| import OAI.Dynamics.TriangleBilliards.SmoothingLp |
|
|
| namespace OAI |
|
|
| universe uAlpha uE |
|
|
| open MeasureTheory Set |
| open scoped ENNReal symmDiff |
| noncomputable section |
| open MeasureTheory Set Filter Function Metric |
| open scoped Topology Convolution ContDiff |
| noncomputable section |
| open MeasureTheory Set |
| open scoped ENNReal |
| noncomputable section |
|
|
| namespace TriangularBilliards.SpatialSmoothing |
| open Filter |
| open scoped Topology BoundedContinuousFunction |
|
|
| lemma rho_fderiv_neg (x : ℂ) : fderiv ℝ rho (-x) = -fderiv ℝ rho x := by |
| have h := (rho_contDiff.differentiable (by simp) (-x)).hasFDerivAt.comp x |
| (hasFDerivAt_id x).neg |
| have he : (fun z : ℂ => rho (-z)) = rho := funext rho_neg |
| change HasFDerivAt (fun z : ℂ => rho (-z)) _ x at h |
| rw [he] at h |
| have hh := h.fderiv |
| simp only [ContinuousLinearMap.comp_neg, ContinuousLinearMap.comp_id] at hh |
| rw [hh] |
| simp |
|
|
| lemma rho_fderiv_integral : (∫ x : ℂ, fderiv ℝ rho x) = 0 := by |
| have h := integral_neg_eq_self (fun x : ℂ => fderiv ℝ rho x) volume |
| simp_rw [rho_fderiv_neg] at h |
| rw [integral_neg] at h |
| ext v |
| have hv := congrArg (fun L : ℂ →L[ℝ] ℝ => L v) h |
| simp only [neg_apply, zero_apply] at hv ⊢ |
| linarith |
|
|
| lemma rho_smulRight_integral (c : ℂ) : |
| (∫ x : ℂ, (fderiv ℝ rho x).smulRight c) = 0 := by |
| have h := ((ContinuousLinearMap.smulRightL ℝ ℂ ℂ).flip c).integral_comp_comm |
| rho_fderiv_integrable |
| change (∫ x : ℂ, (fderiv ℝ rho x).smulRight c) = |
| (∫ x : ℂ, fderiv ℝ rho x).smulRight c at h |
| simpa only [rho_fderiv_integral, ContinuousLinearMap.zero_smulRight] using h |
|
|
| lemma rescaled_gradient_continuous (f : ℂ →ᵇ ℂ) (x : ℂ) (ε : ℝ) : |
| Continuous (fun y : ℂ => (fderiv ℝ rho y).smulRight (f (x - ε • y))) := by |
| let L : (ℂ →L[ℝ] ℝ) →L[ℝ] ℂ →L[ℝ] ℂ →L[ℝ] ℂ := |
| ContinuousLinearMap.smulRightL ℝ ℂ ℂ |
| have h₁ : Continuous (fun y : ℂ => L (fderiv ℝ rho y)) := |
| L.continuous.comp (rho_contDiff.continuous_fderiv (by simp)) |
| have h₂ : Continuous (fun y : ℂ => f (x - ε • y)) := by fun_prop |
| exact h₁.clm_apply h₂ |
|
|
| lemma rescaled_gradient_dominated (f : ℂ →ᵇ ℂ) (x : ℂ) (ε : ℝ) (y : ℂ) : |
| ‖(fderiv ℝ rho y).smulRight (f (x - ε • y))‖ ≤ ‖fderiv ℝ rho y‖ * ‖f‖ := by |
| rw [ContinuousLinearMap.norm_smulRight_apply] |
| exact mul_le_mul_of_nonneg_left (f.norm_coe_le_norm _) (norm_nonneg _) |
|
|
| lemma rescaled_gradient_pointwise (f : ℂ →ᵇ ℂ) (x y : ℂ) : |
| Tendsto (fun ε : ℝ => (fderiv ℝ rho y).smulRight (f (x - ε • y))) |
| (𝓝 0) (𝓝 ((fderiv ℝ rho y).smulRight (f x))) := by |
| let L : ℂ →L[ℝ] ℂ →L[ℝ] ℂ := (ContinuousLinearMap.smulRightL ℝ ℂ ℂ) (fderiv ℝ rho y) |
| have h : Continuous (fun ε : ℝ => L (f (x - ε • y))) := by fun_prop |
| have hL (c : ℂ) : L c = (fderiv ℝ rho y).smulRight c := by |
| ext v |
| rfl |
