| import OAI.MeasureTheory.Falconer.Angular.AngularRestrictedKernel |
| import OAI.MeasureTheory.Falconer.Angular.AngularTransformSplitting |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
|
|
| noncomputable def angularCellTransform (Q : DyadicSquare) (ρ : Measure Plane) |
| (M : ℝ) (β : Plane → UnitCircle → ℂ) (P : CellPath) (z : ℝ × UnitCircle) : ℂ := |
| sourceAngularTransform ρ (Q.descendant P).carrier M β z.1 z.2 |
|
|
| theorem angularCellTransform_measurable (Q : DyadicSquare) (ρ : Measure Plane) |
| [IsFiniteMeasure ρ] (M : ℝ) (β : Plane → UnitCircle → ℂ) |
| (hβ : Measurable (fun z : Plane × UnitCircle => β z.1 z.2)) (P : CellPath) : |
| Measurable (angularCellTransform Q ρ M β P) := |
| measurable_sourceAngularTransform ρ _ M β hβ |
|
|
| theorem sourceAngularTransform_global_bound (ρ : Measure Plane) [IsFiniteMeasure ρ] |
| (X : Set Plane) {M : ℝ} (hM : 0 < M) (β : Plane → UnitCircle → ℂ) |
| {D : ℝ} (hD : 0 ≤ D) (hb : ∀ x u, ‖β x u‖ ≤ D) (r : ℝ) (u : UnitCircle) : |
| ‖sourceAngularTransform ρ X M β r u‖ ≤ D*(ρ univ).toReal/M := by |
| apply (sourceAngularTransform_norm_le ρ X M hM β D (fun x _ => hb x) r u).trans |
| exact div_le_div_of_nonneg_right |
| (mul_le_mul_of_nonneg_left |
| (ENNReal.toReal_mono (measure_ne_top ρ univ) (measure_mono (subset_univ X))) hD) hM.le |
|
|
| noncomputable def angularCellContribution (Q₁ Q₂ : DyadicSquare) (ρ ν : Measure Plane) |
| (A B : CellPath) (n : ℕ) (a v : ℝ) (Ψ : ℝ → ℝ) |
| (Ma₁ Ma₂ Mp₁ Mp₂ : ℝ) (β γ : Plane → UnitCircle → ℂ) |
| (G : (CellPath × CellPath) → (CellPath × CellPath) → Prop) : ℂ := |
| ((Mp₁/Ma₁)^2*(Mp₂/Ma₂)^2) • angularKernelIntegral a v Ψ |
| (angularRestrictedKernel (sourceProductVertices Q₁ Q₂ ρ ν A B n) G |
| (angularCellTransform Q₁ ρ Mp₁ β) (angularCellTransform Q₂ ν Mp₂ γ)) |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
