| import Mathlib |
| 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 |
|
|
