| import OAI.MeasureTheory.Falconer.Graphs.GraphOriginalFubini |
| import OAI.MeasureTheory.Falconer.Graphs.GraphFourTransformExpansion |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
|
|
| theorem sourceGraphFourTransform_original_spatial |
| (ρ : Fin 4 → Measure Plane) [∀ i, IsFiniteMeasure (ρ i)] |
| (X : Fin 4 → Set Plane) (M : Fin 4 → ℝ) |
| (b : Fin 4 → Plane → UnitCircle → ℂ) (W : UnitCircle × UnitCircle → ℂ) |
| (hb : ∀ i, Measurable (fun p : Plane × UnitCircle => b i p.1 p.2)) |
| (hW : Measurable W) (B : Fin 4 → ℝ) (hB : ∀ i, 0 ≤ B i) |
| (hbound : ∀ i x ω, ‖b i x ω‖ ≤ B i) (hw : ∀ p, ‖W p‖ ≤ 1) |
| {Ψ : ℝ → ℝ} (hΨ : Measurable Ψ) |
| (hsupp : ∀ t, Ψ t ≠ 0 → t ∈ Ioo (1/4:ℝ) 4) |
| {C : ℝ} (hC : 0 ≤ C) (hc : ∀ t, |Ψ t| ≤ C) {R : ℝ} (hR : 0 < R) : |
| R⁻¹ • (∫ r : ℝ, ∫ u, ∫ v, Ψ (r/R) • sourceGraphFourTransform ρ X M b W r u v |
| ∂sourceArcLength ∂sourceArcLength) = |
| ((M 0*M 1)*(M 2*M 3))⁻¹ • |
| (∫ x, R⁻¹ • (∫ r : ℝ, ∫ u, ∫ v, Ψ (r/R) • graphFourKernel b W x r u v |
| ∂sourceArcLength ∂sourceArcLength) |
| ∂Measure.pi (fun i => (ρ i).restrict (X i))) := by |
| let μ := Measure.pi (fun i => (ρ i).restrict (X i)) |
| let c := ((M 0*M 1)*(M 2*M 3))⁻¹ |
| have he (r : ℝ) (u v : UnitCircle) : |
| Ψ (r/R) • sourceGraphFourTransform ρ X M b W r u v = |
| c • (∫ x, graphOriginalKernel b W Ψ R (r,(u,v)) x ∂μ) := by |
| rw [sourceGraphFourTransform_endpoint_integral] |
| change Ψ (r/R) • (c • (∫ x, graphFourKernel b W x r u v ∂μ)) = _ |
| rw [smul_comm] |
| congr 1 |
| exact (integral_smul _ _).symm |
| simp_rw [he, integral_smul] |
| rw [smul_comm (R⁻¹) c] |
| congr 1 |
| rw [graphOriginalKernel_spatial_swap μ b W hb hW B hB hbound hw hΨ hsupp hC hc hR] |
| simp only [graphOriginalKernel, integral_smul, μ] |
|
|
| end PlanarFalconer |
|
|
| end OAI |
|
|
