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import OAI.MeasureTheory.FalconerCompletion.Angular.CoordNormBound
import OAI.MeasureTheory.FalconerCompletion.Graphs.GraphSourceFourSpatialDecay
import OAI.MeasureTheory.FalconerCompletion.Sphere.SphereMeasureAeNorm
namespace OAI
/-! Graph energies and pair estimates. -/
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set
open scoped ContDiff ComplexConjugate
variable {V : Type*} [NormedAddCommGroup V] [NormedSpace ℝ V]
lemma graphJetMul {f g : V → ℂ} (hf : ContDiff ℝ ∞ f) (hg : ContDiff ℝ ∞ g)
(m : ℕ) {C D A : ℝ} (hC : 0 ≤ C) (hD : 0 ≤ D) (hA : 0 ≤ A) (w : V)
(hfb : ∀ j ≤ m, ‖iteratedFDeriv ℝ j f w‖ ≤ C*A^j)
(hgb : ∀ j ≤ m, ‖iteratedFDeriv ℝ j g w‖ ≤ D*A^j) :
∀ j ≤ m, ‖iteratedFDeriv ℝ j (fun w => f w*g w) w‖ ≤ (C*D*2^m)*A^j := by
intro j hj
calc
_ ≤ ∑ k ∈ Finset.range (j+1), (j.choose k:ℝ)*‖iteratedFDeriv ℝ k f w‖*
‖iteratedFDeriv ℝ (j-k) g w‖ :=
norm_iteratedFDeriv_mul_le hf hg w (by exact_mod_cast (le_top : (j:ℕ∞)≤⊤))
_ ≤ ∑ k ∈ Finset.range (j+1), (j.choose k:ℝ)*(C*A^k)*(D*A^(j-k)) := by
apply Finset.sum_le_sum
intro k hk
have hkj : k ≤ j := Nat.le_of_lt_succ (Finset.mem_range.mp hk)
gcongr
· exact hfb k (hkj.trans hj)
· exact hgb (j-k) ((Nat.sub_le _ _).trans hj)
_ = C*D*(A+A)^j := by
rw [add_pow,Finset.mul_sum]
apply Finset.sum_congr rfl
intro k hk
ring
_ = (C*D*2^j)*A^j := by rw [show A+A=2*A by ring,mul_pow]; ring
_ ≤ _ := mul_le_mul_of_nonneg_right
(mul_le_mul_of_nonneg_left (pow_le_pow_right₀ (by norm_num : (1:ℝ)≤2) hj)
(mul_nonneg hC hD)) (pow_nonneg hA _)
lemma graphJetConj {f : V → ℂ} {m : ℕ} {C A : ℝ} {w : V}
(h : ∀ j ≤ m, ‖iteratedFDeriv ℝ j f w‖ ≤ C*A^j) :
∀ j ≤ m, ‖iteratedFDeriv ℝ j (fun w => conj (f w)) w‖ ≤ C*A^j := by
intro j hj
change ‖iteratedFDeriv ℝ j (Complex.conjLIE ∘ f) w‖ ≤ _
rw [Complex.conjLIE.norm_iteratedFDeriv_comp_left f w j]
exact h j hj
lemma graphJetPair {θ b c : V → ℂ}
(hθ : ContDiff ℝ ∞ θ) (hb : ContDiff ℝ ∞ b) (hc : ContDiff ℝ ∞ c)
(m : ℕ) {P B C A : ℝ} (hP : 0 ≤ P) (hB : 0 ≤ B) (hC : 0 ≤ C) (hA : 0 ≤ A)
(w : V)
(hdθ : ∀ j ≤ m, ‖iteratedFDeriv ℝ j θ w‖ ≤ P*A^j)
(hdb : ∀ j ≤ m, ‖iteratedFDeriv ℝ j b w‖ ≤ B*A^j)
(hdc : ∀ j ≤ m, ‖iteratedFDeriv ℝ j c w‖ ≤ C*A^j) :
∀ j ≤ m, ‖iteratedFDeriv ℝ j (fun w => θ w*b w*conj (c w)) w‖ ≤
(P*B*2^m*C*2^m)*A^j := by
