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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 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | 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
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