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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 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Angular.SpatialAngularKernelMeasurable
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereShift2
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4PrefixShell
open MeasureTheory Set
open scoped ComplexConjugate
lemma integrable_list_sumND {Q Z : Type*} [MeasurableSpace Z]
(μ : Measure Z) (L : List Q) (F : Q → Z → ℂ)
(hi : ∀ i ∈ L, Integrable (F i) μ) :
Integrable (fun z => (L.map (fun i => F i z)).sum) μ := by
induction L with
| nil => simp
| cons i L ih =>
simp only [List.mem_cons,forall_eq_or_imp] at hi
simp only [List.map_cons,List.sum_cons]
exact hi.1.add (ih hi.2)
lemma integral_list_sumND {Q Z : Type*} [MeasurableSpace Z]
(μ : Measure Z) (L : List Q) (F : Q → Z → ℂ)
(hi : ∀ i ∈ L, Integrable (F i) μ) :
(∫ z, (L.map (fun i => F i z)).sum ∂μ) = (L.map (fun i => ∫ z, F i z ∂μ)).sum := by
induction L with
| nil => simp
| cons i L ih =>
simp only [List.mem_cons,forall_eq_or_imp] at hi
simp only [List.map_cons,List.sum_cons]
rw [integral_add hi.1 (integrable_list_sumND μ L F hi.2), ih hi.2]
variable {d : ℕ}
local notation "V" => EuclideanSpace ℝ (Fin d)
lemma spatial_annulus_pair_boundND (δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K)
(j : ℕ) (scale : ℝ) (z : V × V) :
‖SpatialFourierND.spatialAnnulusKernel d δ K hδ hK j (scale⁻¹ • (z.1-z.2))‖ ≤
SpatialFourierND.spatialFourierCost d δ K hδ hK j := by
rw [← SpatialFourierND.spatialFourierSeparation d δ K hδ hK j scale z.1 z.2]
apply le_trans (norm_integral_le_integral_norm _)
simp only [norm_mul,SpatialFourierND.spatialFourierPhase_norm,Complex.norm_conj,mul_one]
exact le_rfl
lemma spatial_angular_term_integrableND (δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K)
(j : ℕ) (scale : ℝ) (σ : Measure V) (ν : Measure (V × V))
[IsFiniteMeasure σ] [IsFiniteMeasure ν]
(r : ℝ) {ψ : V → ℂ} {b c : V → V → ℂ}
(hψ : Measurable ψ) (hb : Measurable (Function.uncurry b))
(hc : Measurable (Function.uncurry c)) {P B C : ℝ}
(hP : ∀ w, ‖ψ w‖ ≤ P) (hB : ∀ x w, ‖b x w‖ ≤ B)
(hC : ∀ y w, ‖c y w‖ ≤ C) :
Integrable (fun z : V × V =>
SpatialFourierND.spatialAnnulusKernel d δ K hδ hK j (scale⁻¹ • (z.1-z.2)) *
(∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w*b z.1 w*conj (c z.2 w) ∂σ)) ν := by
have hk : Measurable (fun z : V × V =>
SpatialFourierND.spatialAnnulusKernel d δ K hδ hK j (scale⁻¹ • (z.1-z.2))) :=
(SpatialFourierND.spatialAnnulusKernel_smooth d δ K hδ hK j).continuous.measurable.comp
(by fun_prop)
apply Integrable.of_bound (hk.mul (spatial_angular_kernel_measurableND σ r hψ hb hc)).aestronglyMeasurable
(SpatialFourierND.spatialFourierCost d δ K hδ hK j*(P*B*C*σ.real univ))
filter_upwards with z
simp only [Pi.mul_apply,norm_mul]
exact mul_le_mul (spatial_annulus_pair_boundND δ K hδ hK j scale z)
(spatial_angular_kernel_boundND σ r ψ b c hP hB hC z) (norm_nonneg _)
(SpatialFourierND.spatialFourierCost_nonneg d δ K hδ hK j)
theorem finite_spatial_angular_factorizationND {Q : Type*} (L : List Q)
(δ K : ℝ) (hδ : 0 < δ) (hK : 0 < K) (j : ℕ) (scale : ℝ)
(σ ρ ν : Measure V) [IsFiniteMeasure σ] [IsFiniteMeasure ρ] [IsFiniteMeasure ν]
(X Y : Set V) (M N r : ℝ) (a : Q → ℂ) {ψ : V → ℂ}
{b c : Q → V → V → ℂ} (hψ : Measurable ψ)
(hb : ∀ i ∈ L, Measurable (Function.uncurry (b i)))
(hc : ∀ i ∈ L, Measurable (Function.uncurry (c i))) {P : ℝ} {B C : Q → ℝ}
(hP : ∀ w, ‖ψ w‖ ≤ P) (hB : ∀ i ∈ L, ∀ x w, ‖b i x w‖ ≤ B i)
(hC : ∀ i ∈ L, ∀ y w, ‖c i y w‖ ≤ C i) :
((M*N)⁻¹:ℝ) • (∫ z : V × V,
(L.map (fun i => a i*(SpatialFourierND.spatialAnnulusKernel d δ K hδ hK j
(scale⁻¹ • (z.1-z.2)) * (∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w*
b i z.1 w*conj (c i z.2 w) ∂σ)))).sum ∂(ρ.restrict X).prod (ν.restrict Y)) =
(L.map (fun i => a i*(∫ ξ, SpatialFourierND.spatialFourierCoeff d δ K hδ hK j ξ *
∫ w, ψ w*sourceAngularTransformND ρ X M
(fun x w => SpatialFourierND.spatialFourierPhase d scale ξ x*b i x w) r w *
conj (sourceAngularTransformND ν Y N
(fun y w => SpatialFourierND.spatialFourierPhase d scale ξ y*c i y w) r w) ∂σ))).sum := by
rw [integral_list_sumND _ L _ (fun i hi =>
(spatial_angular_term_integrableND δ K hδ hK j scale σ _ r hψ (hb i hi) (hc i hi)
hP (hB i hi) (hC i hi)).const_mul (a i))]
simp only [integral_const_mul,Complex.real_smul,← List.sum_map_mul_left]
apply congrArg List.sum
apply List.map_congr_left
intro i hi
rw [mul_left_comm]
congr 1
exact spatial_angular_factorizationND δ K hδ hK j scale σ ρ ν X Y M N r
hψ (hb i hi) (hc i hi) hP (hB i hi) (hC i hi)
end Attempt4PrefixShell
end GraphLiteralSupport
end
end
end OAI
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