1 x_2 \dots x_{n-1}x_{n}) (y_1 y_2 \dots y_{n-1}y_n) (x_{n} x_{n-1} \dots x_2 x_1) (y_n y_{n-1} \dots y_{2}y_1) \approx
\end{equation}
\[ (y_1 y_2 \dots y_{n-1}y_n) (x_1 x_2 \dots x_{n-1}x_{n}) (y_n y_{n-1} \dots y_{2}y_1) (x_{n} x_{n-1} \dots x_2 x_1),\]
then $M$ is NFB.
\end{sufcon}
Note that for $n=1$ the identity \eqref{eL3} fails on $L_4^1$ and consequently on $L_\ell^1$ for each $\ell \ge 4$.
Let $\pi$ denote the special permutation on $\{1, 2, \dots, n^2\}$ used by Jackson to prove Lemma 5.4 in \cite{MJ}.
The next theorem implies that for each $\ell \ge 5$ the monoid $L_\ell^1$ is NFB \cite{OS}.
\begin{sufcon} \label{L5} \cite[Theorem 2.1]{OS} Let $M$ be a monoid that satisfies Property (C$_5$).
If for each $n>3$, $M$ satisfies the identity
\begin{equation} \label{eL5}
(x_1 x_2 \dots x_{n^2-1}x_{n^2}) \hskip.04in (x^k_{\pi1} x^k_{\pi2} \dots x^k_{\pi n^2}) \hskip.04in (x_{n^2} x_{n^2-1} \dots x_2 x_1) \approx
\end{equation}
\[(x_1 x_2 \dots x_{n^2-1}x_{n^2}) \hskip.04in (x^k_{\pi n^2} \dots x^k_{\pi2} x^k_{\pi1} ) \hskip.04in ( x_{n^2} x_{n^2-1} \dots x_2 x_1)\] for some $k \ge 1$, then $M$ is NFB.
\end{sufcon}
Since $L_4^1$ satisfies $xyxyyx \approx xyxyxy$, it does not satisfy Property (C$_5$). Therefore, Sufficient Condition \ref{L5} cannot be used to establish the non-finite basis property of $L_4^1$.
The following theorem gives us a new sufficient condition under which a monoid is NFB and will be proved in Section \ref{sec:thm}.
\begin{theorem} \label{main}
Let $M$ be a monoid that satisfies Property (C$_4$).
If for each $n>0$, $M$ satisfies the identity
\[{\bf U}_n = (x_1 x_2 \dots x_n) (x_n x_{n-1} \dots x_1) (x_1 x_2 \dots x_n) \approx (x_1 x_2 \dots x_n) (x_1^2 x_2^2 \dots x_n^2) = {\bf V}_n,\]
then $M$ is NFB.
\end{theorem}
If $\tau$ is an equivalence relation on the free semigroup $\mathfrak A^+$ then we say that a word ${\bf u}$ is a {\em $\tau$-term} for a semigroup $S$ if ${\bf u} \tau {\bf v}$ whenever $S$ satisfies ${\bf u} \approx {\bf v}$. Recall \cite{P} that $\bf u$ is an {\em isoterm} for $S$ if ${\bf u} = {\bf v}$ whenever $S$ satisfies ${\bf u} \approx {\bf v}$. If $\bf u$ is an isoterm for $S$ then evidently, $\bf u$ is a $\tau$-term for $S$ for every equivalence relation $\tau$ on $\mathfrak A^+$.
It is shown in \cite{OS} that for $\ell \le 5$ the isoterms for $L_\ell^1$ carry no information about the non-finite basis property of $L_\ell^1$.
However, the non-finite basis property of Lee semigroups $L_\ell$ and Lee monoids $L^1_\ell$ for $\ell \ge 3$ can be established by analyzing $\tau$-terms, where $\tau$ is the equivalence relation on $\mathfrak A^+$ defined by ${\bf u} \tau {\bf v}$ if ${\bf u}$ and $ {\bf v}$ are of the same type.
In particular, Sufficient Condition \ref{L5} is proved in \cite{OS} by analyzing $\tau$-terms for monoids for which all words in $\{x,y\}^+$ of height at most $5$ are $\tau$-terms.
Likewise, we prove Theorem \ref{main} by analyzing $\tau$-terms for monoids for which all words in $\{x,y\}^+$ of height at most $4$ are $\tau$-terms.
In Section \ref{sect:sufconLee}, we prove Sufficient Condition \ref{EL} by analyzing $\tau$-terms for semigroups for which all words in $\{x,y\}^+$ of height at most $3$ are $\tau$-terms.
\section{Lee monoids $L^1_\ell$ are NFB for all $\ell >2$}
We use $\con({\bf u})$ to denote the set of all variables contained in a word ${\bf u}$. Theorem \ref{main} implies the following.
\begin{cor} \label{L41} The monoid $L_4^1 = \langle a,b,1 \mid aa=a, bb=b, abab=0 \rangle$ is NFB.
\end{cor}
\begin{proof} In view of Fact \ref{final}, it is enough to verify that
$L_4^1$ satisfies the identity ${\bf U}_n \approx {\bf V}_n$ for each $n>0$.
Indeed, first notice that each variable appears at least $3$ times in ${\bf U}_n$ and ${\bf V}_n$.
Fix some substitution $\Theta: \mathfrak A \rightarrow L^1_4$. If for some $1 \le i \le n$, the set $\con(\Theta(x_i))$ contains both $a$ and $b$ then both $\Theta({\bf U}_n)$ and $\Theta({\bf V}_n)$ contain $(ab)^{3}$ or $(ba)^{3}$ as a subword and consequently, both are equal to zero. Therefore, we may assume that for each $1 \le i \le n$ we have $\Theta(x_i) \in \{a,b,1\}$.
To avoid some trivial cases we may also assume that $\Theta (x_1 x_2 \dots x_{n-1}x_{n})$ contains both letters $a$ and $b$. Consider two cases.
{\bf Case 1}: $\Theta (x_1 x_2 \dots x_{n-1}x_{n})$ contains $ab$ as a subword.
In this case, both $\Theta({\bf U}_n)$ and $\Theta({\bf V}_n)$ contain $abab$ as a subword and consequently,
both are equal to zero.
{\bf Case 2}: $\Theta (x_1 x_2 \dots x_{n-1}x_{n}) = ba$.
In this case, $\Theta({\bf U}_n) = (ba) (ab) (ba) = ba b a = \Theta({\bf V}_n)$.
\end{proof}
Notice that ${\bf U}_3 \approx {\bf V}_3$ fails on $L_5^1$. Indeed, substitute $x_1 \rightarrow b$, $x_2 \rightarrow a$, $x_3 \rightarrow b$.
\begin{theorem} Lee monoid $L_\ell^1$ is FB if and only if $\ell=2$.
\end{theorem}
\begin{proof}
The 5-element monoid $L_2^1$ was proved to be FB by C. Edmunds \cite{1977}. The 7-element monoid $L_3^1$ is NFB by the result of W. Zhang \cite{WTZ}. The 9-element monoid $L_4^1$ is NFB by Corollary \ref{L41}.
For each $\ell \ge 5$ the monoid $L_\ell^1$ is NFB by the result of the second-named author \cite{OS}.
\end{proof}
\section{Identities of monoids that satisfy Property (C$_4$)}
If a variable $t$ occurs exactly once in a word ${\bf u}$ then we say that $t$ is {\em linear} in ${\bf u}$. If a variable $x$ occurs more than once in ${\bf u}$ then we say that $x$ is {\em non-linear} in ${\bf u}$.
Evidently, $\con({\bf u}) = \lin({\bf u}) \cup \non({\bf u})$ where $\lin({\bf u})$ is the set of all linear variables in $\bf u$ and
$\non({\bf u})$ is the set of all non-linear variables in $\bf u$.
A {\em block} of $\bf u$ is a maximal subword of $\bf u$ that does not contain any linear variables of $\bf u$.
\begin{fact} \label{basic} \cite[Lemma 3.4]{OS} Let $M$ be a monoid that satisfies Property (C$_3$).
If $M \models {\bf u} \approx {\bf v}$ then
(i) $\lin({\bf u}) = \lin({\bf v})$, $\non({\bf u}) = \non({\bf v})$ and the order of occurrences of linear letters in $\bf v$ is the same as in $\bf u$.
(ii) The corresponding blocks in $\bf u$ and $\bf v$ have the same content. In other words, if
\[{\bf u} = {\bf a}_0t_1 {\bf a}_1t_2 \dots t_{m-1}{\bf a}_{m-1} t_m {\bf a}_m,\] where $\non({\bf u}) = \con({\bf a}_0 {\bf a}_1 \dots {\bf a}_{m-1}{\bf a}_m)$ and $\lin({\bf u}) = \{ t_1, \dots, t_m\}$, then
\[{\bf v} = {\bf b}_0t_1 {\bf b}_1t_2 \dots t_{m-1}{\bf b}_{m-1} t_m {\bf b}_m\]
such that $\con({\bf a}_q) = \con({\bf b}_q)$ for each $0 \le q \le m$.
\end{fact}
If $\con({\bf u}) \supseteq \{x_1, \dots, x_n\}$ we write ${\bf u}(x_1, \dots, x_n)$ to refer to the word obtained from ${\bf u}$ by deleting all occurrences of all variables that are not in $\{x_1, \dots, x_n\}$ and say that $\bf u$ {\em deletes} to ${\bf u}(x_1, \dots, x_n)$.
\begin{lemma} \label{2let} \cite[Lemma 3.6]{OS} Let $M$ be a monoid that satisfies Property (C$_4$).
Let ${\bf u}$ be a word with $\non({\bf u}) = \{x,y\}$ such that
(i) ${\bf u}(x,y)$ has height at most $4$;
(ii) every block of $\bf u$ has height at most $3$.
Then $\bf u$ can form an identity of $M$ only with a word of the same type.
\end{lemma}
We use $D_x({\bf u})$ to denote the result of deleting all occurrences of variable $x$ in a word $\bf u$.
The next lemma is a special case of Lemma \ref{manylet}.
\begin{lemma} \label{3letter} Let $M$ be a monoid that satisfies Property (C$_4$).
Let ${\bf u}$ be a word with 3 non-linear variables such that
(I) for each $\{x,y\}\subseteq \con({\bf u})$, the height of ${\bf u}(x,y)$ is at most $4$;
(II) no block of $\bf u$ deletes to $x^+ y^+ x^+ y^+$;
(III) no block of $\bf u$ deletes to $x^+ y^+ x^+ z^+ x^+$;
(IV) If some block $\bf B$ of $\bf u$ deletes to $x^+y^+z^+x^+$ then $\bf u$ satisfies each of the following:
(a) if there is an occurrence of $y$ to the left of $\bf B$ then there is no occurrence of $z$ to the right of $\bf B$;
(b) if there is an occurrence of $z$ to the left of $\bf B$ then there is no occurrence of $y$ to the right of $\bf B$.
Then $\bf u$ can form an identity of $M$ only with a word of the same type.
\end{lemma}
\begin{proof} Suppose that $\non({\bf u}) = \{x,y,z\}$ and that $M$ satisfies ${\bf u} \approx {\bf v}$.
In view of Fact \ref{basic}, in order to prove that ${\bf u}$ and ${\bf v}$ are of the same type, it is enough to show that every block
$\bf B$ of $\bf u$ is of the same type as the corresponding block ${\bf B'}$ of ${\bf v}$.
In view of Condition (I)--(III), modulo duality and renaming variables there are only five possibilities for $\bf B$.
{\bf Case 1:} $\bf B$ involves only $y$ and $z$.
In this case, the words $D_x({\bf u})$ and $D_x({\bf v})$ are of the same type by Lemma \ref{2let}. Therefore, the corresponding blocks of $D_x({\bf u})$ and $D_x({\bf v})$ are also of the same type. In particular,
$\bf B$ is of the same type as the corresponding block $\bf B'$ of $\bf v$.
{\bf Case 2:} ${\bf B} = x^+y^+z^+x^+$.
In this case, in view of Lemma \ref{2let}, we have ${\bf B'}(y,z) = y^+ z^+$, ${\bf B'}(y,x) = x^+y^+ x^+$ and ${\bf B'}(x,z) = x^+ z^+x^+$. So, if $\bf B'$ and $\bf B$ are not of the same type
then ${\bf B'} = x^+y^+x^+z^+x^+$.
Modulo duality, there are four possibilities for the word $\bf u$.
{\bf Subcase 2.1:} Neither $y$ nor $z$ occurs in $\bf u$ to the left of $\bf B$.
In this case, Condition (I) implies the following.
\begin{claim} \label{xyzx}
(a) the occurrences of $y$ can form at most one island (denoted by ${_2y}^+$ if any) to the right of $\bf B$;
(b) the occurrences of $z$ can form at most one island (denoted by ${_2z}^+$ if any) to the right of $\bf B$;
(c) the last occurrence of $x$ in $\bf u$ precedes ${_2y}^+$ and ${_2z}^+$.
\end{claim}
Let $\Theta: \mathfrak A \rightarrow \mathfrak A^*$ be a substitution such that
$\Theta(y) = \Theta(z)=y$, $\Theta(x)=x$ and $\Theta(p)=1$ for each $p \not \in \{x,y\}$. Then $\Theta({\bf v})$ has height at least 5, but in view of Claim \ref{xyzx}, $\Theta({\bf u})$ is either $x^+y^+x^+$ or $x^+y^+x^+y^+$.
This contradicts Property (C$_4$).
{\bf Subcase 2.2:} Both $y$ and $z$ occur in $\bf u$ to the left of $\bf B$.
In this case, Condition (IV) implies that neither $y$ nor $z$ occurs in $\bf u$ to the right of $\bf B$.
Consequently, this case is dual to Subcase 2.1.
{\bf Subcase 2.3:} $y$ occurs in $\bf u$ to the left of $\bf B$ but $z$ does not occur in $\bf u$ to the left of $\bf B$.
In this case, there is no $y$ to the right of $\bf B$ by Condition (I), and no $z$ to the right of $\bf B$ by Condition (IV).
Thus, this case is also dual to Subcase 2.1.
{\bf Subcase 2.4:} $z$ occurs in $\bf u$ to the left of $\bf B$ but $y$ does not occur in $\bf u$ to the left of $\bf B$.
In this case, there is no $z$ to the right of $\bf B$ by Condition (I), and no $y$ to the right of $\bf B$ by Condition (IV).
Thus, this case is also dual to Subcase 2.1.
{\bf Case 3:} ${\bf B} = x^+y^+z^+y^+x^+$.
In this case, in view of Lemma \ref{2let}, we have ${\bf B'}(y,z) = y^+z^+y^+$ and ${\bf B'}(y,x) = x^+ y^+ x^+$.
Therefore, $\bf B'$ must be of the same type as $\bf B$.
{\bf Case 4:} ${\bf B} = x^+y^+x^+z^+$
In this case, in view of Lemma \ref{2let}, we have ${\bf B'}(x,y) = x^+ y^+x^+$ and ${\bf B'}(x,z) = x^+z^+$.
Therefore, $\bf B'$ must be of the same type as $\bf B$.
{\bf Case 5:} ${\bf B} = x^+y^+z^+$
In this case, in view of Lemma \ref{2let}, we have ${\bf B'}(x,y) = x^+ y^+$ and ${\bf B'}(y,z) = y^+ z^+$.
Therefore, $\bf B'$ must be of the same type as $\bf B$.
\end{proof}
\begin{lemma} \label{3let} Two words $\bf u$ and $\bf v$ are of the same type
if and only if $\con({\bf u}) = \con({\bf v})$ and for each set of three variables $\{x,y, z\} \subseteq \con({\bf u})$,
the words ${\bf u}(x,y,z)$ and ${\bf v} (x,y,z)$ are of the same type.
\end{lemma}
\begin{proof} If $\bf u$ and $\bf v$ are of the same type then evidently, $\con({\bf u}) = \con({\bf v})$ and for each
$\mathfrak X \subseteq \con({\bf u})$ the words ${\bf u} (\mathfrak X)$ and ${\bf v} (\mathfrak X)$ are also of the same type.
Now suppose that for each set of three variables $\{x,y, z\} \subseteq \con({\bf u})$,
the words ${\bf u}(x,y,z)$ and ${\bf v} (x,y,z)$ are of the same type.
Then $\bf u$ and $\bf v$ begin with the same letter.
If $\bf u$ and $\bf v$ are not of the same type then ${\bf u}= {\bf a}x y {\bf b}$ and ${\bf u}= {\bf a'}x z {\bf b'}$
for some possibly empty words $\bf a$, $\bf a'$, $\bf b$ and $\bf b'$
such that ${\bf a}x$ and ${\bf a'}x$ have the same type and $\{x,y,z\}$ are pairwise distinct variables.
Then the words ${\bf u}(x,y,z)$ and ${\bf v}(x,y,z)$ are also not of the same type. To avoid a contradiction, we must assume
that $\bf u$ and $\bf v$ are of the same type.
\end{proof}
\begin{lemma} \label{manylet} Let $M$ be a monoid that satisfies Property (C$_4$).
Let ${\bf u}$ be a word such that
(I) for each $\{x,y\}\subseteq \con({\bf u})$, the height of ${\bf u}(x,y)$ is at most $4$;
(II) no block of $\bf u$ deletes to $x^+ y^+ x^+ y^+$;
(III) no block of $\bf u$ deletes to $x^+ y^+ x^+ z^+ x^+$;
(IV) If some block $\bf B$ of $\bf u$ deletes to $x^+y^+z^+x^+$ then $\bf u$ satisfies each of the following:
(a) if there is an occurrence of $y$ to the left of $\bf B$ then there is no occurrence of $z$ to the right of $\bf B$;
(b) if there is an occurrence of $z$ to the left of $\bf B$ then there is no occurrence of $y$ to the right of $\bf B$.
Then $\bf u$ can form an identity of $M$ only with a word of the same type.
\end{lemma}
\begin{proof} Suppose that $M$ satisfies ${\bf u} \approx {\bf v}$.
Let $\bf B$ and $\bf B'$ be the corresponding blocks in $\bf u$ and $\bf v$. Then for each $\{x,y, z\} \subseteq \con({\bf u})$
the words ${\bf B}(x,y,z)$ and ${\bf B'}(x,y,z)$ are of the same type by Lemma \ref{3letter}. Therefore, the words ${\bf B}$ and ${\bf B'}$ are also of the same type by Lemma \ref{3let}.
\end{proof}
\section{Properties of words applicable to ${\bf U}_n$}
\begin{fact} \label{Eu} \cite[Fact 4.2]{OS} Given a word $\bf u$ and a substitution $\Theta: \mathfrak A \rightarrow \mathfrak A^+$, one can rename some variables in $\bf u$ so that the resulting word $E({\bf u})$ has the following properties:
(i) $\con (E({\bf u})) \subseteq \con ({\bf u})$;
(ii) $\Theta(E({\bf u}))$ is of the same type as $\Theta ({\bf u})$;
(iii) for every $x, y \in \con (E({\bf u}))$, if the words $\Theta(x)$ and $\Theta(y)$ are powers of the same variable then $x=y$.
\end{fact}
\begin{lemma} \label{prop1} \cite[Lemma 4.3]{OS} Let $\bf U$ be a word such that for each $\{x,y\}\subseteq \con({\bf U})$,
the height of ${\bf U}(x,y)$ is at most $4$.
Let $\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution which satisfies Property (ii) in Fact \ref{Eu}.
If $\Theta({\bf u}) = {\bf U}$ then $\bf u$ satisfies Condition (I) in Lemma \ref{manylet}, that is,
for each $\{x,y\}\subseteq \con({\bf u})$, the height of ${\bf u}(x,y)$ is at most $4$.
\end{lemma}
A word ${\bf u}$ is called a {\em scattered subword} of a word ${\bf v}$ whenever there exist words ${\bf u}_1, \dots, {\bf u}_k, {\bf v}_0, {\bf v}_1, \dots, , {\bf v}_{k-1}, {\bf v}_k \in \mathfrak A^*$ such that
${\bf u} = {\bf u}_1 \dots {\bf u}_k$ and ${\bf v} = {\bf v}_0 {\bf u}_1 {\bf v}_1\dots {\bf v}_{k-1}{\bf u}_k{\bf v}_k;$
in other terms, this means that one can extract $\bf u$ treated as a sequence of letters from the sequence $\bf v$. For example,
$x_1x_3$ is a scattered subword of $x_1x_2x_3x_2x_1$.
For the rest of this section, for each $n>2$, we use ${\bf U}_n$ to denote a word of the same type as $(x_1 x_2 \dots x_n) (x_n x_{n-1} \dots x_1) (x_1 x_2 \dots x_n) $. The following properties of ${\bf U}_n$ can be easily verified:
$\bullet$ (P1) ${\bf U}_n(x_i, x_j) = x_i^+ x_j^+ x_i^+x_j^+$ for each $1 \le i < j \le n$;
$\bullet$ (P2) ${\bf U}_n(x_i,x_j,x_k)=x_i^+x_j^+x_k^+ x_j^+ x_i^+x_j^+x_k^+$ for each $1 \le i < j < k \le n$;
$\bullet$ (P3) $x_ix_j$ appears exactly twice in ${\bf U}_n$ as a scattered subword and
$x_j x_i$ appears exactly once in ${\bf U}_n$ as a scattered subword for each $1 \le i < j \le n$;
$\bullet$ (P4) ${\bf U}_n$ contains $x^+_n x^+_{n-1} \dots x^+_2 x^+_1$ as a subword between the two scattered subwords $x_i x_j$
for each $1 \le i < j \le n$;
$\bullet$ (P5) the occurrences of $x_1$ and $x_n$ form exactly two islands in ${\bf U}_n$ and for each $1 <i<n$, the occurrences of $x_i$ form exactly three islands in ${\bf U}_n$. We refer to these islands as ${_1x_{1}^+}$, ${_2x_{1}^+}$, ${_1x_{n}^+}$, ${_2x_{n}^+}$, ${_1x_{i}^+}$, ${_2x_{i}^+}$ and ${_3x_{i}^+}$;
$\bullet$ (P6) for each $1 < i < n$, ${\bf U}_n$
contains $x^+_n x^+_{n-1} \dots x^+_{i+2} x^+_{i+1}$ as a subword between ${_1x_{i}^+}$ and ${_2x_{i}^+}$ and, contains
$x^+_{i-1} \dots x^+_1$ as a subword between ${_2x_{i}^+}$ and ${_3x_{i}^+}$. Also, ${\bf U}_n$
contains $x^+_n x^+_{n-1} \dots x^+_{2}$ as a subword between ${_1x_{1}^+}$ and ${_2x_{1}^+}$ and, contains
$x^+_{n-1} \dots x^+_1$ as a subword between ${_1x_{n}^+}$ and ${_2x_{n}^+}$;
$\bullet$ (P7) for each $1 < i \ne j < n$, if $x_j$ occurs in ${\bf U}_n$ to the left of ${_1x_{i}^+}$ or between ${_2x_{i}^+}$ and ${_3x_{i}^+}$ then $j<i$; if $x_j$ occurs in ${\bf U}_n$ between ${_1x_{i}^+}$ and ${_2x_{i}^+}$ or to the right of ${_3x_{i}^+}$
then $j>i$.
\begin{lemma} \label{propii} Let $\bf u$ be a word in less than $n-1$ variables for some $n>2$. Let
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution which satisfies Property (ii) in Fact \ref{Eu}.
If $\Theta({\bf u})={\bf U}_n$, then ${\bf u}$ satisfies Condition (II) in Lemma \ref{manylet}, that is, no block of $\bf u$ deletes to $x^+ y^+ x^+ y^+$.
\end{lemma}
\begin{proof}
Suppose that some block $\bf B$ of ${\bf u}$ deletes to $x^+y^+x^+y^+$, where $\{x,y\}\subseteq \con({\bf u})$. In view of Fact~\ref{Eu}, there are $1\leq i\neq j\leq n$ such that $x_i\in \con(\Theta(x))$ and $x_j \in\con(\Theta(y))$. Since $\Theta({\bf B})$ contains $x_i x_j$ twice as a scattered subword, Property (P3) implies that $i < j$.
Then $\Theta({\bf B})$ contains $x^+_n x^+_{n-1} \dots x^+_2 x^+_1$ between the two scattered subwords $x_i x_j$ by Property (P4).
Since $\con({\bf u})$ involves less than $n-1$ distinct variables there is a variable $t\in\con({\bf B})$ such that
$\Theta(t)$ contains $x_{k+1} x_{k}$ for some $1 < k <n$. In view of Property (P3),
$t$ is linear in $\bf u$. Therefore, there is a linear letter between the two scattered subwords $xy$ in $\bf B$.
A contradiction.
\end{proof}
\begin{lemma} \label{propiii} Let $\bf u$ be a word in less than $n-1$ variables for some $n>2$. Let
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution which satisfies Property (ii) in Fact \ref{Eu}.
If $\Theta({\bf u})={\bf U}_n$, then ${\bf u}$ satisfies Condition (III) in Lemma \ref{manylet}, that is, no block of $\bf u$ deletes to $x^+ y^+ x^+ z^+ x^+$.
\end{lemma}
\begin{proof}
Suppose that some block $\bf B$ of ${\bf u}$ deletes to $x^+y^+x^+z^+x^+$, where $x,y,z$ are three distinct non-linear variables that belong to $\con({\bf u})$.
Due to Fact~\ref{Eu}, there are pairwise distinct $1\leq i,j,k\leq n$ such that $x_i\in\con(\Theta(x))$, $x_j\in\con(\Theta(y))$ and $x_k\in\con(\Theta(z))$. Therefore, the occurrences of $x_i$ form at least three islands in $\Theta({\bf B})$. By Property (P5), $1<i<n$
and the occurrences of $x_i$ form exactly three islands in $\Theta({\bf B})$. Property (P6) implies that $\Theta({\bf B})$ contains $x^+_n x^+_{n-1} \dots x^+_{i+2} x^+_{i+1}$ between the first two islands formed by $x_i$
and $x^+_{i-1} \dots x^+_1$ between the last two islands formed by $x_i$.
Since $\con({\bf u})$ involves less than $n-1$ distinct variables there is a variable $t\in\con({\bf B})$ such that
$\Theta(t)$ contains $x_{r+1} x_{r}$ for some $1 \le r < i-1$ or $i+1 \le r < n$. In view of Property (P3),
$t$ is linear in $\bf u$. Therefore, there is a linear letter in $\bf B$ either between the first two islands formed by $x$ or between the last two islands formed by $x$.
A contradiction.
\end{proof}
\begin{lemma} \label{prop2} Let $\bf u$ be a word in less than $n-1$ variables for some $n>2$ such that some block $\bf B$ of $\bf u$
deletes to $x^+y^+z^+x^+$. Let
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution which satisfies Property (ii) in Fact \ref{Eu}.
If $\Theta({\bf u})={\bf U}_n$, then ${\bf u}$ satisfies Condition (IV) in Lemma \ref{manylet}, that is, each of the following holds:
(a) if there is an occurrence of $y$ to the left of $\bf B$ then there is no occurrence of $z$ to the right of $\bf B$;
(b) if there is an occurrence of $z$ to the left of $\bf B$ then there is no occurrence of $y$ to the right of $\bf B$.
\end{lemma}
\begin{proof} If $\Theta(x)$ is not a power of a variable, then $\Theta(x)$ contains $x_kx_{k+1}$ for some $1 \le k < n$ and $x$ appears twice in $\bf u$ by Property (P3). Since $\bf u$ involves less than $n-1$ variables, there is a linear letter between the two occurrences of $x$ in $\bf u$ due to Property (P4). This contradicts the fact that $\bf B$ is a block of $\bf u$. So, we can assume
that $\Theta(x) = x_i^+$ for some $1 \le i \le n$.
Due to Property (P5), the occurrences of $x_i$ form at most three islands in ${\bf U}_n$. We refer to the two islands formed by $x$ in $\bf B$ as ${_1x^+}$ and ${_2x^+}$.
Since $\Theta$ satisfies Property (ii) in Fact \ref{Eu}, four cases are possible.
{\bf Case 1:} $i=1$ or $i=n$, $\Theta({_1x^+})$ is a subword of ${_1x_{i}^+}$ and $\Theta({_2x^+})$ is a subword of ${_2x_{i}^+}$.
Since $\bf u$ involves less than $n-1$ variables, there is a linear letter between the two occurrences of $x$ in $\bf u$ due to Property (P6).
This contradicts the fact that $\bf B$ is a block of $\bf u$.
{\bf Case 2:} $1 < i < n$, $\Theta({_1x^+})$ is a subword of ${_1x_{i}^+}$ and $\Theta({_2x^+})$ is a subword of ${_3x_{i}^+}$.
Use the same arguments as for Case 1.
{\bf Case 3:} $1 < i < n$, $\Theta({_1x^+})$ is a subword of ${_1x_{i}^+}$ and $\Theta({_2x^+})$ is a subword of ${_2x_{i}^+}$.
In this case, in view of Property (P7), neither $y$ nor $z$ occurs to the left of $\bf B$.
{\bf Case 4:} $1 < i < n$, $\Theta({_1x^+})$ is a subword of ${_2x_{i}^+}$ and $\Theta({_2x^+})$ is a subword of ${_3x_{i}^+}$.
In this case, in view of Property (P7), neither $y$ nor $z$ occurs to the right of $\bf B$.
\end{proof}
\section{Proof of Theorem \ref{main}}\label{sec:thm}
\begin{lemma} \label{nfblemma} \cite[Lemma 5.1]{OS} Let $\tau$ be an equivalence relation on the free semigroup $\mathfrak A^+$ and $S$ be a semigroup.
Suppose that for infinitely many $n$, $S$ satisfies an identity ${\bf U}_n \approx {\bf V}_n$ in at least $n$ variables
such that ${\bf U}_n$ and ${\bf V}_n$ are not $\tau$-related.
Suppose also that for every identity ${\bf u} \approx {\bf v}$ of $S$ in less than $n-1$ variables, every
word $\bf U$ such that ${\bf U} \tau {\bf U}_n$ and every substitution
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ such that $\Theta({\bf u}) = {\bf U}$ we have
${\bf U} \tau \Theta({\bf v})$. Then $S$ is NFB.
\end{lemma}
We use $\con_{2}({\bf u})$ to denote the set of all variables which occur twice in $\bf u$ and $\con_{>2}({\bf u})$ to denote the set of all
variables which occur at least 3 times in $\bf u$. The next lemma is similar to Lemma 4.1 in \cite{OS} (Lemma \ref{redef1} below).
\begin{lemma} \label{redef} Let $\bf u$ and $\bf v$ be two words of the same type such that $\lin({\bf u}) = \lin({\bf v})$,
$\con_2({\bf u}) = \con_2({\bf v})$ and $\con_{>2}({\bf u}) = \con_{>2}({\bf v})$.
Let $\Theta: \mathfrak A \rightarrow \mathfrak A^+$
be a substitution that has the following property:
(*) If $\Theta(x)$ contains more than one variable then $x \in \lin({\bf u}) \cup \con_{2}({\bf u})$.
Then $\Theta({\bf u})$ and $\Theta({\bf v})$ are also of the same type.
\end{lemma}
\begin{proof} Since $\bf u$ and $\bf v$ are of the same type, the following is true.
\begin{claim} \label{yy} Suppose that $y \in \con_2({\bf u})$. If there is an occurrence of $x$ between the two occurrences of $y$ in $\bf u$ then there is an occurrence of $x$ between the two occurrences of $y$ in $\bf v$.
\end{claim}
Since $\bf u$ and $\bf v$ are of the same type, for some $r \ge 1$ and $u_1, \dots,u_r, v_1, \dots, v_r >0$ we have ${\bf u} = c_1^{u_1} c_2^{u_2}\dots c_r^{u_r}$ and ${\bf v} = c_1^{v_1}c_2^{v_2}\dots c_r^{v_r}$,
where $c_1, \dots , c_r \in \mathfrak A$ are such that $c_i \ne c_{i+1}$.
First, let us prove that for each $1 \le i \le r$ the words $\Theta({c}_i^{u_i})$ and $\Theta({c}_i^{v_i})$ are of the same type.
Indeed, If $c_i$ is linear in $\bf u$ (and in $\bf v$) then $u_i=v_i=1$ and $\Theta({c}_i^{u_i}) = \Theta({c}_i^{v_i})$.
If $c_i$ occurs twice in $\bf u$ (and in $\bf v$), then in view of Claim \ref{yy}, either $u_i=v_i=1$ or $u_i=v_i=2$.
In either case, we have $\Theta({c}_i^{u_i}) = \Theta({c}_i^{v_i})$.
If $c_i$ occurs at least 3 times in $\bf u$ (and in $\bf v$) then $\Theta({c}_i^{u_1}) = x^+$ for some variable $x$ and $\Theta({c}_i^{v_i})$ is a power of the same variable.
Since \[\Theta({\bf u}) = \Theta({c}_1^{u_1}{c}_2^{u_2}\dots {c}_r^{u_r} ) = \Theta({c}_1^{u_1}) \Theta({c}_2^{u_2})\dots \Theta({c}_r^{u_r})\] and
\[\Theta({\bf v}) = \Theta({c}_1^{v_1}{c}_2^{v_2}\dots {c}_r^{v_r} ) = \Theta({c}_1^{v_1}) \Theta({c}_2^{v_2})\dots \Theta({ c}_r^{v_r}),\]
we conclude that $\Theta({\bf u})$ and $\Theta({\bf v})$ are of the same type.
\end{proof}
\begin{proof}[Proof of Theorem \ref{main}] Let $\tau$ be the equivalence relation on $\mathfrak A^+$ defined by ${\bf u} \tau {\bf v}$ if $\bf u$ and $\bf v$ are of the same type.
First, notice that the words ${\bf U}_n$ and ${\bf V}_n$ are not of the same type. Indeed, ${\bf U}_n$
contains $x_{n} x_{n-1}$ as a subword but ${\bf V}_n$ does not have this subword.
Now let ${\bf U}$ be of the same type as ${\bf U}_n$.
Let ${\bf u} \approx {\bf v}$ be an identity of $M$ in less than $n-1$ variables and let
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution such that $\Theta({\bf u}) = {\bf U}$.
Notice that $E({\bf u})$ also involves less than $n-1$ variables and $E({\bf u}) \approx E({\bf v})$ is also an identity of $M$.
Due to Property (P1), for each $1\le i < j \le n$ the height of ${\bf U}(x_i, x_j)$ is at most $4$.
So, by Lemma \ref{prop1}, $E({\bf u})$ satisfies Condition (I) in Lemma \ref{manylet}, that is, for each $\{x,y\}\subseteq \con(E({\bf u}))$ the height of $E({\bf u}(x,y))$ is at most 4. Also, $E({\bf u})$ satisfies Conditions (II)--(IV) in Lemma \ref{manylet} by Lemmas \ref{propii}--\ref{prop2}.
Therefore, Lemma \ref{manylet} implies that the word $E({\bf v})$ is of the same type as $E({\bf u})$.
Due to Property (P3), for each $1 \le i \ne j \le n$ the word $x_i x_j$ appears at most twice in $\bf U$ as a scattered subword.
Consequently, the word $E({\bf u})$ and the substitution $\Theta$ satisfy Condition (*) in Lemma \ref{redef}. According to Fact 3.1 in \cite{OS}, Property (C$_4$) implies that the word $x^2$ is an isoterm for $M$. Thus
all other conditions of Lemma \ref{redef} are also met.
Therefore, the word $\Theta(E({\bf v}))$ has the same type as $\Theta(E({\bf u}))$ by Lemma \ref{redef}.
Thus we have
\[ {\bf U} = \Theta({\bf u}) \stackrel{Fact \ref{Eu}}{\tau} \Theta (E ({\bf u})) \stackrel{Lemma \ref{redef}} {\tau} \Theta (E ({\bf v})) \stackrel{Fact \ref{Eu}}{\tau} \Theta({\bf v}).\]
Since $\Theta({\bf v})$ is of the same type as $\bf U$, $M$ is NFB by Lemma \ref{nfblemma}.
\end{proof}
\section{Syntactic version of the sufficient condition of Lee for semigroups}\label{sect:sufconLee}
The following sufficient condition implies that for each $\ell \ge 3$ the semigroup $L_\ell$ is NFB \cite{EL1}.
\begin{fact} \label{EL61} \cite[Theorem 11]{EL} Fix $k \ge 2$.
Let $S$ be a semigroup such that $\var S$ contains $L_3$.
If for each $n \ge 2$, $S$ satisfies the identity
\[x^ky^k_1y^k_2 \dots y^k_nx^k \approx x^ky^k_ny^k_{n-1} \dots y^k_1x^k\]
then $S$ is NFB.
