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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 | 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
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