| import OAI.MeasureTheory.Falconer.Angular.AngularTerminal |
| import OAI.MeasureTheory.Falconer.Fourier.ChartSymbolControl |
| import OAI.MeasureTheory.Falconer.Fourier.SpatialSymbols |
| import OAI.MeasureTheory.Falconer.Geometry.NormalizedInverseFactors |
| import OAI.MeasureTheory.Falconer.Estimates.SpatialEndpointFamily |
|
|
| namespace OAI |
|
|
| /-! # A bounded supremum of the actual admissible endpoint energies |
|
|
| The admissible class records only the existing chart-symbol control and |
| the required support. Its envelope is the real supremum of the original |
| endpoint-transform energies, with zero inserted to ensure nonemptiness. |
| -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
|
|
| def AngularAdmissible {n : ℕ} (k : ℕ) (D : ℝ) |
| (Good : Set (Space (n+2) × Sphere (n+2))) |
| (b : Space (n+2) → Sphere (n+2) → ℂ) : Prop := |
| ChartSymbolControl k b 1 D ∧ ∀ x w, b x w ≠ 0 → (x,w) ∈ Good |
|
|
| theorem AngularAdmissible.spatialMultiplier {n k : ℕ} {D : ℝ} |
| {Good : Set (Space (n+2) × Sphere (n+2))} |
| {b : Space (n+2) → Sphere (n+2) → ℂ} (h : AngularAdmissible k D Good b) |
| (c : Space (n+2) → ℂ) (hc : Measurable c) (hcb : ∀ x, ‖c x‖ ≤ 1) : |
| AngularAdmissible k D Good (fun x w => c x*b x w) := |
| ⟨h.1.spatialMultiplier c hc hcb,fun x w hw => h.2 x w (mul_ne_zero_iff.mp hw).2⟩ |
|
|
| theorem AngularAdmissible.spatialFourier {n k : ℕ} {D : ℝ} |
| {Good : Set (Space (n+2) × Sphere (n+2))} |
| {b : Space (n+2) → Sphere (n+2) → ℂ} (h : AngularAdmissible k D Good b) |
| (s : ℝ) (ξ : Space (n+2)) : |
| AngularAdmissible k D Good (fun x w => spatialFourierPhase s ξ x*b x w) := |
| h.spatialMultiplier _ (by unfold spatialFourierPhase; fun_prop) |
| (fun x => (spatialFourierPhase_norm s ξ x).le) |
|
|
| theorem AngularAdmissible.conj {n k : ℕ} {D : ℝ} |
| {Good : Set (Space (n+2) × Sphere (n+2))} |
| {b : Space (n+2) → Sphere (n+2) → ℂ} (h : AngularAdmissible k D Good b) : |
| AngularAdmissible k D Good (fun x w => star (b x w)) := by |
| refine ⟨h.1.conj,?_⟩ |
| intro x w hw |
| apply h.2 x w |
| intro hzero |
| apply hw |
| change star (b x w) = 0 |
| rw [hzero,star_zero] |
|
|
| noncomputable def angularEnergyValues {n : ℕ} |
| (μ ν : Measure (Space (n+2))) (M P : ℝ) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : Set ℝ := |
| insert 0 {H | ∃ b c, AngularAdmissible k D GoodLeft b ∧ |
| AngularAdmissible k D GoodRight c ∧ H = independentAngularEnergy a v Ψ |
| (fun z => endpointTransform μ M b z.1 z.2) |
| (fun z => endpointTransform ν P c z.1 z.2)} |
|
|
| noncomputable def angularEnvelope {n : ℕ} |
| (μ ν : Measure (Space (n+2))) (M P : ℝ) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : ℝ := |
| sSup (angularEnergyValues μ ν M P k D a v Ψ GoodLeft GoodRight) |
|
|
| noncomputable def angularEnvelopeBound {n : ℕ} |
| (μ ν : Measure (Space (n+2))) (M P a v : ℝ) (Ψ : ℝ → ℝ) : ℝ := |
| (2:ℝ)^(((n+2:ℕ):ℝ)*(v-a))* |
| ((μ.real univ/M)^2*(ν.real univ/P)^2*(sphereMeasure (n+2)).real univ^2)* |
| (PlanarFalconer.sourceAngularShellMeasure v Ψ).real univ |
|
|
| theorem angularEnergyValues_nonempty {n : ℕ} |
| (μ ν : Measure (Space (n+2))) (M P : ℝ) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| (angularEnergyValues μ ν M P k D a v Ψ GoodLeft GoodRight).Nonempty := |
| ⟨0,Or.inl rfl⟩ |
|
|
| theorem angularEnergyValues_bound {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| ∀ H ∈ angularEnergyValues μ ν M P k D a v Ψ GoodLeft GoodRight, |
| H ≤ angularEnvelopeBound μ ν M P a v Ψ := by |
| intro H hH |
| rcases hH with rfl | ⟨b,c,hb,hc,rfl⟩ |
| · unfold angularEnvelopeBound |
| positivity |
| · apply independentAngularEnergy_bound a v Ψ _ _ |
| (div_nonneg measureReal_nonneg hM.le) (div_nonneg measureReal_nonneg hP.le) |
| · intro z |
| simpa only [one_mul] using endpointTransform_norm_le μ hM b hb.1.bound z.1 z.2 |
| · intro z |
| simpa only [one_mul] using endpointTransform_norm_le ν hP c hc.1.bound z.1 z.2 |
|
|
| theorem angularEnergyValues_bddAbove {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| BddAbove (angularEnergyValues μ ν M P k D a v Ψ GoodLeft GoodRight) := |