\end{fact}
In view of Fact \ref{final}, the following sufficient condition is a slight generalization of Fact \ref{EL61}.
\begin{theorem} \label{EL62}
Let $S$ be a semigroup that satisfies Property (C$_3$).
If for infinitely many $n\ge 2$, $S$ satisfies the identity
\[{\bf U}_n= x^ky^k_1y^k_2 \dots y^k_nx^k \approx x^ky^k_ny^k_{n-1} \dots y^k_1x^k ={\bf V}_n\]
for some $k \ge 2$, then $S$ is NFB.
\end{theorem}
The goal of this section is to prove Theorem \ref{EL62} directly using Lemma \ref{nfblemma}.
To this aim, we establish some consequences of Property (C$_3$) for semigroups.
\begin{lemma} \label{sem1} Let $S$ be a semigroup that satisfies Property (C$_3$).
If $S \models {\bf u} \approx {\bf v}$ then
$\con({\bf u}) = \con({\bf v})$ and $\lin({\bf u}) = \lin({\bf v})$.
\end{lemma}
\begin{proof} If $x \in \con({\bf u})$ but $x \not \in \con({\bf v})$ then for some $y \in \con({\bf v})$ and $r>0$
the identity ${\bf u} \approx {\bf v}$ implies $y^r \approx {\bf w}$ such that the height of ${\bf w} \in \{x,y\}^+$ is at least 2.
To avoid a contradiction to Property (C$_1$) we conclude that $\con({\bf u}) = \con({\bf v})$.
If $x$ is linear in $\bf u$ but appears at least twice in $\bf v$ then substitute $xy$ for $x$ and $y$ for all other variables.
Then for some $c+d >0$ the identity ${\bf u} \approx {\bf v}$ implies $y^c x y^d \approx {\bf w}$ such that the height of ${\bf w} \in \{x,y\}^+$ is at least 4. To avoid a
contradiction to Property (C$_3$) we conclude that $\lin({\bf u}) = \lin({\bf v})$.
\end{proof}
\begin{lemma} \label{sem2} Let $S$ be a semigroup that satisfies Property (C$_3$).
If each variable forms only one island in a word $\bf u$ then $\bf u$ can form an identity of $S$ only with a word of the same type.
\end{lemma}
\begin{proof}
Since each variable forms only one island in the word $\bf{u}$ we may assume that ${\bf u}=x^+_1x^+_2\ldots x^+_r$ for some distinct variables $x_1,x_2, \ldots, x_r$. Suppose that $\bf{u}$ forms an identity of $S$ with a word $\bf{v}$. By Lemma \ref{sem1}, $\con(\bf{u})=\con(\bf{v})$. Note that if we replace $x_1$ by $x$ and any other letter by $y$, then the word $\bf{u}$ turns into $x^+y^+$.
Since $S$ satisfies Property (C$_2$) the word $\bf{v}$ also starts with $x_1$.
First, let us prove that each variable forms only one island in the word $\bf{v}$. Suppose the contrary, there is a letter $a\in \con(\bf{u})=\con(\bf{v})$ that forms at least two islands in $\bf{v}$. We replace $a$ by $y$ and other letters by $x$. Thus, the identity $\bf{u}\approx \bf{v}$ implies $\bf{u}'\approx \bf{v}'$.
Notice that the height of ${\bf u'} \in \{x,y\}^+$ is at most $3$ and that $y$ forms only one island in $\bf u'$. On the other hand,
$y$ forms at least two islands in $\bf v'$. Since $S$ satisfies Property (C$_3$), this is impossible.
Second, let us take two consecutive letters $x_i, x_{i+1}\in\con(\bf{u})$ and replace them by $y$, while other letters by $x$. Note that the word $\bf{u}$ transforms into a word ${\bf u}'\in\{x,y\}^+$ of height at most $3$ with one island of $y$. Therefore, by Property (C$_3$) the corresponding word $\bf{v}'$ also has one island of $y$ implying the word $\bf{v}$ has either $x_ix_{i+1}$ or $x_{i+1}x_i$ as a subword. Since $\bf{u}$ and $\bf{v}$ start with the same letter $x_1$ we conclude that $\bf{u}$ and $\bf{v}$ have the same type.
\end{proof}
\begin{lemma} \label{sem3} Let $S$ be a semigroup that satisfies Property (C$_3$). Let $\bf u$ be a word
that begins and ends with $x$ such that $x$ forms exactly two islands in $\bf u$.
Suppose also that $\bf u$ contains a linear letter and that each variable other than $x$ forms only one island in $\bf u$.
Then $\bf u$ can form an identity of $S$ only with a word of the same type.
\end{lemma}
\begin{proof} Suppose that $S \models {\bf u} \approx {\bf v}$.
If we substitute $y$ for all variables in $\con({\bf u}) = \con({\bf v})$ other than $x$ then $\bf u$ turns into $x^+y^+x^+$.
Since $S$ satisfies Property (C$_3$), this implies that $\bf v$ starts and ends with $x$ and $x$ forms exactly two islands in $\bf v$.
Thus we have ${\bf u} = x^+ {\bf a} t {\bf b} x^+$ and ${\bf v} = x^+ {\bf a'} t {\bf b'} x^+$ where $\con ({\bf ab})= \con ({\bf a'b'}) \cap \{x,t\} =\emptyset$.
We have ${\bf b} = y_1^+ y_2^+ \dots y_r^+$ for some $\{y_1, \dots, y_r\} \subseteq \con ({\bf ab})$.
If $y_1$ is not the first letter in ${\bf b'}$ then substitute $xy$ for $t$, $y$ for $y_1$ and $x$ for all other variables.
Then $\bf u$ turns into $x^+ y^+ x^+$ while $\bf v$ becomes a word that contains at least two islands of $y$. To avoid a contradiction
to Property (C$_3$) we conclude that $\bf b'$ starts with $y_1$. If $\bf b'$ does not have $y_1y_2$ as a subword, then substitute $xy$ for $t$, $y$ for $y_1$ and for $y_2$ and $x$ for all other variables. Then $\bf u$ turns into $x^+ y^+ x^+$ while $\bf v$ becomes a word that contains at least two islands of $y$. To avoid a contradiction
to Property (C$_3$) we conclude that $\bf b'$ starts with $y_1^+y_2$. And so on. Eventually we conclude
that the words $\bf b$ and $\bf b'$ are of the same type.
In a similar way, one can show that $\bf a$ and $\bf a'$ are also of the same type. Consequently, $\bf u$ and $\bf v$ are of the same type.
\end{proof}
Finally, in order to prove Theorem \ref{EL62} we need the following statement similar to Lemma \ref{redef}.
\begin{lemma} \label{redef1} \cite[Lemma 4.1]{OS} Let $\bf u$ and $\bf v$ be two words of the same type such that $\lin({\bf u}) = \lin({\bf v})$ and
$\non({\bf u}) = \non({\bf v})$.
Let $\Theta: \mathfrak A \rightarrow \mathfrak A^+$
be a substitution that has the following property:
(*) If $\Theta(x)$ contains more than one variable then $x$ is linear in $\bf u$.
Then $\Theta({\bf u})$ and $\Theta({\bf v})$ are also of the same type.
\end{lemma}
\begin{proof}[Proof of Theorem \ref{EL62}] Let $\tau$ be the equivalence relation on $\mathfrak A^+$ defined by ${\bf u} \tau {\bf v}$ if $\bf u$ and $\bf v$ are of the same type.
First, notice that the words ${\bf U}_n$ and ${\bf V}_n$ are not of the same type. Indeed, ${\bf U}_n$
contains $y_1 y_2$ as a subword but ${\bf V}_n$ does not have this subword.
Now let ${\bf U}$ be of the same type as ${\bf U}_n$.
Let ${\bf u} \approx {\bf v}$ be an identity of $S$ in less than $n$ variables and let
$\Theta: \mathfrak A \rightarrow \mathfrak A^+$ be a substitution such that $\Theta({\bf u}) = {\bf U}$.
Notice that $E({\bf u})$ also involves less than $n$ variables and $E({\bf u}) \approx E({\bf v})$ is also an identity of $S$.
If every variable forms only one island in $E({\bf u})$ then by Lemma \ref{sem2} the word $E({\bf v})$ is of the same type as $E({\bf u})$.
If some variable $x$ forms more than one island in $E({\bf u})$ then in view of Fact \ref{Eu}, $x$ forms exactly two islands in $E({\bf u})$ and
$E({\bf u})$ begins and ends with $x$. Also, $E({\bf u})$ contains a linear letter because it involves less than $n$ variables.
So, in this case, the word $E({\bf v})$ is of the same type as $E({\bf u})$ by Lemma \ref{sem3}.
Therefore, the word $\Theta(E({\bf v}))$ has the same type as $\Theta(E({\bf u}))$ by Lemma \ref{redef1}.
Thus we have
\[ {\bf U} = \Theta({\bf u}) \stackrel{Fact \ref{Eu}}{\tau} \Theta (E ({\bf u})) \stackrel{Lemma \ref{redef1}} {\tau} \Theta (E ({\bf v})) \stackrel{Fact \ref{Eu}}{\tau} \Theta({\bf v}).\]
Since $\Theta({\bf v})$ is of the same type as $\bf U$, $S$ is NFB by Lemma \ref{nfblemma}.
\end{proof}
\subsection*{Acknowledgement} The authors thank an anonymous referee for helpful comments.
| 191,640
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Baseboards are most often smooth and match the floor color. They can be wood or tile. Consequently, most baseboards share similar features.
They"re essentially a small extension of the floor that continues onto the walls. In other words, they connect the wall and floor at a 90º angle.
Most people don"t think twice about them. Instead, they go unnoticed as a simple functional element – these are baseboards in their most basic form.
Basic resources have a functional appearance.
Rustic baseboards are an interesting example of what baseboards can be. They have a country, natural character. The design aims to create a rural, casual ambiance.
Another interesting idea is using small spotlights in your baseboards. As we usually hang lights from our ceilings, this idea might seem a little strange at first.
Lacquered options are a popular choice for homes. They best fit in classical or minimalist settings.
You can also add decor elements to baseboards. Choose from animals, symbols, geometric structures, lines, etc.
A truly unique idea is hinting at a little mouse"s home. If you have children, it"s a great idea to try. You can also use ants or other animals and insects for a realist effect.
All in all, there are countless ways to use baseboards in your home decor.It might interest you...
| 408,189
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TITLE: Apples Across a Bridge (word problem)
QUESTION [3 upvotes]: Here is a fun problem that I am having trouble answering:
Let’s say I live in Town A, and I want to transport 249 apples across a bridge to Town B, but I can only carry 100 at a time. The bridge is 100 feet long, and I will lose one apple every foot I walk in the direction of Town B and it cannot be recovered. I will not lose apples if I walk in the direction of Town A. I am also permitted to set down apples at any point on the bridge and pick them up later. What is the minimum amount of apples I need to get 249 of them across the bridge?
REPLY [0 votes]: I'll give my mathematical model first, so it's easier to figure out if a disagreement comes from an incorrrect argument or a different understanding of the problem.
I consider this a discrete problem, the person will be able to move on the bridge only in integer multiple of $1$ft in either direction. Otherwise, the "I will lose one apple every foot I walk in the direction of Town B" condition becomes hard to interpret without more clarification (for example, if I move $1$ ft in $10$ steps of $0.1$ft each, putting down and taking up some apples on the way, when and how do I loose the $1$ apple).
Then it makes sense to mark the $100$ points of interest on the bridge where the person can stop by their distance in feet from the edge of the bridge towards town A. So point $0$ is the edge of the bridge towards town A, getting apples here incurs no losses. Point $1$ is $1$ft away from it a.s.o., and point $100$ is the edge of the bridge towards town B, where $249$ apples need to be brought to.
First, it is enough to get $252$ apples to point $99$. Then the person can take them in $3$ walks to point $100$, starting for example with $100$, $100$ and $52$ and arriving with $99$, $99$ and $51$, resp., going back empty handed each time for the next batch.
But it is also necessary for any solution to get at least $252$ different apples to point $99$. Because bringing $249$ apples to point $100$ requires at least $3$ trips (with apples) from point $99$ to point $100$, so the person will lose at least $3$ apples that reached point $99$ on those trips. If less then $252$ different apples make it to point $99$, less then $249$ different apples make it to point $100$.
This argument can again be applied to the point $98$: Bringing $255$ apples there is enough to solve the problem, and you need to bring at least $255$ different apples there, in order to solve the problem (aka bring $252$ different apples to point $99$).
This argument continues, increasing the number of apples that need to be brought to point $X-1$ vs. point $X$ by $3$, until you reach point $84=100-16$. The above argument shows that it is necessary and sufficient to bring $297=249+16\times3$ apples there. Since $297=3\times 99$, the above argument just barely works to show that it is necessary and sufficient to bring $300$ apples to point $83$, then make $3$ trips (starting with $100$ apples each time, arriving with $99$ each time) from point $83$ to $84$.
But now we need at least $4$ trips to reach point $83$ from point $82$ bringing $300$ apples there, so we lose at least $4$ apples on that $1$ft trip. So now the number of apples that it is necessary (and sufficient) to bring to point $82$ is increased by $4$ from what it was for point $83$, so that number is $304$.
Again this continues, and we see that we need $308$ apples at point $81$ a.s.o, until we reach point $59=83-24$, which requires $396=300+24\times4$ apples, which is just managable with $4$ trips from point $58$, which needs $400$ apples. From here on we need at least $5$ trips for each $1$ft step, so point $57$ requires $405$ apples,a.s.o.
The next 'step up' in the number of required trips for $1$ft happens between points $38$ and $37$. Point $38=58-20$ requires $500=400+20\times5$ apples, which needs 6 trips from point $37$.
The next 'step up' in the number of required trips for $1$ft happens between points $22$ and $21$. Point $22=38-16$ requires $596=500+16\times6$ apples, which needs 7 trips from point $21$ (since each trip can only arrive with at most $99$ apples, $6$ trips aren't enough, as $596 > 594 = 6\times99$).
The next 'step up' in the number of required trips for $1$ft happens between points $8$ and $7$. Point $8=22-14$ requires $694=596+14\times7$ apples, which needs 8 trips from point $7$.
This means we need to bring (and it is enough to bring) $758=694+8\times8$ apples to point $0$, which is the end if the bridge towards town A. As bringing apples to this point is lossless, that the answer:
The person transporting apples needs to start with at least $758$ apples in town A, and that many apples is enough.
| 208,964
|
EMMAUS, Pa. - From the Lehigh Valley to Bucks County, the toxic chemicals PFAS are a continuing problem in drinking water.
Pennsylvania is set to put enforceable max contamination levels in place. However, some environmental groups say the state's plan is falling short in protecting people's health.
A petition for Pennsylvania's DEP from 28 statewide organizations are calling for stricter regulations of the forever chemicals known as PFAS and PFOAS.
"PFAS are called forever chemicals and that's because they don't break down in the environment," said Tracy Carluccio via a bullhorn outside the agency's Harrisburg headquarters.
Carluccio, the Deputy Director of the Delaware Riverkeeper Network, was part of a handful of environmental groups gathered there.
"It's taken years to get here. It's been an excruciatingly slow process to get where we are today," she said.
The DEP is wrapping up a two-month public comment period on regulating the forever chemicals in public water systems. It's part of Gov. Tom Wolf's PFAS Action Team put in place in 2018 to fight against the forever chemicals.
The compounds never break down and stem in large part from firefighting foam. They're linked to a variety of health issues, including cancer.
Late last year, the Environmental Quality Board recommended 14 parts per trillion for PFAS, and 18 for PFOAS.
Last fall the borough of Emmaus's water was found to have 111 ppt and Bucks County recently sued a host of PFAS manufacturers for knowingly contaminating public water ways. That case is now in federal court.
"If you're just addressing PFOS and PFOAS, you are sort of tackling the tip of a big toxic iceberg," said clean water advocate Stephanie Wein.
Wein, of PennEnvironment, says at least six other similar toxic chemicals have been found throughout the state. They want those regulated and tested too, as well as private wells.
The DEP says it is still early in the process, and things could change but added that those other chemicals will not be added, at least for now. They say they don't have the toxicologists or data to support it.
For DEP to consider regulating additional PFAS chemicals in the future, the agency would need to meet all required elements of the federal Safe Drinking Water Act and Regulatory Review Act, including sufficient occurrence data to demonstrate prevalence of additional chemicals, risk assessment and toxicology data to determine the possible health impacts, data on technical limitations and treatability, and costs and benefits.
However, EPA is beginning a similar effort to regulate PFAS chemicals that could include additional PFAS chemicals, said a DEP spokesman.
They say they don't have jurisdiction over private wells.
Carluccio and crew are pushing for 1ppt and no more than 6 parts per trillion.
"Right now, we don't have safe water. People are drinking water that could harm their health," Carluccio said.
If the rulemaking is finalized as proposed, water systems that serve populations greater than 350 people would have to begin monitoring for PFAS and PFOAS no later than January 1, 2024; systems serving fewer than 350 people would begin monitoring no later than January 1, 2025.
Follow-up, investigative and corrective actions would be triggered as soon as an MCL is exceeded. We would expect all corrective actions to be taken in a timely fashion. If things are delayed, we would escalate enforcement as needed, said a DEP spokesman.
| 65,353
|
\begin{document}
\maketitle
\markboth{Fractional higher derivatives of weak solutions for
Navier-Stokes}{Fractional higher derivatives of weak solutions for
Navier-Stokes}
\renewcommand{\sectionmark}[1]{}
\begin{abstract}
We study weak solutions of the 3D Navier-Stokes equations in whole space
with $L^2$ initial data.
It will be proved that
$\nabla^\alpha u $ is locally integrable
in space-time
for any real $\alpha$ such that
$1< \alpha <3$, which says that almost third derivative is locally integrable.
Up to now, only second derivative $\nabla^2 u$ has been known to
be locally integrable by standard parabolic regularization.
We also present sharp estimates of
those quantities in weak-$L_{loc}^{4/(\alpha+1)}$.
These estimates depend only
on the $L^2$ norm of initial data and integrating domains.
Moreover, they are valid even for $\alpha\geq 3$ as
long as $u$ is smooth.
The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study.
For the local study, we use De Giorgi method with a new pressure decomposition.
To handle non-locality of the fractional Laplacian, we will adopt
some properties of the Hardy space and Maximal functions.
\end{abstract}
\textbf{Mathematics Subject Classification}: 76D05, 35Q30.
\section{Introduction and main result}
\qquad In this paper, any derivative signs ($\nabla,\Delta,(-\Delta)^{\alpha/2},D,\partial$ and etc) denote derivatives in only space variable
$x\in \mathbb{R}^3$ unless time
variable $t\in\mathbb{R}$
is clearly specified.
We study the 3-D Navier-Stokes equations
\begin{equation}\label{navier}
\begin{split}
\partial_tu+(u\cdot\nabla)u+ \nabla P -\Delta u&=0 \quad\mbox{and}\\
\ebdiv u&=0,\quad \quad t\in ( 0,\infty ),\quad x\in \mathbb{R}^3
\end{split}
\end{equation} with $L^2$ initial data
\begin{equation}\label{initial_condition}
u_0\in L^2(\mathbb{R}^3) ,\quad \ebdiv u_0= 0.
\end{equation}
Regularity of weak solutions for the 3D Navier-Stokes equations has long history.
Leray \cite{leray} 1930s and Hopf \cite{hopf} 1950s
proved existence of a global-time
weak solution for any given $L^2$ initial data.
Such Leray-Hopf weak solutions $u$ lie
in $L^\infty(0,\infty;L^2(\mathbb{R}^3))$ and
$\nabla u$ do in $L^2(0,\infty;L^2(\mathbb{R}^3))$ and
satisfy the energy inequality:
\begin{equation*}
\|u(t)\|_{L^2(\mathbb{R}^3)}^2+
2\|\nabla u\|^2_{L^2(0,t;L^2(\mathbb{R}^3))}
\leq \|u_0\|_{L^2(\mathbb{R}^3)}^2 \quad \mbox{ for a.e. } t<\infty.
\end{equation*}
Until now, regularity and uniqueness of such weak solutions are
generally open.\\% while those of local-time smooth solution has been shown
Instead, many criteria which
ensure regularity of weak solutions
have been developed. Among
them the most famous one is Lady{\v{z}}enskaja-Prodi-Serrin Criteria
(\cite{lady},\cite{prodi} and \cite{Serrin}),
which says:
if $u\in L^p((0,T);L^q(\mathbb{R}^3)) $ for
some $p$ and $q$ satisfying $\frac{2}{p}
+\frac{3}{q}=1$ and $p<\infty$, then it is regular. Recently, the limit case $p=\infty$
was established in the paper of
Escauriaza, Ser{\"e}gin and {\v{S}}ver{\'a}k
\cite{seregin}. We may impose similar conditions to derivatives of velocity, vorticity
or pressure. (see
Beale, Kato and Majda \cite{bkm}, Beir{\~a}o da Veiga \cite{beirao} and
Berselli and Galdi \cite{berselli}) Also, many other conditions
exist (e.g. see
Cheskidov and Shvydkoy \cite{Cheski},
Chan \cite{chan}
and
\cite{vas_bjorn}). \\
On the other hand, many efforts have been given to measuring the size
of possible singular set.
This approach has been initiated by
Scheffer \cite{scheffer}. Then, Caffarelli, Kohn and Nirenberg \cite{ckn}
improved the result and showed that possible singular sets have zero Hausdorff measure of one dimension for
certain class of weak solutions (suitable weak
solutions)
satisfying the following additional inequality
\begin{equation}\label{suitable}
\partial_t \frac{|u|^2}{2} + \ebdiv (u \frac{|u|^2}{2}) + \ebdiv (u P) +
|\nabla u |^2 - \Delta\frac{|u|^2}{2} \leq 0
\end{equation} in the sense of distribution.
There are many other proofs of this fact (e.g. see Lin
\cite{lin}, \cite{vas:partial}
and Wolf \cite{wolf}). Similar criteria for interior points with other quantities
can be found in many places (e.g. see Struwe \cite{struwe},
Gustafson, Kang and Tsai \cite{gusta}, Ser{\"e}gin \cite{seregin2}
and Chae, Kang and Lee \cite{chae}).
Also, Robinson and Sadowski \cite{robinson} and Kukavica \cite{kukavica}
studied
box-counting dimensions
of
singular sets.\\
In this paper, our main concern is about space-time $L^p_{(t,x)}=L^p_tL^p_x$
estimates of higher derivatives for weak
solutions assuming only $L^2$
initial data.
$\nabla u\in L^{2}((0,\infty)\times\mathbb{R}^3)$ is obvious from the energy
inequality, and
simple interpolation gives $u\in L^{10/3}$.
For second derivatives of weak solutions,
from standard parabolic regularization theory (see
Lady{\v{z}}enskaja, Solonnikov and
Ural$'$ceva
\cite{LSU}),
we know $\nabla^2 u \in L^{5/4}$ by considering
$(u\cdot\nabla)u$ as a source term.
With different ideas, Constantin \cite{peter} showed
$L^{\frac{4}{3}-\epsilon}$ for any small $\epsilon>0$ in periodic setting,
and later Lions \cite{lions} improved it up to
weak-$L^{\frac{4}{3}}$ (or $ L^{\frac{4}{3},\infty}$)
by assuming $\nabla u_0$ lying in the space
of all bounded measures in $\mathbb{R}^3$.
They used natural structure of the equation with some interpolation technique.
On the other hand, Foia{\c{s}}, Guillop{\'e} and Temam \cite{foias_guillope_tem:higher} and Duff \cite{duff}
obtained other kinds of estimates for higher derivatives
of weak solutions while Giga and Sawada \cite{giga}
and Dong and Du \cite{dong} covered mild solutions. For asymptotic behavior,
we refer Schonbek and Wiegner \cite{Scho_and_Wiegner}.\\
Recently in \cite{vas:higher}, it has been shown that, for any small
$\epsilon > 0$, any integer $d\geq1$ and any smooth solution $u$ on $(0,T)$,
we have bounds of $\nabla^d u$ in
$L_{loc}^{\frac{4}{d+1}-\epsilon}$,
which
depend only on $L^2$ norm of initial data
once we fix $\epsilon$, $d$ and the domain of integration.
It can be considered as a
natural extension of the result of Constantin \cite{peter} for higher derivatives.
But the idea is completely different in the sense that
\cite{vas:higher} used the Galilean invariance of transport part of
the equation and the partial regularity criterion in the version
of \cite{vas:partial}, which re-proved
the famous result of Caffarelli, Kohn and Nirenberg \cite{ckn} by using a parabolic
version of the De Giorgi method \cite{De_Giorgi}.
It is noteworthy that this method
gave full regularity to
the critical Surface Quasi-Geostrophic equation in
\cite{caf_vas}.
The limit
non-linear scaling
$p=\frac{4}{d+1}$
appears from
the following invariance of the Navier-Stokes scaling
$u_\lambda(t,x)=
\lambda u(\lambda^2 t,\lambda x)$:
\begin{equation}\label{best_scaling}
\|\nabla^d u_\lambda\|^p_{L^p}=\lambda^{-1}\|\nabla^d u\|^p_{L^p}.
\end{equation}
In this paper, our main result is better than
the above result of \cite{vas:higher} in the sense of the following three directions.
First, we achieve the limit case weak-$L^{\frac{4}{d+1}}$
(or $ L^{\frac{4}{d+1},\infty}$) as Lions \cite{lions} did for second derivatives.
Second, we make similar bounds for fractional derivatives as well as classical derivatives.
Last, we consider not only smooth solutions but also global-time weak solutions.
These three improvements will give us that
$\nabla^{3-\epsilon}u$, which is almost third derivatives of weak solutions, is locally integrable
on $(0,\infty)\times\mathbb{R}^3$. \\
Our precise result is the following:
\begin{thm}\label{main_thm}
There exist universal constants
$C_{d,\alpha}$ which depend only on integer $d\geq1$ and real $\alpha\in[0,2)$
with the following two properties $(I)$ and $(II)$:\\
(I)
Suppose that we have
a smooth solution $ u $ of \eqref{navier}
on $ (0,T)\times \mathbb{R}^3 $ for some $0<T\leq\infty$
with some initial data \eqref{initial_condition}. Then it satisfies
\begin{equation}\label{main_thm_eq}
\quad\|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u\|
_{L^{p,\infty}(t_0,T;L^{p,\infty}(K))}
\leq C_{d,\alpha}\Big(\|u_0\|_{L^2(\mathbb{R}^3)}^{2}
+ \frac{|K|}{t_0}\Big)^{\frac{1}{p}}
\end{equation}
for any $t_0\in(0,T)$, any integer $d\geq 1$, any $\alpha\in[0,2)$
and any bounded open subset
$K$ of $ \mathbb{R}^3$, where $p = \frac{4}{d+\alpha+1}$
and $|\cdot|=$ the Lebesgue measure in $\mathbb{R}^3$. \\
(II) For any initial data \eqref{initial_condition}, we can construct a
suitable weak solution $u$ of \eqref{navier} on $ (0,\infty)\times \mathbb{R}^3 $ such that
$(-\Delta)^{\frac{\alpha}{2}}\nabla^d u$
is locally integrable in $(0,\infty)\times \mathbb{R}^3$
for $d=1,2$ and for $\alpha\in[0,2)$ with $(d+\alpha)<3$.
Moreover,
the estimate \eqref{main_thm_eq} holds
with $T=\infty$
under the same setting of the above part $(I)$
as long as $(d+\alpha)<3$.
\end{thm}
Let us begin with some simple remarks.
\begin{rem}\label{frac_rem}
For any suitable weak solution $u$,
we can define $(-\Delta)^{\alpha/2}\nabla^d u$
in the sense of distributions $\mathcal{D}^\prime$ for any integer $d\geq0$ and for any real
$\alpha\in[0,2)$:
\begin{equation}\label{fractional_distribution}
<(-\Delta)^{\alpha/2}\nabla^d u;\psi >_{\mathcal{D}^\prime,\mathcal{D}}
= (-1)^{d}\int_{(0,\infty)\times\mathbb{R}^3}u\cdot(-\Delta)^{\alpha/2}\nabla^d \psi\mbox{ } dxdt
\end{equation} for any $\psi\in \mathcal{D}=C_c^\infty((0,\infty)\times\mathbb{R}^3)$
where
$(-\Delta)^{\alpha/2}$ in the right hand side is
the traditional
fractional Laplacian in $\mathbb{R}^3$ defined by the Fourier transform.
Note that
$(-\Delta)^{\alpha/2}\nabla^d \psi$ lies in $L_t^\infty L^2_x$.
Thus, this definition from \eqref{fractional_distribution} makes sense due to
$u\in L_t^\infty L^2_x$. Note also $(-\Delta)^{0}=Id$. For more general extensions
of this fractional Laplacian operator, we
recommend
Silvestre \cite{silve:fractional}.
\end{rem}
\begin{rem}\label{rmk_dissipation of energy}
Since we impose only \eqref{initial_condition} to $u_0$, the estimate \eqref{main_thm_eq}
is a
(quantitative)
regularization result to higher derivatives.
Also, in the proof, we will see that $\|u_0\|_{L^2(\mathbb{R}^3)}^{2}$ in \eqref{main_thm_eq} can be
relaxed to $\|\nabla u\|_{L^2((0,T)\times\mathbb{R}^3)}^{2}$. Thus
it says that any (higher) derivatives can be controlled by having only
$L^2$ estimate of dissipation of energy.
\end{rem}
\begin{rem}\label{weak_L_p}
The result of the part $(I)$ for $\alpha=0$ extends the result of the previous paper \cite{vas:higher}
because for any $0<q<p<\infty$ and any bounded subset $\Omega\subset\mathbb{R}^n$, we have
\begin{equation*}
\|f\|_{L^{q}(\Omega)}\leq C\cdot
\|f\|_{L^{p,\infty}(\Omega)}
\end{equation*} where C depends only
on $p$, $q$, dimension $n$ and Lebesgue measure of $\Omega$ (e.g. see Grafakos \cite{grafakos}).
\end{rem}
\begin{rem}
The assumption ``smoothness'' in the part $(I)$ is about pure differentiability.
For example, the result of the part $(I)$ for $d\geq1$
and $\alpha=0$
holds once we know that $u$ is $d$-times differentiable. In addition,
constants in \eqref{main_thm_eq} are independent of any possible blow-time $T$.
\end{rem}
\begin{rem}
$p=4/(d+\alpha+1)$ is a very interesting relation as mentioned before. Due to this $p$, the estimate \eqref{main_thm_eq}
is
a non-linear estimate while many other $a$ $priori$ estimates are linear.
Also, from the part $(II)$ when $(d+\alpha)$ is very close to $3$,
we can see that almost third derivatives of weak solutions are locally integrable.
Moreover, imagine that the part $(II)$ for $d=\alpha=0$ be true even though
we can NOT prove it here.
This would imply that this weak solution $u$ could lie in $L^{4,\infty}$ which is beyond the
best known
estimate $u\in L^{10/3}$ from $L^2$ initial data. \\
\end{rem}
Before presenting the main ideas, we want to mention that
Caffarelli, Kohn and Nirenberg \cite{ckn}
contains two different kinds of
local regularity criteria. The first one is quantitative, and it says that
if $\| u\|_{L^3(Q(1))}$ and $\| P\|_{L^{3/2}(Q(1))}$ is small,
then $u$ is bounded by some universal constant in $Q({1/2})$.
The second one says that
$u$ is locally bounded near the origin
if $\limsup_{r\rightarrow 0}$ $r^{-1}\|\nabla u\|^2_{L^2(Q(r))}$ is
small. So it is qualitative
in the sense that
the conclusion says not that $u$ is bounded by a universal constant but
that $\sup |u|$ for some local neighborhood is not infinite.\\
On the other hand,
there is a different quantitative local regularity criterion in
\cite{vas:partial}, which showed
that for any $p>1$, there exists $\epsilon_p$ such that
\begin{equation}\label{vas:partial_result}
\mbox{if}\quad
\|u\|_{L^\infty_tL^2_x(Q_1)}+\|\nabla u\|_{L^2_tL^2_x(Q_1)}+
\|P\|_{L^p_tL^1_x(Q_1)}\leq \epsilon_p, \mbox{ then } |u|\leq 1
\mbox { in } Q_{1/2}
\end{equation} Recently, this criterion was used in
\cite{vas:higher} in order to obtain higher derivative estimates.
The main proposition in \cite{vas:higher} says that
if both $ \||\nabla u|^2+|\nabla^2 P|\|_{L^1(Q(1))}$
and some other
quantity about pressure
are small, then $u$ is bounded by $1$
at the origin
once $u$ has a mean zero property in space.
We can observe that $\|\nabla u\|^2_{L^2(Q(1))}$ and $\|\nabla^2 P\|_{L^1(Q(1))}$
have the same best scaling like \eqref{best_scaling} among
all the other quantities which we can obtain from $L^2$ initial data.
However, the other quantity about pressure
has a slightly worse scaling. That is the reason that
the limit case $L^{\frac{4}{d+1},\infty}$ has been missing in \cite{vas:higher}.\\
Here are the main ideas of proof.
First,
in order to obtain the missing limit case
$ L^{\frac{4}{d+1},\infty}$, we will see that it requires an equivalent estimate
of \eqref{vas:partial_result} for $p=1$.
Here we extend this result up to $ p=1$ for some
approximation of the Navier-Stokes (see the proposition
\ref{partial_problem_II_r}).
To obtain this first goal, we will introduce a new pressure decomposition
(see the lemma \ref{lem_pressure_decomposition}),
which will be used in the De Giorgi-type argument. This makes us to
remove the bad scaling term about pressure
in \cite{vas:higher}.
As a result,
by using the Galilean invariance property and some blow-up technique with the standard Navier-Stokes scaling,
we can proceed our local study in order to obtain a better
version of a quantitative partial regularity criterion
for some
approximation of the Navier-Stokes
(see the proposition
\ref{local_study_thm}).
As a result, we can prove $ L^{\frac{4}{d+1},\infty}$ estimate
for classical derivatives ($\alpha=0$ case).\\
Second,
the result for fractional derivatives ($0<\alpha<2$ case) is not obvious at all
because there is no proper interpolation theorem
for $L_{loc}^{p,\infty}$ spaces.
For example, due to the non-locality of the fractional Laplacian operator, the fact $\nabla^2 u\in L_{loc}^{\frac{4}{3},\infty}$
with $\nabla^3 u\in L_{loc}^{1,\infty}$
does not imply the case of fractional derivatives even if we assume $u$ is smooth.
Moreover, even though we assume that
$\nabla^2 u\in L^{\frac{4}{3}}(\mathbb{R}^3)$
and $\nabla^3 u\in L^{1}(\mathbb{R}^3)$ which we can NOT prove here,
the standard interpolation theorem still requires $L^p(\mathbb{R}^3)$ for some $p>1$ (we refer
Bergh and L{\"o}fstr{\"o}m \cite{bergh}).
To overcome
the difficulty,
we will use the Maximal functions
of $u$ which capture its behavior of long-range part.
Unfortunately, second derivatives of pressure, which lie in the Hardy space $\mathcal{H}
\subset L^1(\mathbb{R}^3)$
from
Coifman, Lions, Meyer and Semmes \cite{clms},
do not have an integrable Maximal function since
the Maximal operator is not bounded on $L^1$. In order to handle non-local parts of pressure,
we will use some
property of Hardy space,
which says that some integrable functions play
a similar role of the Maximal function(see \eqref{hardy_property}).\\
Finally, the result $(II)$ for weak solutions comes from
specific approximation of Navier-Stokes equations that
Leray \cite{leray} used in order to construct a global time weak solution
:
$\partial_tu_n+((u_n*\phi_{(1/n)})\cdot\nabla)u_n+ \nabla P_n -\Delta u_n=0$ and $
\ebdiv u_n=0$ where $\phi$ is a fixed mollifier in $\mathbb{R}^3$,
and $\phi_{(1/n)}$ is defined by $\phi_{(1/n)}(\cdot)=n^3\phi(n\mbox{ }\cdot)$.
Main advantage for us of adopting this approximation is that
it has strong existence theory of global-time smooth solutions
$u_n$ for each $n$, and it is well-known that there exists
a suitable weak solution $u$ as a weak limit.
In fact, for any integer $d\geq1$
and for any $\alpha\in[0,2)$,
we will obtain bounds for $u_n$ in the form of \eqref{main_thm_eq}
with $T=\infty$,
which is uniform in $n$.
