import Mathlib import OAI.MeasureTheory.FalconerCompletion.Measures.RadialPhaseIntegral import OAI.MeasureTheory.FalconerCompletion.Sphere.SpherePlaneWave namespace OAI /-! Graph energies and pair estimates. -/ section section namespace Attempt4FullFar open MeasureTheory Set Metric open scoped ContDiff RealInnerProductSpace variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] noncomputable def graphFixedSpatialTermND (Ψ : ℝ → ℂ) (Q : ℝ) (e₁ e₂ : V) (b₁ b₂ : V → ℂ) : ℂ := ∫ u : ℝ, Ψ u*sphereAxisIntegralND (Q*u) e₁ b₁*sphereAxisIntegralND (Q*u) e₂ b₂ lemma canonicalSphereMeasureND_prod_ae_norm : ∀ᵐ z : V×V ∂(canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V), ‖z.1‖=1 ∧ ‖z.2‖=1 := by filter_upwards [(Measure.quasiMeasurePreserving_fst (μ := canonicalSphereMeasureND V) (ν := canonicalSphereMeasureND V)).tendsto_ae.eventually canonicalSphereMeasureND_ae_norm, (Measure.quasiMeasurePreserving_snd (μ := canonicalSphereMeasureND V) (ν := canonicalSphereMeasureND V)).tendsto_ae.eventually canonicalSphereMeasureND_ae_norm] with z h₁ h₂ exact ⟨h₁,h₂⟩ omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma spherePlaneWaveND_norm [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (Λ : ℝ) (e w : V) : ‖spherePlaneWaveND Λ e w‖ = 1 := by simp [spherePlaneWaveND,Complex.norm_exp,Complex.mul_re] lemma graphFixedSpatialTermND_radial (Ψ : ℝ → ℂ) (Q : ℝ) (e₁ e₂ : V) (b₁ b₂ : V → ℂ) (hΨ : Continuous Ψ) (hiΨ : Integrable Ψ) (hb₁ : Continuous b₁) (hb₂ : Continuous b₂) {B₁ B₂ : ℝ} (hB₁ : 0 ≤ B₁) (_hB₂ : 0 ≤ B₂) (hbd₁ : ∀ w, ‖w‖=1 → ‖b₁ w‖ ≤ B₁) (hbd₂ : ∀ w, ‖w‖=1 → ‖b₂ w‖ ≤ B₂) : graphFixedSpatialTermND Ψ Q e₁ e₂ b₁ b₂ = ∫ z : V×V, radialPhaseIntegral Ψ (Q*(inner ℝ z.1 e₁+inner ℝ z.2 e₂))*b₁ z.1*b₂ z.2 ∂(canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V) := by let μ := canonicalSphereMeasureND V let F (u : ℝ) (z : V×V) := Ψ u* ((spherePlaneWaveND (Q*u) e₁ z.1*b₁ z.1)*(spherePlaneWaveND (Q*u) e₂ z.2*b₂ z.2)) have hFc : Continuous (Function.uncurry F) := by dsimp [F,spherePlaneWaveND,Function.uncurry] fun_prop have hmajor := (hiΨ.norm.comp_fst (μ.prod μ)).mul_const (B₁*B₂) have hFi : Integrable (Function.uncurry F) (volume.prod (μ.prod μ)) := by apply hmajor.mono' hFc.aestronglyMeasurable filter_upwards [(Measure.quasiMeasurePreserving_snd (μ := volume) (ν := μ.prod μ)).tendsto_ae.eventually canonicalSphereMeasureND_prod_ae_norm] with p hp simp only [F,Function.uncurry,norm_mul,spherePlaneWaveND_norm,one_mul] gcongr · exact hbd₁ p.2.1 hp.1 · exact hbd₂ p.2.2 hp.2 calc _ = ∫ u : ℝ, ∫ z : V×V, F u z ∂μ.prod μ := by apply integral_congr_ae filter_upwards [] with u rw [show (∫ z, F