exact graphJetMul (hθ.mul hb) (Complex.conjCLE.contDiff.comp hc) m
(by positivity) hC hA w (graphJetMul hθ hb m hP hB hA w hdθ hdb)
(graphJetConj hdc)
lemma graphJet_enlarge {f : V → ℂ} {m : ℕ} {C a A : ℝ} {w : V}
(hC : 0 ≤ C) (ha : 0 ≤ a) (haA : a ≤ A)
(h : ∀ j ≤ m, ‖iteratedFDeriv ℝ j f w‖ ≤ C*a^j) :
∀ j ≤ m, ‖iteratedFDeriv ℝ j f w‖ ≤ C*A^j := by
intro j hj
exact (h j hj).trans (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ ha haA j) hC)
lemma graphCapJet (d m : ℕ) :
∃ C : ℝ, 0 ≤ C ∧ ∀ (δ A : ℝ) (a : Fin d → ℤ), 0 < δ → δ⁻¹ ≤ A →
∀ w, ∀ j ≤ m, ‖iteratedFDeriv ℝ j (fun w => (tensorCapND δ a w:ℂ)) w‖ ≤ C*A^j := by
obtain ⟨C,hC,hb⟩ := tensorCapND_derivative_bound d m
refine ⟨C,hC,?_⟩
intro δ A a hδ hA w j hj
change ‖iteratedFDeriv ℝ j (Complex.ofRealLI ∘ tensorCapND δ a) w‖ ≤ _
rw [Complex.ofRealLI.norm_iteratedFDeriv_comp_left
(tensorCapND_contDiff δ a).contDiffAt (by exact_mod_cast (le_top : (j:ℕ∞)≤⊤))]
exact (hb δ a hδ j hj w).trans
(mul_le_mul_of_nonneg_left (pow_le_pow_left₀ (inv_nonneg.mpr hδ.le) hA j) hC)
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
lemma graph_four_moment_bridge
(μ₁ μ₂ μ₃ μ₄ : Measure V)
[IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄]
(Ψ : ℝ → ℂ) {Q : ℝ} (hQ : 0 < Q)
(θ₁ θ₂ : V → ℂ) (b₁ b₂ b₃ b₄ : V → V → ℂ)
(hi : Integrable (fun p : ℝ×(V×V) => Ψ (p.1/Q)*
(θ₁ p.2.1*sourceAngularTransformND μ₁ univ 1 b₁ p.1 p.2.1*
conj (sourceAngularTransformND μ₂ univ 1 b₂ p.1 p.2.1))*
(θ₂ p.2.2*sourceAngularTransformND μ₃ univ 1 b₃ p.1 p.2.2*
conj (sourceAngularTransformND μ₄ univ 1 b₄ p.1 p.2.2)))
(volume.prod ((canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V)))) :
(Q⁻¹:ℝ) • (∫ p : ℝ×(V×V), Ψ (p.1/Q)*
(θ₁ p.2.1*sourceAngularTransformND μ₁ univ 1 b₁ p.1 p.2.1*
conj (sourceAngularTransformND μ₂ univ 1 b₂ p.1 p.2.1))*
(θ₂ p.2.2*sourceAngularTransformND μ₃ univ 1 b₃ p.1 p.2.2*
conj (sourceAngularTransformND μ₄ univ 1 b₄ p.1 p.2.2))
∂volume.prod ((canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V))) =
∫ u, Ψ u * graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ (Q*u) *
graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ (Q*u) := by
rw [integral_prod _ hi]
have he (r : ℝ) :
(∫ w : V×V, Ψ (r/Q)*
(θ₁ w.1*sourceAngularTransformND μ₁ univ 1 b₁ r w.1*
