openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Angular /ComplexRadialNormalization.lean
| import Mathlib | |
| import OAI.MeasureTheory.FalconerCompletion.Estimates.FarRetainedEqOriginalLabels | |
| namespace OAI | |
| /-! Angular cutoffs and derivative estimates. -/ | |
| section | |
| section | |
| open Set MeasureTheory | |
| open scoped ComplexConjugate RealInnerProductSpace | |
| namespace GraphLiteralSupport | |
| namespace CurrentFalconer | |
| open Set MeasureTheory _root_.OAI.GraphLiteralSupport.Attempt4FullFar | |
| open scoped ComplexConjugate | |
| lemma complex_radial_normalizationND {Ω : ℝ} (hΩ : 0<Ω) (Ψ : ℝ → ℝ) (f : ℝ → ℂ) : | |
| Ω⁻¹ • (∫ r, (Ψ (r/Ω):ℂ)*f r) = ∫ u, (Ψ u:ℂ)*f (Ω*u) := by | |
| have hh := Measure.integral_comp_mul_left (fun r => (Ψ (r/Ω):ℂ)*f r) Ω | |
| simp only [abs_of_pos (inv_pos.mpr hΩ)] at hh | |
| rw [← hh] | |
| apply integral_congr_ae | |
| filter_upwards with u | |
| rw [mul_div_cancel_left₀ u hΩ.ne'] | |
| lemma farAngularEnergyND_normalized {V : Type*} [MeasurableSpace V] (d : ℕ) | |
| (μ : Measure V) [IsFiniteMeasure μ] (a : ℝ) (h : ℕ) (v lam₁ lam₂ : ℝ) | |
| (A D : Finset (Fin d → ℤ)) (U W : (Fin d → ℤ) → ℝ×V → ℂ) | |
| (hU : ∀ i∈A, Measurable (U i)) (hW : ∀ j∈D, Measurable (W j)) | |
| (C E : (Fin d → ℤ) → ℝ) | |
| (hbU : ∀ i∈A, ∀ r, ∀ᵐ w ∂μ, ‖U i (r,w)‖≤C i) | |
| (hbW : ∀ j∈D, ∀ r, ∀ᵐ w ∂μ, ‖W j (r,w)‖≤E j) | |
| {Ψ : ℝ → ℝ} (hmΨ : Measurable Ψ) (hΨ : Integrable Ψ) : | |
| Attempt4Near.farAngularEnergyND d μ a h v lam₁ lam₂ Ψ A D U W = | |
| (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a)))* | |
| ‖∫ p : ℝ×(V×V), (Ψ p.1:ℂ)*Attempt4Near.farRetainedND h A D | |
| (fun i q => U i ((2:ℝ)^v*q.1,q.2)) (fun j q => W j ((2:ℝ)^v*q.1,q.2)) p.1 p.2 | |
| ∂volume.prod (μ.prod μ)‖ := by | |
| let Us := fun i (q : ℝ×V) => U i ((2:ℝ)^v*q.1,q.2) | |
| let Ws := fun j (q : ℝ×V) => W j ((2:ℝ)^v*q.1,q.2) | |
| have hUs : ∀ i∈A, Measurable (Us i) := fun i hi => | |
| (hU i hi).comp ((measurable_const.mul measurable_fst).prodMk measurable_snd) | |
| have hWs : ∀ j∈D, Measurable (Ws j) := fun j hj => | |
| (hW j hj).comp ((measurable_const.mul measurable_fst).prodMk measurable_snd) | |
| have hUb : ∀ i∈A, ∀ r, ∀ᵐ w ∂μ, ‖Us i (r,w)‖≤C i := fun i hi r => hbU i hi ((2:ℝ)^v*r) | |
| have hWb : ∀ j∈D, ∀ r, ∀ᵐ w ∂μ, ‖Ws j (r,w)‖≤E j := fun j hj r => hbW j hj ((2:ℝ)^v*r) | |
| have hi := Attempt4Near.farRetainedND_joint_integrable h μ A D U W hU hW C E hbU hbW | |
| (hmΨ.comp (measurable_id.div_const _)) (hΨ.comp_div (by positivity : (2:ℝ)^v≠0)) | |
| have hj := Attempt4Near.farRetainedND_joint_integrable h μ A D Us Ws hUs hWs C E hUb hWb hmΨ hΨ | |
| rw [Attempt4Near.farAngularEnergyND, | |
| integral_prod (fun p : ℝ×(V×V) => (Ψ (p.1/(2:ℝ)^v):ℂ)*Attempt4Near.farRetainedND h A D U W p.1 p.2) hi, | |
| integral_prod (fun p : ℝ×(V×V) => (Ψ p.1:ℂ)*Attempt4Near.farRetainedND h A D Us Ws p.1 p.2) hj] | |
| simp only [integral_const_mul] | |
| have hn := congrArg norm (complex_radial_normalizationND (by positivity : (0:ℝ)<(2:ℝ)^v) Ψ | |
| (fun r => ∫ z, Attempt4Near.farRetainedND h A D U W r z ∂μ.prod μ)) | |
