| import Mathlib | |
| namespace OAI | |
| /-! Radial pushforward estimates for convex phases. -/ | |
| open MeasureTheory | |
| open scoped NNReal ENNReal ContDiff | |
| noncomputable section | |
| open Filter | |
| open scoped Topology ContDiff | |
| open MeasureTheory Set | |
| open scoped ContDiff FourierTransform InnerProductSpace ENNReal | |
| open MeasureTheory Set Filter Metric | |
| open scoped ContDiff Topology | |
| open Set Filter | |
| open scoped ENNReal InnerProductSpace | |
| open scoped FourierTransform SchwartzMap ENNReal | |
| open scoped ENNReal FourierTransform InnerProductSpace | |
| open MeasureTheory Set Filter | |
| open scoped ContDiff Topology ENNReal | |
| namespace DiagonalExtension.ConvexRadial | |
| abbrev E2 := EuclideanSpace ℝ (Fin 2) | |
| def symPhase (φ : E2 → ℝ) (m dir : E2) (r : ℝ) : ℝ := | |
| φ (m + r • dir) + φ (m - r • dir) | |
| def symPhaseDeriv (φ : E2 → ℝ) (m dir : E2) (r : ℝ) : ℝ := | |
| (fderiv ℝ φ (m + r • dir)) dir - (fderiv ℝ φ (m - r • dir)) dir | |
| lemma hasDerivAt_line (m dir : E2) (r : ℝ) : | |
| HasDerivAt (fun s : ℝ => m + s • dir) dir r := by | |
| simpa only [one_smul, id_eq] using! | |
| ((hasDerivAt_id r).smul_const dir).const_add m | |
| lemma hasDerivAt_line_sub (m dir : E2) (r : ℝ) : | |
| HasDerivAt (fun s : ℝ => m - s • dir) (-dir) r := by | |
| simpa only [one_smul, id_eq, sub_eq_add_neg] using! | |
| ((hasDerivAt_id r).smul_const dir).neg.const_add m | |
| lemma hasDerivAt_symPhase {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| (m dir : E2) (r : ℝ) : | |
| HasDerivAt (symPhase φ m dir) (symPhaseDeriv φ m dir r) r := by | |
| have hd := hφ.differentiable (by simp) | |
| simpa only [symPhase, symPhaseDeriv, map_neg, sub_eq_add_neg, Pi.add_apply, Function.comp_apply] using! | |
| ((hd (m + r • dir)).hasFDerivAt.comp_hasDerivAt r (hasDerivAt_line m dir r)).add | |
| ((hd (m - r • dir)).hasFDerivAt.comp_hasDerivAt r (hasDerivAt_line_sub m dir r)) | |
| lemma hasDerivAt_directional {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| (m dir : E2) (r : ℝ) : | |
| HasDerivAt (fun s : ℝ => (fderiv ℝ φ (m + s • dir)) dir) | |
| ((fderiv ℝ (fderiv ℝ φ) (m + r • dir) dir) dir) r := by | |
| have hd : ContDiff ℝ 1 (fderiv ℝ φ) := hφ.fderiv_right (by exact WithTop.coe_le_coe.mpr (show (1 : ℕ∞) + 1 ≤ ⊤ from le_top)) | |
| have hdf : Differentiable ℝ (fderiv ℝ φ) := hd.differentiable (by norm_num) | |
| have H₀ : HasFDerivAt (fderiv ℝ φ) (fderiv ℝ (fderiv ℝ φ) (m + r • dir)) | |
| (m + r • dir) := (hdf (m + r • dir)).hasFDerivAt | |
| have H₁ : HasDerivAt (fun s : ℝ => fderiv ℝ φ (m + s • dir)) | |
| (fderiv ℝ (fderiv ℝ φ) (m + r • dir) dir) r := | |
| H₀.comp_hasDerivAt r (hasDerivAt_line m dir r) | |
