| import OAI.MeasureTheory.Falconer.Shells.FiniteShellIntegrability |
| import OAI.MeasureTheory.Falconer.Estimates.ScalarInverseTransfer |
|
|
| namespace OAI |
|
|
| /-! # Finite spatial factorization controls the scalar distance shell -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
|
|
| theorem finite_spatial_scalar_shell_le {d : ℕ} {ι : Type*} (S : Finset ι) |
| (a v : ℝ) (Ψ : ℝ → ℝ) (hΨ : Measurable Ψ) (hpos : ∀ x, 0 ≤ Ψ x) |
| (hzero : ∀ x ≤ 0, Ψ x = 0) (hone : ∀ x ∈ Icc (1/2:ℝ) 2, 1 ≤ Ψ x) |
| [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) |
| (F : ℝ → ℂ) (hF : Measurable F) {ε : ℝ} |
| (herr : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| ‖F r-(((2:ℝ)^((v-a)*((d:ℝ)-1)/2):ℝ):ℂ)* |
| (∑ i ∈ S, c i r * |
| spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i) r)‖ ≤ ε) : |
| IntegrableOn (fun r => ‖F r‖^2) (Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1))) ∧ |
| (2:ℝ)^(-a)*(∫ r in Icc ((2:ℝ)^(v-1)) ((2:ℝ)^(v+1)), ‖F r‖^2) ≤ |
| 2*((S.card:ℝ)*∑ i ∈ S, (A i)^2* |
| ((spatialFourierCost (d := d) δ L hδ hL (j i))^2*H i)) + |
| 2*(2:ℝ)^(v-a)*ε^2*(PlanarFalconer.sourceAngularShellMeasure v Ψ).real univ := by |
| exact scalar_shell_of_stationary_approximation d a v Ψ hΨ hpos hzero hone F |
| (fun r => ∑ i ∈ S, c i r * |
| spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i) r) hF |
| (finite_spatial_shell_sq_integrable S v Ψ δ L hδ hL j U V hU hV B C hB hC |
| hbU hbV c A hc hA hbound) |
| herr (finite_spatial_shell_le S a v Ψ δ L hδ hL j U V hU hV B C hB hC |
| hbU hbV c A H hc hA hbound hH) |
|
|
| end Falconer |
|
|
| end OAI |
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|
