| import OAI.MeasureTheory.Falconer.Shells.FiniteSpatialShell |
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| namespace OAI |
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| /-! # Square integrability of the finite stationary sum |
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| The radial coefficients need only be bounded almost everywhere on the |
| source shell. No global bound at zero or negative frequency is required. |
| -/ |
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|
| namespace Falconer |
| open Set MeasureTheory |
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| theorem finite_weighted_sum_sq_integrable {ι : Type*} (S : Finset ι) |
| (μ : Measure ℝ) [IsFiniteMeasure μ] (c F : ι → ℝ → ℂ) |
| (hc : ∀ i ∈ S, Measurable (c i)) (hF : ∀ i ∈ S, Measurable (F i)) |
| (A B : ι → ℝ) (hA : ∀ i ∈ S, 0 ≤ A i) (hB : ∀ i ∈ S, 0 ≤ B i) |
| (hbF : ∀ i ∈ S, ∀ r, ‖F i r‖ ≤ B i) |
| (hbC : ∀ᵐ r ∂μ, ∀ i ∈ S, ‖c i r‖ ≤ A i) : |
| Integrable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) μ := by |
| have hm : Measurable (fun r => ∑ i ∈ S, c i r*F i r) := |
| Finset.measurable_sum S (fun i hi => (hc i hi).mul (hF i hi)) |
| let K := ∑ i ∈ S, A i*B i |
| have hK : 0 ≤ K := Finset.sum_nonneg (fun i hi => mul_nonneg (hA i hi) (hB i hi)) |
| apply Integrable.of_bound (hm.norm.pow_const 2).aestronglyMeasurable (K^2) |
| filter_upwards [hbC] with r hr |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| apply (sq_le_sq₀ (norm_nonneg _) hK).mpr |
| apply (norm_sum_le S (fun i => c i r*F i r)).trans |
| apply Finset.sum_le_sum |
| intro i hi |
| rw [norm_mul] |
| exact mul_le_mul (hr i hi) (hbF i hi r) (norm_nonneg _) (hA i hi) |
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| theorem finite_spatial_shell_sq_integrable {d : ℕ} {ι : Type*} (S : Finset ι) |
| (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 |
| refine finite_weighted_sum_sq_integrable S _ c |
| (fun i => spatialSharedAmplitude (spatialFourierCoeff δ L hδ hL (j i)) (U i) (V i)) hc ?_ |
| A (fun i => ((∫ ξ : Space d, ‖spatialFourierCoeff δ L hδ hL (j i) ξ‖)* |
| (B i*C i))*(sphereMeasure d).real univ) hA ?_ ?_ hbound |
| · 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 mul_nonneg (mul_nonneg (integral_nonneg (fun _ => norm_nonneg _)) |
| (mul_nonneg (hB i hi) (hC i hi))) measureReal_nonneg |
| · intro i hi r |
| exact spatialSharedAmplitude_norm_le _ (spatialFourierCoeff_integrable δ L hδ hL (j i)) |
| (U i) (V i) (hB i hi) (hbU i hi) (hbV i hi) r |
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| end Falconer |
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| end OAI |
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