| import OAI.MeasureTheory.Falconer.Shells.FrequencyShell |
|
|
| namespace OAI |
|
|
| /-! # Covering the actual radial interval by its dyadic frequency annuli -/ |
|
|
| namespace Falconer |
| open Set MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| noncomputable def radialIntervalMeasure (ν : Measure (Sphere d)) (H : ℝ) : Measure (Space d) := |
| Measure.map radialFrequencyMap ((volume.restrict (Icc (-H) H)).prod ν) |
| |
| instance radialIntervalMeasure_finite (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (H : ℝ) : |
| IsFiniteMeasure (radialIntervalMeasure ν H) := by |
| unfold radialIntervalMeasure |
| infer_instance |
| |
| theorem radial_interval_cover_step (n : ℕ) : |
| Icc (-((2:ℝ)^(n+1))) ((2:ℝ)^(n+1)) ⊆ |
| Icc (-((2:ℝ)^n)) ((2:ℝ)^n) ∪ radialFrequencyAnnulus ((2:ℝ)^n) := by |
| intro t ht |
| have hh : |t| ≤ (2:ℝ)^(n+1) := abs_le.mpr ht |
| by_cases hs : |t| ≤ (2:ℝ)^n |
| · exact Or.inl (abs_le.mp hs) |
| · right |
| refine ⟨(le_of_not_ge hs),?_⟩ |
| simpa only [pow_succ,mul_comm] using hh |
| |
| theorem radial_interval_measure_step (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (n : ℕ) : |
| radialIntervalMeasure ν ((2:ℝ)^(n+1)) ≤ |
| radialIntervalMeasure ν ((2:ℝ)^n)+radialFrequencyMeasure ν ((2:ℝ)^n) := by |
| have hvol : volume.restrict (Icc (-((2:ℝ)^(n+1))) ((2:ℝ)^(n+1))) ≤ |
| volume.restrict (Icc (-((2:ℝ)^n)) ((2:ℝ)^n))+ |
| volume.restrict (radialFrequencyAnnulus ((2:ℝ)^n)) := |
| (Measure.restrict_mono (radial_interval_cover_step n) le_rfl).trans (Measure.restrict_union_le _ _) |
| have hm : Measurable (@radialFrequencyMap d) := by unfold radialFrequencyMap; fun_prop |
| have h := Measure.map_mono (Measure.prod_mono hvol (le_refl ν)) hm |
| simpa only [radialIntervalMeasure,radialFrequencyMeasure,Measure.add_prod, |
| Measure.map_add _ _ hm] using h |
| |
| theorem radial_interval_dyadic_cover (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (n : ℕ) : |
| radialIntervalMeasure ν ((2:ℝ)^n) ≤ radialIntervalMeasure ν 1+ |
| ∑ j ∈ Finset.range n, radialFrequencyMeasure ν ((2:ℝ)^j) := by |
| induction n with |
| | zero => simp |
| | succ n ih => |
| calc |
| _ ≤ radialIntervalMeasure ν ((2:ℝ)^n)+radialFrequencyMeasure ν ((2:ℝ)^n) := |
| radial_interval_measure_step ν n |
| _ ≤ (radialIntervalMeasure ν 1+∑ j ∈ Finset.range n, radialFrequencyMeasure ν ((2:ℝ)^j))+ |
| radialFrequencyMeasure ν ((2:ℝ)^n) := add_le_add ih le_rfl |
| _ = _ := by rw [Finset.sum_range_succ,add_assoc] |
| |
| end Falconer |
| |
| end OAI |
| |