conj (sourceAngularTransformND μ₂ univ 1 b₂ r w.1))*
(θ₂ w.2*sourceAngularTransformND μ₃ univ 1 b₃ r w.2*
conj (sourceAngularTransformND μ₄ univ 1 b₄ r w.2))
∂(canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V)) =
Ψ (r/Q)*graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ r * graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ r := by
let F := fun w => θ₁ w*sourceAngularTransformND μ₁ univ 1 b₁ r w*
conj (sourceAngularTransformND μ₂ univ 1 b₂ r w)
let G := fun w => θ₂ w*sourceAngularTransformND μ₃ univ 1 b₃ r w*
conj (sourceAngularTransformND μ₄ univ 1 b₄ r w)
change (∫ w : V×V, Ψ (r/Q)*F w.1*G w.2
∂(canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V)) = _
rw [show (fun w : V×V => Ψ (r/Q)*F w.1*G w.2) =
(fun w : V×V => Ψ (r/Q)*(F w.1*G w.2)) by funext w; ring,
integral_const_mul,integral_prod_mul]
change Ψ (r/Q)*(graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ r * graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ r) = _
ring
simp_rw [he]
have h := Measure.integral_comp_mul_left (fun r => Ψ (r/Q)*
graphPairMoment μ₁ μ₂ b₁ b₂ θ₁ r*graphPairMoment μ₃ μ₄ b₃ b₄ θ₂ r) Q
rw [abs_of_pos (inv_pos.mpr hQ)] at h
rw [←h]
apply integral_congr_ae
filter_upwards with u
rw [mul_div_cancel_left₀ u hQ.ne']
omit [BorelSpace V] in
lemma graph_angular_four_integrable [BorelSpace V] (σ : Measure V) [IsFiniteMeasure σ]
{Ψ : ℝ → ℂ} (hiΨ : Integrable Ψ)
(θ₁ θ₂ : V → ℂ) (hθ₁ : Measurable θ₁) (hθ₂ : Measurable θ₂)
(U₁ U₂ U₃ U₄ : ℝ×V → ℂ)
(hm₁ : Measurable U₁) (hm₂ : Measurable U₂) (hm₃ : Measurable U₃) (hm₄ : Measurable U₄)
{P₁ P₂ B₁ B₂ B₃ B₄ : ℝ}
(hP₁ : 0 ≤ P₁) (hP₂ : 0 ≤ P₂) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂)
(hB₃ : 0 ≤ B₃) (_hB₄ : 0 ≤ B₄)
(hb : ∀ᵐ w ∂σ, ‖θ₁ w‖ ≤ P₁ ∧ ‖θ₂ w‖ ≤ P₂ ∧
∀ r, ‖U₁ (r,w)‖ ≤ B₁ ∧ ‖U₂ (r,w)‖ ≤ B₂ ∧ ‖U₃ (r,w)‖ ≤ B₃ ∧ ‖U₄ (r,w)‖ ≤ B₄) :
Integrable (fun p : ℝ×(V×V) => Ψ p.1*
(θ₁ p.2.1*U₁ (p.1,p.2.1)*conj (U₂ (p.1,p.2.1)))*
(θ₂ p.2.2*U₃ (p.1,p.2.2)*conj (U₄ (p.1,p.2.2))))
(volume.prod (σ.prod σ)) := by
let H := fun w => ‖θ₁ w‖ ≤ P₁ ∧ ‖θ₂ w‖ ≤ P₂ ∧
∀ r, ‖U₁ (r,w)‖ ≤ B₁ ∧ ‖U₂ (r,w)‖ ≤ B₂ ∧ ‖U₃ (r,w)‖ ≤ B₃ ∧ ‖U₄ (r,w)‖ ≤ B₄
change ∀ᵐ w ∂σ, H w at hb
have hprod : ∀ᵐ z : V×V ∂σ.prod σ, H z.1 ∧ H z.2 := by
filter_upwards [Measure.quasiMeasurePreserving_fst.ae hb,
Measure.quasiMeasurePreserving_snd.ae hb] with z h₁ h₂