| simp only [norm_smul,Real.norm_eq_abs,abs_of_pos (by positivity : (0:ℝ)<((2:ℝ)^v)⁻¹)] at hn | |
| calc | |
| _ = (lam₁^2*lam₂^2*(2:ℝ)^((d:ℝ)*(v-a)))* | |
| (((2:ℝ)^v)⁻¹ * ‖∫ r, (Ψ (r/(2:ℝ)^v):ℂ)*(∫ z, Attempt4Near.farRetainedND h A D U W r z ∂μ.prod μ)‖) := by ring | |
| _ = _ := by rw [hn]; rfl | |
| variable {n : ℕ} | |
| theorem physical_farAngularEnergyND_original {Ψ : ℝ → ℝ} (hmΨ : Measurable Ψ) (hΨ : Integrable Ψ) | |
| (a v lam₁ lam₂ : ℝ) | |
| (ρ κ : Measure (EuclideanSpace ℝ (Fin (n+1)))) [IsFiniteMeasure ρ] [IsFiniteMeasure κ] | |
| (h : ℕ) (A D : Finset (Fin (n+1) → ℤ)) (M N : ℝ) | |
| (b c : EuclideanSpace ℝ (Fin (n+1)) → EuclideanSpace ℝ (Fin (n+1)) → ℂ) | |
| (hb : Measurable (Function.uncurry b)) (hc : Measurable (Function.uncurry c)) | |
| {B C : ℝ} (hB : 0≤B) (hC : 0≤C) | |
| (hbd : ∀ x w, ‖w‖=1 → ‖b x w‖≤B) (hcd : ∀ x w, ‖w‖=1 → ‖c x w‖≤C) : | |
| Attempt4Near.farAngularEnergyND (n+1) (canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1)))) | |
| a h v lam₁ lam₂ Ψ A D (originalCellTransformND ρ h M b) (originalCellTransformND κ h N c) = | |
| (lam₁^2*lam₂^2*(2:ℝ)^((n+1:ℕ)*(v-a))/((M*M)*(N*N)))* | |
| ‖∫ z in originalFarRegionND h A D, euclideanFarKernelND Ψ v (scalarPairSymbolND b) (scalarPairSymbolND c) z | |
| ∂(ρ.prod ρ).prod (κ.prod κ)‖ := by | |
| let Bs := fun i => |M⁻¹| *(ρ (ambientGridCell h i)).toReal*B | |
| let Cs := fun j => |N⁻¹| *(κ (ambientGridCell h j)).toReal*C | |
| have hUb : ∀ i∈A, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), | |
| ‖originalCellTransformND ρ h M b i (r,w)‖≤Bs i := by | |
| intro i _ r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm] with w hw | |
| exact sourceAngularTransformND_norm_le ρ _ _ b _ w (fun x => hbd x w hw) | |
| have hWb : ∀ j∈D, ∀ r, ∀ᵐ w ∂canonicalSphereMeasureND (EuclideanSpace ℝ (Fin (n+1))), | |
| ‖originalCellTransformND κ h N c j (r,w)‖≤Cs j := by | |
| intro j _ r | |
| filter_upwards [canonicalSphereMeasureND_ae_norm] with w hw | |
| exact sourceAngularTransformND_norm_le κ _ _ c _ w (fun x => hcd x w hw) | |
| rw [farAngularEnergyND_normalized (n+1) _ a h v lam₁ lam₂ A D (originalCellTransformND ρ h M b) (originalCellTransformND κ h N c) | |
| (fun i _ => sourceAngularTransformND_measurable ρ _ _ hb) | |
| (fun j _ => sourceAngularTransformND_measurable κ _ _ hc) Bs Cs hUb hWb hmΨ hΨ] | |
| rw [← original_far_energy_identityND hmΨ hΨ v ρ κ h A D M N b c hb hc hB hC hbd hcd] | |
| rw [norm_smul,Real.norm_eq_abs,abs_of_nonneg (inv_nonneg.mpr | |
| (mul_nonneg (mul_self_nonneg M) (mul_self_nonneg N)))] | |
| ring | |
| lemma physical_far_profile_coefficientND (R : ℝ) (hR : 0<R) (ma mb K : ℝ) : | |
| (R^(-(mb-ma)))^2*(R^(-(mb-ma)))^2*K/((R^(-mb)*R^(-mb))*(R^(-mb)*R^(-mb))) = | |
| K/(R^(-ma))^4 := by | |
| have hm : R^(-ma)≠0 := ne_of_gt (Real.rpow_pos_of_pos hR _) | |
| have hn : R^(-mb)≠0 := ne_of_gt (Real.rpow_pos_of_pos hR _) | |
| have he : R^(-(mb-ma))=(R^(-ma))⁻¹*R^(-mb) := by | |
| rw [← Real.rpow_neg hR.le,← Real.rpow_add hR] | |
| congr 1 | |
| ring | |
| rw [he] | |
| field_simp | |
| end CurrentFalconer | |
| end GraphLiteralSupport | |
| end | |
| end | |
| end OAI | |