| simpa only [map_zero, add_zero] using H₁.clm_apply (hasDerivAt_const r dir) | |
| lemma radial_deriv_lower {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) | |
| (m : E2) {dir : E2} (hdir : ‖dir‖ = 1) {r : ℝ} (hr : 0 ≤ r) : | |
| 2 * c * r ≤ symPhaseDeriv φ m dir r := by | |
| let F : ℝ → ℝ := fun s => (fderiv ℝ φ (m + s • dir)) dir | |
| have hF (s : ℝ) : HasDerivAt F | |
| ((fderiv ℝ (fderiv ℝ φ) (m + s • dir) dir) dir) s := | |
| hasDerivAt_directional hφ m dir s | |
| have hc (s : ℝ) : c ≤ deriv F s := by | |
| rw [(hF s).deriv] | |
| simpa only [hdir, one_pow, mul_one] using hell (m + s • dir) dir | |
| have H := mul_sub_le_image_sub_of_le_deriv (fun s => (hF s).differentiableAt) | |
| hc (show -r ≤ r by linarith) | |
| simpa only [F, neg_smul, ← sub_eq_add_neg, symPhaseDeriv, sub_neg_eq_add, | |
| ← two_mul, mul_comm c, mul_assoc] using H | |
| lemma symPhase_continuous {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) (m dir : E2) : | |
| Continuous (symPhase φ m dir) := | |
| continuous_iff_continuousAt.mpr (fun r => (hasDerivAt_symPhase hφ m dir r).continuousAt) | |
| lemma symPhase_strictMonoOn {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) | |
| (m : E2) {dir : E2} (hdir : ‖dir‖ = 1) : | |
| StrictMonoOn (symPhase φ m dir) (Ici 0) := by | |
| apply strictMonoOn_of_deriv_pos (convex_Ici (0 : ℝ)) | |
| (symPhase_continuous hφ m dir).continuousOn | |
| intro r hr | |
| rw [interior_Ici, mem_Ioi] at hr | |
| rw [(hasDerivAt_symPhase hφ m dir r).deriv] | |
| exact (mul_pos (mul_pos (by norm_num) hc) hr).trans_le | |
| (radial_deriv_lower hφ hell m hdir hr.le) | |
| lemma radial_lintegral_le {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) | |
| (m : E2) {dir : E2} (hdir : ‖dir‖ = 1) (g : ℝ → ℝ≥0∞) : | |
| ∫⁻ r in Ioi (0 : ℝ), ENNReal.ofReal r * g (symPhase φ m dir r) ≤ | |
| ENNReal.ofReal ((2 * c)⁻¹) * ∫⁻ s : ℝ, g s := by | |
| have hc2 : 0 < 2 * c := mul_pos (by norm_num) hc | |
| have hmono := (symPhase_strictMonoOn hφ hc hell m hdir).mono Ioi_subset_Ici_self | |
| have hchange := lintegral_image_eq_lintegral_abs_deriv_mul measurableSet_Ioi | |
| (fun r (_ : r ∈ Ioi (0 : ℝ)) => (hasDerivAt_symPhase hφ m dir r).hasDerivWithinAt) | |
| hmono.injOn g | |
| have hcoe (r : ℝ) (hr : r ∈ Ioi (0 : ℝ)) : | |
| ENNReal.ofReal r ≤ ENNReal.ofReal ((2 * c)⁻¹) * | |
| ENNReal.ofReal |symPhaseDeriv φ m dir r| := by | |
| rw [← ENNReal.ofReal_mul (inv_nonneg.mpr hc2.le)] | |
| apply ENNReal.ofReal_le_ofReal | |
| calc | |
| r = (2 * c)⁻¹ * (2 * c * r) := by field_simp | |
| _ ≤ (2 * c)⁻¹ * |symPhaseDeriv φ m dir r| := by | |
| apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hc2.le) | |
