File size: 21,602 Bytes
7f37411 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphDescendants
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory
open scoped ContDiff ComplexConjugate
lemma graph_fixed_angular_budget {V : Type*} [NormedAddCommGroup V]
[InnerProductSpace ℝ V] [FiniteDimensional ℝ V]
{ψ : V → ℂ} (hψ : ContDiff ℝ ∞ ψ) (m : ℕ) :
∃ P : ℝ, 0 ≤ P ∧ ∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ P := by
have hb (j : ℕ) := (isCompact_sphere (0:V) 1).exists_bound_of_continuousOn
(hψ.continuous_iteratedFDeriv (by exact_mod_cast (le_top : (j:ℕ∞)≤⊤))).continuousOn
choose C hC using hb
refine ⟨∑ j ∈ Finset.range (m+1), max 0 (C j),Finset.sum_nonneg (fun _ _ => le_max_left _ _),?_⟩
intro j hj w hw
exact ((hC j w (by simpa only [Metric.mem_sphere,dist_zero_right] using hw)).trans
(le_max_right _ _)).trans (Finset.single_le_sum (fun _ _ => le_max_left _ _)
(Finset.mem_range.mpr (by omega)))
lemma graph_cap_with_angular_jet (d m : ℕ) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (δ A : ℝ) (k : Fin d → ℤ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ)
(P : ℝ), 0 < δ → δ⁻¹ ≤ A → 1 ≤ A → 0 ≤ P → ContDiff ℝ ∞ ψ →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ w‖ ≤ P) →
∀ j ≤ m, ∀ w, ‖w‖=1 →
‖iteratedFDeriv ℝ j (fun w => (tensorCapND δ k w:ℂ)*ψ w) w‖ ≤ (C*P*2^m)*A^j := by
obtain ⟨C,hC,hb⟩ := graphCapJet d m
refine ⟨C,hC,?_⟩
intro δ A k ψ P hδ hδA hA hP hψ hdψ j hj w hw
exact graphJetMul (Complex.ofRealCLM.contDiff.comp (tensorCapND_contDiff δ k)) hψ m
hC hP (le_trans (by norm_num) hA) w (fun j hj => hb δ A k hδ hδA w j hj)
(fun j hj => (hdψ j hj w hw).trans
(le_mul_of_one_le_right hP (one_le_pow₀ hA))) j hj
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
theorem graph_cap_four_decay (d : ℕ) {sig c₀ G : ℝ}
(hsig : 0 < sig) (hc₀ : 0 < c₀) (hG : 0 ≤ G) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-p)*R^(30*sig)
let S := R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))
∀ (μ₁ μ₂ μ₃ μ₄ : Measure (EuclideanSpace ℝ (Fin d))),
∀ [IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄],
∀ (Ψ : ℝ → ℂ) (K B P : ℝ) (k₁ k₂ : Fin d → ℤ)
(u₁ u₂ : EuclideanSpace ℝ (Fin d))
(ψ₁ ψ₂ : EuclideanSpace ℝ (Fin d) → ℂ)
(b₁ b₂ b₃ b₄ : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ),
k₁ ∈ sphereCapGridND d δ → k₂ ∈ sphereCapGridND d δ →
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) → 0 ≤ K → 0 ≤ P →
ContDiff ℝ ∞ ψ₁ → ContDiff ℝ ∞ ψ₂ →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ₁ w‖ ≤ P) →
(∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j ψ₂ w‖ ≤ P) →
GraphSymbolJet b₁ m S B → GraphSymbolJet b₂ m S B →
GraphSymbolJet b₃ m S B → GraphSymbolJet b₄ m S B →
(∀ᵐ z ∂(μ₁.prod μ₂).prod (μ₃.prod μ₄),
‖z.1.1-z.1.2‖ ≤ G*(2:ℝ)^(-a) ∧ ‖z.2.1-z.2.2‖ ≤ G*(2:ℝ)^(-a) ∧
