| import OAI.MeasureTheory.Falconer.Angular.AngularCellSpatialIntegral |
| import OAI.MeasureTheory.Falconer.Angular.AngularRestrictedIntegralSum |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| attribute [local instance] Classical.propDecidable |
|
|
| theorem angular_contribution_spatial_sum (Q₁ Q₂ : DyadicSquare) |
| (ρ ν : Measure Plane) [IsFiniteMeasure ρ] [IsFiniteMeasure ν] |
| (A B : CellPath) (n : ℕ) (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) |
| (Ma₁ Ma₂ : ℝ) {Mp₁ Mp₂ : ℝ} (hMp₁ : 0 < Mp₁) (hMp₂ : 0 < Mp₂) |
| (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) |
| (G : (CellPath × CellPath) → (CellPath × CellPath) → Prop) : |
| angularCellContribution Q₁ Q₂ ρ ν A B n a v Ψ Ma₁ Ma₂ Mp₁ Mp₂ β γ G = |
| (((2:ℝ)^(2*(v-a)))/(Ma₁^2*Ma₂^2)) • |
| ∑ row ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n, |
| ∑ col ∈ sourceProductVertices Q₁ Q₂ ρ ν A B n, |
| if G row col then angularCellSpatialIntegral Q₁ Q₂ ρ ν β γ row col v Ψ else 0 := by |
| let := sourceAngularShellMeasure_finite v Ψ hW hw (fun t ht => by |
| by_contra hn |
| have := (hs t hn).2 |
| linarith) |
| unfold angularCellContribution |
| rw [angularRestrictedIntegral_sum a v Ψ _ G _ _ |
| (angularCellTransform_measurable Q₁ ρ Mp₁ β hβ) |
| (angularCellTransform_measurable Q₂ ν Mp₂ γ hγ) |
| (B := D₁*(ρ univ).toReal/Mp₁) (D := D₂*(ν univ).toReal/Mp₂) (by positivity) (by positivity) |
| (fun P z => sourceAngularTransform_global_bound ρ _ hMp₁ β hD₁ hbβ z.1 z.2) |
| (fun P z => sourceAngularTransform_global_bound ν _ hMp₂ γ hD₂ hbγ z.1 z.2)] |
| simp_rw [Finset.smul_sum] |
| apply Finset.sum_congr rfl |
| intro row _ |
| apply Finset.sum_congr rfl |
| intro col _ |
| by_cases hg : G row col |
| · simp only [hg,↓reduceIte] |
| rw [angular_pair_spatial_integral Q₁ Q₂ ρ ν hMp₁ hMp₂ β γ hβ hγ hD₁ hD₂ hbβ hbγ |
| row col a v hΨ hpos hs hW hw,smul_smul,smul_smul] |
| congr 1 |
| simp only [mul_inv_rev,pow_two,div_eq_mul_inv] |
| field_simp |
| · simp [hg] |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
