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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 | import Mathlib
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphChildSquare
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 graphFiniteDirectEstimateND (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 Ψ) (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: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: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, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r) ≤
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
let F := fun (k : κ) (a b : α × β) (p : ℝ × (V × V)) => (Ψ p.1: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)))
have hiF : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (F k a b) (volume.prod (μ.prod μ)) := hi
have hiR : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (fun p => if G k a b then F k a b p else 0) (volume.prod (μ.prod μ)) := by
intro k hk a ha b hb
by_cases h : G k a b
· simpa only [h,ite_true] using hiF k hk a ha b hb
· simp only [h,ite_false]; exact integrable_zero _ _ _
have heF : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b p) =
fun p => (Ψ p.1:ℂ)*graphRetainedDirectND A B K (fun _ _ _ => True) η U W p.1 p.2 := by
funext p
simp only [F,graphRetainedDirectND,ite_true,Finset.mul_sum,mul_assoc]
have heR : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then F k a b p else 0) =
fun p => (Ψ p.1:ℂ)*graphRetainedDirectND A B K G η U W p.1 p.2 := by
funext p
simp only [F,graphRetainedDirectND,Finset.mul_sum,mul_ite,mul_zero,mul_assoc]
have hintF := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B)
(fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiF k hk a ha b hb)))
have hintR := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B)
(fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiR k hk a ha b hb)))
rw [heF] at hintF
rw [heR] at hintR
have hl := graph_integral_localizationND (volume.prod (μ.prod μ)) A B K G F hiF hε hne
rw [heF,heR,integral_prod _ hintF,integral_prod _ hintR] at hl
simp_rw [integral_const_mul] at hl
have he (r : ℝ) := graphExpandedDirectND_integral μ A B K η U W r
(hpart.mono (fun _ h => h.2))
simp_rw [he] at hl
simp only [← Complex.ofReal_mul] at hl
rw [integral_complex_ofReal] at hl
have hb := graphRetainedDirectND_integral_bound μ A B K G η hpart U W hU hW C E hbU hbW
D hrow hcol hΨ hΨpos
have hnon : 0 ≤ ∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r := integral_nonneg (fun r =>
mul_nonneg (mul_nonneg (hΨpos r) (sourceAngularMomentND_nonneg _ _ _))
(sourceAngularMomentND_nonneg _ _ _))
have htt : ‖((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)‖ ≤ ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)‖ + ‖((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)‖ := by
calc
_ = ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ)+(((∫ r, Ψ r*sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ))‖ := by congr 1; abel
_ ≤ _ := norm_add_le _ _
simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hnon] at htt
exact htt.trans (add_le_add hb (by simpa only [mul_assoc] using hl))
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 graphFiniteCrossedEstimateND (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 Ψ) (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: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.2.2)*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)*
((U a.1 (p.1,p.2.1)*W a.2 (p.1,p.2.2))*
conj (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, (∑ 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
let F := fun (k : κ) (a b : α × β) (p : ℝ × (V × V)) => (Ψ p.1: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.2.2)*W b.2 (p.1,p.2.1)))
have hiF : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (F k a b) (volume.prod (μ.prod μ)) := hi
have hiR : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
Integrable (fun p => if G k a b then F k a b p else 0) (volume.prod (μ.prod μ)) := by
intro k hk a ha b hb
by_cases h : G k a b
· simpa only [h,ite_true] using hiF k hk a ha b hb
· simp only [h,ite_false]; exact integrable_zero _ _ _
have heF : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b p) =
fun p => (Ψ p.1:ℂ)*graphRetainedCrossedND A B K (fun _ _ _ => True) η U W p.1 p.2 := by
funext p
simp only [F,graphRetainedCrossedND,ite_true,Finset.mul_sum,mul_assoc]
have heR : (fun p => ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then F k a b p else 0) =
fun p => (Ψ p.1:ℂ)*graphRetainedCrossedND A B K G η U W p.1 p.2 := by
funext p
simp only [F,graphRetainedCrossedND,Finset.mul_sum,mul_ite,mul_zero,mul_assoc]
have hintF := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B)
(fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiF k hk a ha b hb)))
have hintR := integrable_finsetSum K (fun k hk => integrable_finsetSum (A.product B)
(fun a ha => integrable_finsetSum (A.product B) (fun b hb => hiR k hk a ha b hb)))
rw [heF] at hintF
rw [heR] at hintR
have hl := graph_integral_localizationND (volume.prod (μ.prod μ)) A B K G F hiF hε hne
rw [heF,heR,integral_prod _ hintF,integral_prod _ hintR] at hl
simp_rw [integral_const_mul] at hl
have he (r : ℝ) := graphExpandedCrossedND_integral μ A B K η U W r
(hpart.mono (fun _ h => h.2))
simp_rw [he] at hl
simp only [← Complex.ofReal_mul] at hl
rw [integral_complex_ofReal] at hl
have hb := graphRetainedCrossedND_integral_bound μ A B K G η hpart U W hU hW C E hbU hbW
D hrow hcol hΨ hΨpos
have hnon : 0 ≤ ∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 := integral_nonneg (fun r =>
mul_nonneg (hΨpos r) (sq_nonneg _))
have htt : ‖((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)‖ ≤ ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)‖ + ‖((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)‖ := by
calc
_ = ‖(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ)+(((∫ r, Ψ r*‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2 : ℝ):ℂ)-(∫ r, (Ψ r:ℂ)*∫ z, graphRetainedCrossedND A B K G η U W r z ∂μ.prod μ))‖ := by congr 1; abel
_ ≤ _ := norm_add_le _ _
simp only [Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg hnon] at htt
exact htt.trans (add_le_add hb (by simpa only [mul_assoc] using hl))
end Attempt4Near
end
end
end OAI
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