‖(z.1.1-z.1.2)-u₁‖ ≤ G*(2:ℝ)^(-p) ∧ ‖(z.2.1-z.2.2)-u₂‖ ≤ G*(2:ℝ)^(-p)) →
¬(‖u₁-inner ℝ u₁ (sphereCapCenterND δ k₁) • sphereCapCenterND δ k₁‖ ≤ τ ∧
‖u₂-inner ℝ u₂ (sphereCapCenterND δ k₂) • sphereCapCenterND δ k₂‖ ≤ τ ∧
|inner ℝ (sphereCapCenterND δ k₁) u₁+inner ℝ (sphereCapCenterND δ k₂) u₂| ≤ τ) →
‖∫ q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))), Ψ (q.1/(2:ℝ)^v)*
((tensorCapND δ k₁ q.2.1:ℂ)*ψ₁ q.2.1 * sourceAngularTransformND μ₁ univ 1 b₁ q.1 q.2.1 *
conj (sourceAngularTransformND μ₂ univ 1 b₂ q.1 q.2.1))*
((tensorCapND δ k₂ q.2.2:ℂ)*ψ₂ q.2.2 * sourceAngularTransformND μ₃ univ 1 b₃ q.1 q.2.2 *
conj (sourceAngularTransformND μ₄ univ 1 b₄ q.1 q.2.2))
∂volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))‖ ≤
C*K*P^2*B^4*R^(-T)*(2:ℝ)^v*((μ₁.prod μ₂).prod (μ₃.prod μ₄)).real univ := by
obtain ⟨m,C,hC,N₀,hraw⟩ := graph_source_raw_four_decay
(V:=EuclideanSpace ℝ (Fin d)) hsig hc₀ (by positivity : (0:ℝ) ≤ 2*(d+1:ℝ)) hG T
obtain ⟨D,hD,hcap⟩ := graph_cap_with_angular_jet d m
refine ⟨m,C*D^2*((2:ℝ)^m)^6,by positivity,N₀,?_⟩
intro N hN a p v L hh hL hv R δ τ S μ₁ μ₂ μ₃ μ₄ _ _ _ _ Ψ K B P k₁ k₂ u₁ u₂
ψ₁ ψ₂ b₁ b₂ b₃ b₄ hk₁ hk₂ hΨ hsΨ hK hdΨ hK0 hP hψ₁ hψ₂ hdψ₁ hdψ₂
hb₁ hb₂ hb₃ hb₄ hgeo hne
have hδ : 0 < δ := lt_min hc₀ (by positivity)
have hS : 0 ≤ S := by positivity
have hB := hb₁.nonneg
have hδi : 0 ≤ δ⁻¹ := inv_nonneg.mpr hδ.le
let A := 1+S+δ⁻¹
have hSA : S ≤ A := by dsimp [A]; linarith
have hδA : δ⁻¹ ≤ A := by dsimp [A]; linarith
have hA1 : 1 ≤ A := by dsimp [A]; linarith
let θ₁ := fun w => (tensorCapND δ k₁ w:ℂ)*ψ₁ w
let θ₂ := fun w => (tensorCapND δ k₂ w:ℂ)*ψ₂ w
have hθ₁ : ContDiff ℝ ∞ θ₁ := (Complex.ofRealCLM.contDiff.comp (tensorCapND_contDiff δ k₁)).mul hψ₁
have hθ₂ : ContDiff ℝ ∞ θ₂ := (Complex.ofRealCLM.contDiff.comp (tensorCapND_contDiff δ k₂)).mul hψ₂
have hdθ₁ := hcap δ A k₁ ψ₁ P hδ hδA hA1 hP hψ₁ hdψ₁
have hdθ₂ := hcap δ A k₂ ψ₂ P hδ hδA hA1 hP hψ₂ hdψ₂
have hpair₁ := graph_symbol_pair_budget hb₁ hb₂ hθ₁ hS hSA (by positivity : 0 ≤ D*P*2^m) hdθ₁
have hpair₂ := graph_symbol_pair_budget hb₃ hb₄ hθ₂ hS hSA (by positivity : 0 ≤ D*P*2^m) hdθ₂
have hPθ : 0 ≤ D*P*2^m := by positivity
have hnθ₁ (w) (hw : ‖w‖=1) : ‖θ₁ w‖ ≤ D*P*2^m := by
simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hdθ₁ 0 (Nat.zero_le _) w hw
have hnθ₂ (w) (hw : ‖w‖=1) : ‖θ₂ w‖ ≤ D*P*2^m := by
simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hdθ₂ 0 (Nat.zero_le _) w hw
have hs₁ (w) (hw : ‖w‖=1) (hne : θ₁ w ≠ 0) :
‖w-sphereCapCenterND δ k₁‖ ≤ 2*(d+1:ℝ)*δ := by
have hn : tensorCapND δ k₁ w ≠ 0 := by intro h; simp [θ₁,h] at hne
exact sphereCapCenterND_dist hδ hk₁ hw (subset_closure hn)
have hs₂ (w) (hw : ‖w‖=1) (hne : θ₂ w ≠ 0) :
‖w-sphereCapCenterND δ k₂‖ ≤ 2*(d+1:ℝ)*δ := by
have hn : tensorCapND δ k₂ w ≠ 0 := by intro h; simp [θ₂,h] at hne
exact sphereCapCenterND_dist hδ hk₂ hw (subset_closure hn)
