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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 | 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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