exact ⟨h₁,h₂⟩
have hpt : ∀ᵐ p : ℝ×(V×V) ∂volume.prod (σ.prod σ), H p.2.1 ∧ H p.2.2 :=
Measure.quasiMeasurePreserving_snd.ae hprod
let K := (P₁*B₁*B₂)*(P₂*B₃*B₄)
have hm : AEStronglyMeasurable (fun p : ℝ×(V×V) => Ψ p.1*
(θ₁ p.2.1*U₁ (p.1,p.2.1)*conj (U₂ (p.1,p.2.1)))*
(θ₂ p.2.2*U₃ (p.1,p.2.2)*conj (U₄ (p.1,p.2.2)))) (volume.prod (σ.prod σ)) := by
apply (hiΨ.aestronglyMeasurable.comp_quasiMeasurePreserving Measure.quasiMeasurePreserving_fst).mul
(Measurable.aestronglyMeasurable ?_) |>.mul (Measurable.aestronglyMeasurable ?_)
· exact ((hθ₁.comp measurable_snd.fst).mul
(hm₁.comp (measurable_fst.prodMk measurable_snd.fst))).mul
(Complex.continuous_conj.measurable.comp (hm₂.comp (measurable_fst.prodMk measurable_snd.fst)))
· exact ((hθ₂.comp measurable_snd.snd).mul
(hm₃.comp (measurable_fst.prodMk measurable_snd.snd))).mul
(Complex.continuous_conj.measurable.comp (hm₄.comp (measurable_fst.prodMk measurable_snd.snd)))
apply ((hiΨ.norm.comp_fst (σ.prod σ)).mul_const K).mono' hm
filter_upwards [hpt] with p hp
change ‖Ψ p.1 * _ * _‖ ≤ ‖Ψ p.1‖*K
simp only [norm_mul,Complex.norm_conj]
simp only [K]
have hh₁ := (hp.1.2.2 p.1)
have hh₂ := (hp.2.2.2 p.1)
calc
_ ≤ (‖Ψ p.1‖*(P₁*B₁*B₂))*(P₂*B₃*B₄) := by gcongr <;> first | exact hp.1.1 | exact hp.2.2.1 | exact hh₁.1 | exact hh₁.2.1 | exact hh₂.2.2.1 | exact hh₂.2.2.2
_ = _ := by ring
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
noncomputable def graphCellLaw (ρ : Measure V) (P : Set V) (M : ℝ) : Measure V :=
ENNReal.ofReal M⁻¹ • ρ.restrict P
instance graphCellLaw_finite (ρ : Measure V) [IsFiniteMeasure ρ] (P : Set V) (M : ℝ) :
IsFiniteMeasure (graphCellLaw ρ P M) := by
refine ⟨?_⟩
simp only [graphCellLaw,Measure.smul_apply,smul_eq_mul,Measure.restrict_apply_univ]
exact ENNReal.mul_lt_top ENNReal.ofReal_lt_top (measure_lt_top _ _)
omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [BorelSpace V] in
lemma graphCellLaw_ae [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [BorelSpace V]
(ρ : Measure V) {P : Set V} (hP : MeasurableSet P) (M : ℝ) :
∀ᵐ x ∂graphCellLaw ρ P M, x ∈ P := by
exact Measure.ae_smul_measure (ae_restrict_mem hP) _
omit [FiniteDimensional ℝ V] [BorelSpace V] in
lemma graphCellLaw_transform [FiniteDimensional ℝ V] [BorelSpace V]
(ρ : Measure V) (P : Set V) {M : ℝ} (hM : 0 ≤ M)