Since $p=4/(d+\alpha+1)$ is greater than $1$ for the case $(d+\alpha)<3$,
we can know that $(-\Delta)^{\frac{\alpha}{2}}
\nabla^d u $ exists as a locally integrable function from weak-compactness of $L^p$ for $p>1$. \\
\noindent However, to prove \eqref{main_thm_eq} uniformly for the approximation
is nontrivial because our proof is based on local study while
the approximation is not
scaling-invariant with the standard Navier-Stokes scaling:
After the scaling, the advection velocity $u*\phi_{(1/n)}$
depends the original velocity $u$ more non-locally than before.
Moreover, when we consider the case of fractional derivatives of weak solutions, it requires
even Maximal of Maximal functions to handle non-local parts of
the advection velocity
which depends the original velocity
non-locally.\\
The paper is organized as follows. In the next section, preliminaries with
the main propositions
\ref{partial_problem_II_r} and
\ref{local_study_thm}
will be introduced.
Then we prove those propositions \ref{partial_problem_II_r} and \ref{local_study_thm}
in sections \ref{proof_partial_prob_II_r} and
\ref{proof_local study},
respectively.
Finally we will explain how
the proposition \ref{local_study_thm}
implies the part $(II)$ of the theorem
\ref{main_thm}
for $\alpha=0$ and for $0<\alpha<2$ in subsections
\ref{prof_main_thm_II_alpha_0}
and
\ref{prof_main_thm_II_alpha_not_0} respectively while
the part $(I)$ will be covered in the subsection \ref{proof_main_thm_I}.
After that,
the appendix contains some missing proofs of
technical lemmas.\\
\section{Preliminaries, definitions and main propositions}\label{prelim}
We begin this section by fixing some notations and reminding
some well-known results on analysis. After that we will present
definitions of two approximations and two main propositions. In this paper, any derivatives, convolutions and Maximal functions
are with respect to space variable $x\in\mathbb{R}^3$ unless time variable
is specified.\\
\noindent \textbf{Notations for general purpose}\\
We define
$B(r) =\mbox{ the ball in } \mathbb{R}^3$
centered at the origin with radius $r$,
$Q(r) =(-r^2,0)\times B(r)$, the cylinder in $ \mathbb{R}\times\mathbb{R}^3$ and
$B(x;r) =\mbox{ the ball in } \mathbb{R}^3$
centered at x with radius $r$.\\
To the end of this paper, we fix $\phi \in C^{\infty}(\mathbb{R}^3)$ satisfying:\\
\begin{equation*}\begin{split}
\int_{\mathbb{R}^3} &\phi(x)dx = 1,\quad
supp(\phi) \subset B(1),\quad
0 \leq \phi \leq 1\\
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\phi(x)&=1 \mbox{ for } |x|\leq \frac{1}{2} \quad\mbox{ and }\quad
\phi \mbox{ is radial. }
\end{split}\end{equation*}
For real number $r > 0 $ , we define functions $\phi_r \in C^{\infty}(\mathbb{R}^3)$
by $\phi_r(x) =\frac{1}{r^3}\phi(\frac{x}{r})$. Moreover, for $r=0$,
we define $\phi_r=\phi_0=\delta_0$ as the Dirac-delta function, which implies that
the convolution between $\phi_0$ and any function becomes
the function itself.
From the Young's inequality for convolutions, we can observe
\begin{equation}\label{young}
\|f*\phi_r\|_{L^p(B(a))}\leq\|f\|_{L^p(B(a+r))}
\end{equation} due to $supp(\phi_r)\subset B(r)$ for any $p\in[1,\infty]$,
for any $f\in L^p_{loc}$ and for any $a,r>0$.\\
\noindent \textbf{ $L^p$, weak-$L^p$ and Sobolev spaces $W^{n,p}$}\\
Let $K$ be a open subset $K$ of $\mathbb{R}^n$.
For $0<p<\infty $, we define $L^{p}(K)$ by the standard way
with (quasi) norm $\|f\|_{L^{p}(K)}
= (\int_K|f|^pdx)^{(1/p)}$.
From the Banach-–Alaoglu theorem,
any sequence which is bounded in $L^{p}(K)$ for $p\in(1,\infty)$
has a weak limit from some subsequence due to the weak-compactness.\\
Also, for $0<p<\infty $, the weak-$L^p(K)$ space
(or $L^{p,\infty}(K)$) is defined by \\
\begin{equation*}\begin{split}
L^{p,\infty}(K) = \{ f \mbox{ measurable in } K\subset\mathbb{R}^d
\quad: \sup_{\alpha>0}\Big(\alpha^p\cdot|\{|f|>\alpha\}\cap K|\Big)<\infty\}
\end{split}\end{equation*} with (quasi) norm $\|f\|_{L^{p,\infty}(K)}
= \sup_{\alpha>0}\Big(\alpha\cdot|\{|f|>\alpha\}\cap K|^{1/p}\Big)$.
From the Chebyshev's inequality, we have $\|f\|_{L^{p,\infty}(K)}\leq\|f\|_{L^{p}(K)}$ for any $0<p<\infty$.
Also, for $0<q<p<\infty$, $L^{p,\infty}(K)\subset L^{q}(K)$
once $K$ is bounded (refer the remark \ref{weak_L_p} in the beginning).\\
For any integer $n\geq0$ and for any $p\in[1,\infty]$, we denote $W^{n,p}(\mathbb{R}^3)$ and $W^{n,p}(B(r))$
as the standard Sobolev spaces for the whole space $\mathbb{R}^3$ and for any ball $B(r)$ in $\mathbb{R}^3$,
respectively.\\
\noindent \textbf{The Maximal function $\mathcal{M}$ and the Riesz transform $\mathcal{R}_j$}\\
The Maximal function $\mathcal{M}$ in $\mathbb{R}^d$ is
defined by the following standard way:
\begin{equation*}\begin{split}
\mathcal{M}(f)(x)=&\sup_{r>0}\frac{1}{|B(r)|}\int_{B(r)}|f(x+y)|dy.
\end{split}\end{equation*}
Also, we can express this Maximal operator as a
supremum of convolutions: $\mathcal{M}(f)=C\sup_{\delta>0}\Big(\chi_\delta *|f|\Big)$
where $\chi=\mathbf{1}_{\{|x|<1\}}$ is the characteristic function of the unit ball,
and $\chi_\delta(\cdot)=(1/\delta^3)\chi(\cdot/\delta)$.
One of properties of the Maximal function is that
$\mathcal{M}$ is bounded
from $L^{p}(\mathbb{R}^d)$ to $L^{p}(\mathbb{R}^d)$
for $p\in(1,\infty]$ and
from $L^{1}(\mathbb{R}^d)$ to $L^{1,\infty}(\mathbb{R}^d)$.
In this paper,
we denote $\mathcal{M}$ and $\mathcal{M}^{(t)}$ as the Maximal functions
in $\mathbb{R}^3$ and in
$\mathbb{R}^1$, respectively. \\
For $1\leq j\leq 3$, the Riesz Transform $\mathcal{R}_j$ in $\mathbb{R}^3$ is defined by:
\begin{equation*}\begin{split}
\widehat{\mathcal{R}_j(f)}(x)=\mathit{i}\frac{x_j}{|x|}\hat{f}(x)
\end{split}\end{equation*}
for any $f\in\mathcal{S}$ (the Schwartz space).
Moreover we can extend such definition for functions $L^{p}(\mathbb{R}^3)$
for $1<p<\infty$ and it is well-known that
$\mathcal{R}_j$ is bounded in $L^{p}$ for the same range of $p$.\\
\noindent \textbf{The Hardy space $\mathcal{H}$}\\
The Hardy space $\mathcal{H}$ in $\mathbb{R}^3$ is defined by
\begin{equation*}
\mathcal{H}(\mathbb{R}^3)=\{f\in L^1(\mathbb{R}^3)\quad: \quad\sup_{\delta>0}|
\mathcal{P}_\delta * f|\in L^1(\mathbb{R}^3) \}
\end{equation*} where $\mathcal{P}=
C
(1+|x|^2)^{-2}$ is the
Poisson kernel and $\mathcal{P}_\delta$ is defined
by $\mathcal{P}_\delta(\cdot)=\delta^{-3}\mathcal{P}(\cdot/\delta)$.
A norm of
$\mathcal{H}$ is defined by $L^1$ norm of $\sup_{\delta>0}|
\mathcal{P}_\delta * f|$. Thus $\mathcal{H}$
is a subspace of $L^1(\mathbb{R}^3)$ and
$\|f\|_{L^{1}(\mathbb{R}^3)}
\leq\|f\|_{\mathcal{H}(\mathbb{R}^3)}$ for any $f\in\mathcal{H}$.
Moreover,
the Riesz Transform is bounded from $\mathcal{H}$ to $\mathcal{H}$.\\
\noindent One of important applications of the Hardy space is the compensated compactness
(see Coifman, Lions, Meyer and Semmes \cite{clms}). Especially, it says that if $E,B\in L^2(\mathbb{R}^3)$
and $curlE=\ebdiv B=0$ in distribution, then
$E\cdot B\in\mathcal{H}(\mathbb{R}^3)$ and we have
\begin{equation*}
\|E\cdot B\|_{\mathcal{H}(\mathbb{R}^3)}\leq
C\cdot\|E\|_{L^2(\mathbb{R}^3)}\cdot\|B\|_{L^2(\mathbb{R}^3)}
\end{equation*} for some universal constant $C$.
In order to obtain some regularity of second derivative of pressure,
we can combine compensated compactness
with boundedness of the Riesz transform in $\mathcal{H}(\mathbb{R}^3)$.
For example, if $u$ is a weak solution
of the Navier-Stokes \eqref{navier},
then a corresponding pressure $P$ satisfies \begin{equation}\label{pressure_hardy}
\|\nabla^2 P\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}\leq
C\cdot\|\nabla u\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}^2
\end{equation} (see Lions \cite{lions} or the lemma 7 in \cite{vas:higher}).\\
\begin{comment}
Or if $u$ is a solution of \eqref{navier_Problem I-n} in (Problem I-n)
then \begin{equation}\begin{split}\label{pressure_hardy_Problem I-n}
\|\nabla^2 P\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}&\leq
C\|\Delta P\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla (u * \phi_{1/n})\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}
\|\nabla u\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla u\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}^2.\\
\end{split}\end{equation} from $-\Delta P=\ebdiv\ebdiv
\Big( (u * \phi_{1/n})\otimes u$\Big).\\
\end{comment}
\noindent Now it is well known that if we replace the
Poisson kernel $\mathcal{P}$ with any function
$\mathcal{G}\in C^\infty(\mathbb{R}^3)$ with compact support, then we have a constant
$C$ depending only on $\mathcal{G}$ such that
\begin{equation}\begin{split}\label{hardy_property}
\| \sup_{\delta>0}|\mathcal{G}_{\delta}* f|
\|_{L^{1}(\mathbb{R}^3)}
\leq C\| \sup_{\delta>0}|\mathcal{P}_{\delta}* f|
\|_{L^{1}(\mathbb{R}^3)}
= C\|f\|_{\mathcal{H}(\mathbb{R}^3)}
\end{split}\end{equation}
where $\mathcal{G}_\delta(\cdot)=\mathcal{G}(\cdot/\delta)/\delta^3$.
(see Fefferman and Stein \cite{fefferman} or
see Stein \cite{ste:harmonic}, Grafakos \cite{grafakos} for modern texts).
Due to the supremum and the convolution in \eqref{hardy_property},
we can say that
even though the Maximal function
$\sup_{\delta>0}\Big(\chi_\delta *|f|\Big)$ of any non-trivial Hardy space function $f$
is not integrable, there exist at least
integrable functions $\Big(\sup_{\delta>0}\Big|\mathcal{G}_{\delta}* f\Big|\Big)$, which can capture non-local data as
Maximal functions do.
However, note the position of the absolute value sign in \eqref{hardy_property},
which is outside of the convolution while
it is inside of the convolution
for the Maximal function.
It implies that
\eqref{hardy_property} is slightly weaker than the Maximal function
in the sense of controlling non-local data.
This weakness is the reason that we introduce
certain definitions of $\zeta$ and $h^{\alpha}$
in the following.\\
\noindent \textbf{Some notations which will be useful for fractional derivatives
$ {(-\Delta)^{{\alpha}/{2}}}$}\\
The following two definitions of $\zeta$ and $h^{\alpha}$ will be used
only in the proof for fractional derivatives.
We define $\zeta$ by $\zeta(x)=\phi(\frac{x}{2})-\phi(x)$.
Then we have
\begin{equation}\begin{split}\label{property_zeta}
& \zeta \in C^{\infty}(\mathbb{R}^3),\quad supp(\zeta)\subset B(2),
\quad\zeta(x)=0 \mbox{ for } |x|\leq \frac{1}{2} \\
&\mbox{ and }\sum_{j=k}^{\infty}\zeta(\frac{x}{2^j})=1 \mbox{ for }|x|\geq 2^{k}
\mbox{ for any integer } k.
\end{split}\end{equation}
\noindent In addition, we define function $h^{\alpha}$
for $\alpha>0$
by
$h^{\alpha}(x)=\zeta(x)/|x|^{3+\alpha}$.
Also we define $(h^{\alpha})_{\delta}$ and
$(\nabla^{d}{h^{\alpha})_\delta}$ by
$(h^{\alpha})_{\delta}(x)=\delta^{-3}h^{\alpha}(x/\delta)$ and
$(\nabla^{d}{h^{\alpha})_\delta}(x)
=\delta^{-3}(\nabla^{d}{h^{\alpha})}(x/\delta)$
for $\delta>0$ and for positive integer $d$, respectively.
Then they satisfy
\begin{equation}\begin{split}\label{property_h}
& (h^{\alpha})_{\delta} \in C^{\infty}(\mathbb{R}^3),
\quad supp((h^{\alpha})_{\delta})\subset B(2\delta)-
B(\delta/2), \\
&\mbox{ and }\frac{1}{|x|^{3+\alpha}}\cdot\zeta
(\frac{x}{2^j})=\frac{1}{(2^j)^{\alpha }} \cdot (h^{\alpha})_{2^j}(x)
\mbox{ for any integer } j.
\end{split}\end{equation}
\ \\
\noindent \textbf{The definition of the fractional Laplacian
$ {(-\Delta)^{{\alpha}/{2}}}$}\\
For $-3<\alpha\leq2$
and for $f\in \mathcal{S}(\mathbb{R}^3)$
(the Schwartz space), $ (-\Delta)^{\frac{\alpha}{2}}f$
is defined by the Fourier transform:
\begin{equation}\label{fractional_fourier}
\widehat{(-\Delta)^{\frac{\alpha}{2}}f}(\xi)=|\xi|^\alpha \hat{f}(\xi)
\end{equation}
Note that
$(-\Delta)^{0}=Id$.
Especially, for $\alpha\in(0,2)$, the fractional Laplacian can also be defined
by the singular integral for any $f\in\mathcal{S}$:
\begin{equation}\label{fractional_integral}
(-\Delta)^{\frac{\alpha}{2}}f(x)=C_{\alpha}\cdot P.V.\int_{\mathbb{R}^3}
\frac{f(x)-f(y)}{|x-y|^{3+\alpha}}dy.\\
\end{equation}
We introduce two notions of approximations to Navier-Stokes.
The first one or (Problem I-n) is the approximation Leray \cite{leray} used
while the second one or (Problem II-r) will be used in local study after we apply some certain scaling
to (Problem I-n).\\
\noindent \textbf{Definition of {(Problem I-n)}:
the first approximation
to Navier-Stokes}\\
\begin{defn}\label{Problem I-n} Let $n\geq1$ be either an integer or the infinity $\infty$, and let $0<T\leq \infty$.
Suppose that $u_0$ satisfy \eqref{initial_condition}.
We say that $(u,P)\in [C^{\infty}\big((0,T)\times\mathbb{R}^3\big)]^2$
is a solution of {(Problem I-n)} on $(0,T)$ for the data $u_0$
if it satisfies
\begin{equation}\label{navier_Problem I-n}
\begin{split}
\partial_tu+((u*\phi_{{\frac{1}{n}}})\cdot\nabla)u+ \nabla P -\Delta u&=0\\
\ebdiv u&=0 \quad t\in ( 0,T ),\quad x\in \mathbb{R}^3
\end{split}
\end{equation}
and
\begin{equation}\label{initial_condition_Problem I-n}
u(t)\rightarrow u_0*\phi_{\frac{1}{n}} \mbox{ in } L^2 \mbox{-sense as }
t \rightarrow 0.
\end{equation}
\end{defn}
\begin{rem}
When $n=\infty$, \eqref{navier_Problem I-n} is the Navier-Stokes on $(0,T)\times\mathbb{R}^3$ with initial value $u_0$.
\end{rem}
\begin{rem}\label{remark_leray}
If $n$ is not the infinity but an positive integer, then for any given $u_0$ of \eqref{initial_condition}, we have existence
and uniqueness theory of (Problem I-n) on $(0,\infty)$ with the energy equality
\begin{equation}\label{energy_eq_Problem I-n}
\|u(t)\|_{L^2(\mathbb{R}^3)}^2+
2\|\nabla u\|^2_{L^2(0,t;L^2(\mathbb{R}^3))}
= \|u_0*\phi_{\frac{1}{n}}\|_{L^2(\mathbb{R}^3)}^2.\\
\end{equation} for any $t<\infty$
and it is well-known that we can extract a sub-sequence which
converges to a suitable weak
solution $u$ of \eqref{navier} and \eqref{suitable} with the initial data
$u_0$ of \eqref{initial_condition}
by limiting procedure on a sequence of solutions of (Problem I-n)
(see Leray \cite{leray}, or see Lions \cite{lions}, Lemari{\'e}-Rieusset \cite{lemarie} for modern texts).
\end{rem}
\begin{rem}
As mentioned in the introduction section, we can observe that this notion (Problem I-n)
is not invariant under the standard Navier-Stokes scaling $u(t,x)\rightarrow
\epsilon u(\epsilon^2 t,\epsilon x)$
due to the advection velocity $(u*\phi_{1/n})$
unless $n$ is the infinity. \\
\end{rem}
\noindent \textbf{Definition of {(Problem II-r)}:
the second approximation
to Navier-Stokes}\\
\begin{defn}\label{problem II-r} Let $0\leq r<\infty$ be real.
We say that $(u,P)\in [C^{\infty}\big((-4,0)\times\mathbb{R}^3\big)]^2$ is a solution of {(Problem II-r)}
if it satisfies
\begin{equation}\label{navier_Problem II-r}
\begin{split}
\partial_tu+(w
\cdot\nabla)u+ \nabla P -\Delta u&=0\\
\ebdiv u&=0, \quad t\in (-4,0 ), \quad x\in \mathbb{R}^3
\end{split}
\end{equation}
where $w$ is the difference of two functions:
\begin{equation}\begin{split}\label{w_Problem II-n}
w(t,x)
&= w^\prime(t,x) - w^{\prime\prime}(t),
\quad t\in (-4,0 ), x\in \mathbb{R}^3 \\
\end{split}\end{equation} which are defined by u in the following way:
\begin{equation*}\begin{split}
w^\prime(t,x)&=(u*\phi_{r})(t,x) \quad\mbox{ and } \quad
w^{\prime\prime}(t)=\int_{\mathbb{R}^3}\phi(y)(u*\phi_{r})(t,y)dy.
\end{split}\end{equation*}
\end{defn}
\begin{rem}
This notion of {(Problem II-r)} gives us the mean zero property for
the advection velocity $w$: $\int_{\mathbb{R}^3}\phi(x)w(t,x)dx = 0 $ on $(-4,0)$. Also this $w$ is divergent free from the definition.
Moreover, by multiplying $u$ to \eqref{navier_Problem II-r},
we have
\begin{equation}\label{suitable_Problem II-r}
\partial_t \frac{|u|^2}{2} + \ebdiv (w \frac{|u|^2}{2}) + \ebdiv (u P) +
|\nabla u |^2 - \Delta\frac{|u|^2}{2}= 0\\
\end{equation} in classical sense
because our definition needs $u$ to be $C^{\infty}$.
\end{rem}
\begin{rem}
We will introduce some specially designed $\epsilon$-scaling
which is a bridge between (Problem I-n) and (Problem II-r)
(it can be found in \eqref{special designed scaling}).
This scaling
is based on the Galilean invariance in order to obtain
the mean zero property for the velocity $u$:
$\int_{\mathbb{R}^3}\phi(x)u(t,x)dx = 0 $ on $(-4,0)$.
Moreover, after this $\epsilon$-scaling is applied to solutions of (Problem I-n),
the resulting functions will
satisfy not conditions of (Problem II-$\frac{1}{n}$) but
those of (Problem II-$\frac{1}{n\epsilon}$) (it can be found \eqref{special designed scaling result}).
These things will be
stated precisely in the section \ref{proof_main_thm_II}.
\end{rem}
\begin{rem}
When $r=0$, the equation \eqref{navier_Problem II-r} is
the Navier-Stokes on $(-4,0)\times\mathbb{R}^3$
once
we assume the mean zero property for $u$.\\
\end{rem}
Now we present two main local-study propositions which require
the notion of (Problem II-r). These are kinds of partial regularity theorems
for solutions of (Problem II-r).
The main difficulty to prove these two propositions is that
$\bar{\eta}$ and $\bar\delta>0$ should be independent of any $r$ in $[0,\infty)$.
We will prove this independence very carefully, which is the heart of the section
\ref{proof_partial_prob_II_r} and
\ref{proof_local study}.\\
\noindent \textbf{The first local study proposition
for (Problem II-r)}\\
The following one is a quantitative version of partial regularity theorems
which extends that of \cite{vas:partial} up to $p=1$. The proof
will be based on the De Giorgi iteration with a new pressure decomposition
lemma \ref{lem_pressure_decomposition} which will appear later.
\begin{prop}\label{partial_problem_II_r}
There exists a $\bar\delta>0$ with the following property:\\
If u is a solution of (Problem II-r) for some $0\leq r<\infty$ verifying both
\begin{equation*}\begin{split}
&\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4})))}+
\|P\|_{L^1(-2,0;L^{1}(B(1)))}+\| \nabla u\|_{L^{2}(-2,0;L^{2}(B(\frac{5}{4})))}
\leq \bar{\delta}\\
\end{split}\end{equation*}
\begin{equation*}\begin{split}
\mbox{ and }\quad\quad&\| \mathcal{M}(|\nabla u|)
\|_{L^{2}(-2,0;L^{2}(B(2)))}\leq \bar{\delta},
\end{split}\end{equation*}
then we have
\begin{equation*}
|u(t,x)|\leq 1 \mbox{ on } [-\frac{3}{2},0]\times B(\frac{1}{2}).
\end{equation*}\\
\end{prop}
The above proposition, whose proof will appear in the section \ref{proof_partial_prob_II_r},
contains two bad scaling terms
$\| u\|_{L_t^{\infty}L_x^{2}}$ and $\|P\|_{L_t^1L_x^{1}}$,
while the following proposition \ref{local_study_thm} does not have those two.
Instead, the proposition \ref{local_study_thm} will assume
the mean-zero property on $u$ with the additional terms.
We will see later that these additional ones have
the best scaling like $|\nabla u|^2$ (also, see \eqref{best_scaling}).\\
\noindent \textbf{The second local study proposition
for (Problem II-r)}\\
\begin{prop}\label{local_study_thm}
There exists a $\bar{\eta}>0$ and there exist
constants $C_{d,\alpha}$ depending only on $d$ and $\alpha$
with the following property:\\
If $u$ is a solution of (Problem II-r) for some $0\leq r<\infty$ verifying both
\begin{equation}\label{local_study_condition1}
\int_{\mathbb{R}^3}\phi(x)u(t,x)dx = 0
\quad \mbox{ for } t\in(-4,0) \mbox{ and}\\
\end{equation}
\begin{equation}\begin{split}\label{local_study_condition2}
&\int_{-4}^{0}\int_{B(2)}\Big(|\nabla u|^2(t,x)+
|\nabla^2 P|(t,x)+|\mathcal{M}(|\nabla u|)|^2(t,x)\Big)dxdt\leq \bar{\eta},\\
\end{split}\end{equation}
then $|\nabla^d u|\leq C_{d,0}$ on
$Q(\frac{1}{3})=(-(\frac{1}{3})^2,0)\times B(\frac{1}{3})$ for every integer $d\geq 0$.\\
Moreover if we assume further
\begin{equation}\begin{split}\label{local_study_condition3}
&\int_{-4}^{0}\int_{B(2)}
\Big(
|\mathcal{M}(\mathcal{M}(|\nabla u|))|^2+
|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)|^{2/q}\\+
&|\mathcal{M}(|\nabla u|^q)|^{2/q}+
\sum_{m=d}^{d+4} \sup_{\delta>0}(|(\nabla^{m-1}{h^{\alpha})_\delta}
*\nabla^2 P|)
\Big)dxdt\leq \bar{\eta}
\end{split}\end{equation}
for some integer $d\geq 1$ and for some real $\alpha\in(0,2)$
where $q=12/(\alpha+6)$,
then $|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u|\leq C_{d,\alpha}$
on $Q(\frac{1}{6})$ for such $(d,\alpha)$.
\end{prop}
\begin{rem}
For the definitions of $h^{\alpha}$ and $(\nabla^{m-1}{h^{\alpha})_\delta}$, see around \eqref{property_h}.
\end{rem}
The proof will be given in the section \ref{proof_local study}
which will use the conclusion of the previous proposition \ref{partial_problem_II_r}.
Moreover we will use an induction argument and the integral representation
of the fractional Laplacian in order to get estimates
for the fractional case.
The Maximal function term of \eqref{local_study_condition2}
is introduced to estimate non-local part of $u$ while
the Maximal of Maximal function terms of \eqref{local_study_condition3}
is to estimate non-local part of $w$ which is already non-local.
On the other hand, because $\nabla^2 P$ has only $L^1$ integrability,
we can not have $L^1$ Maximal function of $\nabla^2 P$.
Instead, we use
some integrable functions, which is the last term of \eqref{local_study_condition3}.
This term plays the role which captures
non-local information of pressure (see \eqref{hardy_property}).
These will be stated
clearly in sections \ref{proof_local study} and \ref{proof_main_thm_II}.\\
\section{Proof of the first local study proposition \ref{partial_problem_II_r} }\label{proof_partial_prob_II_r}
\qquad This section is devoted to prove the proposition \ref{partial_problem_II_r} which is
a
partial regularity theorem for (Problem II-r).
Remember that we are looking for $\bar\delta$ which must
be independent of $r$. \\
In the first subsection \ref{definition_for_thoerem_partial_problem_II_r},
we present some lemmas about the advection velocity $w$ and
a new pressure
decomposition. After that, two big lemmas \ref{lem_partial_1}
and \ref{lem_partial_2}
in the subsections
\ref{proof_lem_partial_1} and \ref{proof_lem_partial_2}, which give us a control for big $r$ and small $r$ respectively, follow. Then
the actual proof of the proposition
\ref{partial_problem_II_r} will appear in the last subsection \ref{combine_de_giorgi}
where we can combine those two big lemmas.\\
\subsection{A control on the
advection velocity $w$ and a new pressure decomposition}\label{definition_for_thoerem_partial_problem_II_r}
\quad The following lemma
says that
convolution of any functions with $\phi_r$
can be controlled by just one point value of
the Maximal function with some factor of $1/r$.
Of course, it is useful when $r$ is away from $0$.
\begin{lem}\label{convolution_lem}
Let f be an integrable function in $\mathbb{R}^3$.
Then for any integer $d\geq0$,
there exists $C=C(d)$ such that
\begin{equation*}
\|\nabla^d (f*\phi_r)\|_{L^{\infty}(B(2))}\leq \frac{C}{r^d}
\cdot(1+\frac{4}{r})^3\cdot\inf_{x\in B(2)}\mathcal{M}f(x)
\end{equation*} for any $0<r<\infty$.
\end{lem}
\begin{proof}
Let $z,x\in B(2)$. Then
\begin{equation*}\begin{split}
& |\nabla^d (f*\phi_r)(z)| =|(f*\nabla^d \phi_r)(z)|
=|\int_{B(z,r)}f(y)\nabla^d\phi_r(z-y)dy|\\
&\leq\|\nabla^d \phi_r\|_{L^{\infty}}\int_{B(z,r)}|f(y)|dy
=\frac{\|\nabla^d \phi\|_{L^{\infty}}}{r^{d+3}}\int_{B(z,r)}|f(y)|dy \\
&\leq\frac{\|\nabla^d \phi\|_{L^{\infty}}}{r^{d+3}}\frac{(r+4)^3}{(r+4)^3}
\int_{B(x,r+4)}|f(y)|dy
\leq\frac{C}{r^d}
\cdot(1+\frac{4}{r})^3\cdot\mathcal{M}f(x).
\end{split}\end{equation*}
We used $B(z,r)\subset B(x,r+4)$. Then we take $\sup$ in $z$ and $\inf$ in $x$.
Recall that $\phi(\cdot)$ is the fixed function in this paper.\\
\end{proof}
The following corollary is just an application of the previous lemma
to solutions of (Problem II-r).
\begin{cor}\label{convolution_cor}
Let $u$ be a solution of (Problem II-r) for $0<r<\infty$. Then
for any integer $d\geq0$,
there exists $C=C(d)$ such that
\begin{equation*}
\| w\|_{L^2(-4,0;L^{\infty}(B(2)))}\leq {C}
\cdot(1+\frac{4}{r})^3\cdot\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}
\end{equation*} and
\begin{equation*}
\|\nabla^d w\|_{L^2(-4,0;L^{\infty}(B(2)))}\leq \frac{C}{r^{d-1}}
\cdot(1+\frac{4}{r})^3\cdot\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}
\end{equation*} if $d\geq1$.
\end{cor}
\begin{proof}
Recall $\int_{\mathbb{R}^3}w(t,y)\phi(y)dy=0$ and $supp(\phi)\subset B(1)$. Thus for $z\in B(2)$
\begin{equation*}\begin{split}
|w(t,z)|=&\Big|\int_{\mathbb{R}^3}w(t,z)\phi(y)dy-\int_{\mathbb{R}^3}w(t,y)
\phi(y)dy\Big|\\
\leq&\|\nabla w(t,\cdot)\|_{L^{\infty}(B(2))}\int_{\mathbb{R}^3}|z-y|\phi(y)dy\\
\leq&C\|(\nabla u) *\phi_r(t,\cdot)\|_{L^{\infty}(B(2))}\cdot\int_{\mathbb{R}^3}\phi(y)dy\\
\leq&{C}\cdot(1+\frac{4}{r})^3\cdot \inf_{x\in B(2)}\mathcal{M}(|\nabla u|)(t,x).
\end{split}\end{equation*}
For last inequality, we used the lemma \ref{convolution_lem} to $\nabla u$.
For $d\geq1$, use $\nabla^d w =\nabla^{d-1}\Big[(\nabla u)*\phi_r\Big]$.
\end{proof}
To use De Giorgi type argument, we require more notations
which will be used only in this section.
\begin{equation}\begin{split}\label{def_s_k}
\mbox{For real }&k\geq 0, \mbox{ define }\\
B_k &=\mbox{ the ball in } \mathbb{R}^3 \mbox{ centered at the origin with radius }
\frac{1}{2}(1+\frac{1}{2^{3k}}),\\
T_k &= -\frac{1}{2}(3+\frac{1}{2^k}),\\
Q_k &=[T_k,0]\times B_k \quad\mbox{ and}\\
s_k &= \mbox{ distance between } B^c_{k-1} \mbox{ and } B_{k-\frac{5}{6}}\\
&=2^{-3k}\Big((\sqrt{2}-1)2\sqrt{2}\Big).
\end{split}\end{equation} Also we define $s_\infty=0$.
Note that $0<s_1 <\frac{1}{4}$ and the sequence $\{s_k\}_{k=1}^{\infty}$ is strictly decreasing to zero
as k goes to $\infty$.\\
\noindent For each integer $k\geq 0 $, we define and fix a function $\psi_k \in C^{\infty}(\mathbb{R}^3)$ satisfying:\\
\begin{equation}\begin{split}
&\psi_k = 1 \quad\mbox{ in } B_{k-\frac{2}{3}} , \quad\psi_k = 0 \quad\mbox{ in } B_{k-\frac{5}{6}}^C\\
&0\leq \psi_k(x) \leq 1 , \quad
|\nabla\psi_k(x)|\leq C2^{3k} \mbox{ and }
|{\nabla}^2\psi_k(x)|\leq C2^{6k} \mbox{ for }x\in\mathbb{R}^3.
\end{split}\end{equation} This $\psi_k$ plays role of a cut-off function for $B_k$.\\
To prove the proposition \ref{partial_problem_II_r}, We need the following
important lemma about pressure decomposition. Here
we decompose our pressure term into three parts:
a non-local part which depends on $k$,
a local part which depends on $k$ and
a non-local part which does not depend on $k$
and will be absorbed into the velocity component later.
\begin{lem}\label{lem_pressure_decomposition}
There exists a constant ${\Lambda_1}>0$ with
the following property:\\
Suppose $A_{ij}\in L^1(B_0) $ $ 1\leq i,j\leq 3 $
and $ P\in L^1(B_0)$
with $-\Delta P = \sum_{ij}\partial_i \partial_j A_{ij}$ in $B_0$.
Then, there exist a function $P_3 $ with $P_3|_{B_{\frac{2}{3}}} \in L^{\infty} $
such that, for any $k\geq1$, we can decompose $P$ by\\
\begin{equation}\label{pressure_decomposition_expressition}
P = P_{1,k} + P_{2,k} + P_{3}\quad\mbox{ in } B_{\frac{1}{3}},
\end{equation}
and they satisfy
\begin{equation}\label{lem_pressure_decomposition_p1k} \|\nabla P_{1,k}\|_{L^{\infty}(B_{k-\frac{1}{3}}))}
+ \|P_{1,k}\|_{L^{\infty}(B_{k-\frac{1}{3}}))}
\leq {\Lambda_1}
2^{12k} \sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})},
\end{equation}
\begin{equation}\label{lem_pressure_decomposition_p2k} -\Delta P_{2,k} =
\sum_{ij}\partial_i \partial_j (\psi_k A_{ij}) \quad\quad \mbox{in } \mathbb{R}^3 \quad\mbox{ and}
\end{equation}
\begin{equation}\label{lem_pressure_decomposition_p3} \|\nabla P_{3}\|_{L^{\infty}(B_{\frac{2}{3}})} \leq {\Lambda_1}
(\|P\|_{L^1(B_{\frac{1}{6}})} + \sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})}).
\end{equation}
Note that ${\Lambda_1}$ is a totally independent constant. \\
\end{lem}
\begin{proof}
The product rule and the hypothesis imply
\begin{equation*}\begin{split}
-\Delta(\psi_1 P) &= -\psi_1 \Delta P - 2\ebdiv((\nabla \psi_1)P) + P\Delta\psi_1\\
&= \psi_1\sum_{ij}\partial_i \partial_j A_{ij} - 2\ebdiv((\nabla \psi_1)P) + P\Delta\psi_1\\
&= - \Delta P_{1,k} - \Delta P_{2,k} - \Delta P_3
\end{split}
\end{equation*}
where $P_{1,k}$, $ P_{2,k} $ and $ P_3 $ are defined by
\begin{equation*}\begin{split}
- \Delta P_{1,k} &= \sum_{ij}\partial_i \partial_j ((\psi_1 -\psi_k) A_{ij}) \\
- \Delta P_{2,k} &= \sum_{ij}\partial_i \partial_j (\psi_k A_{ij}) \\
- \Delta P_3\mbox{ } &= - \sum_{ij}\partial_j[(\partial_i \psi_1)(A_{ij})]
- \sum_{ij}\partial_i[(\partial_j \psi_1)(A_{ij})] \\&+ \sum_{ij}(\partial_i \partial_j \psi_1)(A_{ij})
- 2\ebdiv((\nabla \psi_1)P) + P\Delta\psi_1.
\end{split}\end{equation*}
$P_{1,k}$ and $P_3$ are defined by the representation formula
${(-\Delta)}^{-1}(f) = \frac{1}{4\pi}(\frac{1}{|x|} * f)$\\
while $P_{2,k}$ by the Riesz transforms.\\
\\
Since $\psi_1 = 1 $ on $ B_{\frac{1}{3}}$, we have $\Delta P = \Delta(\psi_1 P)$ on $ B_{\frac{1}{3}}$.
Thus \eqref{pressure_decomposition_expressition} holds.\\
\\
By definition of $P_{2,k}$, \eqref{lem_pressure_decomposition_p2k} holds.\\
\\
For \eqref{lem_pressure_decomposition_p1k} and \eqref{lem_pressure_decomposition_p3},
it follows the proof of the lemma 3 of \cite{vas:partial} directly.
