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caec230 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 | 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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