| ⟨angularEnvelopeBound μ ν M P a v Ψ, |
| angularEnergyValues_bound μ ν hM hP k D a v Ψ GoodLeft GoodRight⟩ |
|
|
| theorem angularEnvelope_nonneg {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| 0 ≤ angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight := |
| le_csSup (angularEnergyValues_bddAbove μ ν hM hP k D a v Ψ GoodLeft GoodRight) (Or.inl rfl) |
|
|
| theorem angularEnvelope_le_bound {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) : |
| angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight ≤ |
| angularEnvelopeBound μ ν M P a v Ψ := |
| csSup_le (angularEnergyValues_nonempty μ ν M P k D a v Ψ GoodLeft GoodRight) |
| (angularEnergyValues_bound μ ν hM hP k D a v Ψ GoodLeft GoodRight) |
|
|
| theorem admissible_energy_le_angularEnvelope {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) |
| (b c : Space (n+2) → Sphere (n+2) → ℂ) |
| (hb : AngularAdmissible k D GoodLeft b) (hc : AngularAdmissible k D GoodRight c) : |
| independentAngularEnergy a v Ψ |
| (fun z => endpointTransform μ M b z.1 z.2) |
| (fun z => endpointTransform ν P c z.1 z.2) ≤ |
| angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight := |
| le_csSup (angularEnergyValues_bddAbove μ ν hM hP k D a v Ψ GoodLeft GoodRight) |
| (Or.inr ⟨b,c,hb,hc,rfl⟩) |
|
|
| theorem spatial_endpoint_energy_le_envelope {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) (D a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| (GoodLeft GoodRight : Set (Space (n+2) × Sphere (n+2))) |
| (b c : Space (n+2) → Sphere (n+2) → ℂ) |
| (hb : AngularAdmissible k D GoodLeft b) (hc : AngularAdmissible k D GoodRight c) |
| (s : ℝ) (ξ : Space (n+2)) : |
| independentAngularEnergy a v Ψ |
| (fun z => spatialEndpointFamily μ M s b (ξ,z)) |
| (fun z => spatialEndpointFamily ν P s (fun x w => star (c x w)) (ξ,z)) ≤ |
| angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight := by |
| let b' : Space (n+2) → Sphere (n+2) → ℂ := |
| fun x w => spatialFourierPhase s ξ x*b x w |
| let c' : Space (n+2) → Sphere (n+2) → ℂ := |
| fun x w => spatialFourierPhase s ξ x*star (c x w) |
| have hb' : AngularAdmissible k D GoodLeft b' := hb.spatialFourier s ξ |
| have hc' : AngularAdmissible k D GoodRight c' := hc.conj.spatialFourier s ξ |
| change independentAngularEnergy a v Ψ |
| (fun z => endpointTransform μ M b' z.1 z.2) |
| (fun z => endpointTransform ν P c' z.1 z.2) ≤ _ |
| exact admissible_energy_le_angularEnvelope (n := n) μ ν hM hP k D a v Ψ |
| GoodLeft GoodRight b' c' hb' hc' |
|
|
| theorem AngularAdmissible.mono {n k : ℕ} {D E : ℝ} |
| {Good Good' : Set (Space (n+2) × Sphere (n+2))} |
| {b : Space (n+2) → Sphere (n+2) → ℂ} |
| (h : AngularAdmissible k D Good b) (hD : 0 ≤ D) (hDE : D ≤ E) |
| (hGood : Good ⊆ Good') : AngularAdmissible k E Good' b := by |
| refine ⟨⟨h.1.measurable,h.1.bound,h.1.smooth,?_⟩,fun x w hw => hGood (h.2 x w hw)⟩ |
| intro e l hl x y hy u hu |
| exact (h.1.derivative e l hl x y hy u hu).trans |
| (mul_le_mul_of_nonneg_left (pow_le_pow_left₀ hD hDE l) (by norm_num : (0:ℝ) ≤ 1)) |
| |
| theorem angularEnvelope_mono {n : ℕ} |
| (μ ν : Measure (Space (n+2))) [IsFiniteMeasure μ] [IsFiniteMeasure ν] |
| {M P : ℝ} (hM : 0 < M) (hP : 0 < P) (k : ℕ) {D E : ℝ} |
| (hD : 0 ≤ D) (hDE : D ≤ E) (a v : ℝ) (Ψ : ℝ → ℝ) |
| [IsFiniteMeasure (PlanarFalconer.sourceAngularShellMeasure v Ψ)] |
| {GoodLeft GoodRight GoodLeft' GoodRight' : Set (Space (n+2) × Sphere (n+2))} |
| (hLeft : GoodLeft ⊆ GoodLeft') (hRight : GoodRight ⊆ GoodRight') : |
| angularEnvelope μ ν M P k D a v Ψ GoodLeft GoodRight ≤ |
| angularEnvelope μ ν M P k E a v Ψ GoodLeft' GoodRight' := by |
| apply csSup_le_csSup |
| (angularEnergyValues_bddAbove μ ν hM hP k E a v Ψ GoodLeft' GoodRight') |
| (angularEnergyValues_nonempty μ ν M P k D a v Ψ GoodLeft GoodRight) |
| intro H hH |
| rcases hH with rfl | ⟨b,c,hb,hc,rfl⟩ |
| · exact Or.inl rfl |
| · exact Or.inr ⟨b,c,hb.mono hD hDE hLeft,hc.mono hD hDE hRight,rfl⟩ |
| |
| end Falconer |
| |
| end OAI |
| |
