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7f37411 | 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 45 46 47 48 49 50 51 | 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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