(b : V → V → ℂ) (r : ℝ) (w : V) :
sourceAngularTransformND (graphCellLaw ρ P M) univ 1 b r w =
sourceAngularTransformND ρ P M b r w := by
simp only [sourceAngularTransformND,graphCellLaw,Measure.restrict_univ,inv_one,one_smul]
rw [integral_smul_measure,ENNReal.toReal_ofReal (inv_nonneg.mpr hM)]
omit [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [BorelSpace V] in
lemma graphCellLaw_mass [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [BorelSpace V]
(ρ : Measure V) [IsFiniteMeasure ρ] (P : Set V) {M : ℝ} (hM : 0 ≤ M) :
(graphCellLaw ρ P M).real univ = M⁻¹*ρ.real P := by
simp [graphCellLaw,Measure.real,ENNReal.toReal_ofReal (inv_nonneg.mpr hM)]
lemma graph_four_integrable
(μ₁ μ₂ μ₃ μ₄ : Measure V)
[IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄]
{Ψ : ℝ → ℂ} (hiΨ : Integrable Ψ)
(θ₁ θ₂ : V → ℂ) (b₁ b₂ b₃ b₄ : V → V → ℂ)
(hθ₁ : Measurable θ₁) (hθ₂ : Measurable θ₂)
(hb₁ : Measurable (Function.uncurry b₁)) (hb₂ : Measurable (Function.uncurry b₂))
(hb₃ : Measurable (Function.uncurry b₃)) (hb₄ : Measurable (Function.uncurry b₄))
{P₁ P₂ B₁ B₂ B₃ B₄ : ℝ}
(hP₁ : 0 ≤ P₁) (hP₂ : 0 ≤ P₂) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂) (hB₃ : 0 ≤ B₃) (hB₄ : 0 ≤ B₄)
(hdθ₁ : ∀ w, ‖w‖=1 → ‖θ₁ w‖ ≤ P₁) (hdθ₂ : ∀ w, ‖w‖=1 → ‖θ₂ w‖ ≤ P₂)
(hbd₁ : ∀ x w, ‖w‖=1 → ‖b₁ x w‖ ≤ B₁) (hbd₂ : ∀ x w, ‖w‖=1 → ‖b₂ x w‖ ≤ B₂)
(hbd₃ : ∀ x w, ‖w‖=1 → ‖b₃ x w‖ ≤ B₃) (hbd₄ : ∀ x w, ‖w‖=1 → ‖b₄ x w‖ ≤ B₄) :
Integrable (fun p : ℝ×(V×V) => Ψ p.1*
(θ₁ p.2.1*sourceAngularTransformND μ₁ univ 1 b₁ p.1 p.2.1*
conj (sourceAngularTransformND μ₂ univ 1 b₂ p.1 p.2.1))*
(θ₂ p.2.2*sourceAngularTransformND μ₃ univ 1 b₃ p.1 p.2.2*
conj (sourceAngularTransformND μ₄ univ 1 b₄ p.1 p.2.2)))
(volume.prod ((canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V))) := by
apply graph_angular_four_integrable (canonicalSphereMeasureND V) hiΨ θ₁ θ₂ hθ₁ hθ₂
_ _ _ _ (sourceAngularTransformND_measurable μ₁ univ 1 hb₁)
(sourceAngularTransformND_measurable μ₂ univ 1 hb₂)
(sourceAngularTransformND_measurable μ₃ univ 1 hb₃)
(sourceAngularTransformND_measurable μ₄ univ 1 hb₄)
hP₁ hP₂ (mul_nonneg measureReal_nonneg hB₁)
(mul_nonneg measureReal_nonneg hB₂)
(mul_nonneg measureReal_nonneg hB₃)
(mul_nonneg measureReal_nonneg hB₄)
