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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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 | 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
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