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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 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteDirectEstimate
import OAI.MeasureTheory.FalconerCompletion.Recursion.AngularMomentLe
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
theorem graphFiniteCutoffEstimateND (μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) {D : ℝ} (hD : 0 ≤ D)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
{Ψ : ℝ → ℝ} (hΨ : Integrable Ψ) (hΨpos : ∀ r, 0 ≤ Ψ r)
{ψ : V → ℂ} (hmψ : Measurable ψ) (hψ : ∀ᵐ w ∂μ, ‖ψ w‖ ≤ 1)
(hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (fun p : ℝ × (V × V) => (Ψ p.1:ℂ)*(η k p.2:ℂ)*
(((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))*
conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1)))) (volume.prod (μ.prod μ)))
{ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b →
‖∫ p : ℝ × (V × V), (Ψ p.1:ℂ)*(η k p.2:ℂ)*
(((ψ p.2.1*U a.1 (p.1,p.2.1))*W a.2 (p.1,p.2.2))*
conj ((ψ p.2.2*U b.1 (p.1,p.2.2))*W b.2 (p.1,p.2.1))) ∂volume.prod (μ.prod μ)‖ ≤ ε) :
(∫ r, Ψ r*‖∫ w, ψ w*(∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2) ≤
D*∑ i ∈ A, ∑ j ∈ B, (∫ r, Ψ r*sourceAngularMomentND μ (U i) r*
sourceAngularMomentND μ (W j) r) + (K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε := by
have hUb (i) (hi : i ∈ A) (r : ℝ) : ∀ᵐ w ∂μ, ‖ψ w*U i (r,w)‖ ≤ C i := by
filter_upwards [hψ,hbU i hi r] with w hw hu
rw [norm_mul]
exact (mul_le_mul_of_nonneg_right hw (norm_nonneg _)).trans (by simpa using hu)
have h := graphFiniteCrossedEstimateND μ A B K G η hpart (fun i z => ψ z.2*U i z)
W (fun i hi => (hmψ.comp measurable_snd).mul (hU i hi)) hW C E hUb hbW D
hrow hcol hΨ hΨpos hi hε hne
simp_rw [← Finset.mul_sum] at h
apply h.trans
apply add_le_add _ le_rfl
apply mul_le_mul_of_nonneg_left _ hD
apply Finset.sum_le_sum
intro i hi
apply Finset.sum_le_sum
intro j hj
apply integral_mono_of_nonneg (Filter.Eventually.of_forall (fun r => mul_nonneg
(mul_nonneg (hΨpos r) (sourceAngularMomentND_nonneg μ _ r)) (sourceAngularMomentND_nonneg μ _ r)))
(graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ)
filter_upwards with r
exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left
(sourceAngularMomentND_cutoff_le μ (hU i hi) (hbU i hi) ψ hψ r) (hΨpos r))
(sourceAngularMomentND_nonneg μ _ r)
end Attempt4Near
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
theorem graphFiniteHND_bound (d : ℕ) (a p v lam₁ lam₂ : ℝ)
(μ : Measure V) [IsFiniteMeasure μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(η : κ → V × V → ℝ)
(hpart : ∀ᵐ z ∂μ.prod μ, (∀ k ∈ K, 0 ≤ η k z) ∧ ∑ k ∈ K, η k z=1)
(U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(hU : ∀ i ∈ A, Measurable (U i)) (hW : ∀ j ∈ B, Measurable (W j))
(C : α → ℝ) (E : β → ℝ)
(hbU : ∀ i ∈ A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖ ≤ C i)
(hbW : ∀ j ∈ B, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖ ≤ E j) (D : ℝ)
(hrow : ∀ k ∈ K, ∀ a ∈ A.product B, (((A.product B).filter (G k a)).card : ℝ) ≤ D)
(hcol : ∀ k ∈ K, ∀ b ∈ A.product B, (((A.product B).filter (fun a => G k a b)).card : ℝ) ≤ D)
{Ψ : ℝ → ℝ} (hΨ : Integrable (fun r => Ψ (r/(2:ℝ)^v))) (hΨpos : ∀ r, 0 ≤ Ψ r)
(hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (fun p : ℝ × (V × V) => (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)*
((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))*
conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2)))) (volume.prod (μ.prod μ)))
{ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, ¬G k a b →
‖∫ p : ℝ × (V × V), (Ψ (p.1/(2:ℝ)^v):ℂ)*(η k p.2:ℂ)*
((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))*
conj (U b.1 (p.1,p.2.1)*W b.2 (p.1,p.2.2))) ∂volume.prod (μ.prod μ)‖ ≤ ε) :
sourceAngularHND d μ a v Ψ (fun z => lam₁ • ∑ i ∈ A, U i z)
(fun z => lam₂ • ∑ j ∈ B, W j z) ≤
lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*
(∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) +
lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹*
((K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε) := by
have h := graphFiniteDirectEstimateND μ A B K G η hpart U W hU hW C E hbU hbW
D hrow hcol hΨ (fun r => hΨpos (r/(2:ℝ)^v)) hi hε hne
have he : (2:ℝ)^((d:ℝ)*(p-a))*(∑ i ∈ A, ∑ j ∈ B,
sourceAngularHND d μ p v Ψ (U i) (W j)) =
∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j) := by
simp_rw [Finset.mul_sum,← sourceAngularHND_depth d μ a p v Ψ]
have hm := mul_le_mul_of_nonneg_left h
(show 0 ≤ lam₁^2*lam₂^2*((2:ℝ)^((d:ℝ)*(v-a))*((2:ℝ)^v)⁻¹) by positivity)
rw [sourceAngularHND_smul]
have hr : lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(p-a))*D*
(∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ p v Ψ (U i) (W j)) =
lam₁^2*lam₂^2*D*(∑ i ∈ A, ∑ j ∈ B, sourceAngularHND d μ a v Ψ (U i) (W j)) := by
rw [← he]
ring
rw [hr]
simp only [sourceAngularHND,← Finset.mul_sum] at *
nlinarith only [hm]
end Attempt4Near
end
end
end OAI
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