have h := hraw N hN a p v L hh hL hv μ₁ μ₂ μ₃ μ₄ Ψ
(sphereCapCenterND δ k₁) (sphereCapCenterND δ k₂) u₁ u₂ K
(D*P*2^m*B*2^m*B*2^m) (D*P*2^m*B*2^m*B*2^m)
θ₁ θ₂ b₁ b₂ b₃ b₄ (D*P*2^m) (D*P*2^m) B B B B
hΨ hsΨ hK hdΨ (sphereCapCenterND_norm hk₁) (sphereCapCenterND_norm hk₂)
hK0 (by positivity) (by positivity) hθ₁.continuous.measurable hθ₂.continuous.measurable
hb₁.measurable hb₂.measurable hb₃.measurable hb₄.measurable
hPθ hPθ hb₁.nonneg hb₂.nonneg hb₃.nonneg hb₄.nonneg hnθ₁ hnθ₂
hb₁.norm_le hb₂.norm_le hb₃.norm_le hb₄.norm_le hgeo hne
hpair₁.1 hpair₂.1 hpair₁.2 hpair₂.2 hs₁ hs₂
exact h.trans_eq (by ring)
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory
open scoped ComplexConjugate
variable {V : Type*} [MeasurableSpace V]
lemma graph_four_zero_of_ineligible (σ : Measure V) [SFinite σ]
(Ψ : ℝ → ℂ) (η₁ η₂ : V → ℝ) (ψ₁ ψ₂ : V → ℂ)
(U₁ U₂ U₃ U₄ : ℝ×V → ℂ)
(hz : (∀ᵐ w ∂σ, η₁ w=0 ∨ ∀ r, U₁ (r,w)=0) ∨
(∀ᵐ w ∂σ, η₁ w=0 ∨ ∀ r, U₂ (r,w)=0) ∨
(∀ᵐ w ∂σ, η₂ w=0 ∨ ∀ r, U₃ (r,w)=0) ∨
(∀ᵐ w ∂σ, η₂ w=0 ∨ ∀ r, U₄ (r,w)=0)) :
(∫ q : ℝ×(V×V), Ψ q.1*
((η₁ q.2.1:ℂ)*ψ₁ q.2.1*U₁ (q.1,q.2.1)*conj (U₂ (q.1,q.2.1)))*
((η₂ q.2.2:ℂ)*ψ₂ q.2.2*U₃ (q.1,q.2.2)*conj (U₄ (q.1,q.2.2)))
∂volume.prod (σ.prod σ))=0 := by
apply integral_eq_zero_of_ae
rcases hz with hz | hz | hz | hz
· have h : ∀ᵐ z : V×V ∂σ.prod σ, η₁ z.1=0 ∨ ∀ r, U₁ (r,z.1)=0 :=
Measure.quasiMeasurePreserving_fst.ae hz
filter_upwards [Measure.quasiMeasurePreserving_snd.ae h] with q hq
rcases hq with hq | hq <;> simp [hq]
· have h : ∀ᵐ z : V×V ∂σ.prod σ, η₁ z.1=0 ∨ ∀ r, U₂ (r,z.1)=0 :=
Measure.quasiMeasurePreserving_fst.ae hz
filter_upwards [Measure.quasiMeasurePreserving_snd.ae h] with q hq
rcases hq with hq | hq <;> simp [hq]
· have h : ∀ᵐ z : V×V ∂σ.prod σ, η₂ z.2=0 ∨ ∀ r, U₃ (r,z.2)=0 :=
Measure.quasiMeasurePreserving_snd.ae hz
filter_upwards [Measure.quasiMeasurePreserving_snd.ae h] with q hq
rcases hq with hq | hq <;> simp [hq]
· have h : ∀ᵐ z : V×V ∂σ.prod σ, η₂ z.2=0 ∨ ∀ r, U₄ (r,z.2)=0 :=
Measure.quasiMeasurePreserving_snd.ae hz
filter_upwards [Measure.quasiMeasurePreserving_snd.ae h] with q hq
rcases hq with hq | hq <;> simp [hq]
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace
variable {d : ℕ}
lemma graph_cell_cap_ineligible (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ]
{a p : ℕ} {X P : CellPath d} (hP : P ∈ graphChildren ρ a p X)
(M δ : ℝ) (k : Fin d → ℤ) {b : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B)
(hn : P ∉ eligibleCells ρ a p X (tensorCapND δ k) b) :
∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)), tensorCapND δ k w=0 ∨
∀ r, sourceAngularTransformND ρ (treeCell P) M b r w=0 := by
apply graph_uneligible_transform ρ _ (treeCell P) M (tensorCapND δ k) b
(tensorCapND_contDiff δ k).continuous.measurable hb.measurable (by norm_num : (0:ℝ) ≤ 1)