| simp_rw [hL] at h |
| have hh := h.continuousAt (x := (0 : ℝ)) |
| simpa only [zero_smul, sub_zero] using hh.tendsto |
|
|
| /-- Rescaled cancellation for a bounded spatially continuous input. This |
| is the elementary pointwise producer for the small scaled derivative. -/ |
| lemma rescaled_gradient_tendsto (f : ℂ →ᵇ ℂ) (x : ℂ) : |
| Tendsto (fun ε : ℝ => ∫ y : ℂ, |
| (fderiv ℝ rho y).smulRight (f (x - ε • y))) (𝓝 0) (𝓝 0) := by |
| have hi : Integrable (fun y : ℂ => ‖fderiv ℝ rho y‖ * ‖f‖) volume := |
| rho_fderiv_integrable.norm.mul_const ‖f‖ |
| have hlim : Tendsto (fun ε : ℝ => ∫ y : ℂ, |
| (fderiv ℝ rho y).smulRight (f (x - ε • y))) (𝓝 0) |
| (𝓝 (∫ y : ℂ, (fderiv ℝ rho y).smulRight (f x))) := by |
| apply tendsto_integral_filter_of_dominated_convergence |
| (fun y : ℂ => ‖fderiv ℝ rho y‖ * ‖f‖) |
| · exact Eventually.of_forall fun ε => (rescaled_gradient_continuous f x ε).aestronglyMeasurable |
| · exact Eventually.of_forall fun ε => Eventually.of_forall (rescaled_gradient_dominated f x ε) |
| · exact hi |
| · exact Eventually.of_forall (rescaled_gradient_pointwise f x) |
| simpa only [rho_smulRight_integral] using hlim |
|
|
| /-- Exact substitution relating the rescaled derivative to the actual |
| first-derivative convolution kernel. -/ |
| lemma scaled_full_gradient (ε : ℝ) (hε : 0 < ε) (f : ℂ → ℂ) (x : ℂ) : |
| ε • (∫ y : ℂ, (fderiv ℝ (kernel ε) (x-y)).smulRight (f y)) = |
| ∫ y : ℂ, (fderiv ℝ rho y).smulRight (f (x - ε • y)) := by |
| rw [← integral_sub_left_eq_self |
| (fun y : ℂ => (fderiv ℝ (kernel ε) (x-y)).smulRight (f y)) volume x] |
| simp only [sub_sub_cancel] |
| have hs := Measure.integral_comp_smul volume |
| (fun y : ℂ => (fderiv ℝ (kernel ε) y).smulRight (f (x-y))) ε |
| rw [Complex.finrank_real_complex, abs_of_nonneg (inv_nonneg.mpr (sq_nonneg ε))] at hs |
| have he : (∫ y : ℂ, (fderiv ℝ (kernel ε) (ε • y)).smulRight (f (x-ε • y))) = |
| (ε ^ 3)⁻¹ • (∫ y : ℂ, (fderiv ℝ rho y).smulRight (f (x-ε • y))) := by |
| simp_rw [kernel_fderiv, smul_smul, inv_mul_cancel₀ hε.ne', one_smul] |
| have hl (y : ℂ) : ((ε ^ 3)⁻¹ • fderiv ℝ rho y).smulRight (f (x - ε • y)) = |
| (ε ^ 3)⁻¹ • (fderiv ℝ rho y).smulRight (f (x - ε • y)) := by |
| ext v |
| simp only [ContinuousLinearMap.smulRight_apply, smul_apply, smul_eq_mul, |
| smul_smul] |
| simp only [hl, integral_smul] |
| rw [he] at hs |
| have heq := congrArg (fun u : ℂ →L[ℝ] ℂ => ε ^ 3 • u) hs |
| simp only [smul_smul] at heq |
| have h₁ : ε ^ 3 * (ε ^ 3)⁻¹ = 1 := mul_inv_cancel₀ (pow_ne_zero _ hε.ne') |
| have h₂ : ε ^ 3 * (ε ^ 2)⁻¹ = ε := by field_simp |
| simpa only [h₁, h₂, one_smul] using heq.symm |
|
|
| end TriangularBilliards.SpatialSmoothing |
|
|
| namespace TriangularBilliards |
| open Filter SpatialSmoothing |
| open scoped Topology BoundedContinuousFunction |
|
|
| lemma kernel_fderiv_eq_zero {ε : ℝ} (hε : 0 < ε) {x : ℂ} (hx : ε < ‖x‖) : |