u z ∂μ.prod μ) = Ψ u* ((∫ w, spherePlaneWaveND (Q*u) e₁ w*b₁ w ∂μ)* (∫ w, spherePlaneWaveND (Q*u) e₂ w*b₂ w ∂μ)) by dsimp [F] rw [integral_const_mul,integral_prod_mul (fun w => spherePlaneWaveND (Q*u) e₁ w*b₁ w) (fun w => spherePlaneWaveND (Q*u) e₂ w*b₂ w)]] exact mul_assoc _ _ _ _ = ∫ z : V×V, ∫ u : ℝ, F u z ∂volume ∂μ.prod μ := integral_integral_swap hFi _ = _ := by apply integral_congr_ae filter_upwards [] with z unfold radialPhaseIntegral rw [← integral_mul_const,← integral_mul_const] apply integral_congr_ae filter_upwards [] with u have hex : spherePlaneWaveND (Q*u) e₁ z.1*spherePlaneWaveND (Q*u) e₂ z.2 = Complex.exp (Complex.I*(Q*(inner ℝ z.1 e₁+inner ℝ z.2 e₂):ℝ)*(u:ℂ)) := by rw [spherePlaneWaveND,spherePlaneWaveND,← Complex.exp_add] congr 1 push_cast ring dsimp [F] calc _ = (spherePlaneWaveND (Q*u) e₁ z.1*spherePlaneWaveND (Q*u) e₂ z.2)*Ψ u*b₁ z.1*b₂ z.2 := by ring _ = _ := by rw [hex] lemma graphFixedSpatialTermND_swap (Ψ : ℝ → ℂ) (Q : ℝ) (e₁ e₂ : V) (b₁ b₂ : V → ℂ) : graphFixedSpatialTermND Ψ Q e₁ e₂ b₁ b₂ = graphFixedSpatialTermND Ψ Q e₂ e₁ b₂ b₁ := by unfold graphFixedSpatialTermND apply integral_congr_ae filter_upwards [] with u ring omit [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] in lemma graph_center_scalar_error [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (w₀ w d e : V) (hw : ‖w₀‖=1) : |inner ℝ w e-inner ℝ w₀ d| ≤ ‖w-w₀‖*‖e‖+‖e-d‖ := by have he : inner ℝ w e-inner ℝ w₀ d = inner ℝ (w-w₀) e+inner ℝ w₀ (e-d) := by simp only [inner_sub_left,inner_sub_right] ring rw [he] apply (abs_add_le _ _).trans exact add_le_add (abs_real_inner_le_norm _ _) (by simpa [hw] using abs_real_inner_le_norm w₀ (e-d)) end Attempt4FullFar end end section section namespace Attempt4FullFar open MeasureTheory Set Metric open scoped ContDiff RealInnerProductSpace variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] lemma graphFixedSpatialTermND_angular_bound (m : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (Ψ : ℝ → ℂ) (w₀ d e e' : V) (Q τ δ D η A K B B' : ℝ) (b b' : V → ℂ), Continuous Ψ → tsupport Ψ ⊆ Icc (1/4:ℝ) 4 → (∫ u : ℝ, ‖Ψ u‖) ≤ K → ‖w₀‖=1 → 0 < Q → 0 < τ → 0 ≤ δ → 0 ≤ D → 0 ≤ η → 1 ≤ A → 0 ≤ K → 0 ≤ B → 0 ≤ B' → ‖e‖ ≤ D → ‖e-d‖ ≤ η → 2*δ*D+η ≤ τ/2 → τ < ‖d-inner ℝ d w₀ • w₀‖ → ContDiff ℝ ∞ b → (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b w‖ ≤ B*A^j) → (∀ w, ‖w‖=1 → b w ≠ 0 → ‖w-w₀‖ ≤ δ) → (∀ w, ‖w‖=1 → ‖b' w‖ ≤ B') → ‖graphFixedSpatialTermND Ψ Q e e' b b'‖ ≤ C*K*B*B'*((A+1+D/τ)/(Q*τ))^m := by obtain ⟨C,hC,hCb⟩ := sphere_cap_nonstationary_uniform (V := V) m let S := (canonicalSphereMeasureND V).real univ have hS : 0 ≤ S := ENNReal.toReal_nonneg refine ⟨C*4^m*S,by positivity,?