| exact (radial_deriv_lower hφ hell m hdir hr.le).trans (le_abs_self _) | |
| calc | |
| _ ≤ ∫⁻ r in Ioi (0 : ℝ), ENNReal.ofReal ((2 * c)⁻¹) * | |
| (ENNReal.ofReal |symPhaseDeriv φ m dir r| * g (symPhase φ m dir r)) := by | |
| apply lintegral_mono_ae | |
| filter_upwards [ae_restrict_mem measurableSet_Ioi] with r hr | |
| rw [← mul_assoc] | |
| exact mul_le_mul_left (hcoe r hr) _ | |
| _ = ENNReal.ofReal ((2 * c)⁻¹) * | |
| ∫⁻ s in symPhase φ m dir '' Ioi (0 : ℝ), g s := by | |
| rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top, ← hchange] | |
| _ ≤ _ := by gcongr; exact Measure.restrict_le_self | |
| def pairEquiv : (ℝ × ℝ) ≃ᵐ E2 := | |
| MeasurableEquiv.finTwoArrow.symm.trans (MeasurableEquiv.toLp 2 (Fin 2 → ℝ)) | |
| lemma pairEquiv_measurePreserving : MeasurePreserving pairEquiv volume volume := by | |
| exact (PiLp.volume_preserving_toLp (Fin 2)).comp | |
| (volume_preserving_finTwoArrow ℝ).symm | |
| lemma pairEquiv_apply (p : ℝ × ℝ) : pairEquiv p = WithLp.toLp 2 ![p.1, p.2] := rfl | |
| lemma pairEquiv_continuous : Continuous pairEquiv := by | |
| change Continuous (fun p : ℝ × ℝ => WithLp.toLp 2 ![p.1, p.2]) | |
| exact (PiLp.continuous_toLp 2 (fun _ : Fin 2 => ℝ)).comp (by fun_prop) | |
| def direction (θ : ℝ) : E2 := pairEquiv (Real.cos θ, Real.sin θ) | |
| lemma direction_norm (θ : ℝ) : ‖direction θ‖ = 1 := by | |
| have h : ‖direction θ‖ ^ 2 = 1 := by | |
| simp only [EuclideanSpace.real_norm_sq_eq, direction, pairEquiv_apply, | |
| Fin.sum_univ_two, Matrix.cons_val_zero, Matrix.cons_val_one] | |
| nlinarith [Real.sin_sq_add_cos_sq θ] | |
| nlinarith [norm_nonneg (direction θ)] | |
| lemma pairEquiv_polar (r θ : ℝ) : pairEquiv (polarCoord.symm (r, θ)) = r • direction θ := by | |
| ext i | |
| fin_cases i <;> simp [polarCoord_symm_apply, pairEquiv_apply, direction] | |
| lemma polar_lintegral_le {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) | |
| (m : E2) {g : ℝ → ℝ≥0∞} (hg : Measurable g) : | |
| ∫⁻ w : E2, g (φ (m + w) + φ (m - w)) ≤ | |
| ENNReal.ofReal (Real.pi / c) * ∫⁻ s : ℝ, g s := by | |
| let f : ℝ × ℝ → ℝ≥0∞ := fun p => g (φ (m + pairEquiv p) + φ (m - pairEquiv p)) | |
| have hf : Measurable f := hg.comp <| Continuous.measurable <| | |
| (hφ.continuous.comp (continuous_const.add pairEquiv_continuous)).add | |
| (hφ.continuous.comp (continuous_const.sub pairEquiv_continuous)) | |
| have hcomp : (∫⁻ w : E2, g (φ (m + w) + φ (m - w))) = ∫⁻ p, f p := | |
| (pairEquiv_measurePreserving.lintegral_comp_emb pairEquiv.measurableEmbedding _).symm | |
| rw [hcomp, ← lintegral_comp_polarCoord_symm f] | |
| have hmeas : Measurable (fun p : ℝ × ℝ => ENNReal.ofReal p.1 * f (polarCoord.symm p)) := by | |
| apply measurable_fst.ennreal_ofReal.mul | |