For completeness, we present a proof here. Note that $ (\psi_1 -\psi_k) $
is supported in $ (B_{\frac{1}{6}} - B_{k-\frac{2}{3}} )$ and
$ \nabla\psi_1 $ is supported in $ (B_{\frac{1}{6}} - B_{\frac{1}{3}} )$. Thus
for $x\in B_{k-\frac{1}{3}}$,
\begin{equation*}\begin{split}
|P_{1,k}(x)| &= \Bigg|\frac{1}{4\pi}\int_{(B_{\frac{1}{6}} - B_{k-\frac{2}{3}} )}
\frac{1}{|x-y|}\sum_{ij}(\partial_i \partial_j ((\psi_1 -\psi_k) A_{ij}))(y)dy\Bigg|\\
&\leq \sup_{y\in B^C_{k-\frac{2}{3}}}(|\nabla^2_y\frac{1}{|x-y|}|) \cdot \sum_{ij}
\int_{B_{\frac{1}{6}}}|A_{ij}(x)|dy\\
&\leq C\cdot\sup_{y\in B^C_{k-\frac{2}{3}}}(\frac{1}{|x-y|^3}) \cdot
\sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})}
\leq C_1\cdot2^{9k} \cdot
\sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})}.
\end{split}\end{equation*}
We used integration by parts and facts $|x-y| \geq 2^{-3k}$ and $ |(\psi_1 -\psi_k)|\leq 1 $ . \\
\\ In the same way, for $x\in B_{k-\frac{1}{3}}$,
\begin{equation*}\begin{split}
|\nabla P_{1,k}(x)|
&\leq C_2\cdot2^{12k} \cdot
\sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})}.\\
\end{split}\end{equation*}
For $x\in B_{\frac{2}{3}}$,
\begin{equation*}\begin{split}
|\nabla P_{3}(x)|
= & \Bigg|\frac{1}{4\pi}\int_{(B_{\frac{1}{6}} - B_{\frac{1}{3}} )}
(\nabla_y\frac{1}{|x-y|})\Big[-\sum_{ij}\partial_j[(\partial_i \psi_1)(A_{ij})]
- \sum_{ij}\partial_i[(\partial_j \psi_1)(A_{ij})] \\
&\quad + \sum_{ij}(\partial_i \partial_j \psi_1)(A_{ij}))
- 2\ebdiv((\nabla \psi_1)P) + P\Delta\psi_1 \Big]dy\Bigg|\\
\leq& C_3\Big(\sum_{ij}\|A_{ij}\|_{L^1(B_{\frac{1}{6}})}
+ \|P\|_{L^1(B_{\frac{1}{6}})}\Big).
\end{split}\end{equation*}
These prove \eqref{lem_pressure_decomposition_p1k} and \eqref{lem_pressure_decomposition_p3} and we keep the
constant ${\Lambda_1} = max(C_1,C_2,C_3)$ for future use.
\end{proof}
Before presenting De Giori arguments for
large $r$ and small $r$, we need more notations.
In the following two main lemmas \ref{lem_partial_1}
and \ref{lem_partial_2}, $P_3$
will be constructed from solutions $(u,P)$ for (Problem II-r) by
using the previous lemma \ref{lem_pressure_decomposition} and
it will be clearly shown
that $\nabla P_3$ has $L_t^1L_x^{\infty}$ bound. Thus we can define
\begin{equation}\begin{split}\label{def_e_k}
E_k(t) = &\frac{1}{2}(1-2^{-k}) + \int_{-1}^{t}\|\nabla P_3(s,\cdot)
\|_{L^{\infty}(B_{\frac{2}{3}})}ds, \\
& \mbox{ for } t \in[-2,0] \mbox{ and for } k\geq 0.
\end{split}\end{equation}
Note that $E_k$ depends on $t$.
We also define followings like in \cite{vas:partial}
\begin{equation*}\begin{split}
v_k &= (|u|-E_k)_{+},\\
d_k & = \sqrt{\frac{E_k \mathbf{1}_{\{|u|\geq E_k\}}}{|u|}|\nabla|u||^2 +
\frac{v_k}{|u|}|\nabla u|^2} \quad\mbox{ and}\\
U_k &= \sup_{t\in[T_k,0]}\Big(\int_{B_k}|v_k|^2 dx \Big) + \int\int_{Q_k}|d_k|^2 dx dt\\
&= \|v_k\|_{L^{\infty}(T_k,0;L^2(B_k))}^2 + \|d_k\|^2_{L^2(Q_k)}.
\end{split}\end{equation*}
In this way, $P_3$ will be absorbed into $v_k$, which is the key
idea of proof of
this proposition \ref{partial_problem_II_r}. \\
\subsection{De Giorgi argument to get a control for large $r$}\label{proof_lem_partial_1}
The following big lemma says that
we can obtain a certain uniform non-linear estimate
in the form of $W_k\leq C^k\cdot W_{k-1}^\beta $ when $r$ is large.
Then an elementary lemma can give us the conclusion (we will see
the lemma \ref{lem_recursive} later).
On the other hand,
for small $r$, we have the factor of $(1/r^3)$ which blows up
as $r$ goes to zero. This weak point implies that
we still need some extra work after this lemma.
(it will be the next big lemma \ref{lem_partial_2}).
\begin{lem}\label{lem_partial_1}
There exist universal constants ${\delta}_1>0$ and $\bar{C}_1>1$ such that
if u is a solution of (Problem II-r) for some $0<r<\infty$ verifying both
\begin{equation*}\begin{split}
&\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4})))}+
\|P\|_{L^1(-2,0;L^{1}(B(1)))}+\| \nabla u\|_{L^{2}(-2,0;L^{2}(B(\frac{5}{4})))}
\leq {\delta}_1\\ \mbox{ and }
&\| \mathcal{M}(|\nabla u|)\|_{L^{2}(-2,0;L^{2}(B(2)))}\leq {\delta}_1,\\
\end{split}\end{equation*}
then we have
\begin{equation*}
U_k \leq \begin{cases}& (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}} ,\quad \mbox{ for any } k\geq 1 \quad\mbox{ if } r\geq s_{1}\\
&\frac{1}{r^3}\cdot (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}} ,
\quad \mbox{ for any } k\geq 1 \quad\mbox{ if } r< s_{1}. \end{cases}
\end{equation*}
\end{lem}
\begin{rem}
$s_1$ is a pre-fixed constant defined in
\eqref{def_s_k} such that $0<s_1<1/4$, and
$({\delta}_1,\bar{C}_1)$ is independent of any
$ 0<r<\infty$. It will be clear that
the exponent $7/6$ is not optimal and we can make it close to $(4/3)$
arbitrarily. However, any exponent bigger than $1$ is enough for our study.
\end{rem}
\begin{proof}
We assume ${\delta_{1}} <1$. First we claim that there exists
a universal constant ${\Lambda_2}\geq 1$
such that
\begin{equation}\label{w_iu_j}
\||w|\cdot|u|\|_{L^{2}(-2,0;L^{3/2}(B_{\frac{1}{6}}))}\leq {\Lambda_2}\cdot{\delta_{1}} \quad\mbox{for any } 0<r<\infty.
\end{equation}
In order to prove the above claim \eqref{w_iu_j}, we need to separate it into
a large $r$ case and a small $r$ case:\\
\textbf{(I)-large $r$ case.} From the corollary \ref{convolution_cor} if $r\geq s_{1}$, then
\begin{equation}\begin{split}\label{w_large_r}
\| w\|_{L^2(-4,0;L^{\infty}(B(2)))}
&\leq {C}
\cdot(1+\frac{4}{s_{1}})^3\cdot\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}\\
&\leq {C}\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}\leq {C}{\delta_{1}}.
\end{split}\end{equation} Likewise,
\begin{equation}\begin{split}\label{nabla_w_large_r}
\|\nabla w\|_{L^2(-4,0;L^{\infty}(B(2)))}
&\leq {C}{\delta_{1}}.
\end{split}\end{equation}
With Holder's inequality and $B_{\frac{1}{6}}\subset B_{0}=B(1)\subset B(\frac{5}{4})\subset B(2)$,
\begin{equation*}\begin{split}
\||w|\cdot|u|\|_{L^{2}(-2,0;L^{3/2}(B_{\frac{1}{6}}))}&\leq C
\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4}))}\cdot\| w\|_{L^2(-4,0;L^{\infty}(B(2)))}\\
&\leq C \cdot{\delta_{1}}^2\leq C_1 \cdot{\delta_{1}}.
\end{split}\end{equation*} so we obtained \eqref{w_iu_j} for $r\geq s_{1}$.\\
\textbf{(II)-small $r$ case.} On the other hand, if $r< s_{1}$, then
\begin{equation}\begin{split}\label{w_small_r_with_r^3}
\| w\|_{L^2(-4,0;L^{\infty}(B(2)))}&\leq {C}
\cdot(1+\frac{4}{r})^3\cdot\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}\\
&\leq {C}\frac{1}{r^3}\|\mathcal{M}(|\nabla u|)\|_{L^2{(Q(2))}}\leq {C}\frac{1}{r^3}{\delta_{1}}
\end{split}\end{equation} and
\begin{equation}\begin{split}\label{nabla_w_small_r_with_r^3}
\|\nabla w\|_{L^2(-4,0;L^{\infty}(B(2)))}
&\leq {C}\frac{1}{r^3}{\delta_{1}}.
\end{split}\end{equation}
However, it is not enough to prove \eqref{w_iu_j} because $\frac{1}{r^3}$ factor
blows up as $r$ goes to zero. So, instead, we use the idea that $w$ and $u$ are similar if $r$ is small:
\begin{equation*}\begin{split}
\| u\|_{L^{4}(-2,0;L^{3}(B_0))}
&\leq
C\Big(\| u\|_{L^{\infty}(-2,0;L^{2}(B_0))}+
\| \nabla u\|_{L^{2}(-2,0;L^{2}(B_0))}\Big)
\leq C{\delta_{1}}
\end{split}\end{equation*}
and
\begin{equation*}\begin{split}
\| w^{\prime}\|_{L^{4}(-2,0;L^{3}(B_{\frac{1}{6}}))}
&= \| u*\phi_r\|_{L^{4}(-2,0;L^{3}(B_{\frac{1}{6}}))}
\leq\| u\|_{L^{4}(-2,0;L^{3}(B_0))} \leq C{\delta_{1}} \\
\end{split}\end{equation*}
because $u*\phi_r$ in $B_{\frac{1}{6}}$ depends only on $u$ in $B_0$.
(recall that $r\leq s_1$ and $s_1$ is the distance $B^c_{0}$ and $B_{\frac{1}{6}}$
and refer \eqref{young}).
For $w^{\prime\prime}$,
\begin{equation}\begin{split}\label{w_prime_prime_small_r}
\| w^{\prime\prime}\|_{L^\infty(-2,0;L^{\infty}(B(2)))}
&=\| w^{\prime\prime}\|_{L_t^\infty((-2,0))}\\
&=\|\int_{\mathbb{R}^3}\phi(y)(u*\phi_{r})(y)dy\|_{L_t^\infty((-2,0))}\\
&\leq C\|\|u*\phi_{r}\|_{L_x^1(B(1))}\|_{L_t^\infty((-2,0))}\\
&\leq C\|\|u\|_{L_x^1(B(\frac{5}{4}))}\|_{L_t^\infty((-2,0))}\\
&\leq {C}\| u\|_{L^\infty(-2,0;L^{2}(B(\frac{5}{4})))}\\
&\leq {C}{\delta_{1}}\\
\end{split}\end{equation} because $ w^{\prime\prime}$ is a constant in $x$
, $supp(\phi)\subset B(1)$ and $u*\phi_r$ in $B(1)$ depends only on $u$
in $B(1+s_{1})$
which is a subset of $B(\frac{5}{4})$.
As a result, we have
\begin{equation}\begin{split}\label{w_u}
\||w|\cdot|u|\|_{L^{2}(-2,0;L^{3/2}(B_{\frac{1}{6}}))}&\leq C
\| u\|_{L^{4}(-2,0;L^{3}(B(1))}
\cdot\| w\|_{L^4(-2,0;L^{3}(B(\frac{1}{6})))}\\
&\leq C{\delta_{1}}\cdot
\| |w^{\prime}|+|w^{\prime\prime}|\|_{L^4(-2,0;L^{3}(B(\frac{1}{6})))}\\
&\leq C \cdot{\delta_{1}}^2 \leq C_2 \cdot{\delta_{1}}
\end{split}\end{equation} so that we obtained \eqref{w_iu_j} for $r\leq s_{1}$.\\
\noindent
Hence, taking
\begin{equation}\label{def_breve_C}
{\Lambda_2}=\max(C_1,C_2,1),
\end{equation} we have
\eqref{w_iu_j} and ${\Lambda_2}$ is independent of $0<r<\infty$
as long as $\delta_1<1$.
From now on, we assume $\delta_1<1$ sufficiently
small to be $10 \cdot {\Lambda_1}\cdot{\Lambda_2}\cdot\delta_1\leq 1/2$
(Recall that ${\Lambda_1}$ comes from the lemma \ref{lem_pressure_decomposition}). \\
Thanks to the lemma \ref{lem_pressure_decomposition} and \eqref{w_iu_j}, by putting
$A_{ij}=w_iu_j$ we can decompose $P$ by\\
\begin{equation*}
P = P_{1,k} + P_{2,k} + P_{3}\quad\mbox{ in } [-2,0]\times B_{\frac{1}{3}}
\end{equation*} for each $k\geq1$
with following properties:\\
\begin{equation}\begin{split}\label{eq_pressure_decomposition_p1k}
\| |\nabla P_{1,k}|
+ |P_{1,k}|\|_{L^{2}(-2,0;L^{\infty}(B_{k-\frac{1}{3}}))}
&\leq {\Lambda_1} 2^{12k}\sum_{ij}\|w_iu_j\|_{L^{2}(-2,0;L^1(B_{\frac{1}{6}}))}\\
&\leq 9\cdot{\Lambda_1}\cdot{\Lambda_2} \cdot {\delta_{1}}\cdot2^{12k}
\leq 2^{12k} \quad\mbox{ for any } k\geq1,\\
\end{split}\end{equation}
\begin{equation}\label{eq_pressure_decomposition_p2k}
-\Delta P_{2,k} = \sum_{ij} \partial_i \partial_j (\psi_k w_i u_j)
\quad\quad \mbox{in } [-2,0]\times\mathbb{R}^3 \quad\mbox{ for any } k\geq1
\quad \mbox{ and} \\
\end{equation}
\begin{equation}\begin{split}\label{eq_pressure_decomposition_p3}
\|\nabla P_{3}\|_{L^{1}(-2,0;L^{\infty}(B_{\frac{2}{3}}))} &\leq {\Lambda_1}
\Big(\|P\|_{L^{1}(-2,0;L^1(B(1))} + \sum_{ij}\|w_iu_j\|_{L^{2}(-2,0;L^1(B(1))}\Big)\\
&\leq {\Lambda_1}({\delta_{1}}+ 9\cdot{\Lambda_2}\cdot{\delta_{1}})
\leq 10\cdot{\Lambda_1}\cdot {\Lambda_2}\cdot{\delta_{1}}\leq \frac{1}{2}.
\end{split}\end{equation}
\noindent Note that the above \eqref{eq_pressure_decomposition_p3}
enables $E_k$ to be well-defined and it
satisfies $0\leq E_k \leq 1$ (see the definition of $E_k$ in \eqref{def_e_k}).\\
In the following remarks \ref{lem10_39}--\ref{rem_lem10_39}, we gather some easy results, which were obtained in \cite{vas:partial}, without proof. They can be found in
the lemmas
4, 6 and the remark of the lemma 4 of \cite{vas:partial}. Note that any constants $C$ in the following remarks
do not depend on $k$.\\% as long as $k\geq 0.\\
\begin{rem}\label{lem10_39}
For any $k\geq 0$, the function $u$ can be decomposed by
$u=u\frac{v_k}{|u|} + u(1-\frac{v_k}{|u|})$.
Also we have
\begin{equation}\begin{split}\label{d_k}
&\Big|u(1-\frac{v_k}{|u|})\Big|\leq 1,
\quad\frac{v_k}{|u|}|\nabla u|\leq d_k, \quad
\mathbf{1}_{|u|\geq E_k}|\nabla|u||\leq d_k,\\
&|\nabla v_k|\leq d_k \quad\mbox{ and }\quad
\big|\nabla\frac{uv_k}{|u|}\big|\leq 3d_k.
\end{split}\end{equation}
\end{rem}
\begin{rem}\label{lem12_39}
For any $k\geq1$ and
for any $q\geq1$,
\begin{equation*}\begin{split}
\|\mathbf{1}_{v_k>0}\|_{L^q(Q_{k-1})}&\leq C 2^{\frac{10k}{3q}}U^{\frac{5}{3q}}_{k-1} \quad\mbox{ and }\quad
\|\mathbf{1}_{v_k>0}\|_{L^{\infty}(T_{k-1},0;L^q(Q_{k-1})}
\leq C 2^{\frac{2k}{q}}U^{\frac{1}{q}}_{k-1}.\\
\end{split}\end{equation*}
\end{rem}
\begin{rem}\label{rem_lem10_39}
For any $k\geq1$,
$\|v_{k-1}\|_{L^{\frac{10}{3}}(Q_{k-1})}\leq C U_{k-1}^{\frac{1}{2}}.$
\end{rem}
\ \\
From the above remarks \ref{lem10_39}--\ref{rem_lem10_39}, we have for any $1\leq p\leq\frac{10}{3}$,
\begin{equation}\begin{split}\label{raise_of_power}
\|v_k\|_{L^{p}(Q_{k-1})}&=\|v_k\mathbf{1}_{v_k>0}\|_{L^{p}(Q_{k-1})}\\
&\leq \|v_k\|_{L^{\frac{10}{3}}(Q_{k-1})}\cdot\|\mathbf{1}_{v_k>0}\|_{L^{1/(\frac{1}{p}-\frac{3}{10})}(Q_{k-1})}\\
&\leq \|v_{k-1}\|_{L^{\frac{10}{3}}(Q_{k-1})}\cdot C 2^{\frac{10k}{3}\cdot(\frac{1}{p}-\frac{3}{10})}U^{\frac{5}{3}\cdot(\frac{1}{p}-\frac{3}{10})}_{k-1}\\
&\leq C2^{\frac{7k}{3}}U^{\frac{5}{3p}}_{k-1}.\\
\end{split}\end{equation}
Likewise, for any $1\leq p\leq2$,
\begin{equation}\begin{split}\label{raise_of_power2}
\|v_k\|_{L^{\infty}(T_{k-1},0;L^{p}(B_{k-1}))}
&\leq C2^{k}U^{\frac{1}{p}}_{k-1}
\end{split}\end{equation} and
\begin{equation}\begin{split}\label{raise_of_power3}
\|d_k\|_{L^{p}(Q_{k-1})}&
\leq C2^{\frac{5k}{3}}U^{\frac{5}{3p}-\frac{1}{3}}_{k-1}.\\
\end{split}\end{equation}\\
Second, we claim that
for every $k\geq 1$, functions $v_k$ verifies:
\begin{equation}\label{eq_suitable_inequality_for_v_k_prob_II_r}\begin{split}
\partial_t \frac{v_k^2}{2} + &\ebdiv (w \frac{v_k^2}{2}) + d_k^2 - \Delta\frac{v_k^2}{2} \\
&+\ebdiv (u (P_{1,k} + P_{2,k})) + (\frac{v_k}{|u|}-1)u\cdot\nabla (P_{1,k} + P_{2,k}) \leq 0
\end{split}\end{equation}
in $(-2,0)\times B_{\frac{2}{3}}$.\\
\begin{rem}
Note that the above inequality \eqref{eq_suitable_inequality_for_v_k_prob_II_r} does not contain the $P_3$ term. We will see that this fact comes from
the definition of $E_k(t)$ in \eqref{def_e_k}.
\end{rem}
Indeed, observe that
$ \frac{v_k^2}{2} = \frac{|u|^2}{2} + \frac{v_k^2 -|u|^2}{2}$
and note that $E_k$ does not depend on space variable but on time variable.
Thus we can compute,
for time derivatives,
\begin{equation*}\begin{split}
\partial_t&(\frac{v^2_k - |u|^2}{2}) = v_k\partial_{t}v_k - u\partial_{t}u
= v_k\partial_{t}|u| - v_k\partial_t E_k - u\partial_{t}u \\
&=u (\frac{v_k}{|u|} -1)\partial_t u - v_k\partial_t E_k
= u (\frac{v_k}{|u|} -1)\partial_t u - v_k
\|\nabla P_3(t,\cdot)\|_{L^{\infty}(B_{\frac{2}{3}})}
\end{split}\end{equation*} while,
for any space derivatives $\partial_{\alpha}$,
\begin{equation*}\begin{split}
\partial_{\alpha}(\frac{v^2_k - |u|^2}{2})
& =u (\frac{v_k}{|u|} -1)\partial_{\alpha} u.
\end{split}\end{equation*}
Then we follow the same way as the lemma 5 of \cite{vas:partial} did:
First, we multiply \eqref{navier_Problem II-r} by $u (\frac{v_k}{|u|} -1)$,
and then we sum the result and \eqref{suitable_Problem II-r}. We omit the details which can be found in the proof of the lemma 5 of \cite{vas:partial}.
As a result,
we have
\begin{equation*}\begin{split}
0 \geq & \quad\partial_t \frac{v_k^2}{2} + \ebdiv (w \frac{v_k^2}{2}) + d_k^2
- \Delta\frac{v_k^2}{2} +v_k\|\nabla P_3(t,\cdot)\|_{L^{\infty}(B_{\frac{2}{3}})}\\
&+\ebdiv (u P) + (\frac{v_k}{|u|}-1)u\cdot\nabla P \\
=&\quad\partial_t \frac{v_k^2}{2} + \ebdiv (w \frac{v_k^2}{2}) + d_k^2 - \Delta\frac{v_k^2}{2}
+\Big(v_k\|\nabla P_3(t,\cdot)\|_{L^{\infty}(B_{\frac{2}{3}})} + \frac{v_k}{|u|}u\cdot \nabla P_3\Big)\\
&+\ebdiv (u (P_{1,k}+P_{2,k})) + (\frac{v_k}{|u|}-1)u\cdot\nabla (P_{1,k}+P_{2,k}).
\end{split}\end{equation*}
For the last equality, we used the fact $P = P_{1,k} + P_{2,k} + P_{3} $ in $ B_{\frac{1}{3}} $ and
\begin{equation}\label{easyp_3}
\ebdiv (u P_3) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_3= \frac{v_k}{|u|}u\cdot \nabla P_3.
\end{equation}
Thus we proved the claim \eqref{eq_suitable_inequality_for_v_k_prob_II_r}
due to
\begin{equation*}
v_k\|\nabla P_3(t,\cdot)\|_{L^{\infty}(B_{\frac{2}{3}})} + \frac{v_k}{|u|}u\cdot \nabla P_3 \geq 0
\quad \mbox{ on } (-2,0)\times B_{\frac{2}{3}}.
\end{equation*}
For any integer $k$, we introduce a cut-off function $\eta_k(x)\in C^{\infty}(\mathbb{R}^3)$ satisfying
\begin{equation*}\begin{split}
&\eta_k = 1 \quad\mbox{ in } B_{k} \quad , \quad
\eta_k = 0 \quad\mbox{ in } B_{k-\frac{1}{3}}^C\quad , \quad
0\leq \eta_k \leq 1, \\
&|\nabla\eta_k|\leq C2^{3k} \quad \mbox{and} \quad
|{\nabla}^2\eta_k|\leq C2^{6k},
\quad \mbox{ for }\mbox{any }x\in\mathbb{R}^3.
\end{split}\end{equation*}
Multiplying \eqref{eq_suitable_inequality_for_v_k_prob_II_r} by $\eta_k$
and integrating $[\sigma,t]\times
\mathbb{R}^3 $ for $T_{k-1}\leq \sigma\leq T_k\leq t \leq 0$,
\begin{equation*}\begin{split}
&\int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(t,x)|^2}{2}dx
+ \int_{\sigma}^t\int_{\mathbb{R}^3}\eta_k(x)d^2_k(s,x)dxds\\
&\leq \int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(\sigma,x)|^2}{2}dx\\
+&\int_{\sigma}^t\int_{\mathbb{R}^3}(\nabla\eta_k)(x)w(s,x)\frac{|v_k(s,x)|^2}{2}dxds
+\int_{\sigma}^t\int_{\mathbb{R}^3}(\Delta\eta_k)(x)\frac{|v_k(s,x)|^2}{2}dxds\\
-&\int_{\sigma}^t\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u (P_{1,k} + P_{2,k})) + (\frac{v_k}{|u|}-1)u
\cdot\nabla (P_{1,k} + P_{2,k})\Big)(s,x)dxds.
\end{split}\end{equation*}
Integrating in $\sigma\in[T_{k-1},T_k]$ and dividing by
$-(T_{k-1}-T_k)=2^{-(k+1)}$,
\begin{equation*}\begin{split}
&\sup_{t\in[T_k,1]}\Big(\int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(t,x)|^2}{2}dx
+ \int_{T_k}^t\int_{\mathbb{R}^3}\eta_k(x)d^2_k(s,x)dxds\Big)\\
&\leq 2^{k+1}\cdot \int_{T_{k-1}}^{T_k}\int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(\sigma,x)|^2}{2}dx\\
+&\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\nabla\eta_k(x)w(s,x)\frac{|v_k(s,x)|^2}{2}dx\Big|ds
+\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\Delta\eta_k(x)\frac{|v_k(s,x)|^2}{2}dx\Big|ds\\
+&\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u (P_{1,k} + P_{2,k})) + (\frac{v_k}{|u|}-1)u
\cdot\nabla (P_{1,k} + P_{2,k})\Big)(s,x)dx\Big|ds.\\
\end{split}\end{equation*}
From $\eta_k=1$ on $ B_k$,
\begin{equation*}\begin{split}
U_k
&\leq\sup_{t\in[T_k,1]}\Big(\int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(t,x)|^2}{2}dx\Big)
+ \int_{T_k}^0\int_{\mathbb{R}^3}\eta_k(x)d^2_k(s,x)dxds\\
&\leq2\cdot\sup_{t\in[T_k,1]}\Big(\int_{\mathbb{R}^3}\eta_k(x)\frac{|v_k(t,x)|^2}{2}dx
+ \int_{T_k}^t\int_{\mathbb{R}^3}\eta_k(x)d^2_k(s,x)dxds\Big).
\end{split}\end{equation*}
Thus we have
\begin{equation}\begin{split}\label{1234_decompo_1}
&U_k
\leq (I)+(II)+(III)+(IV)
\end{split}\end{equation} where
\begin{equation}\begin{split}\label{1234_decompo_2}
&(I)=C2^{6k}\int_{Q_{k-1}}|v_k(s,x)|^2dxds,\\
&(II)=\int_{Q_{k-1}}|\nabla\eta_k(x)|\cdot|w(s,x)|\cdot|v_k(s,x)|^2dxds,\\
&(III)=2\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u P_{1,k}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{1,k}\Big)(s,x)dx\Big|ds\quad\mbox{ and}\\
&(IV)=2\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u P_{2,k}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{2,k}\Big)(s,x)dx\Big|ds.
\end{split}\end{equation}
For $(I)$, by using \eqref{raise_of_power}, for any $0<r<\infty$,
\begin{equation}\begin{split}\label{(I)}
&(I)= C2^{6k}\|v_k\|^2_{L^{2}(Q_{k-1})}
\leq C2^{10k}U^{\frac{5}{3}}_{k-1}.
\end{split}\end{equation}\\
For $(II)$ with $r\geq s_{1}$, by using \eqref{w_large_r} and \eqref{raise_of_power2},
\begin{equation}\begin{split}\label{(II-1)}
(II)
&\leq C2^{3k}\|w\|_{L^2(-4,0;L^{\infty}(B(2)))}\cdot
\||v_k|^2\|_{L^2(T_{k-1},0;L^{1}(B_{k-1}))}\\
&\leq C2^{3k}{\delta_{1}}\|v_k\|_{L^{\infty}(T_{k-1},0;L^{\frac{6}{5}}(B_{k-1}))}
\cdot\|v_k\|_{L^2(T_{k-1},0;L^{6}(B_{k-1}))}\\
&\leq C2^{4k}{\delta_{1}}U^{\frac{5}{6}}_{k-1}
\cdot\Big(\|v_{k-1}\|_{L^{\infty}(T_{k-1},0;L^{2}(B_{k-1}))}+
\|\nabla v_{k-1}\|_{L^2(T_{k-1},0;L^{2}(B_{k-1}))}\Big)\\
&\leq C2^{4k}\cdot{\delta_{1}}\cdot U^{\frac{5}{6}}_{k-1}
\cdot U^{\frac{1}{2}}_{k-1}\leq C2^{4k}\cdot{\delta_{1}}\cdot U^{\frac{4}{3}}_{k-1}
\leq C2^{4k}\cdot U^{\frac{4}{3}}_{k-1}.
\end{split}\end{equation}
For $r<s_{1}$, follow the above steps using
\eqref{w_small_r_with_r^3} instead of using \eqref{w_large_r}
then we get
\begin{equation}\begin{split}\label{(II-2)}
&(II)\leq C\frac{1}{r^3}2^{4k}\cdot U^{\frac{4}{3}}_{k-1}.
\end{split}\end{equation}
For $(III) $ (non-local pressure term), observe that
\begin{equation*}\begin{split}
\ebdiv (u P_{1,k}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{1,k}
= \frac{v_k}{|u|}u\cdot\nabla P_{1,k}
\end{split}\end{equation*}
because everything is smooth.
Thus, by using \eqref{eq_pressure_decomposition_p1k}
and \eqref{raise_of_power}, for any $0<r<\infty$,
\begin{equation}\begin{split}\label{(III)}
(III)&\leq C\cdot\|\frac{v_k}{|u|}u\cdot\nabla P_{1,k}\|_{L^1(Q_{k-1})}
\leq C\||v_k|\cdot|\nabla P_{1,k}|\|_{L^1(Q_{k-1})}\\
&\leq\|v_k\|_{L^{2}(T_{k-1},0;L^{1}(B_{k-1})))}
\cdot\|\nabla P_{1,k}\|_{L^{2}(T_{k-1},0;L^{\infty}(B_{k-1}))}\\
&\leq\|\mathbf{1}_{v_k>0}\|_{L^{2}(T_{k-1},0;L^{2}(B_{k-1})))}
\|v_k\|_{L^{\infty}(T_{k-1},0;L^{2}(B_{k-1})))}
\cdot 2^{12k}\\
&\leq C2^{\frac{43k}{3}}U^{\frac{5}{6}}_{k-1}
U^{\frac{1}{2}}_{k-1}
\leq C2^{\frac{43k}{3}}U^{\frac{4}{3}}_{k-1}.
\end{split}\end{equation}
For $(IV) $ (local pressure term), as we did for $(III)$, observe
\begin{equation*}\begin{split}
\ebdiv (u P_{2,k}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{2,k}
= \frac{v_k}{|u|}u\cdot\nabla P_{2,k}.
\end{split}\end{equation*}
By definition of $P_{2,k}$, we have
\begin{equation*}\begin{split}
-\Delta P_{2,k}& = \sum_{ij} \partial_i \partial_j (\psi_k w_i u_j)= \sum_{ij} \partial_i ((\partial_j \psi_k) w_i u_j
+\psi_k (\partial_jw_i) u_j)\\
&= \sum_{ij} \partial_i \Big((\partial_j \psi_k) w_i u_j(1-\frac{v_k}{|u|})
+(\partial_j \psi_k) w_i u_j\frac{v_k}{|u|}\\
&\quad\quad+\psi_k (\partial_jw_i) u_j(1-\frac{v_k}{|u|})
+\psi_k (\partial_jw_i) u_j\frac{v_k}{|u|}\Big)\\
\end{split}\end{equation*} and
\begin{equation*}\begin{split}
-\Delta (\nabla P_{2,k})&= \sum_{ij} \partial_i\nabla
\Big((\partial_j \psi_k) w_i u_j(1-\frac{v_k}{|u|})
+(\partial_j \psi_k) w_i u_j\frac{v_k}{|u|}\\
&\quad\quad+\psi_k (\partial_jw_i) u_j(1-\frac{v_k}{|u|})
+\psi_k (\partial_jw_i) u_j\frac{v_k}{|u|}\Big).
\end{split}\end{equation*}\\
\noindent Thus we can decompose $\nabla P_{2,k}$ by the Riesz transform into
\begin{equation*}\begin{split}
\nabla P_{2,k}= G_{1,k}+G_{2,k}+G_{3,k}+G_{4,k}
\end{split}\end{equation*} where
\begin{equation*}\begin{split}
&G_{1,k} =\sum_{ij} (\partial_i\nabla)(-\Delta)^{-1}\Big(
(\partial_j \psi_k) w_i u_j(1-\frac{v_k}{|u|})\Big),\\
&G_{2,k} =\sum_{ij} (\partial_i\nabla)(-\Delta)^{-1}\Big(
(\partial_j \psi_k) w_i u_j\frac{v_k}{|u|}\Big),\\
&G_{3,k} =\sum_{ij} (\partial_i\nabla)(-\Delta)^{-1}\Big(
\psi_k (\partial_jw_i) u_j(1-\frac{v_k}{|u|})\Big)\quad\mbox{ and}\\
&G_{4,k} =\sum_{ij} (\partial_i\nabla)(-\Delta)^{-1}\Big(
\psi_k (\partial_jw_i) u_j\frac{v_k}{|u|}\Big).
\end{split}\end{equation*}
\noindent From $L^p$-boundedness of the Riesz transform with the fact
$supp(\psi_k)\subset B_{k-(5/6)}\subset B_{k-1}$, we have
\begin{equation*}\begin{split}
&\|G_{2,k}\|_{L^2(T_{k-1},0;L^2(\mathbb{R}^3))}
\leq C2^{3k}\|w\|_{L^2(T_{k-1},0;L^{\infty}(B_{k-1}))}
\cdot\|v_k\|_{L^{\infty}(T_{k-1},0;L^2(B_{k-1}))},\\%\mbox{ and}\\
&\|G_{4,k}\|_{L^2(T_{k-1},0;L^2(\mathbb{R}^3))}
\leq C\cdot\|\nabla w\|_{L^2(T_{k-1},0;L^{\infty}(B_{k-1}))}
\cdot\|v_k\|_{L^{\infty}(T_{k-1},0;L^2(B_{k-1}))}.\\
\end{split}\end{equation*}
For any $1<p<\infty$,
\begin{equation*}\begin{split}
&\|G_{1,k}\|_{L^2(T_{k-1},0;L^p(\mathbb{R}^3))}
\leq C_p\cdot2^{3k}\|w\|_{L^2(T_{k-1},0;L^{\infty}(B_{k-1}))}\quad\mbox{ and}\\
&\|G_{3,k}\|_{L^2(T_{k-1},0;L^p(\mathbb{R}^3))}
\leq C_p\cdot\|\nabla w\|_{L^2(T_{k-1},0;L^{\infty}(B_{k-1}))}.
\end{split}\end{equation*}
Therefore, by using \eqref{nabla_w_large_r} and \eqref{nabla_w_small_r_with_r^3}
\begin{equation*}
\||G_{2,k}|+|G_{4,k}|\|_{L^2(T_{k-1},0;L^2(\mathbb{R}^3))}\leq
\begin{cases} &C\cdot2^{3k}\cdot U^{\frac{1}{2}}_{k-1}
,\quad \quad\mbox{ if } r\geq s_{1}\\
&C\cdot2^{3k}\cdot\frac{1}{r^3}\cdot U^{\frac{1}{2}}_{k-1}
,\quad \quad\mbox{ if } r< s_{1}
\end{cases}\end{equation*} and, for any $1<p<\infty$,
\begin{equation*}
\||G_{1,k}|+|G_{3,k}|\|_{L^2(T_{k-1},0;L^p(\mathbb{R}^3))}\leq
\begin{cases} &C_p\cdot2^{3k}
,\quad \quad\mbox{ if } r\geq s_{1}\\
&C_p\cdot2^{3k}\cdot\frac{1}{r^3}
,\quad \quad\mbox{ if } r< s_{1}.