_⟩ intro Ψ w₀ d e e' Q τ δ D η A K B B' b b' hΨ hsΨ hK hw hQ hτ hδ hD hη hA hK0 hB hB' he hed herr hfail hb hbd hs hb' let r := (A+1+D/τ)/(Q*τ) have hr : 0 ≤ r := by dsimp [r]; positivity have hΨi : Integrable Ψ := hΨ.integrable_of_hasCompactSupport (isCompact_Icc.of_isClosed_subset isClosed_closure hsΨ) have hbound : ∀ u : ℝ, ‖Ψ u*sphereAxisIntegralND (Q*u) e b*sphereAxisIntegralND (Q*u) e' b'‖ ≤ ‖Ψ u‖*((C*4^m*S)*B*B'*r^m) := by intro u by_cases hzero : Ψ u=0 · simp [hzero] have hu := hsΨ (subset_closure hzero) have hup : 0 < u := by linarith [hu.1] have hh := hCb w₀ d e (Q*u) τ δ D η A B b hw (mul_pos hQ hup) hτ hδ hD hη hA hB he hed herr hfail hb hbd hs have hr' : (A+1+D/τ)/((Q*u)*τ) ≤ 4*r := by rw [show (A+1+D/τ)/((Q*u)*τ)=r/u by dsimp [r]; ring] apply (div_le_iff₀ hup).2 nlinarith [hu.1] have hhn : ‖sphereAxisIntegralND (Q*u) e b‖ ≤ C*B*(4*r)^m := hh.trans (by gcongr) have hbn : ‖sphereAxisIntegralND (Q*u) e' b'‖ ≤ S*B' := sphereAxisIntegralND_norm_le hb' calc _ = ‖Ψ u‖*‖sphereAxisIntegralND (Q*u) e b‖*‖sphereAxisIntegralND (Q*u) e' b'‖ := by simp only [norm_mul] _ ≤ ‖Ψ u‖*(C*B*(4*r)^m)*(S*B') := by gcongr _ = _ := by rw [mul_pow]; ring have hh := norm_integral_le_of_norm_le (hΨi.norm.mul_const ((C*4^m*S)*B*B'*r^m)) (Filter.Eventually.of_forall hbound) rw [integral_mul_const] at hh apply hh.trans change (∫ u : ℝ, ‖Ψ u‖)*((C*4^m*S)*B*B'*r^m) ≤ _ calc _ ≤ K*((C*4^m*S)*B*B'*r^m) := mul_le_mul_of_nonneg_right hK (by positivity) _ = _ := by ring lemma graphFixedSpatialTermND_radial_bound (m : ℕ) (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ e₁ e₂ : V) (Q τ δ D η K B₁ B₂ : ℝ) (b₁ b₂ : V → ℂ) (hΨ : ContDiff ℝ ∞ Ψ) (hsΨ : tsupport Ψ ⊆ Icc (1/4:ℝ) 4) (hdΨ : ∀ u, ‖iteratedDeriv m Ψ u‖ ≤ K) (hw₁ : ‖w₁‖=1) (hw₂ : ‖w₂‖=1) (hQ : 0 < Q) (hτ : 0 < τ) (hδ : 0 ≤ δ) (_hD : 0 ≤ D) (_hη : 0 ≤ η) (hK : 0 ≤ K) (hB₁ : 0 ≤ B₁) (hB₂ : 0 ≤ B₂) (he₁ : ‖e₁‖ ≤ D) (he₂ : ‖e₂‖ ≤ D) (hed₁ : ‖e₁-d₁‖ ≤ η) (hed₂ : ‖e₂-d₂‖ ≤ η) (herr : 2*δ*D+2*η ≤ τ/2) (hfail : τ < |inner ℝ w₁ d₁+inner ℝ w₂ d₂|) (hb₁ : Continuous b₁) (hb₂ : Continuous b₂) (hbd₁ : ∀ w, ‖w‖=1 → ‖b₁ w‖ ≤ B₁) (hbd₂ : ∀ w, ‖w‖=1 → ‖b₂ w‖ ≤ B₂) (hs₁ : ∀ w, ‖w‖=1 → b₁ w ≠ 0 → ‖w-w₁‖ ≤ δ) (hs₂ : ∀ w, ‖w‖=1 → b₂ w ≠ 0 → ‖w-w₂‖ ≤ δ) : ‖graphFixedSpatialTermND Ψ Q e₁ e₂ b₁ b₂‖ ≤ (4*2^m*((canonicalSphereMeasureND V).real univ)^2)*K*B₁*B₂*(1/(Q*τ))^m := by have hΨi : Integrable Ψ := hΨ.continuous.integrable_of_hasCompactSupport (isCompact_Icc.of_isClosed_subset isClosed_closure hsΨ) rw [graphFixedSpatialTermND_radial Ψ Q e₁ e₂ b₁ b₂ hΨ.continuous hΨi hb₁ hb₂ hB₁ hB₂ hbd₁ hbd₂] have hpoint : ∀ᵐ z : V×V ∂(canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V), ‖radialPhaseIntegral Ψ (Q*(inner ℝ z.1 e₁+inner ℝ z.2 e₂))*b₁ z.1*b₂ z.2‖ ≤ (4*2^m)*K*B₁*B₂*(1/(Q*τ))^m := by filter_upwards [canonicalSphereMeasureND_prod_ae_norm] with z hz by_cases hzero₁ : b₁ z.1=0 · simp only [hzero₁,mul_zero,zero_mul,norm_zero]; positivity by_cases hzero₂ : b₂ z.2=0 · simp only [hzero₂,mul_zero,norm_zero]; positivity have hgeom₁ : |inner ℝ z.1 e₁-inner ℝ w₁ d₁| ≤ δ*D+η := by apply (graph_center_scalar_error w₁ z.1 d₁ e₁ hw₁).trans gcongr exact hs₁ z.1 hz.1 hzero₁ have hgeom₂ : |inner ℝ z.2 e₂-inner ℝ w₂ d₂| ≤ δ*D+η := by apply (graph_center_scalar_error w₂ z.2 d₂ e₂ hw₂).trans gcongr exact hs₂ z.2 hz.2 hzero₂ let s := inner ℝ z.1 e₁+inner ℝ z.2 e₂ let s₀ := inner ℝ w₁ d₁+inner ℝ w₂ d₂ have hdiff : |s-s₀| ≤ 2*δ*D+2*η := by have hre : s-s₀=(inner ℝ z.1 e₁-inner ℝ w₁ d₁)+(inner ℝ z.2 e₂-inner ℝ w₂ d₂) := by dsimp [s,s₀]; ring rw [hre] apply (abs_add_le _ _).trans linarith have hsf : τ/2 < |s| := by have hh := abs_sub_abs_le_abs_sub s₀ s rw [abs_sub_comm] at hh change τ < |s₀| at hfail linarith have hfreq : Q*τ/2 ≤ |Q*s| := by rw [abs_mul,abs_of_pos hQ]; nlinarith have hfp : 0 < |Q*s| := lt_of_lt_of_le (by positivity) hfreq have hrad := radialPhaseIntegral_nonzero hΨ hsΨ (by norm_num) (abs_pos.mp hfp) m hdΨ have hbq : ‖radialPhaseIntegral Ψ (Q*s)‖ ≤ (4*2^m)*K*(1/(Q*τ))^m := by calc _ ≤ K*((4:ℝ)-1/4)/|Q*s|^m := hrad _ ≤ K*4/(Q*τ/2)^m := by apply div_le_div₀ (by positivity) _ (by positivity) _ · nlinarith · exact pow_le_pow_left₀ (by positivity) hfreq m _ = _ := by simp only [div_eq_mul_inv,mul_inv_rev,inv_pow,inv_inv,mul_pow]; ring change ‖radialPhaseIntegral Ψ (Q*s)*b₁ z.1*b₂ z.2‖ ≤ _ calc _ = ‖radialPhaseIntegral Ψ (Q*s)‖*‖b₁ z.1‖*‖b₂ z.2‖ := by simp only [norm_mul] _ ≤ ((4*2^m)*K*(1/(Q*τ))^m)*B₁*B₂ := by gcongr; exact hbd₁ z.1 hz.1; exact hbd₂ z.2 hz.2 _ = _ := by ring have hh := norm_integral_le_of_norm_le_const hpoint apply hh.trans_eq have hmass : ((canonicalSphereMeasureND V).prod (canonicalSphereMeasureND V)).real univ = ((canonicalSphereMeasureND V).real univ)^2 := by rw [Measure.real,← univ_prod_univ,Measure.prod_prod,ENNReal.toReal_mul] simp only [Measure.real,pow_two] rw [hmass] ring end Attempt4FullFar end end section section namespace Attempt4FullFar open MeasureTheory Set Metric open scoped ContDiff RealInnerProductSpace theorem graph_fixed_nonedge_term_uniform (V : Type*) [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (m : ℕ) : ∃ C : ℝ, 0 ≤ C ∧ ∀ (Ψ : ℝ → ℂ) (w₁ w₂ d₁ d₂ e₁ e₂ : V) (Q τ δ D η A K B₁ B₂ : ℝ) (b₁ b₂ : V → ℂ), 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 < Q → 0 < τ → 0 ≤ δ → 0 ≤ D → 0 ≤ η → 1 ≤ A → 0 ≤ K → 0 ≤ B₁ → 0 ≤ B₂ → ‖e₁‖ ≤ D → ‖e₂‖ ≤ D → ‖e₁-d₁‖ ≤ η → ‖e₂-d₂‖ ≤ η → 4*δ*D+2*η ≤ τ/2 → ¬(‖d₁-inner ℝ d₁ w₁ • w₁‖ ≤ τ ∧ ‖d₂-inner ℝ d₂ w₂ • w₂‖ ≤ τ ∧ |inner ℝ w₁ d₁+inner ℝ w₂ d₂| ≤ τ) → ContDiff ℝ ∞ b₁ → ContDiff ℝ ∞ b₂ → (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₁ w‖ ≤ B₁*A^j) → (∀ j ≤ m, ∀ w, ‖w‖=1 → ‖iteratedFDeriv ℝ j b₂ w‖ ≤ B₂*A^j) → (∀ w, ‖w‖=1 → b₁ w ≠ 0 → ‖w-w₁‖ ≤ δ) → (∀ w, ‖w‖=1 → b₂ w ≠ 0 → ‖w-w₂‖ ≤ δ) → ‖graphFixedSpatialTermND Ψ Q e₁ e₂ b₁ b₂‖ ≤ C*K*B₁*B₂*((A+1+D/τ)/(Q*τ))^m := by obtain ⟨Ca,hCa,hCab⟩ := graphFixedSpatialTermND_angular_bound (V := V) m let Cr := 4*2^m*((canonicalSphereMeasureND V).real univ)^2 have hCr : 0 ≤ Cr := by dsimp [Cr]; positivity refine ⟨Ca+Cr,add_nonneg hCa hCr,?_⟩ intro Ψ w₁ w₂ d₁ d₂ e₁ e₂ Q τ δ D η A K B₁ B₂ b₁ b₂ hΨ hsΨ hK hdΨ hw₁ hw₂ hQ hτ hδ hD hη hA hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ herr hne hb₁ hb₂ hd₁ hd₂ hs₁ hs₂ have hbd₁ : ∀ w, ‖w‖=1 → ‖b₁ w‖ ≤ B₁ := by intro w hw simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hd₁ 0 (Nat.zero_le _) w hw have hbd₂ : ∀ w, ‖w‖=1 → ‖b₂ w‖ ≤ B₂ := by intro w hw simpa only [norm_iteratedFDeriv_zero,pow_zero,mul_one] using hd₂ 0 (Nat.zero_le _) w hw have herrA : 2*δ*D+η ≤ τ/2 := by nlinarith [mul_nonneg hδ hD] have herrR : 2*δ*D+2*η ≤ τ/2 := by nlinarith [mul_nonneg hδ hD] have hratio0 : 0 ≤ (A+1+D/τ)/(Q*τ) := by positivity by_cases hfirst : τ < ‖d₁-inner ℝ d₁ w₁ • w₁‖ · have hh := hCab Ψ w₁ d₁ e₁ e₂ Q τ δ D η A K B₁ B₂ b₁ b₂ hΨ.continuous hsΨ hK hw₁ hQ hτ hδ hD hη hA hK0 hB₁ hB₂ he₁ hed₁ herrA hfirst hb₁ hd₁ hs₁ hbd₂ apply hh.trans gcongr; linarith by_cases hsecond : τ < ‖d₂-inner ℝ d₂ w₂ • w₂‖ · have hh := hCab Ψ w₂ d₂ e₂ e₁ Q τ δ D η A K B₂ B₁ b₂ b₁ hΨ.continuous hsΨ hK hw₂ hQ hτ hδ hD hη hA hK0 hB₂ hB₁ he₂ hed₂ herrA hsecond hb₂ hd₂ hs₂ hbd₁ rw [graphFixedSpatialTermND_swap Ψ Q e₁ e₂ b₁ b₂] have hc : Ca*K*B₂*B₁=Ca*K*B₁*B₂ := by ring rw [hc] at hh apply hh.trans gcongr; linarith have hlast : τ < |inner ℝ w₁ d₁+inner ℝ w₂ d₂| := by apply lt_of_not_ge intro hl exact hne ⟨le_of_not_gt hfirst,le_of_not_gt hsecond,hl⟩ have hh := graphFixedSpatialTermND_radial_bound m Ψ w₁ w₂ d₁ d₂ e₁ e₂ Q τ δ D η K B₁ B₂ b₁ b₂ hΨ hsΨ hdΨ hw₁ hw₂ hQ hτ hδ hD hη hK0 hB₁ hB₂ he₁ he₂ hed₁ hed₂ herrR hlast hb₁.continuous hb₂.continuous hbd₁ hbd₂ hs₁ hs₂ change ‖graphFixedSpatialTermND Ψ Q e₁ e₂ b₁ b₂‖ ≤ Cr*K*B₁*B₂*(1/(Q*τ))^m at hh have hratio : 1/(Q*τ) ≤ (A+1+D/τ)/(Q*τ) := by apply div_le_div_of_nonneg_right _ (by positivity) have : 0 ≤ D/τ := div_nonneg hD hτ.le linarith apply hh.trans gcongr; linarith end Attempt4FullFar end end end OAI