hb.nonneg (Filter.Eventually.of_forall (tensorCapND_nonneg δ k))
(Filter.Eventually.of_forall (tensorCapND_le_one δ k))
(canonicalSphereMeasureND_ae_norm.mono (fun w hw x => hb.norm_le x w hw))
exact fun h => hn ((graphChildren_eligible_iff hP).mpr h)
lemma graph_children_transform_split (ρ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ]
{a p : ℕ} (hap : a ≤ p) {X : CellPath d} (hX : X ∈ dyadicPaths a)
{b : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B)
{Ma Mp : ℝ} (hMp : Mp ≠ 0) (r : ℝ) (w : EuclideanSpace ℝ (Fin d)) (hw : ‖w‖=1) :
sourceAngularTransformND ρ (treeCell X) Ma b r w =
(Mp/Ma) • ∑ P ∈ graphChildren ρ a p X, sourceAngularTransformND ρ (treeCell P) Mp b r w := by
have hm : Measurable (fun x => spherePlaneWaveND r x w*b x w) :=
(show Measurable (fun x => spherePlaneWaveND r x w) by unfold spherePlaneWaveND; fun_prop).mul
(hb.measurable.comp (measurable_id.prodMk measurable_const))
have hi : IntegrableOn (fun x => spherePlaneWaveND r x w*b x w) (treeCell X) ρ := by
apply Integrable.of_bound hm.aestronglyMeasurable B
filter_upwards with x
simpa only [norm_mul,spherePlaneWaveND_norm,one_mul] using hb.norm_le x w hw
simp only [sourceAngularTransformND]
rw [graphChildren_integral ρ hap hX _ hi, ←Finset.smul_sum, smul_smul]
congr 1
field_simp
end CurrentFalconer.TreeND
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer.TreeND
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {d : ℕ}
noncomputable def graphDirectCellTerm
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (M₁ M₂ : ℝ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ)
(Ψ : ℝ → ℝ) (v δ : ℝ) (k₁ k₂ : Fin d → ℤ)
(row col : CellPath d × CellPath d) : ℂ :=
∫ q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))),
(Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)*
((sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1 *
sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)*
conj (sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.1 *
sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.2))
∂volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))
noncomputable def graphCrossedCellTerm
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (M₁ M₂ : ℝ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ)
(Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) (v δ : ℝ) (k₁ k₂ : Fin d → ℤ)
(row col : CellPath d × CellPath d) : ℂ :=
∫ q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))),
(Ψ (q.1/(2:ℝ)^v):ℂ)*((tensorCapND δ k₁ q.2.1*tensorCapND δ k₂ q.2.2):ℂ)*
(((ψ q.2.1*sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1) *
sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2)*
conj ((ψ q.2.2*sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.2) *
sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.1))