\end{cases}\end{equation*}
Thus, by using the above estimates
and \eqref{raise_of_power}, for $r\geq s_{1}$ and $p>5$,
\begin{equation*}\begin{split}
(IV)&\leq C\cdot\|\frac{v_k}{|u|}u\cdot\nabla P_{2,k}\|_{L^1(Q_{k-1})}
\leq C\||v_k|\cdot|\nabla P_{2,k}|\|_{L^1(Q_{k-1})}\\
&\leq C\||v_k|\cdot(|G_{1,k}|+|G_{3,k}|)\|_{L^1(Q_{k-1})}
+ C\||v_k|\cdot(|G_{2,k}|+|G_{4,k}|)\|_{L^1(Q_{k-1})}\\
&\leq\|v_k\|_{L^{2}(T_{k-1},0;L^{\frac{p}{p-1}}(B_{k-1})))}
\cdot\||G_{1,k}|+|G_{3,k}|\|_{L^{2}(T_{k-1},0;L^{p}(B_{k-1}))}\\
&\quad\quad+\|v_k\|_{L^{2}(T_{k-1},0;L^{2}(B_{k-1})))}
\cdot\||G_{2,k}|+|G_{4,k}|\|_{L^{2}(T_{k-1},0;L^{2}(B_{k-1}))}\\
&\leq C\cdot C_p\cdot 2^{\frac{16k}{3}}U^{\frac{4p-5}{3p}}_{k-1}.
\end{split}\end{equation*}
\noindent By the same way, for $r< s_{1}$ and $p>5$,
\begin{equation*}\begin{split}
(IV)&\leq C\cdot C_p\cdot\frac{1}{r^3}
2^{\frac{16k}{3}}U^{\frac{4p-5}{3p}}_{k-1}.
\end{split}\end{equation*}
\noindent Thus, by taking $p=10$,
\begin{equation}\label{(IV)}
(IV)\leq \begin{cases}& C\cdot
2^{\frac{16k}{3}}U^{\frac{7}{6}}_{k-1}
,\quad \quad\mbox{ if } r\geq s_{1}\\
& C\cdot\frac{1}{r^3}
2^{\frac{16k}{3}}U^{\frac{7}{6}}_{k-1}
,\quad \quad\mbox{ if } r< s_{1}.
\end{cases}\end{equation}
Finally, combining \eqref{(I)}, \eqref{(II-1)},
\eqref{(II-2)}, \eqref{(III)} and \eqref{(IV)} gives us
\begin{equation*}
(I)+(II)+(III)+(IV)\leq \begin{cases}& C^k\cdot U^{\frac{7}{6}}_{k-1}
,\quad \quad\mbox{ if } r\geq s_{1}\\
& \frac{1}{r^3}\cdot C^k\cdot U^{\frac{7}{6}}_{k-1}
,\quad \quad\mbox{ if } r< s_{1}.
\end{cases}\end{equation*}
\end{proof}
\subsection{De Giorgi argument to get a control for small $r$}\label{proof_lem_partial_2}
The following big lemma makes us be able to avoid
the weak point of the previous lemma \ref{lem_partial_1}
when we handle small $r$ including the case $r=0$.\\
Recall the definition of $s_k$ in \eqref{def_s_k} first. It is the distance
between $B^c_{k-1}$ and $B_{k-\frac{5}{6}}$
and $s_k$ is strictly decreasing to zero as $k\rightarrow\infty$.
For any $0<r<s_{1}$ we
define $k_r$ as
the integer such that $s_{k_r+1}< r\leq s_{k_r}$.
Note that
$k_r$ is integer-valued, $k_r\geq 1$
and is increasing to $\infty$ as $r$ goes to zero.
For the case $r=0$, we simply
define $k_r=k_0=\infty$.\\
\begin{lem}\label{lem_partial_2}
There exist universal constants ${\delta}_2$ and $\bar{C}_2>1$ such that
if u is a solution of (Problem II-r) for some $0\leq r<s_{1}$ verifying both
\begin{equation*}\begin{split}
&\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4}))}+
\|P\|_{L^1(-2,0;L^{1}(B(1))}+\| \nabla u\|_{L^{2}(-2,0;L^{2}(B(\frac{5}{4}))}
\leq {\delta}_2\\
\mbox{ and }
& \| \mathcal{M}(|\nabla u|)\|_{L^{2}(-4,0;L^{2}(B(2)))}\leq {\delta}_2,
\end{split}\end{equation*}
then we have
\begin{equation*}
U_k \leq (\bar{C}_2)^k U_{k-1}^{\frac{7}{6}}\quad\text{ for any integer }
k \mbox{ such that } 1\leq k\leq k_r.
\end{equation*}
\end{lem}
\begin{rem}
Note that ${\delta}_2$ and $\bar{C}_2$ are independent of any $r\in[0, s_1)$
and the exponent $7/6$ is not optimal and we can make it almost $(4/3)$.
\end{rem}
\begin{rem}
This lemma says that even though $r$ is very small, we can make the above
uniform estimate for the first few steps $k\leq k_r$. Moreover, the number $k_r$ of
these steps is increasing to the infinity with a certain rate as $r$ goes to zero. In the subsection \ref{combine_de_giorgi}, we will see that this rate is enough to obtain a uniform estimate
for any small $r$ once we combine two lemmas \ref{lem_partial_1}
and \ref{lem_partial_2}.
\end{rem}
\begin{proof}
In this proof, we can borrow any inequalities
in the proof of the previous lemma \ref{lem_partial_1}
except those which depend on $r$ and blow up as $r$ goes to zero.\\% (having $\frac{1}{r^3}$ factor).\\
Let $0\leq r<s_{1}$ and take any integer k such that $1\leq k\leq k_r$.
Like ${\delta}_1$ of the previous lemma \ref{lem_partial_1}, we assume ${\delta}_2$ so small that
\begin{equation*}\begin{split}
{\delta}_2<1,\quad 10{\Lambda_1} {\Lambda_2}{\delta}_2\leq\frac{1}{2}.
\end{split}\end{equation*}
We begin this proof by decomposing $w^\prime$ by
\begin{equation*}\begin{split}
w^\prime
=u*\phi_r
&=\Big(u(1-\frac{v_k}{|u|})\Big)*\phi_r
+ \Big(u\frac{v_k}{|u|}\Big)*\phi_r =w^{\prime,1} + w^{\prime,2}.
\end{split}\end{equation*}
Thus the advection velocity $w$ has a new decomposition: $w=w^{\prime} -w^{\prime\prime}=
(w^{\prime,1} +w^{\prime,2}) -w^{\prime\prime}
=(w^{\prime,1} -w^{\prime\prime}) +w^{\prime,2} $.
We will verify that $w^{\prime,1} -w^{\prime\prime}$ is bounded and
$w^{\prime,2}$ can be controlled locally.
First, for $w^{\prime,1}$,
\begin{equation}\begin{split}\label{w_1_prime}
|w^{\prime,1}(t,x)|=\Big|\Big(\Big(u(1-\frac{v_k}{|u|})\Big)*\phi_r\Big)(t,x)\Big|
&\leq\|u(1-\frac{v_k}{|u|})(t,\cdot)\|_{L^{\infty}(\mathbb{R}^3)} \leq 1
\end{split}\end{equation} for any $-4\leq t $ and any $x\in\mathbb{R}^3$.
From \eqref{w_prime_prime_small_r},
we still have
\begin{equation}\begin{split}\label{w_prime_prime_small_r_again}
\| w^{\prime\prime}\|_{L^\infty(-2,0;L^{\infty}(B(2)))}\leq {C}\bar{\delta}\leq C.
\end{split}\end{equation}
\noindent Combining above two results,
\begin{equation}\begin{split}\label{w_1_prime_w_prime_prime_small_r}
\||w^{\prime,1}|+|w^{\prime\prime}|\|_{L^\infty(-2,0;L^{\infty}(B(2)))}\leq C.
\end{split}\end{equation}
\noindent For $w^{\prime,2}$, we observe that any $L^p$
norm of $w^{\prime,2}=\Big(u\frac{v_k}{|u|}\Big)*\phi_r$
in $B_{k-\frac{5}{6}}$ is less than or equal to that of $v_k$ in $B_{k-1}$ because
$r\leq s_{k_r}\leq s_{k}$ and $s_k$
is the distance between $B^c_{k-1}$ and $B_{k-\frac{5}{6}}$ (see
\eqref{young}). Thus we have, for any
$1\leq p\leq\infty$,
\begin{equation}\begin{split}\label{w_2_prime_small_r}
\|w^{\prime,2}&\|_{L^{p}(T_{k-1},0;L^{p}(B_{k-\frac{5}{6}}))}
= \| \Big(u\frac{v_k}{|u|}\Big)*\phi_r\|_{L^{p}(T_{k-1},0;L^{p}(B_{k-\frac{5}{6}}))}\\
&= \| |v_k|*\phi_r\|_{L^{p}(T_{k-1},0;L^{p}(B_{k-\frac{5}{6}}))}
\leq\| v_k\|_{L^{p}(Q_{k-1})}.
\end{split}\end{equation}
So, by using \eqref{raise_of_power}, we have
\begin{equation}\begin{split}\label{raise_of_power_ w_prime_2}
&\| w^{\prime,2}\|_{L^{p}(T_{k-1},0;L^{p}(B_{k-\frac{5}{6}}))}
\leq C2^{\frac{7k}{3}}U^{\frac{5}{3p}}_{k-1},
\quad\mbox{ for any } 1\leq p\leq\frac{10}{3}.
\end{split}\end{equation}
\begin{rem}
The above computations says that, for any small $r$,
the advection velocity $w$ can be decomposed into
one bounded part $(w^{\prime,1} -w^{\prime\prime})$ and the other part
$w^{\prime,2}$, which has a good contribution
to the power of $U_{k-1}$.
\end{rem}
Recall that the transpost term estimate \eqref{w_iu_j} is valid for any
$0<r<\infty$. Moreover,
the argument around \eqref{w_u} says that \eqref{w_iu_j} holds even for the case $r=0$.
Thus, for any $r\in[0,s_1)$, we have the same pressure estimates
\eqref{eq_pressure_decomposition_p1k},
\eqref{eq_pressure_decomposition_p2k} and
\eqref{eq_pressure_decomposition_p3}. Thus
we can follow the proof of the previous lemma \ref{lem_partial_1} up to
\eqref{1234_decompo_1} without any single modification.
It remains to control $(I)$--$(IV)$. \\
For $(I)$, \eqref{(I)} holds here too because \eqref{(I)} is independent of $r$.\\
For $(II)$, by using \eqref{w_1_prime_w_prime_prime_small_r}
and \eqref{raise_of_power_ w_prime_2} with
the fact $supp(\eta_k)\subset B_{k-\frac{1}{3}}\subset B_{k-\frac{5}{6}}$, we have
\begin{equation}\begin{split}\label{second_(II)}
&(II)=\||\nabla\eta_k|\cdot|w|\cdot|v_k|^2\|_{L^1(Q_{k-1})}\\
&\leq C2^{3k}\Big(\|(|w^{\prime,1}|+|w^{\prime\prime}|)\cdot|v_k|^2\|_{L^1(Q_{k-1})}
+\||w^{\prime,2}|\cdot|v_k|^2\|_{L^{1}(T_{k-1},0;L^1(B_{k-\frac{5}{6}}))}\Big)\\
&\leq C2^{3k}\|v_k\|^2_{L^2(Q_{k-1})}
+C2^{3k}\|w^{\prime,2}\|_{L^{\frac{10}{3}}(T_{k-1},0;L^\frac{10}{3}(B_{k-\frac{5}{6}}))}\cdot
\||v_k|^2\|_{L^\frac{10}{7}(Q_{k-\frac{5}{6}})}\\
&\leq C2^{\frac{23k}{3}}U^{\frac{5}{3}}_{k-1}
+C2^{10k}U^{\frac{5}{3}}_{k-1}\leq C2^{10k}U^{\frac{5}{3}}_{k-1}.
\end{split}\end{equation}
For $(III)$(non-local pressure term),
we have \eqref{(III)} here too since \eqref{(III)} is independent of $r$.\\
For $(IV)$(local pressure term),
by definition of $P_{2,k}$ and decomposition of $w$,
\begin{equation*}\begin{split}
-\Delta P_{2,k}
&= \sum_{ij} \partial_i \partial_j\Big( \psi_k w_i u_j(1-\frac{v_k}{|u|})
+ \psi_k w_i u_j\frac{v_k}{|u|}\Big)\\
&= \sum_{ij} \partial_i \partial_j\Big( \psi_k (w^{\prime,1}_i-w^{\prime\prime}_i) u_j(1-\frac{v_k}{|u|})+\psi_k w^{\prime,2}_i u_j(1-\frac{v_k}{|u|})\\
&\quad\quad+ \psi_k (w^{\prime,1}_i-
w^{\prime\prime}_i) u_j\frac{v_k}{|u|}+ \psi_k
w^{\prime,2}_i u_j\frac{v_k}{|u|}\quad\Big).
\end{split}\end{equation*}
\noindent Thus we can decompose $ P_{2,k}$ by
\begin{equation*}\begin{split}
P_{2,k}= P_{2,k,1}+P_{2,k,2}+P_{2,k,3}+P_{2,k,4}
\end{split}\end{equation*} where
\begin{equation*}\begin{split}
&P_{2,k,1} =\sum_{ij} (\partial_i\partial_j)(-\Delta)^{-1}\Big(
\psi_k (w^{\prime,1}_i-w^{\prime\prime}_i) u_j(1-\frac{v_k}{|u|})\Big),\\
&P_{2,k,2} =\sum_{ij} (\partial_i\partial_j)(-\Delta)^{-1}\Big(
\psi_k w^{\prime,2}_i u_j(1-\frac{v_k}{|u|})\Big),\\
&P_{2,k,3} =\sum_{ij} (\partial_i\partial_j)(-\Delta)^{-1}\Big(
\psi_k (w^{\prime,1}_i-w^{\prime\prime}_i) u_j\frac{v_k}{|u|}\Big)\quad\mbox{ and}\\
&P_{2,k,4} =\sum_{ij} (\partial_i\partial_j)(-\Delta)^{-1}\Big(
\psi_k w^{\prime,2}_i u_j\frac{v_k}{|u|}\Big).
\end{split}\end{equation*}
By using $\Big|u\Big(1-\frac{v_k}{|u|}\Big)\Big|\leq 1$
and the fact
$\psi_k$ is supported in $B_{k-\frac{5}{6}}$ with
\eqref{w_1_prime_w_prime_prime_small_r},
\begin{equation}\begin{split}\label{second_P_{2,k,1}}
\|P_{2,k,1}\|_{L^p(T_{k-1},0;L^p(\mathbb{R}^3))}&\leq C_p,\quad\mbox{ for } 1<p<\infty\\
\end{split}\end{equation} and,
with
\eqref{raise_of_power_ w_prime_2},
\begin{equation}\begin{split}\label{second_P_{2,k,2}}
\|P_{2,k,2}\|_{L^p(T_{k-1},0;L^p(\mathbb{R}^3))}&\leq
C_p\||\psi_k|\cdot |w^{\prime,2}|\|_{L^p(T_{k-1},0;L^{p}(\mathbb{R}^3)))}\\
&\leq CC_p2^{\frac{7k}{3}}U^{\frac{5}{3p}}_{k-1}\quad\mbox{ for } 1\leq p\leq\frac{10}{3}.\\
\end{split}\end{equation}
Observe that for $i=1,2$,
\begin{equation}\begin{split}\label{second_P_{2,k,1}_P_{2,k,2}}
&\ebdiv \Big(u G_i\Big) + \Big(\frac{v_k}{|u|}-1\Big)u\cdot\nabla G_i
=\ebdiv \Big(v_k\frac{u}{|u|}G_i\Big) -
G_i\ebdiv\Big(\frac{uv_k}{|u|}\Big).
\end{split}\end{equation}
For $P_{2,k,1}$, by using \eqref{d_k}, \eqref{raise_of_power}, \eqref{raise_of_power3},
\eqref{second_P_{2,k,1}_P_{2,k,2}} and \eqref{second_P_{2,k,1}} with $p=10$
\begin{equation}\begin{split}\label{second_P_{2,k,1}_TOTAL}
&\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u P_{2,k,1}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{2,k,1}\Big)(s,x)dx\Big|ds\\
&\leq C^{3k}\|v_k\cdot|P_{2,k,1}|\|_{L^1(Q_{k-1})} +
3\|d_k\cdot|P_{2,k,1}|\|_{L^1(Q_{k-1})}\\
&\leq C^{3k}\|v_k\|_{L^\frac{10}{9}(Q_{k-1})}
\cdot\|P_{2,k,1}\|_{L^{10}(Q_{k-1})} +
3\|d_k\|_{L^\frac{10}{9}(Q_{k-1})}\cdot\|P_{2,k,1}\|_{L^{10}(Q_{k-1})}\\
&\leq C2^{\frac{16k}{3}}U^{\frac{3}{2}}_{k-1}+
C2^{\frac{5k}{3}}U^{\frac{7}{6}}_{k-1}
\leq C2^{\frac{16k}{3}}U^{\frac{7}{6}}_{k-1}.
\end{split}\end{equation}
Likewise, for $P_{2,k,2}$, by using \eqref{second_P_{2,k,2}} instead of \eqref{second_P_{2,k,1}}
\begin{equation}\begin{split}\label{second_P_{2,k,2}_TOTAL}
&\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u P_{2,k,2}) + (\frac{v_k}{|u|}-1)u\cdot\nabla P_{2,k,2}\Big)(s,x)dx\Big|ds\\
&\leq C2^{\frac{23k}{3}}U^{\frac{5}{3}}_{k-1}+
C2^{4k}U^{\frac{4}{3}}_{k-1}
\leq C2^{\frac{23k}{3}}U^{\frac{4}{3}}_{k-1}.
\end{split}\end{equation}
\noindent From definitions of
$P_{2,k,3}$ and $P_{2,k,4}$
with $\ebdiv(w)=0$, we have
\begin{equation*}\begin{split}
-\Delta\nabla (P_{2,k,3}+P_{2,k,4})
&= \sum_{ij} \partial_i \partial_j\nabla\Big( \psi_k w_i u_j\frac{v_k}{|u|}\Big)\\
&= \sum_{ij} \nabla \partial_j
\Big( (\partial_i\psi_k) w_i
u_j\frac{v_k}{|u|}+
\psi_k w_i
\partial_i(u_j\frac{v_k}{|u|})\Big).
\end{split}\end{equation*}
\noindent Then we use the fact
$w=(w^{\prime,1} -w^{\prime\prime}) +w^{\prime,2} $
so that
we can decompose
\begin{equation*}
\nabla(P_{2,k,3}+P_{2,k,4})=H_{1,k}+H_{2,k}+H_{3,k}+H_{4,k}
\end{equation*}
where
\begin{equation*}\begin{split}
H_{1,k}&= \sum_{ij}( \nabla \partial_j)(-\Delta)^{-1}
\Big((\partial_i\psi_k) (w^{\prime,1}_i-w^{\prime\prime}_i)
u_j\frac{v_k}{|u|} \Big),\\
H_{2,k}&= \sum_{ij}( \nabla \partial_j)(-\Delta)^{-1}
\Big((\partial_i\psi_k) w^{\prime,2}_i
u_j\frac{v_k}{|u|} \Big),\\
H_{3,k}&= \sum_{ij}( \nabla \partial_j)(-\Delta)^{-1}
\Big(\psi_k (w^{\prime,1}_i-w^{\prime\prime}_i)
\partial_i(u_j\frac{v_k}{|u|}) \Big)\quad\mbox{ and}\\
H_{4,k}&= \sum_{ij}( \nabla \partial_j)(-\Delta)^{-1}
\Big(\psi_k w^{\prime,2}_i
\partial_i(u_j\frac{v_k}{|u|}) \Big).
\end{split}\end{equation*}
By using $|u|\leq 1+v_k$,
\begin{equation}\begin{split}\label{second_P_{2,k,3}_P_{2,k,4}_TOTAL}
&\int_{T_{k-1}}^{0}\Big|\int_{\mathbb{R}^3}\eta_k(x)
\Big(\ebdiv (u (P_{2,k,3}+P_{2,k,4})) + (\frac{v_k}{|u|}-1)u\cdot
\nabla (P_{2,k,3}+P_{2,k,4})\Big)
dx\Big|ds\\
&\leq C^{3k}\int_{Q_{k-1}}(1+v_k
)\cdot|(P_{2,k,3}+P_{2,k,4})(s,x)|+|\nabla (P_{2,k,3}+P_{2,k,4})
|dxds\\
&\leq C^{3k}\Big(\|P_{2,k,3}\|_{L^1(Q_{k-1})}+\|v_k\cdot|P_{2,k,3}|\|_{L^1(Q_{k-1})}\\
&\quad\quad+\|P_{2,k,4}\|_{L^1(Q_{k-1})}
+\|v_k\cdot|P_{2,k,4}|\|_{L^1(Q_{k-1})}\\
&\quad\quad+\|H_{1,k}\|_{L^1(Q_{k-1})}+\|H_{2,k}\|_{L^1(Q_{k-1})}+\|H_{3,k}\|_{L^1(Q_{k-1})}
+\|H_{4,k}\|_{L^1(Q_{k-1})}\Big).\\
\end{split}\end{equation}
From \eqref{raise_of_power} and \eqref{w_1_prime_w_prime_prime_small_r} with
the Riesz transform,
\begin{equation}\begin{split}\label{second_P_{2,k,3}_TOTAL}
\|P_{2,k,3}\|_{L^1(Q_{k-1})}&
\leq C\|P_{2,k,3}\|_{L^\frac{10}{9}(T_{k-1},0;L^\frac{10}{9}(\mathbb{R}^3))}
\leq C\|v_k\|_{L^\frac{10}{9}(Q_{k-1})}
\leq C2^{\frac{7k}{3}}U^{\frac{3}{2}}_{k-1}.
\end{split}\end{equation}
\noindent Likewise
\begin{equation}\begin{split}\label{second_H_{1,k}_TOTAL}
\|H_{1,k}\|_{L^1(Q_{k-1})}&
\leq C2^{\frac{16k}{3}}U^{\frac{3}{2}}_{k-1}
\end{split}\end{equation} and
\begin{equation}\begin{split}\label{second_v_k_P_{2,k,3}_TOTAL}
\|v_k\cdot|P_{2,k,3}|\|_{L^1(Q_{k-1})}
&\leq \|v_k\|_{L^2(Q_{k-1})}
\|P_{2,k,3}\|_{L^2(Q_{k-1})}\\
&\leq C2^{\frac{7k}{3}}U^{\frac{5}{6}}_{k-1}\cdot
C2^{\frac{7k}{3}}U^{\frac{5}{6}}_{k-1}
\leq C2^{\frac{14k}{3}}U^{\frac{5}{3}}_{k-1}.\\
\end{split}\end{equation}
Using \eqref{raise_of_power}, \eqref{raise_of_power_ w_prime_2},
\eqref{d_k} and \eqref{raise_of_power3}, we have
\begin{equation}\begin{split}\label{second_P_{2,k,4}_TOTAL}
\|P_{2,k,4}\|_{L^1(Q_{k-1})}
&\leq C2^{\frac{14k}{3}}U^{\frac{3}{2}}_{k-1},
\end{split}\end{equation}
\begin{equation}\begin{split}\label{second_H_{2,k}_TOTAL}
\|H_{2,k}\|_{L^1(Q_{k-1})}
&\leq C2^{\frac{23k}{3}}U^{\frac{3}{2}}_{k-1},
\end{split}\end{equation}
\begin{equation}\begin{split}\label{second_v_k_P_{2,k,4}_TOTAL}
\|v_k\cdot|P_{2,k,4}|\|_{L^1(Q_{k-1})}
&\leq C2^{\frac{21k}{3}}U^{\frac{5}{3}}_{k-1},
\end{split}\end{equation}
\begin{equation}\begin{split}\label{second_H_{3,k}_TOTAL}
\|H_{3,k}\|_{L^1(Q_{k-1})}
&\leq C2^{\frac{5k}{3}}U^{\frac{7}{6}}_{k-1}
\end{split}\end{equation} and
\begin{equation}\begin{split}\label{second_H_{4,k}_TOTAL}
\|H_{4,k}\|_{L^1(Q_{k-1})}
&\leq C2^{4k}U^{\frac{7}{6}}_{k-1}.
\end{split}\end{equation}
Combining \eqref{second_P_{2,k,1}_TOTAL},
\eqref{second_P_{2,k,2}_TOTAL} and
\eqref{second_P_{2,k,3}_P_{2,k,4}_TOTAL} together with
\eqref{second_P_{2,k,3}_TOTAL}, $\cdots$,
\eqref{second_H_{4,k}_TOTAL}, we obtain
\begin{equation}\begin{split}\label{second_(IV)}
&(IV)\leq C 2^{\frac{23k}{3}} U^{\frac{7}{6}}_{k-1}.
\end{split}\end{equation}
Finally we combine \eqref{second_(II)} and \eqref{second_(IV)} together with \eqref{(I)} and \eqref{(III)} in the previous lemma
in order to finish the proof of this lemma \ref{lem_partial_2}.
\end{proof}
\subsection{Combining the two De Giorgi arguments}\label{combine_de_giorgi}
First we present one small lemma. Then the actual proof
of the proposition \ref{partial_problem_II_r} will follow.
The following small lemma says that certain non-linear estimates
give zero limit if the initial term is sufficiently small.
This fact is one of key arguments of De Giorgi method.
\begin{lem}\label{lem_recursive}
Let $C>1$ and $\beta>1$. Then there exists a constant $C_0^{*}$
such that for every sequence verifying both $ 0 \leq W_0 < C^{*}_0$
and \begin{equation*}
0\leq W_{k} \leq C^k \cdot W_{k-1}^{\beta} \quad \mbox{ for any } k\geq 1,
\end{equation*} we have $\lim_{k \to\infty} W_k = 0$.
\end{lem}
\begin{proof}
It is quite standard or see the lemma 1 in \cite{vas:partial}.
\end{proof}
Finally we are ready to prove the proposition \ref{partial_problem_II_r}.
\begin{proof}[Proof of proposition \ref{partial_problem_II_r}]
Suppose that u is a solution of (Problem II-r) for some $0\leq r<\infty$ verifying
\begin{equation*}\begin{split}
&\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4})))}+
\|P\|_{L^1(-2,0;L^{1}(B(1)))}+\| \nabla u\|_{L^{2}(-2,0;L^{2}(B(\frac{5}{4})))}
\leq {\delta}\\ \mbox{ and }
&\| \mathcal{M}(|\nabla u|)\|_{L^{2}(-4,0;L^{2}(B(2)))}\leq {\delta}\\
\end{split}\end{equation*} where $\delta$ will be chosen within the proof.\\
From two big lemmas \ref{lem_partial_1} and \ref{lem_partial_2} by assuming
$\delta\leq\min({\delta}_1,\delta_2)$, we have
\begin{equation}\label{before_combine}
U_k \leq
\begin{cases}& (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}} ,
\quad \mbox{ for any } k\geq 1 \quad\mbox{ if } r\geq s_{1}.\\
&\frac{1}{r^3}\cdot (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}
,\quad \mbox{ for any } k\geq 1 \quad\mbox{ if } 0<r< s_{1}.\\%\quad\mbox{ and}\\
&(\bar{C}_2)^k U_{k-1}^{\frac{7}{6}}
\quad\text{ for } k = 1,2,\cdots,k_r \quad\mbox{ if } 0\leq r< s_{1}.
\end{cases}
\end{equation}
Note that $k_r=\infty$ if $r=0$. Thus we can combine
the case $r=0$ with the case $r\geq s_1$ into one estimate:
\begin{equation*}
U_k \leq
(\bar{C}_3)^k U_{k-1}^{\frac{7}{6}}
\quad \mbox{ for any } k\geq 1 \quad\mbox{ if either } r\geq s_1 \mbox{ or } r=0.
\end{equation*} where we define $\bar{C}_3=\max(\bar{C}_1,\bar{C}_2)$.\\
We consider now the case $0<r<s_1$.
Recall that
$s_k=D\cdot 2^{-3k}$ where $D=\Big((\sqrt{2}-1)2\sqrt{2}\Big)>1$
and $s_{k_r+1}< r\leq s_{k_r}$ for any $r\in(0,s_1)$. It gives us
$r\geq D\cdot2^{-3(k_r+1)}$.
Thus if $k\geq k_r$ and if $0<r< s_1$, then the second line in \eqref{before_combine} becomes
\begin{equation}\begin{split}
U_k
&\leq\frac{1}{r^3}\cdot (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}
\leq\frac{2^{9{(k_r+1)}}}{D^3}\cdot (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}\\
&\leq{2^{9{(k+1)}}}\cdot (\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}
\leq({2^{18}}\cdot\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}.
\end{split}\end{equation}
So we have for any $r\in(0,s_1)$,
\begin{equation*}
U_k \leq
\begin{cases}
& ({2^{18}}\cdot\bar{C}_1)^k U_{k-1}^{\frac{7}{6}}
,\quad \mbox{ for any } k\geq k_r. \\%\quad\mbox{ if } r< s_{1}\quad\mbox{ and}\\
&(\bar{C}_2)^k U_{k-1}^{\frac{7}{6}}\quad\text{ for }
k = 1,2,\cdots,k_r.
\end{cases}
\end{equation*}
Define $\bar{C}=\max({2^{18}}\cdot\bar{C}_1,\bar{C}_2,\bar{C}_3)
=\max({2^{18}}\cdot\bar{C}_1,\bar{C}_2)$. Then we can combine
all three cases $r=0$, $0<r<s_1$ and $s_1\leq r<\infty$ into one uniform estimate:
\begin{equation*}
U_k \leq
(\bar{C})^k U_{k-1}^{\frac{7}{6}}
\quad \mbox{ for any } k\geq 1 \quad\mbox{ and for any } 0\leq r<\infty.
\end{equation*}
Finally, by using the recursive lemma
\ref{lem_recursive}, we obtain $C^{*}_0$ such that $U_k\rightarrow 0$ as
$k\rightarrow 0$ whenever $ U_0 < C^{*}_0$. This condition $ U_0 < C^{*}_0$
is achievable once
we assume $\delta$ so small that
${\delta}\leq\sqrt{\frac{C^{*}_0}{2}}$
because
\begin{equation*}\begin{split}
U_0
&\leq \Big(\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4})))}+
\|P\|_{L^1(-2,0;L^{1}(B(1)))}+
\| \nabla u\|_{L^{2}(-2,0;L^{2}(B(\frac{5}{4})))}\big)^2.
\end{split}\end{equation*}
Thus we fix ${\delta}=\min(\sqrt{\frac{C^{*}_0}{2}},\delta_1,\delta_2)$
which does not depend on any $r\in [0,\infty)$.
Observe that for any $k\geq 1$,
\begin{equation*}
\sup_{-\frac{3}{2}\leq t\leq 0}\int_{B(\frac{1}{2})}(|u(t,x)|-1)^2_{+}dx \leq U_k
\end{equation*} from $E_k\leq 1 $ and
$ (-\frac{3}{2},0)\times B(\frac{1}{2})\subset Q_k$. Due to the fact
$U_k\rightarrow 0$, the conclusion of this proposition \ref{partial_problem_II_r} follows.
\end{proof}
\section{Proof of the second local study proposition \ref{local_study_thm}}\label{proof_local study}
First we present technical lemmas, whose proofs will be given in the appendix.
In the subsection \ref{new_step1_2_together}, it will be explained how to apply the previous local study
proposition \ref{partial_problem_II_r} in order to get a $L^\infty$-bound of
the velocity $u$ . Then, the subsections \ref{new_step3} and
\ref{new_step4} will give us $L^\infty$-bounds for classical derivatives $\nabla^d u$
and for fractional derivatives $(-\Delta)^{\alpha/2}\nabla^d u$, respectively.
\subsection{Some lemmas}
The following lemma is an estimate about higher derivatives of pressure
which we can find a similar lemma
in \cite{vas:higher}. However they are different in the sense that
here we require $(n-1)$th order norm of $v_1$ to control $n$th
derivatives of pressure (see \eqref{ineq_parabolic_pressure}) while in \cite{vas:higher} we require one more order,
i.e. $n$th order. This fact follows the divergence structure and it
will be useful for a bootstrap argument in the
subsection \ref{new_step3}
when large $r$ is large (we will see \eqref{step3_second_claim}).
\begin{lem}\label{higher_pressure} Suppose that we have $v_1,v_2\in
(C^\infty(B(1)))^3$ with $\ebdiv v_1=\ebdiv v_2=0$ and $P \in
C^\infty(B(1))$ which satisfy
\begin{equation*}\begin{split}
-\Delta P &=\ebdiv\ebdiv(v_2\otimes v_1)
\end{split}\end{equation*}
on $B(1)\subset \mathbb{R}^3$.\\% for some $0<a<\infty$.\\
\noindent Then, for any $n\geq 2$, $0<b<a<1$ and $1<p<\infty$, we have the
two following estimates:
\begin{equation}\begin{split}\label{ineq_parabolic_pressure}
\|\nabla^n P\|_{L^{p}(B(b))}
&\leq C_{(a,b,n,p)}\Big(
\| v_2 \|_{W^{n-1,p_2}(B(a))}\cdot
\| v_1 \|_{W^{n-1,p_1}(B(a))}\\
&\quad\quad\quad\quad\quad\quad+\| P \|_{L^{1}(B(a))}\Big)
\end{split}\end{equation} where
$\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, and
\begin{equation}\begin{split}\label{ineq_parabolic_pressure2}
\|\nabla^n P\|_{L^{\infty}(B(b))}
&\leq C_{(a,b,n)}\Big(
\| v_2 \|_{W^{n,\infty}(B(a))}\cdot
\| v_1 \|_{W^{n,\infty}(B(a))}\\
&\quad\quad\quad\quad\quad\quad+\| P \|_{L^{1}(B(a))}\Big)
\end{split}\end{equation}
\noindent Note that such constants are independent of any $v_1,v_2$ and $P$. Also,
$\infty$ is allowed for $p_1$ and $p_2$. e.g. if $p_1=\infty$, then $p_2=p$.
\end{lem}
\begin{proof}
See the appendix.
\end{proof}
The following is a local result by using a parabolic regularization.
It will be used in the subsection \ref{new_step3}
to prove
\eqref{step3_first_claim} and \eqref{step3_second_claim}.
\begin{lem}\label{lem_parabolic_v_1_v_2_pressure}
Suppose that we have
smooth solution $(v_1,v_2,P)$ on $Q(1)=(-1,0)\times B(1)$ of
\begin{equation*}\begin{split}
&\partial_t(v_1)+\ebdiv(v_2\otimes v_1)+ \nabla P -\Delta v_1=0\\
&\ebdiv (v_1)=0 \mbox{ and } \ebdiv (v_2)=0.
\end{split}\end{equation*}
Then, for any $n\geq 1$, $0<b<a<1$, $1< p_1<\infty$ and $1<p_2<\infty$, we have
\begin{equation}\begin{split} \label{ineq_parabolic_v_1_v_2}
&\|\nabla^n v_1 \|_{L^{p_1}(-({b})^2,0 ;L^{p_2}(B({b})))}
\leq C_{(a,b,n,p_1,p_2)}
\Big(\|v_2\otimes v_1 \|_{L^{p_1}(-a^2,0;W^{n-1,p_2}(B(a)))}\\&
\quad\quad\quad\quad\quad\quad\quad\quad+
\| v_1 \|_{L^{p_1}(-a^2,0;W^{n-1,p_2}(B(a)))}+
\| P \|_{L^{1}(-a^2,0;L^{1}(B(a)))}\Big)\\
\end{split}\end{equation} where $v_2\otimes v_1$ is the matrix whose
$(i,j)$ component is the product of $j$-th component $v_{2,j}$ of $v_2$
and $i$-th one $v_{1,i}$ of $v_1$ and $\Big(\ebdiv(v_2\otimes v_1)\Big)_i
=\sum_j\partial_j(v_{2,j} v_{1,i})$.\\% =\sum_jv_{2,j}\partial_j v_{1,i}$.\\
Note that such constants are independent of any $v_1,v_2$ and $P$.
\end{lem}
Proof of this lemma \ref{lem_parabolic_v_1_v_2_pressure} is omitted because it
is based on the standard parabolic regularization result (e.g. Solonnikov \cite{Solo})
and precise argument
is essentially contained in \cite{vas:higher}
except that
here we consider
\begin{equation*}\begin{split}
&(v_1)_t+\ebdiv(v_2\otimes v_1)+ \nabla P -\Delta v_1=0\\
\end{split}\end{equation*} while \cite{vas:higher} covered
\begin{equation*}\begin{split}
&(u)_t+\ebdiv(u\otimes u)+ \nabla P -\Delta u=0.\\
\end{split}\end{equation*}
The following lemma will be
used in the subsection \ref{new_step3}
, especially when we prove \eqref{step3_second_claim} for large $r$.
\begin{lem}\label{lem_a_half_upgrading_large_r}
Suppose that we have
smooth solution $(v_1,v_2,P)$ on $Q(1)=(-1,0)\times B(1)$ of
\begin{equation*}\begin{split}
&\partial_t(v_1)+(v_2\cdot\nabla)(v_1)+ \nabla P -\Delta v_1=0\\
&\ebdiv (v_1)=0 \mbox{ and } \ebdiv (v_2)=0.