| fderiv ℝ (kernel ε) x = 0 := by |
| by_contra hn |
| have hm := support_fderiv_subset ℝ (f := kernel ε) hn |
| rw [tsupport, kernel_support hε, closure_ball _ hε.ne', mem_closedBall_zero_iff] at hm |
| exact (not_le_of_gt hx) hm |
| |
| lemma directGradS_eq_full (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| (f : SmoothPhase → ℂ) (z : SmoothPhase) (hz : closedBall z.1.1 ε ⊆ Q.table) : |
| directGradS Q ε f z = ∫ y : ℂ, |
| (fderiv ℝ (kernel ε) (z.1.1 - y)).smulRight (f ((y,z.1.2),z.2)) := by |
| rw [directGradS, ← integral_indicator Q.measurableSet_table] |
| apply integral_congr_ae |
| exact Eventually.of_forall fun y => by |
| by_cases hy : y ∈ Q.table |
| · rw [indicator_of_mem hy] |
| · rw [indicator_of_notMem hy] |
| have hn : ε < ‖z.1.1 - y‖ := by |
| by_contra hh |
| exact hy (hz (by rw [mem_closedBall, dist_comm, dist_eq_norm]; exact not_lt.mp hh)) |
| simp only [kernel_fderiv_eq_zero hε hn, ContinuousLinearMap.zero_smulRight] |
| |
| lemma reflectedGradS_zero_of_coord (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| (f : SmoothPhase → ℂ) (z : SmoothPhase) |
| (hz : Q.coordinateBound * ε < Q.basis.coord (i+2) z.1.1) : |
| reflectedGradS Q i ε f z = 0 := by |
| apply integral_eq_zero_of_ae |
| filter_upwards [ae_restrict_mem Q.measurableSet_table] with y hy |
| have hn : ε < ‖z.1.1 - wallReflection Q i y‖ := by |
| by_contra hh |
| exact (not_le_of_gt hz) (Q.coord_small_of_reflected_near i hy (not_lt.mp hh)) |
| simp only [kernel_fderiv_eq_zero hε hn, ContinuousLinearMap.zero_smulRight, Pi.zero_apply] |
| |
| lemma eventually_interior_gradient (Q : Triangle) (f : SmoothPhase → ℂ) |
| (z : SmoothPhase) (hz : z.1.1 ∈ Q.table) : |
| ∀ᶠ ε in 𝓝[>] (0 : ℝ), reflectedSmoothingGradient Q ε f z = |
| ∫ y : ℂ, (fderiv ℝ (kernel ε) (z.1.1-y)).smulRight (f ((y,z.1.2),z.2)) := by |
| obtain ⟨r, hr, hball⟩ := Metric.isOpen_iff.mp Q.isOpen_table _ hz |
| have hcoords : ∀ᶠ ε in 𝓝 (0 : ℝ), ∀ i : Fin 3, |
| Q.coordinateBound * ε < Q.basis.coord (i+2) z.1.1 := by |
| rw [Filter.eventually_all] |
| intro i |
| have hp := (Q.table_iff_coords z.1.1).mp hz (i+2) |
| have hc : Continuous (fun ε : ℝ => Q.coordinateBound * ε) := by fun_prop |
| simpa only [mul_zero] using hc.continuousAt.eventually (gt_mem_nhds (by simpa only [mul_zero] using hp)) |
| filter_upwards [nhdsWithin_le_nhds hcoords, |
| nhdsWithin_le_nhds (gt_mem_nhds hr), self_mem_nhdsWithin] with ε hc hεr hε |
| have hε' : 0 < ε := hε |
| have hfull := directGradS_eq_full Q hε' f z (fun y hy => hball (by |
| rw [mem_ball] |
| exact (mem_closedBall.mp hy).trans_lt hεr)) |
| change directGradS Q ε f z + ∑ i : Fin 3, reflectedGradS Q i ε f z = _ |
| simp only [reflectedGradS_zero_of_coord Q _ hε' f z (hc _), Finset.sum_const_zero, add_zero] |
| exact hfull |
|
|
| lemma scaled_gradient_pointwise (Q : Triangle) (f : DoublePhase →ᵇ ℂ) |
