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b17ead8 | 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 40 41 42 43 44 | 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
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