∂volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))))
lemma graphDirectCellTerm_eq (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (M₁ M₂ : ℝ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ)
(Ψ : ℝ → ℝ) (v δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) :
graphDirectCellTerm ρ κ M₁ M₂ b c Ψ v δ k₁ k₂ row col =
∫ q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))), (Ψ (q.1/(2:ℝ)^v):ℂ)*
((tensorCapND δ k₁ q.2.1:ℂ)*1*sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1 *
conj (sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.1))*
((tensorCapND δ k₂ q.2.2:ℂ)*1*sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2 *
conj (sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.2))
∂volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))) := by
apply integral_congr_ae
filter_upwards with q
simp only [map_mul]
ring
lemma graphCrossedCellTerm_eq (ρ κ : Measure (EuclideanSpace ℝ (Fin d))) (M₁ M₂ : ℝ)
(b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ)
(Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ)
(v δ : ℝ) (k₁ k₂ : Fin d → ℤ) (row col : CellPath d × CellPath d) :
graphCrossedCellTerm ρ κ M₁ M₂ b c Ψ ψ v δ k₁ k₂ row col =
∫ q : ℝ×((EuclideanSpace ℝ (Fin d))×(EuclideanSpace ℝ (Fin d))), (Ψ (q.1/(2:ℝ)^v):ℂ)*
((tensorCapND δ k₁ q.2.1:ℂ)*ψ q.2.1*sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2.1 *
conj (sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2.1))*
((tensorCapND δ k₂ q.2.2:ℂ)*conj (ψ q.2.2)*sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2.2 *
conj (sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2.2))
∂volume.prod ((canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d))).prod
(canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))) := by
apply integral_congr_ae
filter_upwards with q
simp only [map_mul]
ring
lemma graph_direct_ineligible_zero
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
{a p : ℕ} {X Y : CellPath d} {row col : CellPath d × CellPath d}
(hr : row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y))
(hc : col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y))
(M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hcj : GraphSymbolJet c m s B)
(Ψ : ℝ → ℝ) (v δ : ℝ) (k₁ k₂ : Fin d → ℤ)
(hin : ¬(row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧
col ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c))) :
graphDirectCellTerm ρ κ M₁ M₂ b c Ψ v δ k₁ k₂ row col = 0 := by
rw [graphDirectCellTerm_eq]
apply graph_four_zero_of_ineligible (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))
(fun r => (Ψ (r/(2:ℝ)^v):ℂ)) (tensorCapND δ k₁) (tensorCapND δ k₂)
(fun _ => 1) (fun _ => 1)
(fun q => sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2)
(fun q => sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2)