\end{split}\end{equation*}
Then, for any $n\geq 0$ and $0<b<a<1$, we have
\begin{equation*}\begin{split}
\|\nabla^n {v_1} &\|_{L^{\infty}(-{({b})}^2,0 ;L^{1}(B{({b})}))}
\leq\\ & C_{(a,b,n)}\Big[
\Big(\| v_2\|_{L^2(-{{a}^2},0;W^{n,\infty}(B({a})))} +1\Big)
\cdot
\| {v_1} \|_{L^{2}(-{a}^2,0;W^{n,{1}}(B{(a)}))}\\
&\quad\quad\quad\quad\quad\quad\quad\quad+ \|\nabla^{n+1}P \|_{L^{1}(-{a}^2,0;L^{1}(B{(a)}))} \Big]
\end{split}\end{equation*}
and, for any ${p}\geq 1$,
\begin{equation*}\begin{split}
&\|\nabla^{n} {v_1} \|^{p+\frac{1}{2}}_{L^{\infty}(-{({b})}^2,0 ;L^{p+\frac{1}{2}}(B{({b})}))}\leq\\
& \quad\quad C_{(a,b,n,p)}\Big[
\Big(\| v_2\|_{L^2(-{{a}^2},0;W^{n,\infty}(B({a})))} +1\Big)
\cdot
\|{v_1} \|_{L^{2}(-{({a})}^2,0;W^{n,2p}(B{({a})}))}\\
&\quad\quad\quad\quad\quad\quad+ \|\nabla^{n+1} P\|_{L^{1}(-{({a})}^2,0;L^{2p}(B{({a})}))}
\Big]\cdot
\| {v_1} \|^{p-\frac{1}{2}}_{L^{\infty}(-{({a})}^2,0;W^{n,p}(B{(a)}))}.
\end{split}\end{equation*}
Note that such constants are independent of any $v_1,v_2$ and $P$.
\end{lem}
\begin{proof}
See the appendix.
\end{proof}
The following non-local version of Sobolev-type lemmas will be useful
when we handle fractional derivatives by Maximal
functions. We will see
in the subsection \ref{new_step4}
that the power $({1+\frac{3}{p}})$ of $M$
on the right hand side of the following estimate is very important
to obtain a required estimate \eqref{step4_third_claim}.
\begin{lem}\label{lem_Maximal 2.5 or 4}
Let $M_0>0$ and $1\leq p <\infty$. Then there exist $C=C(M_0,p)$ with
the following property: \\
For any $M\geq M_0$ and
for any $f\in C^1(\mathbb{R}^3)$
such that $ \int_{\mathbb{R}^3}\phi(x)f(x)dx=0$, we have
\begin{equation*}
\|f\|_{L^p(B(M))}\leq CM^{1+\frac{3}{p}}\cdot\Big(
\|\mathcal{M}(|\nabla f|^p)\|^{1/p}_{L^{1}(B(1))}
+\|\nabla f\|_{L^1(B(2))}\Big).
\end{equation*}
\end{lem}
\begin{proof}
See the appendix.
\end{proof}
With the above lemmas, we are ready to prove the proposition \ref{local_study_thm}.
\begin{proof} [Proof of proposition \ref{local_study_thm}]
We divide this proof into three stages.\\
Stage 1 in subsection \ref{new_step1_2_together}: First, we will obtain a $L_t^{\infty}L_x^2$-local bound for $u$
by using the mean-zero property of $u$ and $w$.
Then, a $L^\infty$-local bound of $u$
follows thanks to the first local study
proposition \ref{partial_problem_II_r}.\\
Stage 2 in subsection \ref{new_step3}: We will get a $L^\infty$-local bound for $\nabla^d u$ for $d\geq1$ by using
an induction argument with
a boot-strapping.
This is not obvious especially when $r$ is large because $w$
depends a non-local part of $u$ while our knowledge about the $L^\infty$-bound of
$u$ from the stage 1 is only local.\\
Stage 3 in subsection \ref{new_step4}: We will achieve a $L^\infty$-local bound for $(-\Delta)^{\alpha/2}\nabla^d u$ for $d\geq1$ with $0<\alpha<2$ from the integral representation
of the fractional Laplacian. The non-locality of this fractional operator
will let us to adopt more complicated conditions (see \eqref{local_study_condition3}).\\
\subsection{Stage 1: to obtain $L^{\infty}$-local bound for $u$.}\label{new_step1_2_together}
First we suppose that $u$ satisfies all conditions of the proposition \ref{local_study_thm}
without \eqref{local_study_condition3} (The condition \eqref{local_study_condition3} will be assumed only at the stage 3). Our goal is to find a sufficiently small $\bar{\eta}$ which is independent of
$r\in[0,\infty)$.\\
Assume $\bar{\eta}\leq 1$ and
define $\bar{r}_0=\frac{1}{4}$ for this subsection.
From \eqref{local_study_condition1}, we get
\begin{equation*}
\|u\|_{L^2(-4,0;L^{6}(B(2)))}\leq C\|\nabla u\|_{L^2(-4,0;L^{2}(B(2)))}
\leq {C}\cdot\bar{\eta}.
\end{equation*}
From the corollary \ref{convolution_cor}, if $r\geq\bar{r}_0$, then
\begin{equation*}\begin{split}
\| w\|_{L^2(-4,0;L^{\infty}(B(2)))}&
\leq {C}\cdot\bar{\eta}.
\end{split}\end{equation*}
On the other hand, if $0\leq r<\bar{r}_0$, then
\begin{equation*}\begin{split}
\| w^{\prime}\|_{L^2(-4,0;L^{6}(B(\frac{7}{4})))}
&\leq {C}\| u\|_{L^2(-4,0;L^{6}(B(2)))}
\leq {C}\bar{\eta}
\end{split}\end{equation*} because $\phi_r$ is supported
in $B(r)\subset B(\bar{r})$, and $w=u*\phi_r$ (see \eqref{young}).
For $w^{\prime\prime}$,
\begin{equation*}\begin{split}
&\| w^{\prime\prime}\|_{L^2(-4,0;L^{\infty}(B(2)))}
\leq\|\|u*\phi_{r}\|_{L^1(B(1))}\|_{L^2((-4,0))}
\leq\|\|u\|_{L^1(B(2))}\|_{L^2((-4,0))}\\
&\leq {C}\| u\|_{L^2(-4,0;L^{6}(B(2)))}
\leq {C}\bar{\eta}.
\end{split}\end{equation*}
Thus $\| w\|_{L^2(-4,0;L^{6}(B(\frac{7}{4})))}
\leq {C}\bar{\eta}$ if $r<\bar{r}_0$ from $w= w^{\prime} +w^{\prime\prime}$.\\
In sum, for any $0\leq r<\infty$,
\begin{equation}\label{step1_w}
\| w\|_{L^2(-4,0;L^{6}(B(\frac{7}{4})))}\leq {C}\bar{\eta}.
\end{equation}
Since the equation \eqref{navier_Problem II-r} depends only on $\nabla P$,
without loss of generality, we may assume
$\int_{\mathbb{R}^3}\phi(x)P(t,x)=0$ for $t\in(-4,0)$. Then with the mean zero property \eqref{local_study_condition1} of $u$, we have
\begin{equation*}\label{step1_nabla_p}
\|\int_{\mathbb{R}^3}\phi(x)\nabla P(\cdot,x) dx\|_{L^1(-4,0)}\leq C \bar{\eta}^{\frac{1}{2}}
\end{equation*} after integration in $x$.\\
From Sobolev,
\begin{equation*}\begin{split}
\|\nabla P\|
_{L^1(-4,0;L^{\frac{3}{2}}(B(\frac{7}{4}))}
&\leq C \bar{\eta}^{\frac{1}{2}}\\
\mbox{ and } \quad \| P\|_{L^1(-4,0;L^{3}(B(\frac{7}{4}))}
&
\leq C \bar{\eta}^{\frac{1}{2}}\\
\end{split}
\end{equation*}
Then we follows step 1 and step 2 of the proof of the proposition 10 in \cite{vas:higher},
we can obtain
\begin{equation*}
\| u\|_{L^{\infty}(-3,0;L^{\frac{3}{2}}(B(\frac{6}{4})))}
\leq {C}\bar{\eta}^{\frac{1}{3}}.\\
\end{equation*} and then
\begin{equation*}
\| u\|_{L^{\infty}(-2,0;L^{2}(B(\frac{5}{4})))}\leq {C}\bar{\eta}^{\frac{1}{4}}
\end{equation*} for $0\leq r<\infty$. Details are omitted.\\
Finally, by taking $0<\bar\eta<1$ such that $C\bar\eta^\frac{1}{4}\leq\bar\delta$,
we have all assumptions of the proposition \ref{partial_problem_II_r}. As a result,
we have
$|u(t,x)|\leq 1 \mbox{ on } [-\frac{3}{2},0]\times B(\frac{1}{2})$.\\
\subsection{Stage 2: to obtain $L^\infty$ local bound for $\nabla^d u$.}\label{new_step3}
Here we cover only classical derivatives, i.e. $\alpha=0$.
For any integer $d\geq1$, our goal is to find $C_{d,0}$ such that
$|((-\Delta)^{\frac{0}{2}}\nabla^d) u(t,x)|=|\nabla^d u(t,x)|\leq C_{d,0}$ on
$(-(\frac{1}{3})^2,0)\times(B(\frac{1}{3}))$. \\
We define a strictly decreasing sequence of balls
and parabolic cylinders from
$(-(\frac{1}{2})^2,0)\times B(\frac{1}{2})$
to $(-(\frac{1}{3})^2,0)\times(B(\frac{1}{3}))$ by
\begin{equation*}\begin{split}
&\bar{B}_n =B(\frac{1}{3}+\frac{1}{6}\cdot 2^{-n})=B(l_n) \\
&\bar{Q}_n=(-(\frac{1}{3}+\frac{1}{6}\cdot 2^{-n})^2,0)\times \bar{B}_n
=(-(l_n)^2,0)\times \bar{B}_n
\end{split}\end{equation*} where $l_n=\frac{1}{3}+\frac{1}{6}\cdot 2^{-n}$.\\
First we claim in order to cover the small $r$ case:\\
There exist two positive sequences
$\{\bar{r}_n\}_{n=0}^{\infty}$
and $\{C_{n,small}\}_{n=0}^{\infty}$ such that for any integer $n\geq0$
and for any $r\in [0,\bar{r}_n)$,
\begin{equation}\begin{split}\label{step3_first_claim}
&\| \nabla^n u\|_{L^{\infty}(\bar{Q}_{11n})}\leq C_{n,small}.
\end{split}\end{equation}
Indeed, from the previous subsection \ref{new_step1_2_together} (the stage 1), \eqref{step3_first_claim} holds for $n=0$
by taking $\bar{r}_0=1$ and $C_{0,small}=1$.
We define $\bar{r}_n= $ distance between
$B_{11n}$ and $(B_{11n-1})^c$ for $n \geq 1$.
Then $\{\bar{r}_n\}_{n=0}^{\infty}$ is decreasing
to zero as $n$ goes to $\infty$. Moreover,
we
can control $w$ by $u$
as long as $0\leq r<\bar{r}_n$: for any $n\geq 1$,
\begin{equation}\begin{split}\label{intermidiate balls}
&\|w\|_{L^{p_1}(-(l_{m})^2,0;L^{p_2}({\bar{B}_{m}}))}\leq
\Big(\|u\|_{L^{p_1}(-(l_{m-1})^2,0;L^{p_2}({\bar{B}_{m-1}}))} + C\Big) \quad\mbox{ and}\\
&\|\nabla^{k}w\|_{L^{p_1}(-(l_{m})^2,0;L^{p_2}({\bar{B}_{m}}))}\leq
\|\nabla^{k}u\|_{L^{p_1}(-(l_{m-1})^2,0;L^{p_2}({\bar{B}_{m-1}}))}
\end{split}\end{equation} for any integer $m$ such that
$m\leq 11\cdot n$, for any $ k\geq1$
and for any $p_1\in [1,\infty]$ and $p_2\in [1,\infty]$ (see \eqref{young}).\\% $1\leq p_1,p_2\leq\infty$.\\
We will use an induction with a boot-strapping.
First we fix $d\geq 1$ and suppose that \eqref{step3_first_claim} is true up to $n=(d-1)$. It implies
for any $r\in [0,\bar{r}_{d-1})$
\begin{equation*}\begin{split}
\|u\|_{L^{\infty}(-l_{s}^2,0;W^{d-1,{\infty}} (\bar{B}_{s}))}
\leq C
\end{split}\end{equation*} where $s=11(d-1)$.
We want to
show that \eqref{step3_first_claim} is also true for the case $n=d$.\\
\noindent From \eqref{intermidiate balls},
$\|w\|_{L^{\infty }(-l_{s+{1 }}^2,0;W^{{ d-1 } ,{ \infty}} (\bar{B}_{s+{ 1 }}))}\leq C$ and,
From the lemma \ref{lem_parabolic_v_1_v_2_pressure} with
$v_2=w$ and $v_1=u$,\quad
$\|u\|_{L^{16}(-l_{s+{ 2}}^2,0;W^{{ d } ,{32 }} (\bar{B}_{s+{ 2 }}))}\leq C$.
Then, we use \eqref{intermidiate balls}
and the lemma \ref{lem_parabolic_v_1_v_2_pressure} in turn:
\begin{equation*}\begin{split}
&\rightarrow w\in{L^{16}(-l_{s+{ 3}}^2,0;W^{{ d } ,{32 }} (\bar{B}_{s+{3 }}))}
\rightarrow
u\in{L^{8}(-l_{s+{ 4}}^2,0;W^{{ d+1 } ,{16 }} (\bar{B}_{s+{ 4 }}))}\\
&\rightarrow w\in{L^{8}W^{{ d+1 } ,{16 }} }
\rightarrow ... \rightarrow
u\in{L^{2}W^{{ d+3 } ,{4 }} }
\end{split}\end{equation*} Then, from Sobolev,
\begin{equation*}\begin{split}
&\rightarrow u\in{L^{2}W^{{ d+2 } ,{\infty}} }\rightarrow
w\in{L^{2}(-l_{s+{ 9}}^2,0;W^{{ d+2 } ,{\infty}} (\bar{B}_{s+{ 9 }}))}.
\end{split}\end{equation*}
This estimate gives us
\begin{equation*}\begin{split}
&\Delta(\nabla^d u) ,
\ebdiv(\nabla^d(w\otimes u))\mbox{ and }
\nabla(\nabla^dP) \in{L^{1}(-l_{s+{ 10}}^2,0;L^{\infty}(\bar{B}_{s+{10 }}))}
\end{split}\end{equation*}
where
we used
\eqref{ineq_parabolic_pressure2}
for the pressure term. Thus
\begin{equation*}\begin{split}
&\partial_t(\nabla^d u)\in{L^{1}(-l_{s+{ 10}}^2,0;L^{\infty}(\bar{B}_{s+{10 }}))}.
\end{split}\end{equation*}
Finally, we obtain that for any $r\in [0,\bar{r}_{d})$
\begin{equation*}\begin{split}
&\|\nabla^d u\|_{L^{\infty}(-l_{s+{ 11}}^2,0;L^{\infty}(\bar{B}_{s+{11 }}))}\leq C.
\end{split}\end{equation*} where $C$ depends only on $d$.
By the induction argument, we showed the above claim \eqref{step3_first_claim}. \\
Now we introduce the second claim:\\
There exist a sequences
$\{C_{n,large}\}_{n=0}^{\infty}$ such that for any integer $n\geq0$
and for any $r\geq\bar{r}_n$,
\begin{equation}\begin{split}\label{step3_second_claim}
&\| \nabla^n u\|_{L^{\infty}(\bar{Q}_{21\cdot n})}\leq C_{n,large} \\
\end{split}\end{equation} where $\bar{r}_n$ comes from previous claim
\eqref{step3_first_claim}.\\
Before proving the above second claim \eqref{step3_second_claim},
we need the following two observations \textbf{(I),(II)}
from the lemmas \ref{lem_parabolic_v_1_v_2_pressure} and
\ref{higher_pressure}:\\
\textbf{(I).} From the corollary \ref{convolution_cor} for any $n\geq 0$,
if $r\geq\bar{r}_n$, then
\begin{equation*}\begin{split}\label{nabla_n_w_large_r}
\| w\|_{L^2(-4,0;W^{n,\infty}(B(2)))}&
\leq {C_n}.
\end{split}\end{equation*}
We use \eqref{ineq_parabolic_v_1_v_2} in the lemma \ref{lem_parabolic_v_1_v_2_pressure}
with $v_1=u$ and $v_2=w$.
Then it
becomes
\begin{equation}\begin{split} \label{ineq_nabla_n_u_large_r}
\|\nabla^n u \|_{L^{p_1}(-(l_m)^2,0 ;L^{p_2}(\bar{B}_m))}
&\leq C_{(m,n,p_2)}
\Big(\| u \|_{L^{\frac{2p_1}{2-p_1}}
(-(l_{m-1})^2,0;W^{n-1,p_2}({\bar{B}_{m-1}}))}+
1\Big)\\
\end{split}\end{equation}
for $n\geq 1$, $m\geq 1$, $1< p_1\leq 2$ and $1<p_2<\infty$. (For
the case $p_1=2$, we may interpret $\frac{2p_1}{2-p_1}=\infty$.)\\
\textbf{(II).} Moreover, \eqref{ineq_parabolic_pressure} in the lemma
\ref{higher_pressure} becomes
\begin{equation}\begin{split} \label{ineq_nabla_n_pressure_large_r}
\|\nabla^n P\|_{L^{1}(-(l_m)^2,0;L^{p}(\bar{B}_m))}
&\leq C_{(m,n,p)}\Big(
\| u \|_{L^{2}(-(l_{m-1})^2,0;W^{n-1,p}({\bar{B}_{m-1}}))}+1\Big)
\end{split}\end{equation} for $n\geq 2$ and $1<p<\infty$.\\
Now we are ready to prove the second claim \eqref{step3_second_claim}
by an induction with a boot-strapping. From the previous subsection
\ref{new_step1_2_together} (the stage 1), \eqref{step3_second_claim} holds for $n=0$
with $C_{0,large}=1$. Fix $d\geq 1$ and suppose that we
have \eqref{step3_second_claim} up to $n=(d-1)$. It implies
for any $r\geq\bar{r}_{d-1}$
\begin{equation*}\begin{split}
\|u\|_{L^{\infty}(-l_s^2,0;W^{d-1,{\infty}} (\bar{B}_{s}))}
\leq C_{d-1,large}
\end{split}\end{equation*} where $s=21(d-1)$.
We want to
show \eqref{step3_second_claim} for $n=d$.\\
\noindent By using \eqref{ineq_nabla_n_u_large_r} with $n=d, p_1=2$ and $p_2=11$,
\begin{equation*}\begin{split}
\|u\|_{L^{ 2}(-l_{s+{1 }}^2,0;W^{{ d } ,{11 }} (\bar{B}_{s+{ 1 }}))}\leq C
\end{split}\end{equation*}
and,
from \eqref{ineq_nabla_n_pressure_large_r} with $n=d+1,m=0$ and $p=11$,
\begin{equation*}\begin{split}
\|\nabla^{d+{1 }}P\|_{L^{1 }(-l_{s+{ 2}}^2,0;L^{ 11 } (\bar{B}_{s+{ 2}}))}\leq C.
\end{split}\end{equation*}
Combining the above two results with the
lemma \ref{lem_a_half_upgrading_large_r} for $v_1=u$ and $v_2=w$
, we can have increased integrability in space by $0.5$ up to $6$:
\begin{equation*}\begin{split}
&\|u\|_{L^{\infty }(-l_{s+{3 }}^2,0;W^{{ d } ,{ 1}} (\bar{B}_{s+{ 3 }}))}\leq C,\\
&\|u\|_{L^{\infty }(-l_{s+{ 4}}^2,0;W^{{ d } ,{1.5 }} (\bar{B}_{s+{ 4 }}))}\leq C,\\
&\quad\quad\quad\quad\quad\quad\cdots, \quad\mbox{ and} \\
&\|u\|_{L^{\infty }(-l_{s+{ 13}}^2,0;W^{{ d } ,{6 }} (\bar{B}_{s+{ 13 }}))}\leq C.\\
\end{split}\end{equation*}
By using \eqref{ineq_nabla_n_u_large_r} and \eqref{ineq_nabla_n_pressure_large_r}
again, we have
\begin{equation*}\begin{split}
&\|u\|_{L^{ 2}(-l_{s+{14 }}^2,0;W^{{d+1 } ,{ 6}} (\bar{B}_{s+{ 14 }}))}\leq C
\quad\mbox{and}\\
&\|\nabla^{d+{ 2 }}P\|_{L^{1 }(-l_{s+{ 15}}^2,0;L^{6 } (\bar{B}_{s+{ 15 }}))}\leq C.
\end{split}\end{equation*}
Combining the above two results with the lemma \ref{lem_a_half_upgrading_large_r} again
, we have
\begin{equation*}\begin{split}
&\|u\|_{L^{\infty }(-l_{s+{16 }}^2,0;W^{{ d+1 } ,{ 1}} (\bar{B}_{s+{ 16 }}))}\leq C,\\
&\quad\quad\quad\quad\quad\quad\cdots,\quad\mbox{ and} \\
&\|u\|_{L^{\infty }(-l_{s+{21 }}^2,0;W^{{ d+1} ,{ 3.5}} (\bar{B}_{s+{21 }}))}\leq C.\\
\end{split}\end{equation*}
Finally, from Sobolev's inequality,
\begin{equation*}\begin{split}
&\|\nabla^d u\|_{L^{\infty }(-l_{s+{21 }}^2,0;L^{\infty} (\bar{B}_{s+{ 21 }}))}
\leq C
\end{split}\end{equation*} where $C$ depends
only $d$ not $u$ nor $r$ as long as $r\geq \bar{r}_d$.
From induction, we proved second claim \eqref{step3_second_claim}.\\
Define for any $n\geq0$, $C_{n,0} = \max({C_{n,small},C_{n,large}})$ where
$C_{n,small}$ and $C_{n,large}$ come from \eqref{step3_first_claim}
and \eqref{step3_second_claim} respectively.
Then we have:
\begin{equation}\begin{split}\label{step3_conclusion}
&\| \nabla^n u\|_{L^{\infty}(Q(\frac{1}{3}))}\leq C_{n,0} \\
\end{split}\end{equation} for any $n\geq0$ and for any $0\leq r<\infty$
because $Q(\frac{1}{3})\subset \bar{Q}_{n}$.
It ends this stage 2.\\
\subsection{Stage 3: to obtain $L^\infty$ local bound for $(-\Delta)^{\alpha/2}\nabla^d u$.}\label{new_step4}
From now on, we
assume further that $(u,P)$ satisfies \eqref{local_study_condition3} as well as
all the other conditions of the proposition \ref{local_study_thm}.
In the following proof, we will not
divide the proof into a small $r$ part and a large $r$ part.\\% as you will see.
Fix an integer $d\geq 1$ and
a real $\alpha$ with $0<\alpha<2$.
i.e. any constant which will appear may depend $d$ and $\alpha$.
But they will be independent of any $ r\in [0,\infty)$ and any solution $(u,P)$.\\
First, we claim:\\
There exists a constant $C=C({d,\alpha})$
such that
\begin{equation}\begin{split}\label{step4_first_claim}
|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u(t,x)|
\leq C({d,\alpha})+\Big|\int_{|y|\geq {(1/6)}}
\frac{\nabla^{d} u(t,x-y)}{|y|^{3+\alpha}}dy\Big|
\end{split}\end{equation} for $ |x|\leq (1/6)$ and
for $ -(1/3)^2\leq t\leq 0$.\\% for $-{(a_{n+1})}^2\leq t\leq 0$.\\
\noindent To prove \eqref{step4_first_claim},
we first recall the Taylor expansion of any $C^2$ function $f$ at $x$:
$f(y)-f(x)=(\nabla f)(x)\cdot(y-x)+R(x,y)$, and we have
an error estimate $|R|\leq C|x-y|^2\cdot\|\nabla^2 f\|_{L^\infty(B(x;|x-y|))}$.
Note that if we integrate the first order term $(\nabla f)(x)\cdot(y-x)$
in $y$ on any sphere with the center $x$, we have zero by symmetry.
As a result, if we
take any $x$ and $t$ for $ |x|\leq (1/6)$ and
for $ -(1/3)^2\leq t\leq 0$ respectively, then we have
\begin{equation*}\begin{split}
|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u(t,x)|&
=\Big|P.V.\int_{\mathbb{R}^3}\frac{\nabla^d u(t,x)-\nabla^d u(t,y)}{|x-y|^{3+\alpha}}dy\Big|\\
&\leq \sup_{z\in B((1/3))}(|\nabla^{d+2}u(t,z)|)\cdot\int_{|x-y|< (1/6)}\frac{1}{|x-y|^{3+\alpha-2}}dy\\
&\quad+ \sup_{z\in B((1/3))}(|\nabla^d u(t,z)|)\cdot
\int_{|x-y|\geq (1/6)}\frac{1}{|x-y|^{3+\alpha}}dy\\
&\quad+\Big|\int_{|x-y|\geq (1/6)}\frac{\nabla^d u(t,y)}{|x-y|^{3+\alpha}}dy\Big|\\
&\leq C({d,\alpha}) +\Big|\int_{|y|\geq (1/6)}
\frac{\nabla^d u(t,x-y)}{|y|^{3+\alpha}}dy\Big|
\end{split}\end{equation*}
where we used
the result \eqref{step3_conclusion} of the previous
subsection \ref{new_step3} (the stage 2)
together with the Taylor expansion of
$\nabla^d u(t,\cdot)$ at $x$
in order to reduce singularity by $2$ at the origin $x=y$. We
proved the first claim \eqref{step4_first_claim}.\\
Second, we claim:\\
There exists $C=C({d,\alpha})$ such that
\begin{equation}\begin{split}\label{step4_second_claim}
\Big|\int_{|y|\geq {{(1/6)}}}\frac{\nabla^{d} u(t,x-y)}
{|y|^{3+\alpha}}dy\Big|
\leq C({d,\alpha})+\sum_{j=k}^{\infty}(\frac{1}{2^{\alpha}})^j\cdot
|({(h^{\alpha})_{2^j}}*\nabla^d u)(t,x)|\\
\end{split}\end{equation}
for $ |x|\leq (1/6)$ and
for $ -(1/3)^2\leq t\leq 0$ where
$k$ is the integer such that
$2^k\leq (1/6)< 2^{k+1}$.(i.e. from now on, we fix $ k=-3$). Recall that $ h^\alpha$ is defined around \eqref{property_h}.\\%(k may be negative)\\
\noindent To prove the above second claim \eqref{step4_second_claim}: (Recall \eqref{property_zeta}
and \eqref{property_h})
\begin{equation*}\begin{split}
&\Big|\int_{|y|\geq {{(1/6)}}}\frac{\nabla^{d} u(t,x-y)}
{|y|^{3+\alpha}}dy\Big|=
\Big|\int_{|y|\geq {{(1/6)}}}\sum_{j=k}^{\infty}\zeta
(\frac{y}{2^j})\frac{\nabla^{d} u(t,x-y)}
{|y|^{3+\alpha}}dy\Big|\\
&=\Big|\int_{|y|\geq {{(1/6)}}}\sum_{j=k}^{\infty}
\frac{1}{(2^j)^{\alpha }} \cdot {(h^{\alpha})_{2^j}}(y)\nabla^{d} u(t,x-y)dy\Big|\\
\end{split}\end{equation*}
\begin{equation*}\begin{split}
&\leq\sum_{j=k}^{k+1}\frac{1}{(2^j)^{\alpha }}
\cdot\Big|\int_{|y|\geq {{(1/6)}}} {(h^{\alpha})_{2^j}}(y)\nabla^{d} u(t,x-y)dy\Big|\\
&\quad+\sum_{j=k+2}^{\infty}\frac{1}{(2^j)^{\alpha }}
\cdot\Big|\int_{|y|\geq {{(1/6)}}} {(h^{\alpha})_{2^j}}(y)\nabla^{d} u(t,x-y)dy\Big|\\
&=(I)+(II).
\end{split}\end{equation*}
\noindent For $(I)$,
\begin{equation*}\begin{split}
(I)&\leq\sum_{j=k}^{k+1}\frac{1}{(2^j)^{\alpha }}
\cdot\Big(\Big|\int_{\mathbb{R}^3} {(h^{\alpha})_{2^j}}(y)\nabla^{d} u(t,x-y)dy\Big|\\
&\quad\quad\quad\quad\quad\quad\quad\quad +
\int_{|y|\leq {{(1/6)}}} |{(h^{\alpha})_{2^j}}(y)|\cdot|\nabla^{d} u(t,x-y)|dy\Big)\\
&\leq\sum_{j=k}^{k+1}\frac{1}{(2^j)^{\alpha }}
\Big(| ({(h^{\alpha})_{2^j}}*\nabla^{d} u)(t,x)|
+ C\cdot \sup_{z\in B(1/3)}|\nabla^{d} u(t,z)|\Big)\\
&= \sum_{j=k}^{k+1}(\frac{1}{2^{\alpha}})^j\cdot| ({(h^{\alpha})_{2^j}}
*\nabla^{d} u)(t,x)|+C({d,\alpha}).
\end{split}\end{equation*}
\noindent For $(II)$, by using $supp( h^{\alpha}_{2^j})
\subset (B(2^{j-1}))^C\subset (B(1/6))^C$
for any $j\geq k+2$,
\begin{equation*}\begin{split}
(II)&=
\sum_{j=k+2}^{\infty}\frac{1}{(2^j)^{\alpha }}
\cdot\Big|\int_{\mathbb{R}^3} {(h^{\alpha})_{2^j}}(y)\nabla^{d} u(t,x-y)dy\Big|\\
&= \sum_{j=k+2}^{\infty}(\frac{1}{2^{\alpha}})^j
\cdot| ({(h^{\alpha})_{2^j}}*\nabla^{d} u)(t,x)|.
\end{split}\end{equation*} We showed the second claim
\eqref{step4_second_claim}.\\
Third, we claim:\\%that
There exists $C=C({d,\alpha})$ such that
\begin{equation}\begin{split}\label{step4_third_claim}
\|{(h^{\alpha})_M}*\nabla^d u\|_{L^{\infty}( -(1/6)^2,0;L^1(B(1/6)))}
\leq C({d,\alpha})\cdot M^{1-d}\\
\end{split}\end{equation} for any $M\geq 2^k$. (Recall
$k= -3$.)\\% is the integer such that
\noindent To prove the above third claim \eqref{step4_third_claim},
take a convolution first with
$ \nabla^d[{(h^{\alpha})_M}]$ into the equation
\eqref{navier_Problem II-r}.
Then we have
\begin{equation*}\begin{split}
(\nabla^d [{(h^{{\alpha}})_M}]*u)_t &+(\nabla^d [{(h^{{\alpha}})_M}]*\Big((w\cdot\nabla)u\Big))\\&
+(\nabla^d [{(h^{{\alpha}})_M}]*\nabla P) -(\nabla^d [{(h^{{\alpha}})_M}]*\Delta u)=0
\end{split}\end{equation*} so that
\begin{equation*}\begin{split}
( \nabla^{d-1}[{(h^{{\alpha}})_M}]&*\nabla u)_t +(\nabla^d {[(h^{{\alpha}})_M}]*\Big((w\cdot\nabla)u\Big))\\& +(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla^2 P) -\Delta(
\nabla^{d-1}
[{(h^{{\alpha}})_M}]* \nabla u)=0.
\end{split}\end{equation*}
\noindent Define a cut-off $\Phi(t,x)$ by
\begin{equation*}\begin{split}
&0\leq\Phi(x)\leq 1 \quad , \quad
supp(\Phi)\subset (-4,0)\times B({2})\\
&\Phi(t,x) = 1 \mbox{ for } (t,x)\in (-(1/6)^2,0)\times B({(1/6)}).
\end{split}
\end{equation*}
\noindent Multiply $\Phi(t,x)\frac{(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u)(t,x)}{|(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u) (t,x)|}$, then
integrate in $x$:
\begin{equation*}\begin{split}
&\frac{d}{dt}\int_{\mathbb{R}^3}\Phi(t,x)|(\nabla^{d-1}[{(h^{{\alpha}})_M}]*
\nabla u) (t,x)|dx\\
&\leq\int_{\mathbb{R}^3}(|\partial_t\Phi(t,x)|+|\Delta\Phi(t,x)|)
|(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u) (t,x)|dx \\
&\quad\quad+\int_{\mathbb{R}^3}|\Phi(t,x)||(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla^2 P)|dx \\
&\quad\quad+\int_{\mathbb{R}^3}|\Phi(t,x)||\nabla^d [{(h^{{\alpha}})_M}]*
\Big((w\cdot\nabla)u\Big)|dx.
\end{split}\end{equation*}
\noindent Then integrating on $[-4,t]$ for any $t\in[-(1/6),0]$ gives
\begin{equation*}\begin{split}
&\|{(h^{{\alpha}})_M}*\nabla^d u\|_{L^{\infty}( -(1/6)^2,0;L^1(B(1/6)))}\\
&=\|\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u\|_{L^{\infty}( -(1/6)^2,0;L^1(B(1/6)))}\\
&\leq C\Big(\|\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u\|_{L^{1}( -4,0;L^1(B(2)))} \\
&+ \|\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla^2 P\|_{L^{1}( -4,0;L^1(B(2)))} \\
&+ \|\nabla^d [{(h^{{\alpha}})_M}]*\Big((w\cdot\nabla)u\Big)\|
_{L^{1}( -4,0;L^1(B(2)))}\Big) \\
&=(I) + (II)+ (III).
\end{split}\end{equation*}
For $(I)$, we use simple observations
$\nabla^m[(f)_\delta]=\delta^{-m}\cdot(\nabla^mf)_\delta$
and \\$|(f)_\delta*\nabla u|(x)\leq C_f\cdot\mathcal{M}(|\nabla u|)(x) $
for any $f\in C^\infty_0(\mathbb{R}^3)$ so that
\begin{equation*}\begin{split}
|(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u)(t,x)|&
=M^{-(d-1)}\cdot|((\nabla^{d-1}{h^{{\alpha}})_M}*\nabla u)(t,x)|\\
&\leq C\cdot M^{-(d-1)}\cdot\mathcal{M}(|\nabla u|)(t,x)
\end{split}\end{equation*} for any $0<M<\infty$ so that
\begin{equation*}\begin{split}
(I)&=\|(\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla u)\|_{L^{1}( -4,0;L^1(B(2)))}\\
&\leq C\cdot M^{-(d-1)}\cdot\|\mathcal{M}(|\nabla u|)\|_{L^{1}( -4,0;L^1(B(2)))}\\
&\leq C\cdot M^{-(d-1)}\cdot\|\mathcal{M}(|\nabla u|)\|_{L^{2}( -4,0;L^2(B(2)))}
\leq C\cdot M^{1-d}
\end{split}\end{equation*} for any $0<M<\infty$.\\
For $(II)$, we use our global information about pressure in \eqref{local_study_condition3}
thanks to the property of the Hardy space \eqref{hardy_property}:
\begin{equation}\begin{split}\label{pressure_hardy_used}
(II)&=\|\nabla^{d-1}[{(h^{{\alpha}})_M}]*\nabla^2 P\|_{L^{1}( -4,0;L^1(B(2)))}\\
&=M^{-(d-1)}\cdot\|(\nabla^{d-1}{h^{{\alpha}})_M}*\nabla^2 P\|_{L^{1}( -4,0;L^1(B(2)))}\\
&\leq M^{-(d-1)}\cdot
\| \sup_{\delta>0}(|(\nabla^{d-1}{h^{{\alpha}})_\delta}*\nabla^2 P|)\|_{L^{1}(-4,0;L^{1}(B(2)))}\\&
\leq C \cdot M^{1-d}
\end{split}\end{equation} for any $0<M<\infty$.\\
For $(III)$, we use following useful facts \textbf{(1, }$\cdots$\textbf{, 5)}:\\
\noindent \textbf{1.} From $supp((h^{{\alpha}})_M)\subset B(2M)$,
\begin{equation*}\begin{split}
\quad&\|\nabla^d [{(h^{{\alpha}})_M}]*\Big((w\cdot\nabla)u\Big)(t,\cdot)\|
_{L^1(B(2))} \\
&\leq\int_{B(2)}
\int_{\mathbb{R}^3}\Big|\Big((w\cdot\nabla)u\Big)(t,y)\cdot (\nabla^d [{(h^{{\alpha}})_M}])(x-y)\Big|dy
dx\\
&\leq\int_{B(2M+2)}\Big|\Big((w\cdot\nabla)u\Big)(t,y)\Big|\cdot\Big[
\int_{{B(2)}}| (\nabla^d [{(h^{{\alpha}})_M}])(x-y)|dx \Big]
dy\\
&\leq C\|\nabla^d [{(h^{{\alpha}})_M}]\|_{L^{\infty}(\mathbb{R}^3)}\cdot
\|\Big((w\cdot\nabla)u\Big)(t,\cdot)\|_{L^1{(B(2M+2))}}
\\
&\leq C \cdot\frac{1}{M^{3+d}}\cdot
\|\Big((w\cdot\nabla)u\Big)(t,\cdot)\|_{L^1{(B(2M+2))}}
\\
&\leq C \cdot\frac{1}{M^{3+d}}\cdot
\|w(t,\cdot)\|_{L^{q^\prime}{(B(2M+2))}}
\cdot\|\nabla u(t,\cdot)\|_{L^q{(B(2M+2))}}
\end{split}\end{equation*} where $q=12/(\alpha +6)$ and $1/q+1/q^\prime = 1$.\\
Note: Because $0<\alpha<2$, we know $12/8<q<2$.
\begin{equation*}\begin{split}
\textbf{2.}\quad& \|w(t,\cdot)\|_{L^{q}{(B(2M+2))}}\\
&\quad\leq
CM^{1+\frac{3}{q}}\cdot\Big(
\|\mathcal{M}(|\nabla w|^q)(t,\cdot)\|^{1/q}_{L^{1}(B(1))}
+\|\nabla w(t,\cdot)\|_{L^1(B(2))}\Big)\\
&\quad\leq
CM^{1+\frac{3}{q}}\cdot\Big(
\|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)(t,\cdot)\|^{1/q}_{L^{1}(B(1))}
+\|\mathcal{M}(|\nabla u|)(t,\cdot)\|_{L^1(B(2))}\Big)\\
\end{split}\end{equation*} for any $M\geq 2^k$.\\
\noindent For the first inequality, we used the lemma \ref{lem_Maximal 2.5 or 4}
and for the second one, we used the fact
$|\nabla w (t,x)|=|(\nabla u * \phi_r)(t,x)|\leq
C|\mathcal{M}(|\nabla u|)(t,x)|$ where
$C$ is independent of $0\leq r<\infty$. (For $r>0$, it follows
definitions of the convolution and the Maximal function while
for $r=0$, it follows the Lebesgue differentiation theorem with
continuity of $\nabla u$.)
