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b17ead8 | 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 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 | import OAI.MeasureTheory.Falconer.Fourier.FourierEnergy
import Mathlib.Analysis.SpecialFunctions.Pow.Integral
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic
namespace OAI
namespace PlanarFalconer
open Set Metric MeasureTheory
open scoped ENNReal
noncomputable def sineKernel (v θ : ℝ) : ℝ≥0∞ :=
(ENNReal.ofReal (|Real.sin θ|))^(-v)
theorem sine_power_domination {v x : ℝ} (hv : 0 < v) (hx : |x| ≤ Real.pi/2) :
‖|Real.sin x|^(-v)‖ ≤ (2/Real.pi)^(-v) * |x|^(-v) := by
by_cases hz : x = 0
· subst x
simp [Real.zero_rpow (neg_ne_zero.mpr hv.ne')]
· have hxpos : 0 < |x| := abs_pos.mpr hz
have hc : 0 < 2/Real.pi := by positivity
have hs := Real.mul_abs_le_abs_sin hx
rw [Real.norm_eq_abs,abs_of_nonneg (Real.rpow_nonneg (abs_nonneg _) _)]
calc
|Real.sin x|^(-v) ≤ ((2/Real.pi)*|x|)^(-v) :=
Real.rpow_le_rpow_of_nonpos (mul_pos hc hxpos) hs (by linarith)
_ = _ := Real.mul_rpow hc.le hxpos.le
theorem intervalIntegrable_sine_power {v : ℝ} (hv : 0 < v) (hv1 : v < 1)
(a b : ℝ) : IntervalIntegrable (fun θ : ℝ => |Real.sin θ|^(-v)) volume a b := by
have hi : IntegrableOn (fun θ : ℝ => |Real.sin θ|^(-v)) (ball 0 (Real.pi/2)) := by
apply integrableOn_ball_of_norm_le_rpow (by simp) (by simpa using hv1)
(C := (2/Real.pi)^(-v))
· filter_upwards [self_mem_ae_restrict measurableSet_ball] with x hx
rw [mem_ball_zero_iff] at hx
simpa only [Real.norm_eq_abs] using sine_power_domination hv hx.le
· have hm : Measurable (fun θ : ℝ => |Real.sin θ|^(-v)) := by fun_prop
exact hm.aestronglyMeasurable
have hj : IntervalIntegrable (fun θ : ℝ => |Real.sin θ|^(-v)) volume
(-Real.pi/2) (Real.pi/2) := by
rw [intervalIntegrable_iff_integrableOn_Ioo_of_le (by linarith [Real.pi_pos])]
simpa only [Real.ball_eq_Ioo, zero_sub, zero_add, neg_div] using hi
have hp : Function.Periodic (fun θ : ℝ => |Real.sin θ|^(-v)) Real.pi := by
intro θ
simp
apply hp.intervalIntegrable Real.pi_ne_zero (t := -Real.pi/2) _ a b
simpa only [show -Real.pi / 2 + Real.pi = Real.pi / 2 by ring] using hj
theorem sine_ne_zero_ae : ∀ᵐ θ : ℝ ∂volume, Real.sin θ ≠ 0 := by
have he : {θ : ℝ | Real.sin θ = 0} = range (fun n : ℤ => (n : ℝ)*Real.pi) := by
ext θ
simp only [mem_ofPred_eq,mem_range,Real.sin_eq_zero_iff]
rw [ae_iff]
simp only [not_not]
change volume {θ : ℝ | Real.sin θ = 0} = 0
rw [he]
exact (countable_range _).measure_zero volume
theorem sineKernel_lintegral_eq {v : ℝ} (hv : 0 < v) (hv1 : v < 1) (a b : ℝ) :
(∫⁻ θ : ℝ in Ioc a b, sineKernel v θ) =
ENNReal.ofReal (∫ θ : ℝ in Ioc a b, |Real.sin θ|^(-v)) := by
rw [ofReal_integral_eq_lintegral_ofReal (intervalIntegrable_sine_power hv hv1 a b).1
(Filter.Eventually.of_forall (fun θ => Real.rpow_nonneg (abs_nonneg _) _))]
apply lintegral_congr_ae
filter_upwards [ae_restrict_of_ae sine_ne_zero_ae] with θ hθ