(fun q => sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2)
(fun q => sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2)
have hr := Finset.mem_product.mp hr
have hc := Finset.mem_product.mp hc
by_cases h₁ : row.1 ∈ eligibleCells ρ a p X (tensorCapND δ k₁) b
· by_cases h₂ : col.1 ∈ eligibleCells ρ a p X (tensorCapND δ k₁) b
· by_cases h₃ : row.2 ∈ eligibleCells κ a p Y (tensorCapND δ k₂) c
· exact Or.inr (Or.inr (Or.inr (graph_cell_cap_ineligible κ hc.2 M₂ δ k₂ hcj
(fun h₄ => hin ⟨Finset.mem_product.mpr ⟨h₁,h₃⟩,Finset.mem_product.mpr ⟨h₂,h₄⟩⟩))))
· exact Or.inr (Or.inr (Or.inl (graph_cell_cap_ineligible κ hr.2 M₂ δ k₂ hcj h₃)))
· exact Or.inr (Or.inl (graph_cell_cap_ineligible ρ hc.1 M₁ δ k₁ hb h₂))
· exact Or.inl (graph_cell_cap_ineligible ρ hr.1 M₁ δ k₁ hb h₁)
lemma graph_crossed_ineligible_zero
(ρ κ : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ]
{a p : ℕ} {X Y : CellPath d} {row col : CellPath d × CellPath d}
(hr : row ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y))
(hc : col ∈ (graphChildren ρ a p X).product (graphChildren κ a p Y))
(M₁ M₂ : ℝ) {b c : EuclideanSpace ℝ (Fin d) → EuclideanSpace ℝ (Fin d) → ℂ}
{m : ℕ} {s B : ℝ} (hb : GraphSymbolJet b m s B) (hcj : GraphSymbolJet c m s B)
(Ψ : ℝ → ℝ) (ψ : EuclideanSpace ℝ (Fin d) → ℂ) (v δ : ℝ) (k₁ k₂ : Fin d → ℤ)
(hin : ¬(row ∈ (eligibleCells ρ a p X (tensorCapND δ k₁) b).product
(eligibleCells κ a p Y (tensorCapND δ k₂) c) ∧
col ∈ (eligibleCells ρ a p X (tensorCapND δ k₂) b).product
(eligibleCells κ a p Y (tensorCapND δ k₁) c))) :
graphCrossedCellTerm ρ κ M₁ M₂ b c Ψ ψ v δ k₁ k₂ row col = 0 := by
rw [graphCrossedCellTerm_eq]
apply graph_four_zero_of_ineligible (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin d)))
(fun r => (Ψ (r/(2:ℝ)^v):ℂ)) (tensorCapND δ k₁) (tensorCapND δ k₂)
ψ (fun w => conj (ψ w))
(fun q => sourceAngularTransformND ρ (treeCell row.1) M₁ b q.1 q.2)
(fun q => sourceAngularTransformND κ (treeCell col.2) M₂ c q.1 q.2)
(fun q => sourceAngularTransformND κ (treeCell row.2) M₂ c q.1 q.2)
(fun q => sourceAngularTransformND ρ (treeCell col.1) M₁ b q.1 q.2)
have hr := Finset.mem_product.mp hr
have hc := Finset.mem_product.mp hc
by_cases h₁ : row.1 ∈ eligibleCells ρ a p X (tensorCapND δ k₁) b
· by_cases h₂ : col.2 ∈ eligibleCells κ a p Y (tensorCapND δ k₁) c
· by_cases h₃ : row.2 ∈ eligibleCells κ a p Y (tensorCapND δ k₂) c
· exact Or.inr (Or.inr (Or.inr (graph_cell_cap_ineligible ρ hc.1 M₁ δ k₂ hb
(fun h₄ => hin ⟨Finset.mem_product.mpr ⟨h₁,h₃⟩,Finset.mem_product.mpr ⟨h₄,h₂⟩⟩))))
· exact Or.inr (Or.inr (Or.inl (graph_cell_cap_ineligible κ hr.2 M₂ δ k₂ hcj h₃)))
· exact Or.inr (Or.inl (graph_cell_cap_ineligible κ hc.2 M₂ δ k₁ hcj h₂))
· exact Or.inl (graph_cell_cap_ineligible ρ hr.1 M₁ δ k₁ hb h₁)
end CurrentFalconer.TreeND
end
end
end OAI
|