So, for any $M\geq 2^k$, from \eqref{local_study_condition3},
\begin{equation*}\begin{split}
&\|w\|_{L^2(-4,0;L^q(B(2M+2)))}\\
&\quad\leq
CM^{1+\frac{3}{q}}\Big(
\|\|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)\|^{1/q}_{L_x^{1}(B(1))}\|
_{L_t^2(-4,0)}\\
&\quad\quad\quad\quad\quad\quad+\|\|\mathcal{M}(|\nabla u|)\|_{L_x^1(B(2))}\|
_{L_t^2(-4,0)}\Big)\\
&\quad\leq
CM^{1+\frac{3}{q}}\Big(
\|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)\|^{1/q}_{L^{2/q}(-4,0;L^1(B(2)))}\\
&\quad\quad\quad\quad\quad\quad+\|\|\mathcal{M}(|\nabla u|)\|_{L^2(-4,0;L^1(B(2)))}\Big)\\
&\quad\leq
CM^{1+\frac{3}{q}}.
\end{split}\end{equation*}
Before stating the third fact, we needs the following two observations:\\
From standard Sobolev-Poincare inequality on balls (e.g. see
Saloff-Coste \cite{sobolev}), we have $C$ such that
\begin{equation}\begin{split}\label{Sobolev-Poincare inequality}
\|f-\Bar{f}\|_{L^{3q/(3-q)}(B(M))}\leq C\cdot\|\nabla f\|_{L^q(B(M))}
\end{split}\end{equation} for any $0 <M <\infty$ and for any $f$
whose derivatives are in $L^q_{loc}(\mathbb{R}^3)$ where $\Bar{f}
=\int_{B}fdx/|B|$ is
the mean value on $B$. Note that $C$ is independent of $M$.\\
On the other hand, once we fix $M_0>0$, then there exist $C=C(M_0)$ with
the following property: \\
For any $p$ with $1\leq p<\infty$, for any $M\geq M_0$
and for any $f\in L^p_{loc}(\mathbb{R}^3)$, we have
\begin{equation}\label{lem_Maximal q}
\|f\|_{L^p(B(M))}\leq CM^{\frac{3}{p}}\cdot
\|\mathcal{M}(|f|^p)\|^{1/p}_{L^{1}(B(2))}
\end{equation}
To prove \eqref{lem_Maximal q}, it is enough to show that
\begin{equation*}
\|g\|_{L^1(B(M))}\leq CM^{3}\cdot
\|\mathcal{M}(g)\|_{L^{1}(B(2))}
\end{equation*}
For any $z\in B(2)$,
\begin{equation*}\begin{split}
\int_{B(M)} |g(x)|dx&=\frac{(M+2)^3}{(M+2)^3}\cdot\int_{B(M+2)} |g(z+x)|dx\\
&\leq (M+2)^3\mathcal{M}(g)(z)
\leq C_{M_0}M^{3}\mathcal{M}(g)(z)
\end{split}\end{equation*} Then we take integral on $z\in B(2)$.\\
Now we states the third fact.
\begin{equation*}\begin{split}
\textbf{3.} \quad& \|w(t,\cdot)\|_{L^{3q/(3-q)}{(B(2M+2))}}\\
&\quad\quad\quad\leq C\cdot\|\nabla w(t,\cdot)\|_{L^q(B(2M+2))}+
\|\Bar{w}(t,\cdot)\|_{L^{3q/(3-q)}(B(2M+2))}\\
&\quad\quad\quad\leq
C\cdot M^{3/q} \cdot \|\mathcal{M}(|\nabla w|^q)(t,\cdot)\|
_{L^1(B(2))}^{1/q}
\\&\quad\quad\quad\quad+
CM^{-3}\|w(t,\cdot)\|_{L^1(B(2M+2))}\cdot CM^{3\cdot\frac{3-q}{3q}}\\
&\quad\quad\quad\leq
C\cdot M^{3/q} \cdot \|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)(t,\cdot)\|
_{L^1(B(2))}^{1/q}
\\&\quad\quad\quad\quad+
CM^{\frac{3}{q}-4}\|w(t,\cdot)\|_{L^1(B(2M+2))}\\
&\quad\quad\quad\leq
C\cdot M^{3/q} \cdot \|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)(t,\cdot)\|
_{L^1(B(2))}^{1/q}
\\&\quad\quad\quad\quad+
CM^{\frac{3}{q}-4} CM^{1+\frac{3}{1}}\cdot\Big(
\|\mathcal{M}(|\nabla w|^1)(t,\cdot)\|^{1/1}_{L^{1}(B(1))}
+\|\nabla w(t,\cdot)\|_{L^1(B(2))}\Big)\\
&\quad\quad\quad\leq
C\cdot M^{3/q} \cdot \|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)(t,\cdot)\|
_{L^1(B(2))}^{1/q}
\\&\quad\quad\quad\quad+
CM^{\frac{3}{q}}\Big(
\|\mathcal{M}(|\mathcal{M}(|\nabla u|)|)(t,\cdot)\|_{L^{1}(B(1))}
+\|\mathcal{M}(|\nabla u|)(t,\cdot)\|_{L^1(B(2))}\Big)\\
\end{split}\end{equation*}
we used \eqref{Sobolev-Poincare inequality} for the first inequality,
\eqref{lem_Maximal q} and definition of mean value
for the second one and
$|\nabla w (t,x)|\leq
C|\mathcal{M}(|\nabla u|)(t,x)|$ and the lemma \ref{lem_Maximal 2.5 or 4}
for fourth and fifth ones respectively. So, by taking $L^2$-norm on time $[-4,0]$ with \eqref{local_study_condition3},
\begin{equation*}\begin{split}
&\|w\|_{L^2(-4,0;L^{\frac{3q}{3-q}}(B(2M+2)))}
\leq
CM^{\frac{3}{q}}
\end{split}\end{equation*} for any $M\geq 2^k$.
\begin{equation*}\begin{split}
\textbf{4.}\quad& \|w(t,\cdot)\|_{L^{q\prime}{(B(2M+2))}}\leq
\|w(t,\cdot)\|^{\theta}_{L^{q}{(B(2M+2))}}\cdot
\|w(t,\cdot)\|^{1-\theta}_{L^{3q/(3-q)}{(B(2M+2))}}\quad\quad
\end{split}\end{equation*} where $q^\prime=q/(q-1)$ and $\theta=(4q-6)/q$.\\
Note: Because $12/8<q<2$, we have $0<\theta<1$.
So, for any $M\geq 2^k$,
\begin{equation*}\begin{split}
&\|w\|_{L^2(-4,0;L^{q\prime}(B(2M+2)))}\\&\quad\leq
\|w\|^{\theta}_{L^2(-4,0;L^{q}{(B(2M+2)))}}\cdot
\|w\|^{1-\theta}_{L^2(-4,0;L^{3q/(3-q)}{(B(2M+2)))}}\\
&\leq C\cdot (M^{1+(3/q)})^{\theta}(M^{3/q})^{1-\theta}
= C\cdot M^{4-\frac{3}{q} }.
\end{split}\end{equation*}
\noindent \textbf{5.} From \eqref{lem_Maximal q}, for any $M\geq 2^k$,
\begin{equation*}\begin{split}
\quad& \|\nabla u(t,\cdot)\|_{L^q{(B(2M+2))}}
\leq C\cdot M^{3/q} \cdot \|\mathcal{M}(|\nabla u|^q)(t,\cdot)\|_{L^1(B(2))}^{1/q}.
\end{split}\end{equation*} So, for any $M\geq 2^k$, from \eqref{local_study_condition3},
\begin{equation*}\begin{split}
\|\nabla u\|_{L^2(-4,0;L^q(B(2M+2)))}
&\leq C\cdot M^{3/q} \cdot \|\|\mathcal{M}(|\nabla u|^q)\|
_{L_x^1(B(2))}^{1/q}\|_{L_t^2(-4,0)}\\
&\leq C\cdot M^{3/q} \cdot \|\mathcal{M}(|\nabla u|^q)\|
_{L^{2/q}(-4,0;L^1(B(2)))}^{1/q}
\leq C\cdot M^{3/q}.
\end{split}\end{equation*}
\noindent Using above five results $\textbf{(1, }\cdots\textbf{, 5)}$ all together, we have for any $M\geq 2^k$,
\begin{equation*}\begin{split}
(III)
&\leq C \cdot\frac{1}{M^{3+d}}\cdot
\|w\|_{L^{2}( -4,0;L^{q^{\prime}}(B(2M+2)))}
\|\nabla u\|_{L^{2}( -4,0;L^q(B(2M+2)))}\\
&\leq C\cdot\frac{1}{M^{3+d}}\cdot M^{4-(3/q)}\cdot M^{3/q}
= C\cdot M^{1-d}
\end{split}\end{equation*}
which proved the above third claim \eqref{step4_third_claim}.\\
Finally we combine three claims
\eqref{step4_first_claim}, \eqref{step4_second_claim}
and \eqref{step4_third_claim}:
\begin{equation*}\begin{split}
&\|(-\Delta)^{\frac{\alpha}{2}}
\nabla^d u\|_{L^\infty(-{(1/6)}^2,0;
L^{1}(B((1/6))))}\\
&\quad\leq \|
C\Big(1+\Big|\int_{|y|\geq {(1/6)}}
\frac{\nabla^{d} u(\cdot_t,\cdot_x-y)}{|y|^{3+\alpha}}dy\Big|\Big)\|
_{L^\infty(-{(1/6)}^2,0;
L^{1}(B((1/6))))}\\
&\quad\leq C+C\sum_{j=k}^{\infty}(\frac{1}{2^{{\alpha}}})^j\cdot \|
|({(h^{{\alpha}})_{2^j}}*\nabla^d u)(\cdot_t,\cdot_x)|\|
_{L^\infty(-{(1/6)}^2,0;
L^{1}(B((1/6))))}\\
&\quad\leq C+C\sum_{j=k}^{\infty}(\frac{1}{2^{{\alpha}}})^j
\cdot (2^j)^{1-d}
\leq C+C\sum_{j=k}^{\infty}(\frac{1}{2^{d+\alpha-1}})^j
\leq C
\end{split}\end{equation*} because $d+\alpha-1>0$
from $d\geq 1$ and $\alpha>0$.\\
\noindent By the exact same way, we can also prove that
\begin{equation*}
\|(-\Delta)^{\frac{\alpha}{2}}
\nabla^m u\|_{L^\infty(-{(1/6)}^2,0;
L^{1}(B((1/6))))}\leq C
\end{equation*} for $m=d+1, ... ,d+4$. By repeated uses of Sobolev's inequality,
\begin{equation*}
\|(-\Delta)^{\frac{\alpha}{2}}
\nabla^d u\|_{L^\infty(-{(1/6)}^2,0;
L^{\infty}(B((1/6))))}\leq C({d,\alpha})
\end{equation*} and it finishes this proof of the proposition \ref{local_study_thm}.
\end{proof}
\section{Proof of the main theorem \ref{main_thm}}\label{proof_main_thm_II}
We begin this section by presenting one small lemma about pivot quantities.
After that,
the subsection \ref{prof_main_thm_II_alpha_0} covers the part (II) for $\alpha=0$
while the subsection \ref{prof_main_thm_II_alpha_not_0} does the part (II) for $0<\alpha<2$.
Finally the part (I) for $0\leq\alpha<2$ follows in the subsection \ref{proof_main_thm_I}.
\subsection{$L^1$ Pivot quantities}
The following lemma says that $L^1$ space-time norm of our pivot quantities can be controlled by $L^2$ space norm of the initial data. These things have
the best scaling like $|\nabla u|^2$ and $|\nabla^2 P|$ among all other $a$ $priori$ quantities from $L^2$ initial data (also see \eqref{best_scaling}).
\begin{lem}\label{lemma7_problem I-n}
There exist constant $C>0$ and $C_{d,\alpha}$
for integer $d\geq1$ and real $\alpha\in(0,2)$ with the following property:\\
If $(u,P)$ is a solution of (Problem I-n) for some $1\leq n \leq \infty$, then we have
\begin{equation*}
\int_0^{\infty}\int_{\mathbb{R}^3}\big(|\nabla u(t,x)|^2 +
|\nabla^2P(t,x)| + |\mathcal{M}(|\nabla u|)(t,x)|^2\big)dxdt
\leq C\|u_0\|^2_{L^2(\mathbb{R}^3)}
\end{equation*} and
\begin{equation*}\begin{split}
\int_0^{\infty}\int_{\mathbb{R}^3}&\Big(
|\mathcal{M}(\mathcal{M}(|\nabla u|))|^2+|\mathcal{M}(|\nabla u|^q)|^{2/q}+
|\mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)|^{2/q}\\
&+
\sum_{m=d}^{d+4} \sup_{\delta>0}(|(\nabla^{m-1}{h^{\alpha})_\delta}*\nabla^2 P|)
\Big)dxdt
\leq C_{d,\alpha}\|u_0\|^2_{L^2(\mathbb{R}^3)}
\end{split}\end{equation*} for any integer $d\geq1$ and any real $\alpha\in(0,2)$
where $q=q(\alpha)$ is defined by $12/(\alpha+6)$.
\begin{rem}
The definitions of $h^{\alpha}$ and
$(\nabla^{m-1}{h^{\alpha})_\delta}$ can be found around \eqref{property_h}.
\end{rem}
\begin{rem}
In the following proof, we will see that every quantity in the left hand sides of the above two estimates can be controlled by
dissipation of energy $\|\nabla u\|_{L^2((0,\infty)\times\mathbb{R}^3)}^{2}$
only.
It explains the latter part of the remark \ref{rmk_dissipation of energy}.
\end{rem}
\end{lem}
\begin{proof}
From \eqref{energy_eq_Problem I-n},
\begin{equation*}\begin{split}
\|\nabla u\|^2_{L^2(0,\infty;L^2(\mathbb{R}^3))}
&\leq \|u_0*\phi_{\frac{1}{n}}\|_{L^2(\mathbb{R}^3)}^2\leq \|u_0\|_{L^2(\mathbb{R}^3)}^2.
\end{split}\end{equation*}
\noindent For the pressure term,
we use boundedness of the Riesz transform on Hardy space $\mathcal{H}$
and compensated compactness result in
Coifman, Lions, Meyer and Semmes \cite{clms}:
\begin{equation}\begin{split}\label{pressure_hardy_Problem I-n}
\|\nabla^2 P\|_{L^1(0,\infty;L^1(\mathbb{R}^3))}&\leq
\|\nabla^2 P\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}
\leq C\|\Delta P\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}\\
&=\|\ebdiv\ebdiv
\Big( (u * \phi_{1/n})\otimes u\Big)\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla (u * \phi_{1/n})\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}
\|\nabla u\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla u\|_{L^2(0,\infty;L^2(\mathbb{R}^3))}^2
\leq C\|u_0\|_{L^2(\mathbb{R}^3)}^2.
\end{split}\end{equation}
\noindent For Maximal functions,
\begin{equation*}\begin{split}
\| \mathcal{M}(\mathcal{M}(|\nabla u|))\|^2_{L^2(0,\infty;L^2(\mathbb{R}^3))}&\leq
C\cdot\|\mathcal{M}(|\nabla u|)\|^2_{L^2(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla u\|^2_{L^2(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|u_0\|^2_{L^2(\mathbb{R}^3)}.\\
\end{split}\end{equation*}
\noindent Let $d\geq1$ and $0<\alpha<2$ and take $q=12/(\alpha+6)$. From
$1<(2/q)<(4/3)$,
\begin{equation*}\begin{split}
\|\mathcal{M}(|\nabla u|^q)\|^{2/q}_{L^{2/q}(0,\infty;L^{2/q}(\mathbb{R}^3))}
&\leq C\cdot\||\nabla u|^q\|^{2/q}_{L^{2/q}(0,\infty;L^{2/q}(\mathbb{R}^3))}\\
&= C\cdot\|\nabla u\|^2_{L^{2}(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|u_0\|^{2}_{L^2(\mathbb{R}^3)}
\end{split}\end{equation*} and
\begin{equation*}\begin{split}
\| \mathcal{M}(|\mathcal{M}(|\nabla u|)|^q)
\|^{2/q}_{L^{2/q}(0,\infty;L^{2/q}(\mathbb{R}^3))}
&\leq C\cdot\||\mathcal{M}(|\nabla u|)|^q\|^{2/q}_{L^{2/q}(0,\infty;L^{2/q}(\mathbb{R}^3))}\\
&\leq C\cdot\|\mathcal{M}(|\nabla u|)\|^{2}_{L^{2}(0,\infty;L^{2}(\mathbb{R}^3))}\\
&\leq C\cdot\|\nabla u\|^2_{L^{2}(0,\infty;L^2(\mathbb{R}^3))}\\
&\leq C\cdot\|u_0\|^{2}_{L^2(\mathbb{R}^3)}
\end{split}\end{equation*}
where $C$ depends
only
on $\alpha$.\\
\noindent Thanks to the property of Hardy space \eqref{hardy_property}
with \eqref{pressure_hardy_Problem I-n}, we have
\begin{equation*}\begin{split}
\sum_{m=d}^{d+4}\| \sup_{\delta>0}(|(\nabla^{m-1}{h^{\alpha})_\delta}
*\nabla^2 P|)\|_{L^{1}(0,\infty;L^{1}(\mathbb{R}^3))}
&\leq\sum_{m=d}^{d+4}C\|\nabla^2 P
\|_{L^1(0,\infty;\mathcal{H}(\mathbb{R}^3))}\\
&\leq C\|u_0\|_{L^2(\mathbb{R}^3)}^2
\end{split}\end{equation*} where the above $C$ depends only on $d$ and $\alpha$.
\end{proof}
We are ready to prove the main theorem \ref{main_thm}.
\begin{rem}\label{n_infty_rem}
In the following subsections \ref{prof_main_thm_II_alpha_0}
and \ref{prof_main_thm_II_alpha_not_0}, we consider solutions of
{(Problem I-n)} for positive integers $n$.
However it will be clear that
every computation in these subsections can also be verified for the case $n=\infty$
once we assume that the smooth solution $u$ of the Navier-Stokes exists.
This $n=\infty$ case (the original Navier-Stokes)
will be covered in the subsection \ref{proof_main_thm_I}. \\
\end{rem}
We focus on the $\alpha=0$ case of the part (II) first.\\
\subsection{Proof of theorem \ref{main_thm} part (II)
for $\alpha=0$ case}\label{prof_main_thm_II_alpha_0}
\begin{proof}[Proof of proposition \ref{main_thm} part (II) for the $\alpha=0$ case]
\ \\
Let any $u_0$ of \eqref{initial_condition} be given.
From the Leray's construction,
there exists the $C^{\infty}$ solution sequence $\{u_n\}_{n=1}^{\infty}$ of
{(Problem I-n)} on $(0,\infty)$ with
corresponding pressures $\{P_n\}_{n=1}^{\infty}$.
From now on, our goal is to make an estimate for $\nabla^d u_n$ which is uniform in $n$.\\
\noindent For each $n$, $\epsilon>0$, $t>0$ and $x\in\mathbb{R}^3$,
define a new flow $X_{n,\epsilon}(\cdot,t,x)$ by solving
\begin{equation*}\begin{split}
&\frac{\partial X_{n,\epsilon} }{\partial s}
(s,t,x) = u_{n}*\phi_{\frac{1}{n}}*\phi_{\epsilon}(s,X_{n,\epsilon}(s,t,x))
\quad \mbox{ for } s\in[0,t],\\
&X_{n,\epsilon}(t,t,x) =x.
\end{split}
\end{equation*}
For convenience, we define $F_n(t,x)$ and $g_n(t)$.
\begin{equation*}\begin{split}
F_n(t,x)& = \big(|\nabla u_n|^2 + |\nabla^2P_n|
+ |\mathcal{M}(\nabla u_n)|^2\big)(t,x),\quad
g_n(t) = \int_{\mathbb{R}^3}F_n(t,x)dx.
\end{split}\end{equation*}
We
define for $n$, $t>0$ and $0<4\epsilon^2\leq t$
\begin{equation*}
\Omega_{n,\epsilon,t} = \{ x\in \mathbb{R}^3\quad |\quad \frac{1}{\epsilon}
\int_{t-4{\epsilon}^2}^{t}\int_{B({2\epsilon})}F_n(s,X_{n,\epsilon}
(s,t,x)+y)dyds\leq{\bar{\eta}}\}
\end{equation*} where $\bar{\eta}$ comes from the proposition
\ref{local_study_thm}.
We measure size of $(\Omega_{n,\epsilon,t})^C$:
\begin{equation}\begin{split}\label{omega_estimate}
|(\Omega_{n,\epsilon,t})^C|
&= |\{ x\in \mathbb{R}^3\quad |\quad \frac{1}{\epsilon}
\int_{t-4{\epsilon}^2}^{t}\int_{B({2\epsilon})}F_n(s,X_{n,\epsilon}(s,t,x)+y)dyds > {\bar{\eta}}\}|\\
&\leq\frac{1}{{\bar{\eta}}}\int_{\mathbb{R}^3}\Big(\frac{1}{\epsilon}
\int_{t-4{\epsilon}^2}^{t}\int_{B({2\epsilon})}F_n(s,X_{n,\epsilon}(s,t,x)+y)dyds\Big)dx\\
&=\frac{1}{{\bar{\eta}}\epsilon}\Big(\int_{B({2\epsilon})}
\int_{-4{\epsilon}^2}^{0}\int_{\mathbb{R}^3}F_n(t+s,X_{n,\epsilon}(t+s,t,x)+y)dxdsdy\Big)\\
&=\frac{1}{{\bar{\eta}}\epsilon}\Big(\int_{B({2\epsilon})}
\int_{-4{\epsilon}^2}^{0}\int_{\mathbb{R}^3}F_n(t+s,z+y)dzdsdy\Big)\\
&\leq\frac{1}{{\bar{\eta}}\epsilon}\Big(\int_{B({2\epsilon})}1 dy\Big)\Big(
\int_{-4{\epsilon}^2}^{0}\int_{\mathbb{R}^3}F_n(t+s,\bar{z})d\bar{z}ds\Big)\\
&\leq\frac{C\epsilon^2}{{\bar{\eta}}}\Big(
\int_{-4{\epsilon}^2}^{0}\int_{\mathbb{R}^3}F_n(t+s,\bar{z})d\bar{z}ds\Big)\\
&\leq C\frac{\epsilon^4}{\bar{\eta}}\Big(\frac{1}{4\epsilon^2}
\int_{-4\epsilon^2}^0g_n(t+s)ds\Big)
\leq \epsilon^4\mathcal{M}^{(t)}
(\frac{C}{\bar{\eta}}g_n\cdot \mathbf{1}_{(0,\infty)})(t)
=\epsilon^4 \tilde{g}_n(t)\
\end{split}\end{equation} where $\tilde{g}_n= \mathcal{M}^{(t)}
(\frac{C}{\bar{\eta}}g_n\cdot \mathbf{1}_{(0,\infty)})$ and $\mathcal{M}^{(t)}$ is
the Maximal function in $\mathbb{R}^1$. For the third inequality, we used the fact
that $X_{n,\epsilon}(\cdot,t,x)$ is incompressible.
From the fact that the Maximal operator is bounded from $L^1$ to $L^{1,\infty}$ together with
the lemma \ref{lemma7_problem I-n},
$\|\tilde{g}_n(\cdot)\|_{L^{1,\infty}(0,\infty)}\leq
\frac{C}{\bar{\eta}}\|g_n(\cdot)\|_{L^{1}(0,\infty)}\leq
\frac{C}{\bar{\eta}}\|u_0\|^2_{L^2(\mathbb{R}^3)}$.\\
Now we fix $n, t, \epsilon$ and $ x$ with $n\geq1$, $0<t<\infty$ , $0<4\epsilon^2\leq t$ and $x\in\Omega_{n,\epsilon,t}$.
We define ${v}, {Q}$ on $(-4,\infty)\times\mathbb{R}^3$ by
using the Galilean invariance:
\begin{equation}\begin{split}\label{special designed scaling}
{v}(s,y) =& \epsilon u_n(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y)\\
&- \epsilon (u_n*\phi_{\epsilon})(t+\epsilon^2s,X_{n,\epsilon}(t+\epsilon^2s,t,x))\\
{Q}(s,y) =& \epsilon^2 P_n(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y)\\
&+ \epsilon y \partial_s[ (u_n*\phi_{\epsilon})(t+\epsilon^2s,
X_{n,\epsilon}(t+\epsilon^2s,t,x))].
\end{split}\end{equation}
\begin{rem}
This specially designed
$\epsilon$-scaling will give the mean zero property to both the velocity
and the advection
velocity of the resulting equation \eqref{special designed scaling result}.
\end{rem}
\noindent Let us denote $\Box$ and $\Diamond$ by $\Box= \big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y\big)$ and
$\Diamond=\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)\big)$,
respectively.
Then the chain rule gives us
\begin{equation*}\begin{split}
& \partial_s{v}(s,y) =
\epsilon^3\partial_t(u_n)(\Box)
+ \epsilon^3 \big((u_{n}*\phi_{\frac{1}{n}}*\phi_{\epsilon})(\Diamond)\cdot\nabla\big)u_n(\Box)
- \epsilon \partial_s[(u_{n}*\phi_{\epsilon})(\Diamond)],\\
&\big({v} *_y \phi_{\frac{1}{n\epsilon}}\big)(s,y)
=\epsilon (u_n*\phi_{\frac{1}{n}})(\Box)-\epsilon(u_{n}*\phi_{\epsilon})(\Diamond),\\
&\int_{\mathbb{R}^3}\big({v} *_y \phi_{\frac{1}{n\epsilon}}\big)(s,z)\phi(z)dz
=\epsilon (u_n*\phi_{\frac{1}{n}}*\phi_{\epsilon})(\Diamond)-\epsilon(u_{n}*\phi_{\epsilon})(\Diamond),\\
&\bigg(\Big(\big({v} *_y \phi_{\frac{1}{n\epsilon}}\big)(s,y) -
\int_{\mathbb{R}^3}\big({v} *_y \phi_{\frac{1}{n\epsilon}}\big)(s,z)\phi(z)dz\Big)\cdot\nabla\bigg)
{v}(s,y)=\\
&\epsilon^3 \Big((u_n*\phi_{\frac{1}{n}})(\Box)\cdot\nabla\Big)u_n(\Box)
-\epsilon^3 \Big((u_n*\phi_{\frac{1}{n}}*\phi_{\epsilon})(\Diamond)\cdot\nabla\Big)u_n(\Box),\\
&-\Delta_y {v}(s,y) = -\epsilon^3\Delta_y u_n(\Box) \mbox{ and} \\
&\nabla_y{Q}(s,y)= \epsilon^3 \nabla P_n(\Box)
+ \epsilon \partial_s[ (u_n*\phi_{\epsilon})(\Diamond))].
\end{split}\end{equation*}
\noindent Thus, for $(s,y)\in (-4,\infty)\times\mathbb{R}^3$,
\begin{equation}\begin{split}\label{special designed scaling result}
\Big[\partial_s{v}+
\bigg(\Big(\big({v} * \phi_{\frac{1}{n\epsilon}}\big) -
\int\big({v} * \phi_{\frac{1}{n\epsilon}}\big)\phi \Big)\cdot\nabla\bigg)
{v} +\nabla{Q}-\Delta {v}\Big](s,y)=0.
\end{split}\end{equation} As a result,
$({v}(\cdot_s,\cdot_y),{Q}(\cdot_s,\cdot_y))$ is a solution of (Problem II-$\frac{1}{n\epsilon}$).\\
From definition of the Maximal function, we can verify that
$|\mathcal{M}(\nabla {v})|^2$
behaves like $|\nabla v|^2$ under the scaling
in the following sense:
\begin{equation}\begin{split}\label{Maximal_scaling}
\mathcal{M}(\nabla {v})(s,y)
&=\sup_{M>0}\frac{C}{M^3}\int_{B(M)}\epsilon^2(\nabla {u_n})
\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon (y+z)\big)dz\\
&=\sup_{\epsilon M>0}\frac{C}{\epsilon^3M^3}\int_{B(\epsilon M)}\epsilon^2(\nabla {u_n})
\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y+\bar{z}\big)d\bar{z}\\
&=\epsilon^2\mathcal{M}(\nabla {u_n})
(\Box)
\end{split}\end{equation} As a result,
\begin{equation*}\begin{split}
&\int_{-4}^{0}\int_{B(2)}\big(|\nabla {v}(s,y)|^2 +
|\nabla^2{Q}(s,y)| + |\mathcal{M}(\nabla {v})(s,y)|^2\big)dyds \\
&=\epsilon^4\int_{-4}^{0}\int_{B(2)}\Big[|\nabla {u_n}|^2 +
|\nabla^2{P_n}| + |\mathcal{M}(\nabla {u_n})|^2\Big](\Box)dyds \\
&=\epsilon^4\int_{-4}^{0}\int_{B(2)}F_n(\Box)dyds \\
&=\epsilon^{-1}\int_{t-4\epsilon^2}^{t}\int_{B(2\epsilon)}F_n
\big(s,X_{n,\epsilon}(s,t,x)+y\big)dyds
\leq {\bar{\eta}}
\end{split}\end{equation*} where the first equality comes from the definition of $
(v,Q)$ and the second one follows the change of variable
$\big(t+\epsilon^2 s,\epsilon y\big)\rightarrow (s,y)$.
Moreover, it satisfies
\begin{equation}\label{local_study_condition_satisfied}\begin{split}
&\int_{\mathbb{R}^3}\phi(z){v}(s,z)dz = 0, \quad -4<s<0.
\end{split}\end{equation}
So $(v,Q)$ satisfies all conditions (\ref{local_study_condition1},
\ref{local_study_condition2})
in the proposition \ref{local_study_thm} with $r=1/(n\epsilon)\in[0,\infty)$. \\
The conclusion of the proposition \ref{local_study_thm} implies that if $x\in\Omega_{n,\epsilon,t}$ for some $n,t$ and $\epsilon$ such that
$4\epsilon^2\leq t$ then
$|\nabla^d {v}(0,0)|\leq C_{d}.$
As a result, using $\nabla^d {v}(0,0)=
\epsilon^{d+1} \nabla^d u_n(t,x)$ for any integer $d\geq 1$,
we have
\begin{equation*}
|\{x\in \mathbb{R}^3 | \quad|\nabla^d u_n(t,x)|>
\frac{C_{d}}{\epsilon^{d+1}}\}|\leq|\Omega_{n,\epsilon,t}^C|
\leq \epsilon^4\cdot\tilde{g}_n(t).\
\end{equation*}
\noindent Let $K$ be any open bounded subset in $\mathbb{R}^3$.
Also define $p=4/(d+1)$. Then for any $t>0$,
\begin{equation*}
\beta^p\cdot\Big|\{x\in K: |
(\nabla^d u_n)(t,x)|
> \beta\}\Big|\leq
\begin{cases}& \beta^p\cdot|K|
,\quad\mbox{ if } \beta\leq C\cdot t^{-2/p}\\
& C\cdot \tilde{g}_n(t) ,
\quad\mbox{ if } \beta>C\cdot t^{-2/p}.\\
\end{cases}
\end{equation*} Thus,
\begin{equation*}
\|(\nabla^d u_n)(t,\cdot)\|^p_{L^{p,\infty}(K)}
\leq C\cdot\max\big( \tilde{g}_n(t),\frac{|K|}{t^2} \big)
\end{equation*}
\noindent We pick any $t_0>0$. If we take ${L^{1,\infty}(t_0,T)}$-norm
to the above inequality, then we obtain
\begin{equation}\begin{split}\label{integer_estimate}
\quad\|\nabla^d u_n\|^p_{L^{p,\infty}(t_0,\infty;L^{p,\infty}(K))}
&\leq C\Big(\| \tilde{g}_n\|_{L^{1,\infty}{(0,\infty)}} + |K|\cdot\|\frac{1}
{{|\cdot|}^2}\|_{L^{1,\infty}(t_0,\infty)}\Big)\\
&\leq C\Big(\|u_0\|^2_{L^{2}(\mathbb{R}^3)} +
\frac{|K|}{t_0}\Big)
\end{split}\end{equation} where $C$ depends only on $d\geq1$.\\
We observe that
the above
estimate is uniform in $n$.
It is well known that both $\nabla u$ and $\nabla^2 u$ are locally integrable functions for any suitable weak solution $u$
which can be obtained by a limiting argument of $u_n$ (e.g. see
Lions \cite{lions}). Thus, the
above estimates \eqref{integer_estimate} holds
even for $u$ with $d=1,2$.\\
\begin{rem}
In fact, for the case $d=1$, the above estimate says $\nabla u\in L^{2,\infty}_{loc}$, which is useless because
we know a better estimate $\nabla u\in L^2$.
\end{rem}
\begin{rem}
For $d \geq 3$,
the above estimate \eqref{integer_estimate} does not give us any
direct information about higher derivatives $\nabla^d u$ of a weak
solution $u$
because full regularity of weak solutions is still open, so
$\nabla^d u$ may not be locally integrable for $d\geq3$.
Instead,
the only thing we can say is that, for $d\geq3$, higher derivatives $\nabla^d u_n$
of a Leray's approximation $u_n$ have $L^{4/(d+1),\infty}_{loc}$ bounds which are uniform in $n\geq1$.
\end{rem}
\end{proof}
From now on, we will prove the $0<\alpha<2$ case of the part (II).
\subsection{Proof of theorem \ref{main_thm} part (II)
for $0<\alpha<2$ case}\label{prof_main_thm_II_alpha_not_0}
\begin{proof}[Proof of proposition \ref{main_thm} part (II) for the $0<\alpha<2$ case]
\ \\
We fix $d\geq 1$ and $0<\alpha<2$.
