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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 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | import Mathlib
import OAI.MeasureTheory.Falconer.Fourier.KernelFourierIdentity
import OAI.MeasureTheory.Falconer.Fourier.MeasureKernelConvolution
import OAI.MeasureTheory.Falconer.Model
import OAI.MeasureTheory.Falconer.Shells.ShellFrequencyEnergy
import OAI.MeasureTheory.Falconer.Shells.ShellKernels
import OAI.MeasureTheory.Falconer.Shells.Shells
import OAI.MeasureTheory.Falconer.Shells.SmoothShellPartition
import OAI.MeasureTheory.FalconerCompletion.Fourier.SignedVariationFinite
namespace OAI
/-! Angular cutoffs and derivative estimates. -/
open _root_.OAI.PlanarFalconer
section
section
namespace CurrentFalconer.MonotoneShell
open MeasureTheory Set Filter PlanarFalconer
open scoped Topology ENNReal FourierTransform
lemma inverse_add (f g : ℝ → ℂ) (hf : Integrable f) (hg : Integrable g) :
sourceInverseKernel (f+g) = sourceInverseKernel f+sourceInverseKernel g := by
have he : 𝓕 (f+g) = 𝓕 f+𝓕 g :=
VectorFourier.fourierIntegral_add Real.continuous_fourierChar (innerSL ℝ).continuous₂ hf hg
funext x
simp only [sourceInverseKernel,he,Pi.add_apply,smul_add]
lemma inverse_smul (c : ℂ) (f : ℝ → ℂ) :
sourceInverseKernel (c • f) = c • sourceInverseKernel f := by
have he : 𝓕 (c • f) = c • 𝓕 f :=
VectorFourier.fourierIntegral_const_smul Real.fourierChar volume (innerₗ ℝ) f c
funext x
simp only [sourceInverseKernel,he,Pi.smul_apply]
exact smul_comm _ _ _
lemma inverse_sub (f g : ℝ → ℂ) (hf : Integrable f) (hg : Integrable g) :
sourceInverseKernel (f-g) = sourceInverseKernel f-sourceInverseKernel g := by
have he : f-g = f+(-1:ℂ) • g := by simp [sub_eq_add_neg]
rw [he,inverse_add _ _ hf (hg.smul _),inverse_smul]
simp [sub_eq_add_neg]
noncomputable def complexMultiplier (ν : ComplexMeasure ℝ) (n : ℕ) (r : ℝ) : ℂ :=
(sourceShellBand n r : ℂ)*complexChar ν r
lemma complexMultiplier_continuous (ν : ComplexMeasure ℝ) (n : ℕ) : Continuous (complexMultiplier ν n) :=
(Complex.continuous_ofReal.comp (sourceShellBand_smooth n).continuous).mul (complexChar_continuous ν)
lemma complexMultiplier_zero (ν : ComplexMeasure ℝ) (n : ℕ) (r : ℝ)
(hr : r ∉ dyadicShell (n+1)) : complexMultiplier ν n r = 0 := by
have he : sourceShellBand n r = 0 := by
by_contra hn
exact hr (sourceShellBand_support n hn)
simp [complexMultiplier,he]
lemma complexMultiplier_memLp (ν : ComplexMeasure ℝ) (n : ℕ) (p : ℝ≥0∞) :
MemLp (complexMultiplier ν n) p volume :=
(complexMultiplier_continuous ν n).memLp_of_hasCompactSupport
(HasCompactSupport.intro (dyadicShell_compact (n+1)) (complexMultiplier_zero ν n))
lemma complexMultiplier_decompose (ν : ComplexMeasure ℝ) (n : ℕ) :
complexMultiplier ν n =
(sourceShellMultiplier (ComplexMeasure.re ν).toJordanDecomposition.posPart n -
sourceShellMultiplier (ComplexMeasure.re ν).toJordanDecomposition.negPart n) +
Complex.I • (sourceShellMultiplier (ComplexMeasure.im ν).toJordanDecomposition.posPart n -
sourceShellMultiplier (ComplexMeasure.im ν).toJordanDecomposition.negPart n) := by
funext r
simp only [complexMultiplier,complexChar,signedChar,sourceShellMultiplier,Pi.add_apply,Pi.sub_apply,Pi.smul_apply,smul_eq_mul]
ring
lemma complexMultiplier_positive (μ : FiniteMeasure ℝ) (n : ℕ) :
complexMultiplier (complexOfPositive μ) n = sourceShellMultiplier (μ : Measure ℝ) n := by
funext r
simp [complexMultiplier,sourceShellMultiplier,complexChar_positive]
lemma complexMultiplier_sub (μ ν : ComplexMeasure ℝ) [IsFiniteMeasure μ.variation] [IsFiniteMeasure ν.variation]
(n : ℕ) : complexMultiplier (μ-ν) n = complexMultiplier μ n-complexMultiplier ν n := by
funext r
simp [complexMultiplier,complexChar_sub,mul_sub]
end CurrentFalconer.MonotoneShell
end
end
section
section
namespace CurrentFalconer.MonotoneShell
open MeasureTheory Set Filter PlanarFalconer
open scoped Topology ENNReal
noncomputable def signedConv (s : SignedMeasure ℝ) (n : ℕ) (x : ℝ) : ℂ :=
kernelMeasureConvolution (sourceShellKernel n) s.toJordanDecomposition.posPart x -
kernelMeasureConvolution (sourceShellKernel n) s.toJordanDecomposition.negPart x
noncomputable def complexConv (ν : ComplexMeasure ℝ) (n : ℕ) (x : ℝ) : ℂ :=
signedConv (ComplexMeasure.re ν) n x+Complex.I*signedConv (ComplexMeasure.im ν) n x
lemma signedConv_L1 (s : SignedMeasure ℝ) (n : ℕ) : Integrable (signedConv s n) ∧
(∫ x, ‖signedConv s n x‖) ≤ s.variation.real univ * ∫ x, ‖sourceShellKernel 0 x‖ := by
have hp := shellKernelMeasureConvolution_L1 s.toJordanDecomposition.posPart n
have hn := shellKernelMeasureConvolution_L1 s.toJordanDecomposition.negPart n
refine ⟨hp.1.sub hn.1,?_⟩
calc
_ ≤ ∫ x, ‖kernelMeasureConvolution (sourceShellKernel n) s.toJordanDecomposition.posPart x‖ +
‖kernelMeasureConvolution (sourceShellKernel n) s.toJordanDecomposition.negPart x‖ :=
integral_mono (hp.1.sub hn.1).norm (hp.1.norm.add hn.1.norm) (fun x => norm_sub_le _ _)
_ = _ := integral_add hp.1.norm hn.1.norm
_ ≤ s.toJordanDecomposition.posPart.real univ * (∫ x, ‖sourceShellKernel 0 x‖) +
s.toJordanDecomposition.negPart.real univ * (∫ x, ‖sourceShellKernel 0 x‖) := add_le_add hp.2 hn.2
_ = _ := by rw [←s.totalVariation_eq_variation,SignedMeasure.totalVariation,measureReal_add_apply,add_mul]
lemma complexConv_integrable (ν : ComplexMeasure ℝ) (n : ℕ) : Integrable (complexConv ν n) :=
(signedConv_L1 _ n).1.add ((signedConv_L1 _ n).1.const_mul Complex.I)
lemma complexConv_L1_bound (ν : ComplexMeasure ℝ) [IsFiniteMeasure ν.variation] (n : ℕ) :
(∫ x, ‖complexConv ν n x‖) ≤ (2*ν.variation.real univ) * ∫ x, ‖sourceShellKernel 0 x‖ := by
have hr := signedConv_L1 (ComplexMeasure.re ν) n
have hi := signedConv_L1 (ComplexMeasure.im ν) n
have hrv : (ComplexMeasure.re ν).variation.real univ ≤ ν.variation.real univ :=
ENNReal.toReal_mono (measure_ne_top _ _) (real_variation_le ν univ)
have hiv : (ComplexMeasure.im ν).variation.real univ ≤ ν.variation.real univ :=
ENNReal.toReal_mono (measure_ne_top _ _) (imag_variation_le ν univ)
calc
_ ≤ ∫ x, ‖signedConv (ComplexMeasure.re ν) n x‖ +
‖Complex.I*signedConv (ComplexMeasure.im ν) n x‖ :=
integral_mono (complexConv_integrable ν n).norm
(hr.1.norm.add (hi.1.const_mul Complex.I).norm) (fun x => norm_add_le _ _)
_ = (∫ x, ‖signedConv (ComplexMeasure.re ν) n x‖) +
(∫ x, ‖signedConv (ComplexMeasure.im ν) n x‖) := by
rw [integral_add hr.1.norm (hi.1.const_mul Complex.I).norm]
simp only [norm_mul,Complex.norm_I,one_mul]
_ ≤ ((ComplexMeasure.re ν).variation.real univ+(ComplexMeasure.im ν).variation.real univ) *
(∫ x, ‖sourceShellKernel 0 x‖) := by
rw [add_mul]
exact add_le_add hr.2 hi.2
_ ≤ _ := mul_le_mul_of_nonneg_right (by linarith) (integral_nonneg (fun x => norm_nonneg _))
lemma complexConv_inverse (ν : ComplexMeasure ℝ) (n : ℕ) :
complexConv ν n = sourceInverseKernel (complexMultiplier ν n) := by
have hi (μ : Measure ℝ) [IsFiniteMeasure μ] : Integrable (sourceShellMultiplier μ n) :=
memLp_one_iff_integrable.mp (sourceShellMultiplier_memLp μ n 1)
rw [complexMultiplier_decompose,inverse_add _ _ ((hi _).sub (hi _)) (((hi _).sub (hi _)).smul Complex.I),
inverse_smul,inverse_sub _ _ (hi _) (hi _),inverse_sub _ _ (hi _) (hi _)]
funext x
simp only [complexConv,signedConv,Pi.add_apply,Pi.sub_apply,Pi.smul_apply,smul_eq_mul,
shellKernelMeasureConvolution_fourier]
lemma complexConv_positive (μ : FiniteMeasure ℝ) (n : ℕ) :
complexConv (complexOfPositive μ) n = kernelMeasureConvolution (sourceShellKernel n) (μ : Measure ℝ) := by
rw [complexConv_inverse,complexMultiplier_positive]
funext x
exact (shellKernelMeasureConvolution_fourier μ n x).symm
lemma complexConv_sub (μ ν : ComplexMeasure ℝ) [IsFiniteMeasure μ.variation] [IsFiniteMeasure ν.variation]
(n : ℕ) : complexConv (μ-ν) n = complexConv μ n-complexConv ν n := by
simp only [complexConv_inverse,complexMultiplier_sub]
exact inverse_sub _ _ (memLp_one_iff_integrable.mp (complexMultiplier_memLp μ n 1))
(memLp_one_iff_integrable.mp (complexMultiplier_memLp ν n 1))
end CurrentFalconer.MonotoneShell
end
end
end OAI
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