filter_upwards [canonicalSphereMeasureND_ae_norm (V:=V)] with w hw
refine ⟨hdθ₁ w hw,hdθ₂ w hw,fun r => ?_⟩
constructor
· simpa [Measure.real] using sourceAngularTransformND_norm_le μ₁ univ 1 b₁ r w (fun x => hbd₁ x w hw)
constructor
· simpa [Measure.real] using sourceAngularTransformND_norm_le μ₂ univ 1 b₂ r w (fun x => hbd₂ x w hw)
constructor
· simpa [Measure.real] using sourceAngularTransformND_norm_le μ₃ univ 1 b₃ r w (fun x => hbd₃ x w hw)
· simpa [Measure.real] using sourceAngularTransformND_norm_le μ₄ univ 1 b₄ r w (fun x => hbd₄ x w hw)
end CurrentFalconer
end
end
section
section
open Set MeasureTheory
open scoped ComplexConjugate RealInnerProductSpace
namespace CurrentFalconer
open Set MeasureTheory Attempt4FullFar
open scoped ComplexConjugate RealInnerProductSpace ContDiff
variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
theorem graph_source_raw_four_decay
{sig c₀ Ccap G : ℝ} (hsig : 0 < sig) (hc₀ : 0 < c₀)
(hCcap : 0 ≤ Ccap) (hG : 0 ≤ G) (T : ℝ) :
∃ m : ℕ, ∃ C : ℝ, 0 ≤ C ∧ ∃ N₀ : ℕ, ∀ N ≥ N₀,
∀ (a p v L : ℝ), p-a ≤ v-p+2*sig*N → L ≤ v-p+2*sig*N → 0 ≤ v-p →
let R := (2:ℝ)^N
let δ := min c₀ (((2:ℝ)^(p-a))⁻¹*R^(10*sig))
let τ := (2:ℝ)^(-p)*R^(30*sig)
let A := 1+R^sig*max 1 ((2:ℝ)^L*R^(-50*sig))+δ⁻¹
∀ (μ₁ μ₂ μ₃ μ₄ : Measure V),
∀ [IsFiniteMeasure μ₁] [IsFiniteMeasure μ₂] [IsFiniteMeasure μ₃] [IsFiniteMeasure μ₄],
∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ : V) (K J₁ J₂ : ℝ)
(θ₁ θ₂ : V → ℂ) (b₁ b₂ b₃ b₄ : V → V → ℂ)
(P₁ P₂ B₁ B₂ B₃ B₄ : ℝ),
ContDiff ℝ ∞ Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 →
(∫ u : ℝ, ‖Ψ u‖) ≤ K → (∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) →
‖w₁‖=1 → ‖w₂‖=1 → 0 ≤ K → 0 ≤ J₁ → 0 ≤ J₂ →
Measurable θ₁ → Measurable θ₂ →
Measurable (Function.uncurry b₁) → Measurable (Function.uncurry b₂) →
Measurable (Function.uncurry b₃) → Measurable (Function.uncurry b₄) →
0 ≤ P₁ → 0 ≤ P₂ → 0 ≤ B₁ → 0 ≤ B₂ → 0 ≤ B₃ → 0 ≤ B₄ →
(∀ w, ‖w‖=1 → ‖θ₁ w‖ ≤ P₁) → (∀ w, ‖w‖=1 → ‖θ₂ w‖ ≤ P₂) →
(∀ x w, ‖w‖=1 → ‖b₁ x w‖ ≤ B₁) → (∀ x w, ‖w‖=1 → ‖b₂ x w‖ ≤ B₂) →
(∀ x w, ‖w‖=1 → ‖b₃ x w‖ ≤ B₃) → (∀ x w, ‖w‖=1 → ‖b₄ x w‖ ≤ B₄) →
(∀ᵐ z ∂(μ₁.prod μ₂).prod (μ₃.prod μ₄),
‖z.1.1-z.1.2‖ ≤ G*(2:ℝ)^(-a) ∧ ‖z.2.1-z.2.2‖ ≤ G*(2:ℝ)^(-a) ∧
‖(z.1.1-z.1.2)-d₁‖ ≤ G*(2:ℝ)^(-p) ∧ ‖(z.2.1-z.2.2)-d₂‖ ≤ G*(2:ℝ)^(-p)) →
¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧
‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) →
(∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₁ b₁ b₂ z)) →
(∀ z, ContDiff ℝ ∞ (graphPairAmplitude θ₂ b₃ b₄ z)) →
(∀ z j, j ≤ m → ∀ w, ‖w‖=1 →
‖iteratedFDeriv ℝ j (graphPairAmplitude θ₁ b₁ b₂ z) w‖ ≤ J₁*A^j) →
(∀ z j, j ≤ m → ∀ w, ‖w‖=1 →
‖iteratedFDeriv ℝ j (graphPairAmplitude θ₂ b₃ b₄ z) w‖ ≤ J₂*A^j) →
(∀ w, ‖w‖=1 → θ₁ w ≠ 0 → ‖w-w₁‖ ≤ Ccap*δ) →
(∀ w, ‖w‖=1 → θ₂ w ≠ 0 → ‖w-w₂‖ ≤ Ccap*δ) →
‖∫ p : ℝ×(V×V), Ψ (p.1/(2:ℝ)^v)*
(θ₁ p.2.1*sourceAngularTransformND μ₁ univ 1 b₁ p.1 p.2.1*
conj (sourceAngularTransformND μ₂ univ 1 b₂ p.1 p.2.1))*
(θ₂ p.2.2*sourceAngularTransformND μ₃ univ 1 b₃ p.1 p.2.2*
conj (sourceAngularTransformND μ₄ univ 1 b₄ p.1 p.2.2))
∂volume.prod ((canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V))‖ ≤
C*K*J₁*J₂*R^(-T)*(2:ℝ)^v*((μ₁.prod μ₂).prod (μ₃.prod μ₄)).real univ := by
obtain ⟨m,C,hC,N₀,hN₀⟩ := graph_source_four_spatial_decay (V:=V) hsig hc₀ hCcap hG T
refine ⟨m,C,hC,N₀,?_⟩
intro N hN a p v L hh hL hq R δ τ A μ₁ μ₂ μ₃ μ₄ _ _ _ _
Ψ w₁ w₂ d₁ d₂ K J₁ J₂ θ₁ θ₂ b₁ b₂ b₃ b₄ P₁ P₂ B₁ B₂ B₃ B₄
hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄
hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄ hgeo hne
ha₁ ha₂ hd₁ hd₂ hs₁ hs₂
have hiΨ : Integrable Ψ := hΨ.continuous.integrable_of_hasCompactSupport
(isCompact_Icc.of_isClosed_subset (isClosed_tsupport Ψ) hsΨ)
have hQ : (0:ℝ)<(2:ℝ)^v := by positivity
have hi := graph_four_integrable μ₁ μ₂ μ₃ μ₄ (hiΨ.comp_div hQ.ne') θ₁ θ₂
b₁ b₂ b₃ b₄ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄ hP₁ hP₂ hB₁ hB₂ hB₃ hB₄
hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄
have he := graph_four_moment_bridge μ₁ μ₂ μ₃ μ₄ Ψ hQ θ₁ θ₂ b₁ b₂ b₃ b₄ hi
have hd := hN₀ N hN a p v L hh hL hq μ₁ μ₂ μ₃ μ₄ Ψ w₁ w₂ d₁ d₂
K J₁ J₂ θ₁ θ₂ b₁ b₂ b₃ b₄ P₁ P₂ B₁ B₂ B₃ B₄
hΨ hsΨ hK hdΨ hw₁ hw₂ hK0 hJ₁ hJ₂ hθ₁ hθ₂ hb₁ hb₂ hb₃ hb₄
hP₁ hP₂ hB₁ hB₂ hB₃ hB₄ hdθ₁ hdθ₂ hbd₁ hbd₂ hbd₃ hbd₄ hgeo hne
ha₁ ha₂ hd₁ hd₂ hs₁ hs₂
rw [←he,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hQ)] at hd
have hz := mul_le_mul_of_nonneg_left hd hQ.le
rw [←mul_assoc,mul_inv_cancel₀ hQ.ne',one_mul] at hz
exact hz.trans_eq (by ring)
end CurrentFalconer
end
end
end OAI
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