openAI-math-dataset / lean /OAI /MeasureTheory /FalconerCompletion /Graphs /ComplexOfPositiveFinite.lean
| import Mathlib | |
| import OAI.MeasureTheory.Falconer.Fourier.MeasureKernelConvolution | |
| import OAI.MeasureTheory.Falconer.Fourier.SchwartzLpTests | |
| import OAI.MeasureTheory.Falconer.Graphs.SourceSpatialEnergy | |
| import OAI.MeasureTheory.Falconer.Model | |
| import OAI.MeasureTheory.Falconer.Shells.ShellKernels | |
| import OAI.MeasureTheory.FalconerCompletion.Fourier.BandTermSubOfLe | |
| namespace OAI | |
| /-! Graph energies and pair estimates. -/ | |
| open _root_.OAI.PlanarFalconer | |
| section | |
| section | |
| namespace CurrentFalconer.MonotoneShell | |
| open MeasureTheory Set Filter PlanarFalconer | |
| open scoped Topology ENNReal FourierTransform SchwartzMap | |
| instance complexOfPositive_finite (μ : FiniteMeasure ℝ) : IsFiniteMeasure (complexOfPositive μ).variation := by | |
| rw [complexOfPositive_variation] | |
| infer_instance | |
| lemma complexConv_memLp_two (ν : ComplexMeasure ℝ) (n : ℕ) : MemLp (complexConv ν n) 2 volume := by | |
| rw [complexConv_inverse] | |
| exact sourceInverseKernel_memLp_two _ (memLp_one_iff_integrable.mp (complexMultiplier_memLp ν n 1)) | |
| (complexMultiplier_memLp ν n 2) | |
| lemma complexConv_energy (ν : ComplexMeasure ℝ) (n : ℕ) : | |
| (∫ x, ‖complexConv ν n x‖^2) ≤ (2*Real.pi)⁻¹ * complexShellEnergy ν (n+1) := by | |
| rw [complexConv_inverse,sourceInverseKernel_plancherel _ | |
| (memLp_one_iff_integrable.mp (complexMultiplier_memLp ν n 1)) (complexMultiplier_memLp ν n 2)] | |
| exact mul_le_mul_of_nonneg_left (complexMultiplier_energy_le ν n) (by positivity) | |
| noncomputable def complexBand1 (ν : ComplexMeasure ℝ) (n : ℕ) : Lp (α:=ℝ) ℂ 1 volume := | |
| (memLp_one_iff_integrable.mpr (complexConv_integrable ν n)).toLp (complexConv ν n) | |
| noncomputable def complexBand2 (ν : ComplexMeasure ℝ) (n : ℕ) : Lp (α:=ℝ) ℂ 2 volume := | |
| (complexConv_memLp_two ν n).toLp (complexConv ν n) | |
| lemma complexBand1_norm (ν : ComplexMeasure ℝ) [IsFiniteMeasure ν.variation] (n : ℕ) : | |
| ‖complexBand1 ν n‖ ≤ (2*ν.variation.real univ) * ∫ x, ‖sourceShellKernel 0 x‖ := by | |
| rw [complexBand1,Lp.norm_toLp, | |
| eLpNorm_one_eq_lintegral_enorm (complexConv_integrable ν n).aestronglyMeasurable, | |
| ←integral_norm_eq_lintegral_enorm (complexConv_integrable ν n).aestronglyMeasurable] | |
| exact complexConv_L1_bound ν n | |
| lemma complexBand2_norm (ν : ComplexMeasure ℝ) (n : ℕ) : | |
| ‖complexBand2 ν n‖ ≤ Real.sqrt ((2*Real.pi)⁻¹) * Real.sqrt (complexShellEnergy ν (n+1)) := by | |
| have he : ‖complexBand2 ν n‖^2 ≤ (2*Real.pi)⁻¹ * complexShellEnergy ν (n+1) := by | |
| rw [complexBand2,←integral_sq_norm_toLp] | |
| exact complexConv_energy ν n | |
| have hs := Real.sqrt_le_sqrt he | |
| simpa only [Real.sqrt_sq (norm_nonneg _),Real.sqrt_mul (by positivity : 0 ≤ (2*Real.pi)⁻¹)] using hs | |
| lemma complex_band_recombine (μ : FiniteMeasure ℝ) (ν : ComplexMeasure ℝ) [IsFiniteMeasure ν.variation] | |
| (n : ℕ) (φ : 𝓢(ℝ,ℂ)) : | |
| schwartzLpTest 1 φ (complexBand1 (complexOfPositive μ-ν) n) + | |
| schwartzLpTest 2 φ (complexBand2 ν n) = schwartzLpTest 1 φ (bandTerm n (μ : Measure ℝ)) := by | |
| rw [complexBand1,complexBand2,schwartzLpTest_toLp,schwartzLpTest_toLp,schwartz_bandTerm, | |
| complexConv_sub,complexConv_positive] | |
| have ht := (φ.memLp_top volume).integrable_mul | |
| (memLp_one_iff_integrable.mpr (shellKernelMeasureConvolution_L1 (μ : Measure ℝ) n).1) | |
| have hv := (φ.memLp_top volume).integrable_mul | |
| (memLp_one_iff_integrable.mpr (complexConv_integrable ν n)) | |
| change Integrable (fun x => φ x * kernelMeasureConvolution (sourceShellKernel n) (μ : Measure ℝ) x) at ht | |
| change Integrable (fun x => φ x * complexConv ν n x) at hv | |
| simp only [Pi.sub_apply,mul_sub] | |
| rw [integral_sub ht hv,sub_add_cancel] | |
| end CurrentFalconer.MonotoneShell | |
| end | |
| end | |
| end OAI | |