| 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 | |