openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /SpatialAngularKernelMeasurable.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.SpatialInnerBump | |
| import OAI.MeasureTheory.FalconerCompletion.Graphs.SpherePlaneWaveSub3 | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace Attempt4PrefixShell | |
| open MeasureTheory Set | |
| open scoped ComplexConjugate | |
| variable {d : ℕ} | |
| local notation "V" => EuclideanSpace ℝ (Fin d) | |
| lemma spatial_angular_kernel_measurableND (σ : Measure V) [SFinite σ] (r : ℝ) | |
| {ψ : V → ℂ} {b c : V → V → ℂ} (hψ : Measurable ψ) | |
| (hb : Measurable (Function.uncurry b)) (hc : Measurable (Function.uncurry c)) : | |
| Measurable (fun z : V × V => ∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w* | |
| b z.1 w*conj (c z.2 w) ∂σ) := by | |
| have hm : Measurable (fun z : (V × V) × V => ψ z.2*spherePlaneWaveND r (z.1.1-z.1.2) z.2* | |
| b z.1.1 z.2*conj (c z.1.2 z.2)) := by | |
| exact (((hψ.comp measurable_snd).mul | |
| (by unfold spherePlaneWaveND; fun_prop)).mul | |
| (hb.comp (measurable_fst.fst.prodMk measurable_snd))).mul | |
| (Complex.continuous_conj.measurable.comp | |
| (hc.comp (measurable_fst.snd.prodMk measurable_snd))) | |
| exact hm.stronglyMeasurable.integral_prod_right'.measurable | |
| lemma spatial_angular_kernel_boundND (σ : Measure V) [IsFiniteMeasure σ] (r : ℝ) | |
| (ψ : V → ℂ) (b c : V → V → ℂ) {P B C : ℝ} | |
| (hP : ∀ w, ‖ψ w‖ ≤ P) (hB : ∀ x w, ‖b x w‖ ≤ B) | |
| (hC : ∀ y w, ‖c y w‖ ≤ C) (z : V × V) : | |
| ‖∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w*b z.1 w*conj (c z.2 w) ∂σ‖ ≤ | |
| P*B*C*σ.real univ := by | |
| have hP₀ : 0 ≤ P := (norm_nonneg (ψ 0)).trans (hP 0) | |
| have hB₀ : 0 ≤ B := (norm_nonneg (b 0 0)).trans (hB 0 0) | |
| apply norm_integral_le_of_norm_le_const | |
| filter_upwards with w | |
| simp only [norm_mul,spherePlaneWaveND_norm,Complex.norm_conj,mul_one] | |
| exact mul_le_mul (mul_le_mul (hP w) (hB z.1 w) (norm_nonneg _) hP₀) | |
| (hC z.2 w) (norm_nonneg _) (mul_nonneg hP₀ hB₀) | |
| theorem spatial_angular_factorizationND (δ L : ℝ) (hδ : 0 < δ) (hL : 0 < L) | |
| (j : ℕ) (scale : ℝ) (σ ρ ν : Measure V) | |
| [IsFiniteMeasure σ] [IsFiniteMeasure ρ] [IsFiniteMeasure ν] | |
| (X Y : Set V) (M N 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) : | |
| ((M*N)⁻¹:ℝ) • (∫ z : V × V, | |
| SpatialFourierND.spatialAnnulusKernel d δ L hδ hL 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) | |
| ∂(ρ.restrict X).prod (ν.restrict Y)) = | |
| ∫ ξ, SpatialFourierND.spatialFourierCoeff d δ L hδ hL j ξ * | |
| ∫ w, ψ w*sourceAngularTransformND ρ X M | |
| (fun x w => SpatialFourierND.spatialFourierPhase d scale ξ x*b x w) r w * | |
| conj (sourceAngularTransformND ν Y N | |
| (fun y w => SpatialFourierND.spatialFourierPhase d scale ξ y*c y w) r w) ∂σ := by | |
| rw [SpatialFourierND.spatialFourier_integral_swap d δ L hδ hL j scale _ _ | |
| (spatial_angular_kernel_measurableND σ r hψ hb hc) | |
| (spatial_angular_kernel_boundND σ r ψ b c hP hB hC)] | |
| rw [← integral_smul] | |
| apply integral_congr_ae | |
| filter_upwards with ξ | |
| let p := SpatialFourierND.spatialFourierPhase d scale ξ | |
| have hp : Measurable p := by | |
| dsimp [p,SpatialFourierND.spatialFourierPhase] | |
| fun_prop | |
| have hpn (x : V) : ‖p x‖=1 := SpatialFourierND.spatialFourierPhase_norm d scale ξ x | |
| have hf := angular_pair_factorizationND σ ρ ν X Y M N r | |
| (b := fun x w => p x*b x w) (c := fun y w => p y*c y w) hψ | |
| ((hp.comp measurable_fst).mul hb) ((hp.comp measurable_fst).mul hc) | |
| hP (fun x w => by simpa only [norm_mul,hpn,one_mul] using hB x w) | |
| (fun y w => by simpa only [norm_mul,hpn,one_mul] using hC y w) | |
| have he : (∫ z : V × V, SpatialFourierND.spatialPairPhase d scale ξ z * | |
| (∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w*b z.1 w*conj (c z.2 w) ∂σ) | |
| ∂(ρ.restrict X).prod (ν.restrict Y)) = | |
| ∫ z : V × V, (∫ w, ψ w*spherePlaneWaveND r (z.1-z.2) w* | |
| (p z.1*b z.1 w)*conj (p z.2*c z.2 w) ∂σ) | |
| ∂(ρ.restrict X).prod (ν.restrict Y) := by | |
| apply integral_congr_ae | |
| filter_upwards with z | |
| rw [← integral_const_mul] | |
| apply integral_congr_ae | |
| filter_upwards with w | |
| simp only [SpatialFourierND.spatialPairPhase,p,map_mul] | |
| ring | |
| rw [he] | |
| simp only [Complex.real_smul] at hf ⊢ | |
| rw [mul_left_comm, hf] | |
| end Attempt4PrefixShell | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |