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import OAI.MeasureTheory.FalconerCompletion.Graphs.AngularMoment
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphFiniteDirectEstimate
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
noncomputable def graphRetainedDirectND (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) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) : ℂ :=
∑ k ∈ K, (η k z:ℂ)*(∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then (U a.1 (r,z.1)*W a.2 (r,z.2))*
conj (U b.1 (r,z.1)*W b.2 (r,z.2)) else 0)
lemma graphRetainedDirectND_angular_bound (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)
(r : ℝ) :
‖∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi := graphChildSquareND_integrable μ A B U W hU hW C E hbU hbW r
have hb := norm_integral_le_of_norm_le (hi.const_mul D)
(show ∀ᵐ z ∂μ.prod μ, ‖graphRetainedDirectND A B K G η U W r z‖ ≤
D*graphChildSquareND A B U W r z from by
filter_upwards [hpart] with z hz
have hh := graph_partition_product_schurND A B K G
(fun i => U i (r,z.1)) (fun i => U i (r,z.1))
(fun j => W j (r,z.2)) (fun j => W j (r,z.2)) (fun k => η k z) hz.1 hz.2 D hrow hcol
convert hh using 1
· rfl
· dsimp [graphChildSquareND]
ring)
rw [integral_const_mul,graphChildSquareND_integral μ A B U W hU hW C E hbU hbW] at hb
exact hb
theorem graphRetainedDirectND_integral_bound (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) :
‖∫ r, (Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, ∫ r, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r := by
have hi (i) (hi : i ∈ A) (j) (hj : j ∈ B) :=
graph_weighted_moments_integrableND μ (hU i hi) (hW j hj) (hbU i hi) (hbW j hj) hΨ
have hs := (integrable_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))).const_mul D
have hh := norm_integral_le_of_norm_le hs
(show ∀ᵐ r : ℝ, ‖(Ψ r:ℂ)*∫ z, graphRetainedDirectND A B K G η U W r z ∂μ.prod μ‖ ≤
D*∑ i ∈ A, ∑ j ∈ B, Ψ r*sourceAngularMomentND μ (U i) r*sourceAngularMomentND μ (W j) r from by
filter_upwards with r
rw [norm_mul,Complex.norm_real,Real.norm_eq_abs,abs_of_nonneg (hΨpos r)]
have hp := mul_le_mul_of_nonneg_left (graphRetainedDirectND_angular_bound
μ A B K G η hpart U W hU hW C E hbU hbW D hrow hcol r) (hΨpos r)
simpa only [Finset.mul_sum,mul_assoc,mul_left_comm,mul_comm] using hp)
rw [integral_const_mul,integral_finsetSum A (fun i hiA => integrable_finsetSum B
(fun j hjB => hi i hiA j hjB))] at hh
apply hh.trans_eq
congr 1
apply Finset.sum_congr rfl
intro i hiA
exact integral_finsetSum B (fun j hjB => hi i hiA j hjB)
end Attempt4Near
end GraphLiteralSupport
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4Near
open Set MeasureTheory
open scoped ComplexConjugate
variable {V α β κ : Type*} [MeasurableSpace V]
lemma graph_four_sumND (A : Finset α) (B : Finset β)
(U₁ U₂ : α → ℂ) (W₁ W₂ : β → ℂ) :
(∑ a ∈ A.product B, ∑ b ∈ A.product B,
(U₁ a.1*W₁ a.2)*conj (U₂ b.1*W₂ b.2)) =
((∑ i ∈ A, U₁ i)*(∑ j ∈ B, W₁ j))*conj ((∑ i ∈ A, U₂ i)*(∑ j ∈ B, W₂ j)) := by
rw [← Finset.sum_mul_sum,← map_sum]
simp_rw [Finset.product_eq_sprod,Finset.sum_product,← Finset.sum_mul_sum]
omit [MeasurableSpace V] in
lemma graphRetainedDirectND_complete (A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) :
graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z =
((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ) := by
simp only [graphRetainedDirectND,ite_true]
rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul,
graph_four_sumND A B (fun i => U i (r,z.1)) (fun i => U i (r,z.1))
(fun j => W j (r,z.2)) (fun j => W j (r,z.2)),
Complex.mul_conj,Complex.normSq_eq_norm_sq,norm_mul,mul_pow]
omit [MeasurableSpace V] in
lemma graphRetainedCrossedND_complete (A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (z : V × V) (hpart : ∑ k ∈ K, η k z=1) :
graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z =
((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))*
conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) := by
simp only [graphRetainedCrossedND,ite_true]
rw [← Finset.sum_mul,← Complex.ofReal_sum,hpart,Complex.ofReal_one,one_mul]
rw [graph_four_sumND A B (fun i => U i (r,z.1)) (fun i => U i (r,z.2))
(fun j => W j (r,z.2)) (fun j => W j (r,z.1))]
simp only [map_mul,starRingEnd_self_apply]
ring
lemma graphExpandedDirectND_integral (μ : Measure V) [SFinite μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) :
(∫ z, graphRetainedDirectND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) =
((sourceAngularMomentND μ (fun z => ∑ i ∈ A, U i z) r*
sourceAngularMomentND μ (fun z => ∑ j ∈ B, W j z) r:ℝ):ℂ) := by
calc
_ = ∫ z : V × V, ((‖∑ i ∈ A, U i (r,z.1)‖^2*‖∑ j ∈ B, W j (r,z.2)‖^2:ℝ):ℂ)
∂μ.prod μ := integral_congr_ae (by
filter_upwards [hpart] with z hz
exact graphRetainedDirectND_complete A B K η U W r z hz)
_ = _ := by
rw [integral_complex_ofReal,integral_prod_mul
(fun w => ‖∑ i ∈ A, U i (r,w)‖^2) (fun w => ‖∑ j ∈ B, W j (r,w)‖^2)]
rfl
lemma graphExpandedCrossedND_integral (μ : Measure V) [SFinite μ]
(A : Finset α) (B : Finset β) (K : Finset κ)
(η : κ → V × V → ℝ) (U : α → ℝ × V → ℂ) (W : β → ℝ × V → ℂ)
(r : ℝ) (hpart : ∀ᵐ z ∂μ.prod μ, ∑ k ∈ K, η k z=1) :
(∫ z, graphRetainedCrossedND A B K (fun _ _ _ => True) η U W r z ∂μ.prod μ) =
((‖∫ w, (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)) ∂μ‖^2:ℝ):ℂ) := by
calc
_ = ∫ z : V × V, ((∑ i ∈ A, U i (r,z.1))*conj (∑ j ∈ B, W j (r,z.1)))*
conj ((∑ i ∈ A, U i (r,z.2))*conj (∑ j ∈ B, W j (r,z.2))) ∂μ.prod μ :=
integral_congr_ae (by
filter_upwards [hpart] with z hz
exact graphRetainedCrossedND_complete A B K η U W r z hz)
_ = _ := by
rw [integral_prod_mul
(fun w => (∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)))
(fun w => conj ((∑ i ∈ A, U i (r,w))*conj (∑ j ∈ B, W j (r,w)))),
integral_conj,Complex.mul_conj,Complex.normSq_eq_norm_sq]
end Attempt4Near
end GraphLiteralSupport
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
namespace Attempt4Near
open MeasureTheory
open scoped BigOperators
variable {Z α β κ : Type*} [MeasurableSpace Z]
lemma graph_integral_sum_truncateND (μ : Measure Z) (S : Finset α)
(G : α → Prop) [DecidablePred G] (F : α → Z → ℂ)
(hi : ∀ i ∈ S, Integrable (F i) μ) {ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ i ∈ S, ¬G i → ‖∫ z, F i z ∂μ‖ ≤ ε) :
‖(∫ z, ∑ i ∈ S, F i z ∂μ)-(∫ z, ∑ i ∈ S, if G i then F i z else 0 ∂μ)‖ ≤
S.card*ε := by
have hit (i : α) (hiS : i ∈ S) : Integrable (fun z => if G i then F i z else 0) μ := by
by_cases h : G i
· simpa only [h,ite_true] using hi i hiS
· simp only [h,ite_false]; exact integrable_zero _ _ _
rw [integral_finsetSum S hi,integral_finsetSum S hit]
have he : (∑ i ∈ S, ∫ z, if G i then F i z else 0 ∂μ) =
∑ i ∈ S.filter G, ∫ z, F i z ∂μ := by
rw [Finset.sum_filter]
apply Finset.sum_congr rfl
intro i _
by_cases h : G i <;> simp [h]
rw [he]
exact graph_discarded_sumND S G (fun i => ∫ z, F i z ∂μ) hε hne
lemma graph_integral_localizationND (μ : Measure Z) (A : Finset α) (B : Finset β) (K : Finset κ)
(G : κ → (α × β) → (α × β) → Prop) [∀ k, DecidableRel (G k)]
(F : κ → (α × β) → (α × β) → Z → ℂ)
(hi : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B, Integrable (F k a b) μ)
{ε : ℝ} (hε : 0 ≤ ε)
(hne : ∀ k ∈ K, ∀ a ∈ A.product B, ∀ b ∈ A.product B,
¬G k a b → ‖∫ z, F k a b z ∂μ‖ ≤ ε) :
‖(∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B, F k a b z ∂μ)-
(∫ z, ∑ k ∈ K, ∑ a ∈ A.product B, ∑ b ∈ A.product B,
if G k a b then F k a b z else 0 ∂μ)‖ ≤
(K.card:ℝ)*((A.card:ℝ)*B.card)^2*ε := by
let S := K.product ((A.product B).product (A.product B))
have h := graph_integral_sum_truncateND μ S
(fun z => G z.1 z.2.1 z.2.2) (fun z => F z.1 z.2.1 z.2.2)
(fun z hz => hi z.1 (Finset.mem_product.mp hz).1
z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1
z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2) hε
(fun z hz => hne z.1 (Finset.mem_product.mp hz).1
z.2.1 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).1
z.2.2 (Finset.mem_product.mp (Finset.mem_product.mp hz).2).2)
simpa only [S,Finset.product_eq_sprod,Finset.sum_product,Finset.card_product,
Nat.cast_mul,pow_two,mul_assoc] using h
end Attempt4Near
end GraphLiteralSupport
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace GraphLiteralSupport
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 GraphLiteralSupport
end
end
end OAI
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