| import OAI.MeasureTheory.Falconer.Angular.AngularFourSpatialIntegral |
| import OAI.MeasureTheory.Falconer.Angular.AngularCellKernels |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ComplexConjugate |
|
|
| noncomputable def angularCellSpatialIntegral (Q₁ Q₂ : DyadicSquare) |
| (ρ ν : Measure Plane) (β γ : Plane → UnitCircle → ℂ) |
| (row col : CellPath × CellPath) (v : ℝ) (Ψ : ℝ → ℝ) : ℂ := |
| ∫ x, ((2:ℝ)^v)⁻¹ • (∫ r : ℝ, ∫ u, ∫ w, Ψ (r/(2:ℝ)^v) • |
| graphFourKernel ![β,β,γ,γ] (fun _ => 1) x r u w |
| ∂sourceArcLength ∂sourceArcLength) |
| ∂Measure.pi (fun i => (![ρ,ρ,ν,ν] i).restrict |
| (![ (Q₁.descendant row.1).carrier, (Q₁.descendant col.1).carrier, |
| (Q₂.descendant row.2).carrier, (Q₂.descendant col.2).carrier] i)) |
|
|
| theorem angular_pair_spatial_integral (Q₁ Q₂ : DyadicSquare) |
| (ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] |
| {M₁ M₂ : ℝ} (hM₁ : 0 < M₁) (hM₂ : 0 < M₂) |
| (β γ : Plane → UnitCircle → ℂ) |
| (hβ : Measurable (fun z : Plane × UnitCircle => β z.1 z.2)) |
| (hγ : Measurable (fun z : Plane × UnitCircle => γ z.1 z.2)) |
| {D₁ D₂ : ℝ} (hD₁ : 0 ≤ D₁) (hD₂ : 0 ≤ D₂) |
| (hbβ : ∀ x u, ‖β x u‖ ≤ D₁) (hbγ : ∀ x u, ‖γ x u‖ ≤ D₂) |
| (row col : CellPath × CellPath) (a v : ℝ) {Ψ : ℝ → ℝ} |
| (hΨ : Measurable Ψ) (hpos : ∀ t, 0 ≤ Ψ t) |
| (hs : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) |
| {W : ℝ} (hW : 0 ≤ W) (hw : ∀ t, Ψ t ≤ W) : |
| angularKernelIntegral a v Ψ (fun z => angularPairTerm |
| (fun P => angularCellTransform Q₁ ρ M₁ β P (z.1,z.2.1)) |
| (fun Q => angularCellTransform Q₂ ν M₂ γ Q (z.1,z.2.2)) row col) = |
| ((2:ℝ)^(2*(v-a))) • ((M₁^2*M₂^2)⁻¹ • |
| angularCellSpatialIntegral Q₁ Q₂ ρ ν β γ row col v Ψ) := by |
| let := sourceAngularShellMeasure_finite v Ψ hW hw (fun t ht => by |
| by_contra hn |
| have := (hs t hn).2 |
| linarith) |
| have hz : ∀ t ≤ 0, Ψ t = 0 := by |
| intro t ht |
| by_contra hn |
| have := (hs t hn).1 |
| linarith |
| let U := angularCellTransform Q₁ ρ M₁ β |
| let V := angularCellTransform Q₂ ν M₂ γ |
| have hu := angularCellTransform_measurable Q₁ ρ M₁ β hβ |
| have hv := angularCellTransform_measurable Q₂ ν M₂ γ hγ |
| have hm : Measurable (fun z : ℝ × (UnitCircle × UnitCircle) => |
| angularPairTerm (fun P => U P (z.1,z.2.1)) (fun Q => V Q (z.1,z.2.2)) row col) := by |
| unfold angularPairTerm |
| fun_prop |
| have hb : ∀ z : ℝ × (UnitCircle × UnitCircle), |
| ‖angularPairTerm (fun P => U P (z.1,z.2.1)) (fun Q => V Q (z.1,z.2.2)) row col‖ ≤ |
| (D₁*(ρ univ).toReal/M₁)^2*(D₂*(ν univ).toReal/M₂)^2 := by |
| intro z |
| simp only [angularPairTerm,norm_mul,Complex.norm_conj,pow_two] |
| apply mul_le_mul |
| (mul_le_mul (sourceAngularTransform_global_bound ρ _ hM₁ β hD₁ hbβ _ _) |
| (sourceAngularTransform_global_bound ρ _ hM₁ β hD₁ hbβ _ _) (norm_nonneg _) (by positivity)) |
| (mul_le_mul (sourceAngularTransform_global_bound ν _ hM₂ γ hD₂ hbγ _ _) |
| (sourceAngularTransform_global_bound ν _ hM₂ γ hD₂ hbγ _ _) (norm_nonneg _) (by positivity)) |
| (by positivity) (by positivity) |
| rw [angularKernelIntegral_original a v hΨ hpos hz _ hm hb] |
| let ρs : Fin 4 → Measure Plane := ![ρ,ρ,ν,ν] |
| let bs : Fin 4 → Plane → UnitCircle → ℂ := ![β,β,γ,γ] |
| let Ds : Fin 4 → ℝ := ![D₁,D₁,D₂,D₂] |
| let X : Fin 4 → Set Plane := ![(Q₁.descendant row.1).carrier, (Q₁.descendant col.1).carrier, |
| (Q₂.descendant row.2).carrier, (Q₂.descendant col.2).carrier] |
| let : ∀ index, IsFiniteMeasure (ρs index) := by |
| intro index |
| fin_cases index <;> dsimp [ρs] <;> infer_instance |
| have he := sourceGraphFourTransform_original_spatial ρs X ![M₁,M₁,M₂,M₂] bs (fun _ => 1) |
| (by intro l; fin_cases l <;> assumption) measurable_const Ds |
| (by intro l; fin_cases l <;> assumption) |
| (by intro l; fin_cases l <;> assumption) (by intro p; simp) |
| hΨ hs hW (fun t => by rw [abs_of_nonneg (hpos t)]; exact hw t) |
| (by positivity : (0:ℝ) < (2:ℝ)^v) |
| congr 1 |
| simpa only [sourceGraphFourTransform,angularPairTerm,angularCellTransform, |
| angularCellSpatialIntegral,ρs,bs,Ds,X,Matrix.cons_val_zero,Matrix.cons_val_one, |
| Matrix.cons_val_two,Matrix.cons_val_three,Matrix.head_cons,Matrix.tail_cons, |
| one_mul,pow_two,U,V,angularCellTransform] using he |
|
|
| end PlanarFalconer |
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| end OAI |
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