| import OAI.MeasureTheory.Falconer.Measures.FrequencyMeasure |
| import OAI.MeasureTheory.Falconer.Slicing.GaussianLocalization |
|
|
| namespace OAI |
|
|
| /-! # Gaussian localization for the actual frequency-shell law -/ |
|
|
| namespace Falconer |
| open Set Metric MeasureTheory |
| open scoped ENNReal |
| variable {d : ℕ} |
|
|
| theorem measurableSet_radialFrequencyAnnulus (B : ℝ) : MeasurableSet (radialFrequencyAnnulus B) := |
| (isClosed_le continuous_const continuous_abs).measurableSet.inter |
| (isClosed_le continuous_abs continuous_const).measurableSet |
| |
| instance radialFrequencyMeasure_finite (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (B : ℝ) : |
| IsFiniteMeasure (radialFrequencyMeasure ν B) := by |
| have hsub : radialFrequencyAnnulus B ⊆ Icc (-2*|B|) (2*|B|) := by |
| intro t ht |
| have hab : |t| ≤ 2*|B| := ht.2.trans (by nlinarith [le_abs_self B]) |
| simpa only [mem_Icc,neg_mul] using abs_le.mp hab |
| have : IsFiniteMeasure (volume.restrict (radialFrequencyAnnulus B)) := |
| ⟨by |
| rw [Measure.restrict_apply_univ] |
| apply (measure_mono hsub).trans_lt |
| rw [Real.volume_Icc] |
| exact ENNReal.ofReal_lt_top⟩ |
| unfold radialFrequencyMeasure |
| infer_instance |
| |
| theorem radial_frequency_support (ν : Measure (Sphere d)) [IsFiniteMeasure ν] (B : ℝ) : |
| ∀ᵐ v ∂radialFrequencyMeasure ν B, B ≤ ‖v‖ ∧ ‖v‖ ≤ 2*B := by |
| have hm : Measurable (@radialFrequencyMap d) := by unfold radialFrequencyMap; fun_prop |
| rw [radialFrequencyMeasure,ae_map_iff (p := fun v : Space d => B ≤ ‖v‖ ∧ ‖v‖ ≤ 2*B) hm.aemeasurable |
| ((isClosed_le continuous_const continuous_norm).measurableSet.inter |
| (isClosed_le continuous_norm continuous_const).measurableSet)] |
| apply (Measure.ae_prod_iff_ae_ae (by measurability)).mpr |
| filter_upwards [self_mem_ae_restrict (measurableSet_radialFrequencyAnnulus B)] with t ht |
| change B ≤ |t| ∧ |t| ≤ 2*B at ht |
| exact Filter.Eventually.of_forall (fun w => by |
| simpa only [radialFrequencyMap,norm_smul,Real.norm_eq_abs,sphere_norm,mul_one] using ht) |
| |
| theorem radial_frequency_polynomial_balls (ν : Measure (Sphere d)) [IsFiniteMeasure ν] |
| {B R L s : ℝ} (hB : 1 ≤ B) (hBR : B ≤ R) (hL : 1 ≤ L) (hs : 0 ≤ s) |
| (hmass : ν univ ≤ 1) |
| (hball : ∀ x : Space d, ∀ r : ℝ, 1/R ≤ r → r ≤ 1 → |
| ν {w : Sphere d | dist (w:Space d) x < r} ≤ ENNReal.ofReal (L*r^s)) : |
| ∀ x : Space d, ∀ r : ℝ, 1 ≤ r → |
| radialFrequencyMeasure ν B (ball x r) ≤ |
| ENNReal.ofReal ((8*(2:ℝ)^s*L/B^s)*r^(s+1)) := by |
| have hB₀ : 0 < B := zero_lt_one.trans_le hB |
| have hR₀ : 0 < R := hB₀.trans_le hBR |
| intro x r hr |
| have hr₀ : 0 < r := zero_lt_one.trans_le hr |
| have hscale : 1/R ≤ 2*r/B := by |
| have h₁ : 1/R ≤ 1/B := one_div_le_one_div_of_le hB₀ hBR |
| exact h₁.trans ((div_le_div_iff_of_pos_right hB₀).mpr (by linarith)) |
| have h := radial_frequency_ball_bound ν hB₀ hr₀ hL hs hmass hscale hball x |
| have he : 8*r*L*(2*r/B)^s = (8*(2:ℝ)^s*L/B^s)*r^(s+1) := by |
| rw [Real.div_rpow (by positivity) hB₀.le, |
| Real.mul_rpow (by norm_num : (0:ℝ) ≤ 2) hr₀.le, |
| Real.rpow_add hr₀,Real.rpow_one] |
| ring |
| simpa only [he] using h |
| |
| theorem radial_frequency_gaussian (ν : Measure (Sphere d)) [IsFiniteMeasure ν] |
| {B R L s : ℝ} {k : ℕ} (hB : 1 ≤ B) (hBR : B ≤ R) (hL : 1 ≤ L) |
| (hs : 0 ≤ s) (hsk : s+1 ≤ k) (hmass : ν univ ≤ 1) |
| (hball : ∀ x : Space d, ∀ r : ℝ, 1/R ≤ r → r ≤ 1 → |
| ν {w : Sphere d | dist (w:Space d) x < r} ≤ ENNReal.ofReal (L*r^s)) |
| (u : Space d) : |
| (∫ v, Real.exp (-2*(dist u v)^2) ∂radialFrequencyMeasure ν B) ≤ |
| Real.exp 1*(8*(2:ℝ)^s*L/B^s)*gaussianBallConstant k* |
| Real.exp (-‖u‖^2/(16*B^2)) := by |
| exact gaussian_localized_mass (radialFrequencyMeasure ν B) hB |
| (by have : 0 < B := zero_lt_one.trans_le hB; positivity) hsk |
| ((radial_frequency_support ν B).mono (fun _ h => h.2)) |
| (radial_frequency_polynomial_balls ν hB hBR hL hs hmass hball) u |
| |
| end Falconer |
| |
| end OAI |
| |