exact ENNReal.ofReal_rpow_of_pos (abs_pos.mpr hθ)
theorem sineKernel_lintegral_lt_top {v : ℝ} (hv : 0 < v) (hv1 : v < 1) (a b : ℝ) :
(∫⁻ θ : ℝ in Ioc a b, sineKernel v θ) < ∞ := by
rw [sineKernel_lintegral_eq hv hv1]
exact ENNReal.ofReal_lt_top
theorem one_le_sineKernel {v : ℝ} (hv : 0 ≤ v) (θ : ℝ) : 1 ≤ sineKernel v θ := by
have hbase : ENNReal.ofReal |Real.sin θ| ≤ 1 := by
exact ENNReal.ofReal_le_one.mpr (Real.abs_sin_le_one θ)
simpa only [sineKernel,ENNReal.rpow_zero] using
ENNReal.rpow_le_rpow_of_exponent_ge hbase (show -v ≤ 0 by linarith)
noncomputable def angularEnergyConstant (v : ℝ) : ℝ≥0∞ :=
∫⁻ θ : ℝ in Ioc 0 (2*Real.pi), sineKernel v θ
theorem angularEnergyConstant_pos {v : ℝ} (hv : 0 ≤ v) :
0 < angularEnergyConstant v := by
have h := lintegral_mono (μ := volume.restrict (Ioc 0 (2*Real.pi))) (one_le_sineKernel hv)
simp only [lintegral_const,one_mul,Measure.restrict_apply MeasurableSet.univ,
univ_inter,Real.volume_Ioc,sub_zero] at h
exact (ENNReal.ofReal_pos.mpr (mul_pos (by norm_num) Real.pi_pos)).trans_le h
theorem angularEnergyConstant_lt_top {v : ℝ} (hv : 0 < v) (hv1 : v < 1) :
angularEnergyConstant v < ∞ := sineKernel_lintegral_lt_top hv hv1 _ _
theorem sine_shift_ne_zero_ae (c : ℝ) :
∀ᵐ θ : ℝ ∂volume, Real.sin (θ-c) ≠ 0 := by
rw [ae_iff]
simp only [not_not]
apply measure_mono_null (t := range (fun n : ℤ => (n : ℝ)*Real.pi+c))
· intro θ hθ
obtain ⟨n,hn⟩ := Real.sin_eq_zero_iff.mp hθ
exact ⟨n,by linarith⟩
· exact (countable_range _).measure_zero volume
theorem sineKernel_shift_integral {v : ℝ} (hv : 0 < v) (hv1 : v < 1) (c : ℝ) :
(∫⁻ θ : ℝ in Ioc 0 (2*Real.pi), sineKernel v (θ-c)) = angularEnergyConstant v := by
let f : ℝ → ℝ := fun θ => |Real.sin θ|^(-v)
have hp : Function.Periodic f (2*Real.pi) := by intro θ; simp [f]
have hi : IntervalIntegrable (fun θ => f (θ-c)) volume 0 (2*Real.pi) := by
simpa only [f, neg_add_cancel, sub_add_cancel] using
(intervalIntegrable_sine_power hv hv1 (-c) (2*Real.pi-c)).comp_sub_right c
have hnon : ∀ θ, 0 ≤ f θ := fun θ => Real.rpow_nonneg (abs_nonneg _) _
have he : (∫⁻ θ : ℝ in Ioc 0 (2*Real.pi), sineKernel v (θ-c)) =
ENNReal.ofReal (∫ θ : ℝ in Ioc 0 (2*Real.pi), f (θ-c)) := by
rw [ofReal_integral_eq_lintegral_ofReal hi.1 (Filter.Eventually.of_forall (fun θ => hnon (θ-c)))]
apply lintegral_congr_ae
filter_upwards [ae_restrict_of_ae (sine_shift_ne_zero_ae c)] with θ hθ
exact ENNReal.ofReal_rpow_of_pos (abs_pos.mpr hθ)
rw [he,angularEnergyConstant,sineKernel_lintegral_eq hv hv1]
congr 1
rw [← intervalIntegral.integral_of_le (by positivity : (0 : ℝ) ≤ 2*Real.pi),
intervalIntegral.integral_comp_sub_right]
rw [zero_sub]
change (∫ x in -c..2*Real.pi-c, f x) = ∫ x in Ioc 0 (2*Real.pi), f x
rw [show 2*Real.pi-c = -c+2*Real.pi by ring,
hp.intervalIntegral_add_eq (-c) 0,zero_add,
intervalIntegral.integral_of_le (by positivity : (0 : ℝ) ≤ 2*Real.pi)]
end PlanarFalconer
end OAI
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