| (z : DoublePhase) (hz : z.1.1 ∈ Q.table) : |
| Tendsto (fun ε : ℝ => ε • reflectedSmoothingGradient Q ε f z) |
| (𝓝[>] 0) (𝓝 0) := by |
| let g : ℂ →ᵇ ℂ := f.compContinuous ⟨fun y => ((y,z.1.2),z.2), by fun_prop⟩ |
| have h := (rescaled_gradient_tendsto g z.1.1).mono_left (nhdsWithin_le_nhds (s := Ioi (0 : ℝ))) |
| apply h.congr' |
| filter_upwards [eventually_interior_gradient Q f z hz, self_mem_nhdsWithin] with ε hε hp |
| rw [hε] |
| exact (scaled_full_gradient ε hp (fun y => f ((y,z.1.2),z.2)) z.1.1).symm |
| |
| end TriangularBilliards |
| |
| namespace TriangularBilliards |
| open Filter SpatialSmoothing |
| open scoped Topology BoundedContinuousFunction |
| |
| lemma norm_integral_le_kernel_bound {α : Type uAlpha} {E : Type uE} [MeasurableSpace α] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] {μ : Measure α} |
| (F : α → E) {K : α → ℝ≥0∞} {H C : ℝ} (hH : 0 ≤ H) (hC : 0 ≤ C) |
| (hF : ∀ x, ‖F x‖ₑ ≤ K x * ENNReal.ofReal H) |
| (hK : ∫⁻ x, K x ∂μ ≤ ENNReal.ofReal C) : |
| ‖∫ x, F x ∂μ‖ ≤ C * H := by |
| apply (ENNReal.ofReal_le_ofReal_iff (mul_nonneg hC hH)).mp |
| rw [ofReal_norm, ENNReal.ofReal_mul hC] |
| calc |
| ‖∫ x, F x ∂μ‖ₑ ≤ ∫⁻ x, ‖F x‖ₑ ∂μ := enorm_integral_le_lintegral_enorm F |
| _ ≤ ∫⁻ x, K x * ENNReal.ofReal H ∂μ := lintegral_mono hF |
| _ = (∫⁻ x, K x ∂μ) * ENNReal.ofReal H := lintegral_mul_const' _ _ ENNReal.ofReal_ne_top |
| _ ≤ _ := by gcongr |
|
|
| lemma directGradS_norm_bound (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : ∀ z, ‖f z‖ ≤ H) |
| (z : DoublePhase) : ‖directGradS Q ε f z‖ ≤ (derivativeMass / ε) * H := by |
| apply norm_integral_le_kernel_bound (μ := volume.restrict Q.table) |
| (K := fun y => ENNReal.ofReal ‖fderiv ℝ (kernel ε) (z.1.1-y)‖) _ hH |
| (div_nonneg derivativeMass_nonneg hε.le) |
| · intro y |
| rw [← ofReal_norm, ContinuousLinearMap.norm_smulRight_apply, |
| ENNReal.ofReal_mul (norm_nonneg _)] |
| exact mul_le_mul_right (ENNReal.ofReal_le_ofReal (hf ((y,z.1.2),z.2))) _ |
| · exact derivative_kernel_row_bound Q hε z.1.1 |
|
|
| lemma reflectedGradS_norm_bound (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : ∀ z, ‖f z‖ ≤ H) |
| (z : DoublePhase) : ‖reflectedGradS Q i ε f z‖ ≤ (derivativeMass / ε) * H := by |
| apply norm_integral_le_kernel_bound (μ := volume.restrict Q.table) |
| (K := fun y => ENNReal.ofReal ‖fderiv ℝ (kernel ε) (z.1.1 - wallReflection Q i y)‖) _ hH |
| (div_nonneg derivativeMass_nonneg hε.le) |
| · intro y |
| rw [← ofReal_norm, ContinuousLinearMap.norm_smulRight_apply, |
| ENNReal.ofReal_mul (norm_nonneg _)] |
| exact mul_le_mul_right (ENNReal.ofReal_le_ofReal (hf ((y,reflectedDirection Q i z.1.2),z.2+1))) _ |
| · exact reflected_derivative_kernel_row_bound Q i hε z.1.1 |
|
|
| lemma scaled_gradient_norm_bound (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} {H : ℝ} (hH : 0 ≤ H) (hf : ∀ z, ‖f z‖ ≤ H) |