| apply hf.comp | |
| change Measurable (fun p : ℝ × ℝ => (p.1 * Real.cos p.2, p.1 * Real.sin p.2)) | |
| fun_prop | |
| rw [polarCoord_target] | |
| change (∫⁻ p : ℝ × ℝ, ENNReal.ofReal p.1 * f (polarCoord.symm p) | |
| ∂(volume.prod volume).restrict (Ioi (0 : ℝ) ×ˢ Ioo (-Real.pi) Real.pi)) ≤ _ | |
| rw [← Measure.prod_restrict] | |
| rw [lintegral_prod_symm _ hmeas.aemeasurable] | |
| change (∫⁻ θ in Ioo (-Real.pi) Real.pi, | |
| ∫⁻ r in Ioi (0 : ℝ), ENNReal.ofReal r * f (polarCoord.symm (r, θ))) ≤ _ | |
| calc | |
| _ = ∫⁻ θ in Ioo (-Real.pi) Real.pi, ∫⁻ r in Ioi (0 : ℝ), | |
| ENNReal.ofReal r * g (symPhase φ m (direction θ) r) := by | |
| simp only [f, pairEquiv_polar, symPhase] | |
| _ ≤ ∫⁻ _θ in Ioo (-Real.pi) Real.pi, | |
| ENNReal.ofReal ((2 * c)⁻¹) * ∫⁻ s : ℝ, g s := by | |
| apply lintegral_mono | |
| intro θ | |
| exact radial_lintegral_le hφ hc hell m (direction_norm θ) g | |
| _ = _ := by | |
| rw [lintegral_const, Measure.restrict_apply_univ, Real.volume_Ioo] | |
| rw [mul_comm _ (ENNReal.ofReal _), ← mul_assoc, | |
| ← ENNReal.ofReal_mul (by linarith [Real.pi_pos] : 0 ≤ Real.pi - -Real.pi)] | |
| congr 1 | |
| congr 1 | |
| field_simp | |
| ring | |
| lemma pair_graph_lintegral_le {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) | |
| {G : E2 × ℝ → ℝ≥0∞} (hG : Measurable G) : | |
| ∫⁻ ξ : E2, ∫⁻ η : E2, G (ξ + η, φ ξ + φ η) ≤ | |
| ENNReal.ofReal (Real.pi / c) * ∫⁻ u : E2 × ℝ, G u := by | |
| have hswap : AEMeasurable | |
| (Function.uncurry (fun ξ η : E2 => G (ξ + η, φ ξ + φ η))) | |
| (volume.prod volume) := by | |
| apply Measurable.aemeasurable | |
| exact hG.comp ((continuous_fst.add continuous_snd).prodMk | |
| ((hφ.continuous.comp continuous_fst).add (hφ.continuous.comp continuous_snd))).measurable | |
| have hswap₂ : AEMeasurable | |
| (Function.uncurry (fun η m : E2 => G (m, φ (m - η) + φ η))) | |
| (volume.prod volume) := by | |
| apply Measurable.aemeasurable | |
| exact hG.comp (continuous_snd.prodMk | |
| ((hφ.continuous.comp (continuous_snd.sub continuous_fst)).add | |
| (hφ.continuous.comp continuous_fst))).measurable | |
| calc | |
| _ = ∫⁻ η : E2, ∫⁻ ξ : E2, G (ξ + η, φ ξ + φ η) := lintegral_lintegral_swap hswap | |
| _ = ∫⁻ η : E2, ∫⁻ m : E2, G (m, φ (m - η) + φ η) := by | |
| apply lintegral_congr | |
| intro η | |
| have hm : Measurable (fun m : E2 => G (m, φ (m - η) + φ η)) := by | |
| apply hG.comp | |
| exact (continuous_id.prodMk | |
| ((hφ.continuous.comp (continuous_id.sub continuous_const)).add | |
| continuous_const)).measurable | |
| have H := (measurePreserving_add_right (volume : Measure E2) η).lintegral_comp hm | |
| simpa only [add_sub_cancel_right] using H | |
