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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 | 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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