| 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 | |