| import OAI.MeasureTheory.Falconer.Shells.SpatialSharedShell |
|
|
| namespace OAI |
|
|
| /-! # Finite summation of the stationary shell terms |
|
|
| The finite inverse order and separated factor count contribute only their |
| explicit finite cardinality. No supremum or interchange with a variable |
| measure is hidden in this step. |
| -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
|
|
| private theorem norm_sum_sq_le_card {ι : Type*} (S : Finset ι) (f : ι → ℂ) : |
| ‖∑ i ∈ S, f i‖^2 ≤ (S.card:ℝ)*∑ i ∈ S, ‖f i‖^2 := by |
| apply (pow_le_pow_left₀ (norm_nonneg _) (norm_sum_le S f) 2).trans |
| simpa only [one_mul,one_pow,Finset.sum_const,nsmul_eq_mul,mul_one] |
| using Finset.sum_mul_sq_le_sq_mul_sq S (fun _ => (1:ℝ)) (fun i => ‖f i‖) |
|
|
| theorem finite_shell_sum_sq_integrable {α ι : Type*} [MeasurableSpace α] |
| (μ : Measure α) (S : Finset ι) (c F : ι → α → ℂ) (C : ι → ℝ) |
| (hc : ∀ i ∈ S, Measurable (c i)) (hF : ∀ i ∈ S, Measurable (F i)) |
| (hFi : ∀ i ∈ S, Integrable (fun r => ‖F i r‖^2) μ) |
| (hC : ∀ i ∈ S, 0 ≤ C i) |
| (hbound : ∀ᵐ r ∂μ, ∀ i ∈ S, ‖c i r‖ ≤ C i) : |
| Integrable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) μ := by |
| have hr : Integrable (fun r => (S.card:ℝ)*∑ i ∈ S, (C i)^2*‖F i r‖^2) μ := |
| (integrable_finsetSum S (fun i hi => (hFi i hi).const_mul ((C i)^2))).const_mul _ |
| have hm : Measurable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) := |
| (Finset.measurable_sum S (fun i hi => (hc i hi).mul (hF i hi))).norm.pow_const 2 |
| apply hr.mono' hm.aestronglyMeasurable |
| filter_upwards [hbound] with r h |
| rw [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| apply (norm_sum_sq_le_card S (fun i => c i r*F i r)).trans |
| apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _) |
| apply Finset.sum_le_sum |
| intro i hi |
| rw [norm_mul,mul_pow] |
| exact mul_le_mul_of_nonneg_right |
| ((sq_le_sq₀ (norm_nonneg _) (hC i hi)).mpr (h i hi)) (sq_nonneg _) |
|
|
| theorem finite_shell_sum_le {ι : Type*} (S : Finset ι) (v : ℝ) (Ψ : ℝ → ℝ) |
| (c F : ι → ℝ → ℂ) (C E : ι → ℝ) |
| (hc : ∀ i ∈ S, Measurable (c i)) (hF : ∀ i ∈ S, Measurable (F i)) |
| (hFi : ∀ i ∈ S, Integrable (fun r => ‖F i r‖^2) |
| (PlanarFalconer.sourceAngularShellMeasure v Ψ)) |
| (hC : ∀ i ∈ S, 0 ≤ C i) |
| (hbound : ∀ᵐ r ∂PlanarFalconer.sourceAngularShellMeasure v Ψ, |
| ∀ i ∈ S, ‖c i r‖ ≤ C i) |
| {P : ℝ} (hP : 0 ≤ P) |
| (hE : ∀ i ∈ S, P*(∫ r, ‖F i r‖^2 ∂PlanarFalconer.sourceAngularShellMeasure v Ψ) ≤ E i) : |
| P*(∫ r, ‖∑ i ∈ S, c i r*F i r‖^2 |
| ∂PlanarFalconer.sourceAngularShellMeasure v Ψ) ≤ |
| (S.card:ℝ)*∑ i ∈ S, (C i)^2*E i := by |
| let μ := PlanarFalconer.sourceAngularShellMeasure v Ψ |
| have hr : Integrable (fun r => (S.card:ℝ)*∑ i ∈ S, (C i)^2*‖F i r‖^2) μ := |
| (integrable_finsetSum S (fun i hi => (hFi i hi).const_mul ((C i)^2))).const_mul _ |
| have hm : Measurable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) := |
| (Finset.measurable_sum S (fun i hi => (hc i hi).mul (hF i hi))).norm.pow_const 2 |
| have hb : ∀ᵐ r ∂μ, ‖∑ i ∈ S, c i r*F i r‖^2 ≤ |
| (S.card:ℝ)*∑ i ∈ S, (C i)^2*‖F i r‖^2 := by |
| filter_upwards [hbound] with r hr |
| apply (norm_sum_sq_le_card S (fun i => c i r*F i r)).trans |
| apply mul_le_mul_of_nonneg_left _ (Nat.cast_nonneg _) |
| apply Finset.sum_le_sum |
| intro i hi |
| rw [norm_mul,mul_pow] |
| exact mul_le_mul_of_nonneg_right |
| ((sq_le_sq₀ (norm_nonneg _) (hC i hi)).mpr (hr i hi)) (sq_nonneg _) |
| have hi : Integrable (fun r => ‖∑ i ∈ S, c i r*F i r‖^2) μ := by |
| apply hr.mono' hm.aestronglyMeasurable |
| filter_upwards [hb] with r h |
| rwa [Real.norm_eq_abs,abs_of_nonneg (sq_nonneg _)] |
| calc |
| _ ≤ P*(∫ r, (S.card:ℝ)*∑ i ∈ S, (C i)^2*‖F i r‖^2 ∂μ) := |
| mul_le_mul_of_nonneg_left (integral_mono_ae hi hr hb) hP |
| _ = (S.card:ℝ)*∑ i ∈ S, |
| (C i)^2*(P*(∫ r, ‖F i r‖^2 ∂μ)) := by |
| rw [integral_const_mul,integral_finsetSum S (fun i hi => (hFi i hi).const_mul _)] |
| simp only [integral_const_mul,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (Finset.sum_le_sum (fun i hi => mul_le_mul_of_nonneg_left (hE i hi) (sq_nonneg _))) |
| (Nat.cast_nonneg _) |
|
|
| end Falconer |
|
|
| end OAI |
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|