Then,
for any positive integer $n$, any $t>0$ and $x\in\mathbb{R}^3$, we denote $F_n(t,x)$ in this time by:
\begin{equation*}\begin{split}
F_n(t,x) = \Big(&|\nabla u_n(t,x)|^2 + |\nabla^2P_n(t,x)| +
|\mathcal{M}(\nabla u_n)(t,x)|^2\\&+
|\mathcal{M}(\mathcal{M}(|\nabla u_n|))|^2+
(\mathcal{M}(|\mathcal{M}(|\nabla u_n|)|^q))^{2/q}\\+
&|\mathcal{M}(|\nabla u_n|^q)|^{2/q}+
\sum_{m=d}^{d+4} \sup_{\delta>0}(|(\nabla^{m-1}{h^{\alpha})_\delta}*\nabla^2 P|)
\Big).
\end{split}\end{equation*}
We use the same definitions for $g_n$, $\tilde{g}_n$, $X_{n,\epsilon}$ and $\Omega_{n,\epsilon,t}$
of the previous section \ref{prof_main_thm_II_alpha_0} for the case $\alpha=0$.
Note that they depend on $d$ and
$\alpha$, and we have
$\|\tilde{g}_n\|_{L^{1,\infty}(0,\infty)}\leq
\frac{C_{d,\alpha}}{\bar{\eta}}\cdot\|u_0\|^2_{L^2(\mathbb{R}^3)}$
from the lemma \ref{lemma7_problem I-n}. \\
Now we pick any $x\in\Omega_{n,\epsilon,t}$ and any $\epsilon$ such that
$4\epsilon^2\leq t$, and define $v$ and $Q$
as the previous section \ref{prof_main_thm_II_alpha_0} (see \eqref{special designed scaling}).\\
\noindent In order to follow the previous subsection \ref{prof_main_thm_II_alpha_0},
only thing which remains is to verify if every quantity in
$F_n(t,x)$
has the same scaling with $|\nabla v|^2$ after the transform
\eqref{special designed scaling}.
For Maximal of Maximal functions,
\begin{equation*}\begin{split}
&\mathcal{M}(\mathcal{M}(|\nabla v|))(s,y)\\
&=\sup_{M>0}\frac{C}{M^3}\int_{B(M)}\mathcal{M}(|\nabla v|)(s,y+z)dz\\
&=\sup_{M>0}\frac{C}{M^3}\int_{B(M)}\epsilon^2\mathcal{M}
(|\nabla u_n|)\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon (y+z)\big)dz\\
&=\epsilon^2\mathcal{M}(\mathcal{M}(|\nabla u_n|))(\Box).\\
\end{split}\end{equation*} where $\Box=\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y\big)$
and we used the idea of \eqref{Maximal_scaling}
for second and third equalities. Likewise,
$\mathcal{M}(|\nabla v|^q)(s,y)
=\epsilon^{2q}\cdot\mathcal{M}(|\nabla u_n|^q)(\Box)$
and
$\mathcal{M}(|\mathcal{M}(|\nabla v|)|^q)(s,y)
=\epsilon^{2q}\cdot\mathcal{M}(|\mathcal{M}(|\nabla u_n|)|^q)(\Box)$. \\
\noindent Also, we have for any function $\mathcal{G}\in C_0^\infty$,
\begin{equation*}\begin{split}
\sup_{\delta>0}&(|{\mathcal{G}_{\delta}}*\nabla^2 Q|)(s,y)=
\sup_{\delta>0}\Big|\int_{\mathbb{R}^3}\frac{1}{\delta^{3}}{\mathcal{G}}(\frac{z}{\delta})\cdot
(\nabla^2 Q)(s,y-z)dz\Big|\\
&=
\sup_{\delta>0}\Big|\int_{\mathbb{R}^3}\frac{\epsilon^4}{\delta^{3}}{\mathcal{G}}(\frac{z}{\delta})\cdot
(\nabla^2 P_n)\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon (y-z)\big)dz\Big|\\
&=
\sup_{\delta>0}\Big|\int_{\mathbb{R}^3}\frac{\epsilon^4}{\epsilon^3
\delta^{3}}{\mathcal{G}}(\frac{z}{\epsilon\delta})\cdot
(\nabla^2 P_n)\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y-z\big)dz\Big|
\end{split}\end{equation*}
\begin{equation*}\begin{split}
&=
\sup_{\epsilon\delta>0}\Big|\int_{\mathbb{R}^3}\epsilon^4
{\mathcal{G}}_{\epsilon\delta}(z)\cdot
(\nabla^2 P_n)\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y-z\big)dz\Big|\\
&=
\sup_{\epsilon\delta>0}\epsilon^4\Big|
\Big({\mathcal{G}}_{\epsilon\delta}*
(\nabla^2 P_n)\Big)\big(t+\epsilon^2 s,X_{n,\epsilon}(t+\epsilon^2 s,t,x)+\epsilon y\big)\Big|\\
&=\epsilon^4
\sup_{\delta>0}\Big|
{\mathcal{G}_{\delta}}*
(\nabla^2 P_n)\Big|(\Box).
\end{split}\end{equation*} Thus by taking $\mathcal{G}=(\nabla^{m-1}{h^{\alpha}})$,
we have \begin{equation*}\begin{split}
\sup_{\delta>0}(|{(\nabla^{m-1}{h^{\alpha})_\delta}}*\nabla^2 Q|)(s,y)
&=\epsilon^4
\sup_{\delta>0}\Big|
{(\nabla^{m-1}{h^{\alpha})_\delta}}*
(\nabla^2 P_n)\Big|\big(\Box\big).
\end{split}\end{equation*}
\noindent As a result,
we have
\begin{equation*}\begin{split}
&\int_{-4}^{0}\int_{B(2)}\Big[|\nabla {v}|^2 +
|\nabla^2{Q}| + |\mathcal{M}(\nabla {v})|^2+\\
&\quad+|\mathcal{M}(\mathcal{M}(|\nabla v|))|^2+
|\mathcal{M}(|\mathcal{M}(|\nabla v|)|^q)|^{q/2}\\
&\quad+|\mathcal{M}(|\nabla v|^q)|^{2/q}+
\sum_{m=d}^{d+4} \sup_{\delta>0}(
|{(\nabla^{m-1}{h^{\alpha})_\delta}}*\nabla^2 Q|)
\Big](s,y)
dyds \\
&=\epsilon^4\int_{-4}^{0}\int_{B(2)}F_n(\Box)dyds \\
&=\epsilon^{-1}\int_{t-4\epsilon^2}^{t}\int_{B(2\epsilon)}F_n
\big(s,X_{n,\epsilon}(s,t,x)+y\big)dyds
\leq {\bar{\eta}}.\\
\end{split}\end{equation*}
\noindent Then $(v,Q)$ satisfies condition
\eqref{local_study_condition3} as well as
\eqref{local_study_condition1} and
\eqref{local_study_condition2}
of the proposition \ref{local_study_thm}
with $r=1/(n\epsilon)\in[0,\infty)$. In sum if $x\in\Omega_{n,\epsilon,t}$
and if $4\epsilon^2\leq t$, then
\begin{equation*}
| (-\Delta)^{\alpha/2}\nabla^d {v}(0,0)|\leq C_{d,\alpha}.
\end{equation*}
Because $u_n$ is a smooth solution of (Problem I-n),
$(-\Delta)^{\alpha/2}\nabla^d u_n$ is not only a distribution
but also a locally integrable function.
Indeed, from a boot-strapping argument, it is
easy to show that $\nabla^d u_n(t)$ has a good behavior at infinity
which is required
in order to use the integral
representation \eqref{fractional_integral} pointwise. For example,
$(C^2\cap W^{2,\infty})$ is enough (For a better approach, see Silvestre \cite{silve:fractional}). Also it can be easily verified that
the resulting function $(-\Delta)^{\alpha/2}[\nabla^d u_n(t,\cdot)](x)$
from the integral representation \eqref{fractional_integral}
satisfies the definition in the remark \ref{frac_rem}. \\
\noindent
As a result, it makes sense to talk about pointwise values
of $(-\Delta)^{\alpha/2}\nabla^d u_n$. Thus,
from the simple observation:
for any integer $d\geq 1$ and any real $0<\alpha<2$,
\begin{equation*}
(-\Delta)^{\alpha/2}\nabla^d {v}(0,0)=
\epsilon^{d+\alpha+1} (-\Delta)^{\alpha/2}\nabla^d u_n(t,x),
\end{equation*}
we can
deduce the following set inclusion:
\begin{equation}\label{fractional_inclusion}
\{x\in \mathbb{R}^3 | \quad|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u_n(t,x)|>
\frac{C_{d,\alpha}}{\epsilon^{d+\alpha +1}}\}\qquad\subset\qquad\Omega_{n,\epsilon,t}^C.
\end{equation}
\noindent Thus we have
for any $0<t<\infty$ and for any $0<4\epsilon^2\leq t$
\begin{equation*}
|\{x\in \mathbb{R}^3 | \quad|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u_n(t,x)|>
\frac{C_{d,\alpha}}{\epsilon^{d+\alpha +1}}\}|\leq|\Omega_{n,\epsilon,t}^C|
\leq \epsilon^4\cdot \tilde{g}_n(t).
\end{equation*}
\noindent Define $p=4/(d+\alpha+1)$. Like we did for case $\alpha=0$, we obtain
\begin{equation*}\begin{split}
\|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u_n\|^p_{L^{p,\infty}
(t_0,\infty;L^{p,\infty}(K))}
&\leq C\Big(\|u_0\|^2_{L^{2}(\mathbb{R}^3)} +
\frac{|K|}{t_0}\Big)
\end{split}\end{equation*}
for any integer $n,d\geq1$, for any real $\alpha\in(0,2)$, for
any bounded open subset $K$ of $\mathbb{R}^3$ and for any $t_0\in(0,\infty)$ where $C$ depends only on $d$ and $\alpha$.\\
If we restrict further $(d+\alpha)<3$, then
$ p = \frac{4}{d+\alpha+1}>1 $. This implies
$(-\Delta)^{\alpha/2}\nabla^d u_n\in L^q_{loc}((t_0,\infty)\times K)$
for every $q$ between $1$ and $p$,
and the norm is uniformly bounded in $n$.
Thus,
from weak-compactness of $L^q$ for $q>1$,
we conclude that
if $u$ is a suitable weak solution obtained by a limiting argument
of $u_n$, then any higher derivatives
$(-\Delta)^{\alpha/2}\nabla^d u$,
which is defined in the remark \ref{frac_rem},
lie in $L^1_{loc}$
as long as $(d+\alpha)<3$ with the same estimate
\begin{equation}\begin{split}\label{final_comment}
\|(-\Delta)^{\frac{\alpha}{2}}\nabla^d u\|^p_{L^{p,\infty}
(t_0,\infty;L^{p,\infty}(K))}
&\leq C_{d,\alpha}\Big(\|u_0\|^2_{L^{2}(\mathbb{R}^3)} +
\frac{|K|}{t_0}\Big).
\end{split}\end{equation}
\end{proof}
\subsection{Proof of theorem \ref{main_thm} part (I)}\label{proof_main_thm_I}
\begin{proof}[Proof of proposition \ref{main_thm} part (I)]
Suppose that $(u,P)$ is a smooth solution of the Navier-Stokes equations
\eqref{navier}
on $(0,T)$ with \eqref{initial_condition}. Then it satisfies all conditions of
{(Problem I-n)} for $n=\infty$ on $(0,T)$.
As we mentioned at the remark \ref{n_infty_rem}, we follow
every steps in the subsections
\ref{prof_main_thm_II_alpha_0}
and \ref{prof_main_thm_II_alpha_not_0} except each last arguments
which impose $d<3$ or $(d+\alpha)<3$.
Indeed,
under the scaling \eqref{special designed scaling},
the resulting function $(v,Q)$ is a solution for (Problem II-r)
for $r=0$.\\
\noindent
Recall that $u$ is smooth by assumption.
As a result, we do NOT have
any restriction like $d<3$ or $(d+\alpha)<3$ at this time because
we do not need any limiting argument any more
which requires a weak-compactness.
Thus, we obtain \eqref{final_comment}
for any integer $d\geq1$,
for any real $\alpha\in[0,2)$
and for any $t_0\in(0,T)$.
It finishes
the proof of the part $(I)$ of the main theorem \ref{main_thm}.
\end{proof}
\section*{A. Appendix: proofs of some technical lemmas}\label{appendix}
\begin{proof} [proof for lemma \ref{higher_pressure}]
Fix $(n,a,b,p)$ such that $n\geq 2$, $0<b<a<1$ and $1<p<\infty$.
Let $\alpha$ be any multi index such that
$|\alpha|=n$ and $D^{\alpha}=\partial_{\alpha_1} \partial_{\alpha_2}
D^{\beta}$ where $\beta$ is a multi index with $|\beta|=n-2$.\\
Observe that from $\ebdiv(v_2)=0$ and $\ebdiv(v_1)=0$,
\begin{equation*}\begin{split}
-\Delta(D^{\alpha}P) &=\ebdiv\ebdiv D^{\alpha}(v_2\otimes v_1)\\
&=D^{\alpha}\Big(\sum_{ij} (\partial_j v_{2,i})(\partial_iv_{1,j})\Big)\\
&=\partial_{\alpha_1} \partial_{\alpha_2}
H\\
\end{split}\end{equation*} where $H=D^{\beta}\Big(\sum_{ij}
(\partial_j v_{2,i})(\partial_iv_{1,j})\Big)$
and $v_k=(v_{k,1},v_{k,2},v_{k,3})$ for $k=1,2$.\\
\noindent Then for any $(p_1,p_2)$ such that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$
\begin{equation*}\begin{split}
\|H\|_{L^{p}(B(a))}
&\leq C
\| v_2 \|_{W^{n-1,p_2}(B(a))}\cdot
\| v_1 \|_{W^{n-1,p_1}(B(a))}
\end{split}\end{equation*}
where $C$ is independent of choice of $p_1$ and $p_2$
and
\begin{equation*}\begin{split}
\|H\|_{W^{1,{\infty}}(B(a))}&
\leq C
\| v_2 \|_{W^{n,\infty}(B(a))}\cdot
\| v_1 \|_{W^{n,\infty}(B(a))}.
\end{split}\end{equation*}
Fix a function $\psi \in C^{\infty}(\mathbb{R}^3)$ satisfying:
\begin{equation*}\begin{split}
&\psi = 1 \quad\mbox{ in } B(b+\frac{a-b}{3}),\quad
\psi = 0 \quad\mbox{ in } (B(b+\frac{2(a-b)}{3}))^C \mbox{ and }
0\leq \psi \leq 1.
\end{split}\end{equation*}
\noindent We decompose $D^{\alpha}P$ by using $\psi$:
\begin{equation*}\begin{split}
-\Delta({\psi} D^{\alpha}P)
&= -{\psi} \Delta D^{\alpha}P
- 2\ebdiv((\nabla {\psi})(D^{\alpha}P)) +
(D^{\alpha} P)\Delta{\psi}\\
&= {\psi}\partial_{\alpha_1} \partial_{\alpha_2}H
- 2\ebdiv((\nabla {\psi})(D^{\alpha}P)) +
(D^{\alpha} P)\Delta{\psi}\\
&= - \Delta Q_{1} - \Delta Q_{2} - \Delta Q_{3}\\
\end{split}
\end{equation*} where
\begin{equation*}\begin{split}
- \Delta Q_{1} &= \partial_{\alpha_1} \partial_{\alpha_2} ({\psi}H), \\
- \Delta Q_{2} &= - \partial_{\alpha_2}[(\partial_{\alpha_1} {\psi})(H)]
-\partial_{\alpha_1}[(\partial_{\alpha_2} {\psi})(H)]
+ (\partial_{\alpha_1} \partial_{\alpha_2} {\psi})(H) \quad\mbox{ and} \\
- \Delta Q_{3}\mbox{ } &= - 2\ebdiv((\nabla {\psi})(D^{\alpha}P)) +
(D^{\alpha} P)\Delta{\psi}.
\end{split}\end{equation*}
Here $Q_{2}$ and $Q_{3}$ are defined by the representation formula
${(-\Delta)}^{-1}(f) = \frac{1}{4\pi}(\frac{1}{|x|} * f)$\\
while $Q_{1}$ by the Riesz transforms.\\
\noindent Then, by the Riesz transform,
\begin{equation*}\begin{split}
\|Q_{1}\|_{L^{p}(B(b))}
&\leq
C\|\psi H\|_{L^{p}(\mathbb{R}^3)}
\leq C
\|H\|_{L^{p}(B(a))}\\
&\leq C
\| v_2 \|_{W^{n-1,p_2}(B(a))}\cdot
\| v_1 \|_{W^{n-1,p_1}(B(a))}.\\
\end{split}\end{equation*}
\noindent Moreover, using Sobolev,
\begin{equation*}\begin{split}
\|Q_{1}\|_{L^\infty(B(b))}
&\leq
C\Big(\|Q_{1}\|_{L^4(B(b))}+\|\nabla Q_{1}\|_{L^4(B(b))}\Big)\\
&\leq C
\|H\|_{W^{1,{4}}(B(a))}
\leq C
\|H\|_{W^{1,{\infty}}(B(a))}\\
&\leq C
\| v_2 \|_{W^{n,\infty}(B(a))}\cdot
\| v_1 \|_{W^{n,\infty}(B(a))}.\\
\end{split}\end{equation*}
\noindent For $x\in B(b)$,
\begin{equation*}\begin{split}
|Q_2(x)|
&= \Bigg|\frac{1}{4\pi}\int_{(B(b+\frac{2(a-b)}{3})-B(b+\frac{a-b}{3}))}
\frac{1}{|x-y|} \Big( \partial_{\alpha_2}[(\partial_{\alpha_1} {\psi})(H)](y)\\
&\quad\quad\quad\quad -\partial_{\alpha_1}[(\partial_{\alpha_2} {\psi})(H)](y)
+ (\partial_{\alpha_1} \partial_{\alpha_2} {\psi})(H)(y)\Big)dy\Bigg|\\
&\leq 2\|\nabla \psi\|_{L^\infty}\cdot
\sup_{y\in B(b+\frac{a-b}{3})^C}(|\nabla_y\frac{1}{|x-y|}|)
\cdot \|H\|_{L^1(B(a))}\\
&\quad\quad+ \|\nabla^2 \psi\|_{L^\infty}\cdot
\sup_{y\in B(b+\frac{a-b}{3})^C}(|\frac{1}{|x-y|}|)
\cdot \|H\|_{L^1(B(a))}\\
&\leq C\cdot \|H\|_{L^1(B(a))}\\
\end{split}\end{equation*} because $|x-y|\geq (a-b)/3$.
Likewise, for $x\in B(b)$,
\begin{equation*}\begin{split}
|Q_3(x)|
&\leq C\Big(\sum_{k=0}^{n}\|\nabla^{k+1} \psi\|_{L^\infty}\Big)\cdot
\Big(\sum_{k=0}^{n}\sup_{y\in B(b+\frac{a-b}{3})^C}
|\nabla^{k+1}_y\frac{1}{|x-y|}|\Big)
\cdot \|P\|_{L^1(B(a))}\\
&+ C\Big(\sum_{k=0}^{n}\|\nabla^{k+2} \psi\|_{L^\infty}\Big)\cdot
\Big(\sum_{k=0}^{n}\sup_{y\in B(b+\frac{a-b}{3})^C}
|\nabla^{k}_y\frac{1}{|x-y|}|\Big)
\cdot \|P\|_{L^1(B(a))}\\
&\leq C\cdot \|P\|_{L^1(B(a))}.
\end{split}\end{equation*}
\noindent Finally,
\begin{equation*}\begin{split}
\|\nabla^n &P\|_{L^{p}(B(b))}
\leq \|Q_1\|_{L^{p}(B(b))}+
C\||Q_2|+|Q_3|\|_{L^{\infty}(B(b))}\\
&\leq C\cdot
\|H\|_{L^p(B(a))} +
C\cdot \|H\|_{L^1(B(a))}
+C\cdot \|P\|_{L^1(B(a))}\\
&\leq C_{a,b,p,n}\Big(
\| v_2 \|_{W^{n-1,p_2}(B(a))}\cdot
\| v_1 \|_{W^{n-1,p_1}(B(a))}
+\cdot \|P\|_{L^1(B(a))}\Big)
\end{split}\end{equation*} and
\begin{equation*}\begin{split}
\|\nabla^n &P\|_{L^{\infty}(B(b))}
\leq \||Q_1|+|Q_2|+|Q_3|\|_{L^{\infty}(B(b))}\\
&\leq C\cdot
\|H\|_{W^{1,\infty}(B(a))} +
C\cdot \|H\|_{L^1(B(a))}
+C\cdot \|P\|_{L^1(B(a))}\\
&\leq C_{a,b,n}\Big(
\| v_2 \|_{W^{n,\infty}(B(a))}\cdot
\| v_1 \|_{W^{n,\infty}(B(a))}
+ \|P\|_{L^1(B(a))}\Big).
\end{split}\end{equation*}
\end{proof}
\begin{proof} [proof for lemma \ref{lem_a_half_upgrading_large_r}]
We fix $(n,a,b)$ such that $n\geq 0$ and $0<b<a<1$
and let $\alpha$ be a multi index with $|\alpha|=n$. Then,
by taking $D^{\alpha}$ to \eqref{navier_Problem II-r}, we have
\begin{equation}
\begin{split}\label{eq_d_alpha_large_r}
0
=&\partial_t(D^{\alpha}{v_1})+\sum_{\beta\leq\alpha, |\beta|>0}\binom{\alpha}{\beta}
((D^{\beta}{v_2})
\cdot\nabla)(D^{\alpha-\beta}{v_1} )+({v_2}
\cdot\nabla)(D^{\alpha}{v_1} )\\&
\quad\quad\quad\quad\quad
\quad\quad\quad\quad\quad
+ \nabla(D^{\alpha} P) -\Delta(D^{\alpha} {v_1} ).
\end{split}
\end{equation}
\noindent We define $\Phi(t,x)\in C^\infty$ by
$0\leq\Phi\leq 1,
\Phi = 1 \mbox{ on } Q_{{b}} \mbox{ and }
\Phi = 0 \mbox{ on } Q_{{a}}^C.$
We observe that, for $p\geq \frac{1}{2}$ and for $f\in C^\infty$,
\begin{equation*}\begin{split}
&({p}+\frac{1}{2})|f|^{{p}-\frac{3}{2}}f\cdot\partial_{x} f = \partial_{x}|f|^{{p}+\frac{1}{2}}
\mbox{ and } ({p}+\frac{1}{2})|f|^{{p}-\frac{3}{2}}f\cdot\Delta f
\leq \Delta(|f|^{{p}+\frac{1}{2}}) .
\end{split}\end{equation*} which can be verified by direct computations
with the fact $|\nabla f|\geq |\nabla|f||$.\\
\noindent Now
we multiply $(p+\frac{1}{2}){\Phi}\frac{D^{\alpha}v_1}
{|D^{\alpha}v_1|^{(3/2)-p}}$ to \eqref{eq_d_alpha_large_r},
use the above observation
and integrate in x. Then we have for any $p\geq \frac{1}{2}$,\\
\begin{equation*}\begin{split}
&\frac{d}{dt}\int_{\mathbb{R}^3}{\Phi}(t,x)|D^{\alpha}{v_1} (t,x)|^{p+\frac{1}{2}}dx\\
&\leq\int_{\mathbb{R}^3}(|\partial_t{\Phi}(t,x)|+|\Delta{\Phi}(t,x)|)
|D^{\alpha}{v_1} (t,x)|^{p+\frac{1}{2}}dx \\
&\quad+(p+\frac{1}{2})\int_{\mathbb{R}^3}|\nabla D^{\alpha}P(t,x)||D^{\alpha}{v_1} (t,x)|^{p-\frac{1}{2}}dx \\
&\quad+(p+\frac{1}{2})\sum_{\beta\leq\alpha, |\beta|>0}\binom{\alpha}{\beta}
\int_{\mathbb{R}^3}\Big|(D^{\beta}{v_2}(t,x)
\cdot\nabla)D^{\alpha-\beta}{v_1} (t,x)\Big||D^{\alpha}{v_1} (t,x)|^{p-\frac{1}{2}}dx \\
&\quad\quad -\int_{\mathbb{R}^3}{\Phi}(t,x)(v_2(t,x)
\cdot\nabla)(|D^{\alpha}{v_1} (t,x)|^{p+\frac{1}{2}})dx \\
\end{split}
\end{equation*}
\begin{equation*}\begin{split}
&\leq C\||\nabla^n {v_1} (t,\cdot)|^{p+\frac{1}{2}}\|_{L^{1}(B{(a)})} \\
&\quad+C\|\nabla^{n+1} P(t,\cdot)\|_{L^{2p}(B{(a)})}
\cdot\||\nabla^n {v_1} (t,\cdot)|^{p-\frac{1}{2}}\|_{L^{\frac{2p}{2p-1}}(B{(a)})} \\
&+C \| {v_2} (t,\cdot)\|_{W^{n,\infty}(B{(a)})}\cdot
\|{v_1} (t,\cdot)\|_{W^{n,p+\frac{1}{2}}(B{(a)})}\cdot
\||\nabla^n {v_1} (t,\cdot)|^{p-\frac{1}{2}}\|_{L^{\frac{p+\frac{1}{2}}{p-\frac{1}{2}}}(B{(a)})} \\
&-\int_{\mathbb{R}^3}{\Phi}(t,x)\ebdiv\Big({v_2} (t,x)
\otimes|D^{\alpha}{v_1} (t,x)|^{p+\frac{1}{2}}\Big)dx \\
\end{split}
\end{equation*}
\begin{equation*}\begin{split}
&\leq C\|{v_1} (t,\cdot)\|^{p+\frac{1}{2}}_{W^{n,p+\frac{1}{2}}(B{(a)})}\\
&\quad+C\|\nabla^{n+1} P(t,\cdot)\|_{L^{2p}(B{(a)})}
\cdot\|\nabla^n {v_1} (t,\cdot)\|^{p-\frac{1}{2}}_{L^{p}(B{(a)})} \\
&\quad+C \| {v_2} (t,\cdot)\|_{W^{n,\infty}(B{(a)})}
\cdot\|{v_1} (t,\cdot)\|^{p+\frac{1}{2}}_{W^{n,p+\frac{1}{2}}(B{(a)})} \\
&\quad+C\| {v_2} (t,\cdot)\|_{L^{\infty}(B{(a)})}\cdot
\|\nabla^n {v_1} (t,\cdot)\|^{p+\frac{1}{2}}_{L^{p+\frac{1}{2}}(B{(a)})}.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad
\end{split}
\end{equation*}
\noindent Then integrating on $[-a^2,t]$ for any $t\in[-b^2,0]$ gives
\begin{equation*}\begin{split}
&\|D^{\alpha} {v_1} \|^{p+\frac{1}{2}}_{L^{\infty}(-{({b})}^2,0 ;L^{p+\frac{1}{2}}(B{({b})}))}\\
&\leq C\|{v_1} \|^{p+\frac{1}{2}}
_{L^{p+\frac{1}{2}}(-{({a})}^2,0;W^{n,p+\frac{1}{2}}(B{({a})}))}\\
&\quad+C\|\nabla^{n+1} P\|_{L^{1}(-{({a})}^2,0;L^{2p}(B{({a})}))}\cdot
\|\nabla^n {v_1} \|^{p-\frac{1}{2}}_{L^{\infty}(-{({a})}^2,0;L^{p}(B{(a)}))}\\
&\quad+C \| {v_2} \|_{L^{2}(-{({a})}^2,0;W^{n,\infty}(B{({a})}))}
\cdot\|{v_1} \|^{p+\frac{1}{2}}_{L^{2p+1}(-{({a})}^2,0;W^{n,p+\frac{1}{2}}(B{({a})}))} \\
&\quad+C\| {v_2} \|_{L^{2}(-{({a})}^2,0;L^{\infty}(B{(a)}))}\cdot
\|\nabla^n {v_1} \|^{p+\frac{1}{2}}_{L^{2p+1}(-{({a})}^2,0;L^{p+\frac{1}{2}}(B{(a)}))}\\
\end{split}
\end{equation*}
\noindent Thus for the case $p=1/2$, we have
\begin{equation*}\begin{split}
&\|D^{\alpha} {v_1} \|_{L^{\infty}(-{({b})}^2,0 ;L^{1}(B{({b})}))}\\
&\leq C\Big[
\Big(\| v_2\|_{L^2(-{{a}^2},0;W^{n,\infty}(B({a})))} +1\Big)
\cdot
\| {v_1} \|_{L^{2}(-{a}^2,0;W^{n,{1}}(B{(a)}))}\\
&\quad\quad\quad\quad\quad\quad\quad\quad+ \|\nabla^{n+1}P
\|_{L^{1}(-{a}^2,0;L^{1}(B{(a)}))} \Big]
\end{split}\end{equation*}
while, for the case $p\geq 1$, we have
\begin{equation*}\begin{split}
&\|D^{\alpha} {v_1} \|^{p+\frac{1}{2}}_{L^{\infty}(-{({b})}^2,0 ;L^{p+\frac{1}{2}}(B{({b})}))}\\
&\leq C
\Big(\| v_2\|_{L^2(-{{a}^2},0;W^{n,\infty}(B({a})))} +1\Big)
\\&\quad\quad\quad\cdot
\Big(
\|{v_1} \|^{\frac{1}{p+\frac{1}{2}}}
_{L^{2}(-{({a})}^2,0;W^{n,2p}(B{({a})}))}
\cdot
\|{v_1} \|^{1-\frac{1}{p+\frac{1}{2}}}_{L^{\infty}(-{({a})}^2,0;W^{n,p}(B{({a})}))}\Big)^{p+\frac{1}{2}}\\
&\quad\quad\quad+C\|\nabla^{n+1} P\|_{L^{1}(-{({a})}^2,0;L^{2p}(B{({a})}))}\cdot
\| {v_1} \|^{p-\frac{1}{2}}_{L^{\infty}(-{({a})}^2,0;W^{n,p}(B{(a)}))}\\
&\leq C_{a,b,n,p}\Big[
\Big(\| v_2\|_{L^2(-{{a}^2},0;W^{n,\infty}(B({a})))} +1\Big)
\cdot
\|{v_1} \|_{L^{2}(-{({a})}^2,0;W^{n,2p}(B{({a})}))}\\
&\quad\quad\quad\quad\quad\quad+ \|\nabla^{n+1} P\|_{L^{1}(-{({a})}^2,0;L^{2p}(B{({a})}))}
\Big]\cdot
\| {v_1} \|^{p-\frac{1}{2}}_{L^{\infty}(-{({a})}^2,0;W^{n,p}(B{(a)}))}.
\end{split}
\end{equation*}
\end{proof}
\begin{proof}[proof for lemma \ref{lem_Maximal 2.5 or 4}]
Fix any $M_0>0$ and $1\leq p<\infty$ first. Then, for any $M\geq M_0$ and
for any $f\in C^1(\mathbb{R}^3)$
such that $ \int_{\mathbb{R}^3}\phi(x)f(x)dx=0$,
we have
\begin{equation*}\begin{split}
&\|f\|_{L^p(B(M))}=
\Big(\int_{B(M)}\Big|\int_{\mathbb{R}^3}(f(x)-f(y))
\phi(y)dy\Big|^{p}dx\Big)^{1/p}\\
&\leq C\Big(\int_{B(M)}\Big(\int_{B(1)}\Big|f(x)-f(y)\Big|
dy\Big)^{p}dx\Big)^{1/p}\\
&\leq C\Big(\int_{B(M)}\Big(\int_{B(1)}
\int_0^1\Big|(\nabla f)((1-t)x+ty)\cdot(x-y)\Big|dt
dy\Big)^{p}dx\Big)^{1/p}
\end{split}\end{equation*}
\begin{equation*}\begin{split}
&\leq C(M+1)\Big(\int_{B(M)}\Big(\int_{B(1)}
\int_0^1\Big|(\nabla f)((1-t)x+ty)\Big|dt
dy\Big)^{p}dx\Big)^{1/p}\\
&\leq C(M+1)\Big(\int_{B(M)}\Big(\int_{B(1)}
\int_0^{\frac{M}{M+1}}\Big|(\nabla f)((1-t)x+ty)\Big|dt
dy\Big)^{p}dx\Big)^{1/p}\\
&\quad + C(M+1)\Big(\int_{B(M)}\Big(\int_{B(1)}
\int_{\frac{M}{M+1}}^1\Big|(\nabla f)((1-t)x+ty)\Big|dt
dy\Big)^{p}dx\Big)^{1/p}\\
&=(I)+(II)
\end{split}\end{equation*} where we used $x\in B(M)$ and $y\in B(1)$.\\
\noindent For $(I)$,
\begin{equation*}\begin{split}
&(I)
\leq C_{M_0}\Big(\int_{B(1)}
\int_0^{\frac{M}{M+1}}\Big(\int_{B(M)}
\Big|(\nabla f)((1-t)x+ty)\Big|^{p}dx\Big)^{1/p}dt
dy\Big)\\
&\leq C_{M_0}\cdot M
\int_0^{\frac{M}{M+1}}\frac{1}{(1-t)^{3/p}}\Big(\int_{B((1-t)M+1)}
\Big|(\nabla f)(z)\Big|^{p}dz\Big)^{1/p}dt\\
&\leq C_{M_0}\cdot M
\int_0^{\frac{M}{M+1}}\frac{1}{(1-t)^{3/p}}
\Big(
\int_{B(1)}
\int_{B((1-t)M+2)}
\Big|(\nabla f)(z+u)\Big|^{p}dzdu\Big)^{1/p}dt\\
&\leq C_{M_0}\cdot M
\int_0^{\frac{M}{M+1}}\frac{((1-t)M+2)^{3/p}}{(1-t)^{3/p}}\Big(
\int_{B(1)}
\mathcal{M}(|\nabla f|^{p})(u)du\Big)^{1/p}dt\\
&\leq C_{M_0,p}\cdot M \cdot\|\mathcal{M}(|\nabla f|^{p})\|^{1/p}_{L^1(B(1))}
\int_0^{\frac{M}{M+1}}\Big(M^{3/p}+\frac{1}{(1-t)^{3/p}}\Big)
dt
\\
&\leq C_{M_0,p}\cdot M \cdot\|\mathcal{M}(|\nabla f|^{p})\|^{1/p}_{L^1(B(1))}
\Big(M^{3/p}+\int_{\frac{1}{M+1}}^{1}\frac{1}{s^{3/p}}
ds\Big)
\\
&\leq C_{M_0}\cdot M \cdot\|\mathcal{M}(|\nabla f|^{p})\|^{1/p}_{L^1(B(1))}
\Big(M^{3/p}+{(M+1)}^{3/p}
\Big)
\\
&\leq C_{M_0,p}\cdot M^{1+\frac{3}{p}}
\cdot\|\mathcal{M}(|\nabla f|^{p})\|^{1/p}_{L^1(B(1))}
\end{split}\end{equation*} where we used
an integral version of the Minkoski's inequality
and $(1+M)\leq C_{M_0}\cdot M$ from $M\geq M_0$
for the first inequality.\\
\noindent For $(II)$, observe that if $\frac{M}{M+1}\leq t\leq 1$, then
$0\leq 1-t\leq \frac{1}{M+1}$ and
\begin{equation*}\begin{split}
|(1-t)x+ty|\leq (1-t)\cdot|x|+t|y|\leq \frac{M}{M+1}+1\leq 2
\end{split}\end{equation*} because $x\in B(M)$ and $y\in B(1)$. Thus
\begin{equation*}\begin{split}
(II)
& \leq
C_{M_0}\cdot M\Big(\int_{B(M)}\Big(
\int_{\frac{M}{M+1}}^1\frac{1}{t^3}\int_{B(2)}\Big|(\nabla f)(z)\Big|dz
dt\Big)^{p}dx\Big)^{1/p}\\
& \leq
C_{M_0}\cdot M\cdot M^{3/p}\cdot
\int_{B(2)}|(\nabla f)(z)|dz
\cdot
\int_{\frac{M}{M+1}}^1\frac{1}{t^3}
dt\\
& \leq
C_{M_0} M^{1+\frac{3}{p}}\cdot
\|\nabla f\|_{L^1(B(2))}.
\end{split}\end{equation*}
\end{proof}
\begin{ack}
The second author was partially supported by both the NSF
and the EPSRC Science and Innovation award to the Oxford Centre
for Nonlinear PDE (EP/E035027/1).
\end{ack}
\bibliographystyle{plain}
\bibliography{Fractional_NS_Choi_Vasseur}
\end{document}
| 454
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Closing or Destroying A Session
When you are finished with a session in php is there a way to destroy or close it like the Session.Abandon command is asp? I know you can use session_destroy or session_unset to destroy the data or variables, but this does not destroy the session.Wordpress Auto Installer
session_destroy will destroy the session...
the session automatically gets destroyed once the browser is killed.. it also gets destroyed by the server after a particular period has lapsed.. this period can be defined in your php.ini file...i dunno...
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