| import OAI.MeasureTheory.Falconer.Shells.FiniteShellSum | |
| import OAI.MeasureTheory.Falconer.Estimates.SpatialSharedRegularity | |
| namespace OAI | |
| /-! # Finite separated inverse terms in the independent angular shell | |
| The spatial coefficient is the actual inverse-distance Fourier coefficient. | |
| All radial and spatial integrability is derived from the bounded endpoint | |
| families. The only energy hypotheses concern the two independent angles. | |
| -/ | |
| namespace Falconer | |
| open Set MeasureTheory | |
| variable {d : ℕ} | |
| theorem finite_spatial_shell_sum_integrable {ι : Type*} (S : Finset ι) | |
| (_a v : ℝ) (Ψ : ℝ → ℝ) | |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] | |
| (δ L : ℝ) (hδ : 0 < δ) (hL : 0 < L) (j : ι → ℕ) | |
| (U V : ι → Space d × (ℝ × Sphere d) → ℂ) | |
| (hU : ∀ i ∈ S, Measurable (U i)) (hV : ∀ i ∈ S, Measurable (V i)) | |
| (B C : ι → ℝ) (hB : ∀ i ∈ S, 0 ≤ B i) (hC : ∀ i ∈ S, 0 ≤ C i) | |
| (hbU : ∀ i ∈ S, ∀ z, ‖U i z‖ ≤ B i) | |
| (hbV : ∀ i ∈ S, ∀ z, ‖V i z‖ ≤ C i) | |
| (c : ι → ℝ → ℂ) (A : ι → ℝ) | |
| (hc : ∀ i ∈ S, Measurable (c i)) (hA : ∀ i ∈ S, 0 ≤ A i) | |
| (hbound : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, | |
| ∀ i ∈ S, ‖c i r‖ ≤ A i) : | |
| Integrable (fun r => ‖∑ i ∈ S, c i r * | |
| spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i) r‖^2) | |
| (PlanarFalconer.sourceAngularShellMeasure v Ψ) := by | |
| apply finite_shell_sum_sq_integrable (PlanarFalconer.sourceAngularShellMeasure v Ψ) S c | |
| (fun i => spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i)) A hc | |
| · intro i hi | |
| exact spatialSharedAmplitude_measurable _ | |
| (spatialFourierCoeff (d := d) δ L hδ hL (j i)).continuous.measurable | |
| (U i) (V i) (hU i hi) (hV i hi) | |
| · intro i hi | |
| exact spatialSharedAmplitude_sq_integrable _ | |
| (spatialFourierCoeff (d := d) δ L hδ hL (j i)).continuous.measurable | |
| (spatialFourierCoeff_integrable δ L hδ hL (j i)) | |
| (U i) (V i) (hU i hi) (hV i hi) (hB i hi) (hC i hi) | |
| (hbU i hi) (hbV i hi) _ | |
| · exact hA | |
| · exact hbound | |
| theorem finite_spatial_shell_le {ι : Type*} (S : Finset ι) | |
| (a v : ℝ) (Ψ : ℝ → ℝ) | |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] | |
| (δ L : ℝ) (hδ : 0 < δ) (hL : 0 < L) (j : ι → ℕ) | |
| (U V : ι → Space d × (ℝ × Sphere d) → ℂ) | |
| (hU : ∀ i ∈ S, Measurable (U i)) (hV : ∀ i ∈ S, Measurable (V i)) | |
| (B C : ι → ℝ) (hB : ∀ i ∈ S, 0 ≤ B i) (hC : ∀ i ∈ S, 0 ≤ C i) | |
| (hbU : ∀ i ∈ S, ∀ z, ‖U i z‖ ≤ B i) | |
| (hbV : ∀ i ∈ S, ∀ z, ‖V i z‖ ≤ C i) | |
| (c : ι → ℝ → ℂ) (A H : ι → ℝ) | |
| (hc : ∀ i ∈ S, Measurable (c i)) (hA : ∀ i ∈ S, 0 ≤ A i) | |
| (hbound : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, | |
| ∀ i ∈ S, ‖c i r‖ ≤ A i) | |
| (hH : ∀ i ∈ S, ∀ ξ : Space d, independentAngularEnergy a v Ψ | |
| (fun p => U i (ξ,p)) (fun p => V i (ξ,p)) ≤ H i) : | |
| (2:ℝ)^((d:ℝ)*(v-a)) * | |
| (∫ r, ‖∑ i ∈ S, c i r * | |
| spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i) r‖^2 | |
| ∂PlanarFalconer.sourceAngularShellMeasure v Ψ) ≤ | |
| (S.card:ℝ)*∑ i ∈ S, (A i)^2* | |
| ((spatialFourierCost (d := d) δ L hδ hL (j i))^2*H i) := by | |
| apply finite_shell_sum_le S v Ψ c | |
| (fun i => spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i)) | |
| A (fun i => (spatialFourierCost (d := d) δ L hδ hL (j i))^2*H i) hc | |
| · intro i hi | |
| exact spatialSharedAmplitude_measurable _ | |
| (spatialFourierCoeff (d := d) δ L hδ hL (j i)).continuous.measurable | |
| (U i) (V i) (hU i hi) (hV i hi) | |
| · intro i hi | |
| exact spatialSharedAmplitude_sq_integrable _ | |
| (spatialFourierCoeff (d := d) δ L hδ hL (j i)).continuous.measurable | |
| (spatialFourierCoeff_integrable δ L hδ hL (j i)) | |
| (U i) (V i) (hU i hi) (hV i hi) (hB i hi) (hC i hi) | |
| (hbU i hi) (hbV i hi) _ | |
| · exact hA | |
| · exact hbound | |
| · positivity | |
| · intro i hi | |
| exact spatial_fourier_shared_shell_le a v Ψ δ L hδ hL (j i) | |
| (U i) (V i) (hU i hi) (hV i hi) (hB i hi) (hC i hi) | |
| (hbU i hi) (hbV i hi) (H i) (hH i hi) | |
| end Falconer | |
| end OAI | |