| 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 |
|
|