| _ = ∫⁻ m : E2, ∫⁻ η : E2, G (m, φ (m - η) + φ η) := lintegral_lintegral_swap hswap₂ | |
| _ = ∫⁻ m : E2, ∫⁻ w : E2, | |
| G (m, φ ((2 : ℝ)⁻¹ • m + w) + φ ((2 : ℝ)⁻¹ • m - w)) := by | |
| apply lintegral_congr | |
| intro m | |
| have hm : Measurable (fun η : E2 => G (m, φ (m - η) + φ η)) := by | |
| apply hG.comp | |
| exact (continuous_const.prodMk | |
| ((hφ.continuous.comp (continuous_const.sub continuous_id)).add hφ.continuous)).measurable | |
| have H := ((measurePreserving_add_left (volume : Measure E2) ((2 : ℝ)⁻¹ • m)).comp | |
| (Measure.measurePreserving_neg volume)).lintegral_comp hm | |
| symm | |
| convert H using 1 | |
| congr 1 | |
| ext w | |
| dsimp only [Function.comp_apply] | |
| have he : m - ((2 : ℝ)⁻¹ • m + -w) = (2 : ℝ)⁻¹ • m + w := by module | |
| simp only [he, sub_eq_add_neg] | |
| _ ≤ ∫⁻ m : E2, ENNReal.ofReal (Real.pi / c) * ∫⁻ s : ℝ, G (m, s) := by | |
| apply lintegral_mono | |
| intro m | |
| exact polar_lintegral_le hφ hc hell ((2 : ℝ)⁻¹ • m) | |
| (hG.comp (measurable_const.prodMk measurable_id)) | |
| _ = _ := by | |
| rw [lintegral_const_mul' _ _ ENNReal.ofReal_ne_top] | |
| congr 1 | |
| exact (lintegral_prod G hG.aemeasurable).symm | |
| def pairGraph (φ : E2 → ℝ) (z : E2 × E2) : E2 × ℝ := | |
| (z.1 + z.2, φ z.1 + φ z.2) | |
| lemma pairGraph_continuous {φ : E2 → ℝ} (hφ : Continuous φ) : Continuous (pairGraph φ) := | |
| (continuous_fst.add continuous_snd).prodMk | |
| ((hφ.comp continuous_fst).add (hφ.comp continuous_snd)) | |
| lemma pairGraph_map_le {φ : E2 → ℝ} (hφ : ContDiff ℝ ∞ φ) | |
| {c : ℝ} (hc : 0 < c) (hell : ∀ x v : E2, | |
| c * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ φ) x v) v) : | |
| Measure.map (pairGraph φ) (volume.prod volume) ≤ | |
| ENNReal.ofReal (Real.pi / c) • (volume : Measure (E2 × ℝ)) := by | |
| refine Measure.le_iff.mpr fun s hs => ?_ | |
| let G : E2 × ℝ → ℝ≥0∞ := s.indicator (fun _ => 1) | |
| have hG : Measurable G := measurable_const.indicator hs | |
| have H := pair_graph_lintegral_le hφ hc hell hG | |
| have heq : (∫⁻ ξ : E2, ∫⁻ η : E2, G (ξ + η, φ ξ + φ η)) = | |
| Measure.map (pairGraph φ) (volume.prod volume) s := by | |
| change (∫⁻ ξ : E2, ∫⁻ η : E2, G (pairGraph φ (ξ, η))) = _ | |
| rw [← lintegral_prod (μ := volume) (ν := volume) (fun z : E2 × E2 => G (pairGraph φ z)) | |
| (hG.comp (pairGraph_continuous hφ.continuous).measurable).aemeasurable] | |
| rw [← lintegral_map hG (pairGraph_continuous hφ.continuous).measurable] | |
| simp [G, hs] | |
| rw [heq] at H | |
| simpa only [G, lintegral_indicator hs, lintegral_const, Measure.restrict_apply_univ, | |
| one_mul, Measure.smul_apply, smul_eq_mul] using H | |
| end DiagonalExtension.ConvexRadial | |
| end | |
| end OAI | |