| import OAI.MeasureTheory.Falconer.Angular.CircleTests |
| import OAI.MeasureTheory.Falconer.Radial.RadialShellMeasure |
|
|
| namespace OAI |
|
|
| namespace PlanarFalconer |
| open Set MeasureTheory |
| open scoped ENNReal NNReal |
|
|
| noncomputable def sourceAngularTransform (ρ : Measure Plane) (X : Set Plane) (M : ℝ) |
| (b : Plane → UnitCircle → ℂ) (r : ℝ) (ω : UnitCircle) : ℂ := |
| M⁻¹ • ∫ x in X, Complex.exp (Complex.I * (r : ℂ) * (inner ℝ (ω : Plane) x : ℂ)) * b x ω ∂ρ |
|
|
| noncomputable def sourceAngularMoment (U : ℝ × UnitCircle → ℂ) (r : ℝ) : ℝ≥0∞ := |
| ∫⁻ ω, ‖U (r, ω)‖ₑ^2 ∂sourceArcLength |
|
|
| noncomputable def sourceAngularH (a v : ℝ) (Ψ : ℝ → ℝ) (U V : ℝ × UnitCircle → ℂ) : ℝ≥0∞ := |
| ENNReal.ofReal ((2:ℝ)^(2*(v-a))) * ∫⁻ r, |
| sourceAngularMoment U r * sourceAngularMoment V r ∂sourceAngularShellMeasure v Ψ |
|
|
| theorem measurable_sourceAngularMoment (U : ℝ × UnitCircle → ℂ) (hU : Measurable U) : |
| Measurable (sourceAngularMoment U) := (hU.enorm.pow_const 2).lintegral_prod_right' |
|
|
| theorem sourceAngularH_two_profile (a v : ℝ) (Ψ : ℝ → ℝ) |
| (U V : ℝ × UnitCircle → ℂ) (hU : Measurable U) (hV : Measurable V) : |
| sourceAngularH a v Ψ U V ≤ |
| (sourceAngularH a v Ψ U U)^(1/2 : ℝ) * (sourceAngularH a v Ψ V V)^(1/2 : ℝ) := by |
| let C := ENNReal.ofReal ((2:ℝ)^(2*(v-a))) |
| have hC0 : C ≠ 0 := (ENNReal.ofReal_pos.mpr (Real.rpow_pos_of_pos (by norm_num) _)).ne' |
| have hCt : C ≠ ∞ := ENNReal.ofReal_ne_top |
| have hCs : C^(1/2 : ℝ)*C^(1/2 : ℝ) = C := by |
| rw [← ENNReal.rpow_add _ _ hC0 hCt] |
| norm_num |
| have hpq : (2:ℝ).HolderConjugate 2 := Real.HolderConjugate.two_two |
| have hh := ENNReal.lintegral_mul_le_Lp_mul_Lq (sourceAngularShellMeasure v Ψ) hpq |
| (measurable_sourceAngularMoment U hU).aemeasurable |
| (measurable_sourceAngularMoment V hV).aemeasurable |
| simp only [ENNReal.rpow_two, pow_two] at hh |
| unfold sourceAngularH |
| change C * _ ≤ (C * _)^(1/2 : ℝ) * (C * _)^(1/2 : ℝ) |
| apply (mul_le_mul_right hh C).trans_eq |
| rw [ENNReal.mul_rpow_of_nonneg _ _ (by norm_num : (0:ℝ) ≤ 1/2), |
| ENNReal.mul_rpow_of_nonneg _ _ (by norm_num : (0:ℝ) ≤ 1/2)] |
| calc |
| _ = (C^(1/2 : ℝ)*C^(1/2 : ℝ)) * _ := by rw [hCs] |
| _ = _ := by ac_rfl |
|
|
| end PlanarFalconer |
|
|
| end OAI |
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|