| (z : DoublePhase) : ‖ε • reflectedSmoothingGradient Q ε f z‖ ≤ 4 * derivativeMass * H := by |
| have hb : ‖reflectedSmoothingGradient Q ε f z‖ ≤ 4 * (derivativeMass / ε) * H := by |
| change ‖directGradS Q ε f z + ∑ i : Fin 3, reflectedGradS Q i ε f z‖ ≤ _ |
| calc |
| _ ≤ ‖directGradS Q ε f z‖ + ∑ i : Fin 3, ‖reflectedGradS Q i ε f z‖ := |
| (norm_add_le _ _).trans (add_le_add le_rfl (norm_sum_le _ _)) |
| _ ≤ (derivativeMass / ε) * H + ∑ _i : Fin 3, (derivativeMass / ε) * H := by |
| gcongr |
| · exact directGradS_norm_bound Q hε hH hf z |
| · exact reflectedGradS_norm_bound Q _ hε hH hf z |
| _ = _ := by simp; ring |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos hε] |
| calc |
| _ ≤ ε * (4 * (derivativeMass / ε) * H) := mul_le_mul_of_nonneg_left hb hε.le |
| _ = _ := by field_simp |
|
|
| lemma ae_double_position_in_table (Q : Triangle) : |
| ∀ᵐ z ∂doubleMeasure Q, z.1.1 ∈ Q.table := by |
| exact (measurePreserving_fst (μ := phaseMeasure Q) (ν := parityMeasure)).quasiMeasurePreserving.ae |
| (ae_position_in_table Q) |
|
|
| /-- The small-scaled-derivative estimate on the dense class of bounded |
| continuous fields, for the actual reflected convolution kernel. -/ |
| lemma scaled_gradient_integral_tendsto (Q : Triangle) (f : DoublePhase →ᵇ ℂ) : |
| Tendsto (fun ε : ℝ => ∫ z, ‖ε • reflectedSmoothingGradient Q ε f z‖ ^ 2 ∂doubleMeasure Q) |
| (𝓝[>] 0) (𝓝 0) := by |
| have hlim : Tendsto (fun ε : ℝ => ∫ z, ‖ε • reflectedSmoothingGradient Q ε f z‖ ^ 2 ∂doubleMeasure Q) |
| (𝓝[>] 0) (𝓝 (∫ _ : DoublePhase, (0 : ℝ) ∂doubleMeasure Q)) := by |
| apply tendsto_integral_filter_of_norm_le_const |
| · exact Eventually.of_forall fun ε => |
| (((reflectedSmoothingGradient_stronglyMeasurable Q ε f.continuous.stronglyMeasurable).const_smul ε).measurable.norm.pow_const 2).aestronglyMeasurable |
| · refine ⟨(4 * derivativeMass * ‖f‖) ^ 2, ?_⟩ |
| filter_upwards [self_mem_nhdsWithin] with ε hε |
| exact Eventually.of_forall fun z => by |
| rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)] |
| exact pow_le_pow_left₀ (norm_nonneg _) (scaled_gradient_norm_bound Q hε (norm_nonneg f) f.norm_coe_le_norm z) 2 |
| · filter_upwards [ae_double_position_in_table Q] with z hz |
| simpa only [norm_zero, OfNat.ofNat_ne_zero, ne_eq, not_false_eq_true, zero_pow] using |
| ((scaled_gradient_pointwise Q f z hz).norm.pow 2) |
| simpa only [integral_zero] using hlim |
|
|
| end TriangularBilliards |
|
|
| namespace TriangularBilliards |
| open Filter SpatialSmoothing |
| open scoped Topology BoundedContinuousFunction |
|
|
| lemma integrable_slice_ae (Q : Triangle) {f : DoublePhase → ℂ} |
| (hf : Integrable f (doubleMeasure Q)) : |
| ∀ᵐ z ∂doubleMeasure Q, Integrable (fun y => f ((y,z.1.2),z.2)) (volume.restrict Q.table) := by |
| let μ : Measure ℂ := (volume Q.table)⁻¹ • volume.restrict Q.table |
| have hμ : IsProbabilityMeasure μ := ⟨Q.normalized_area_univ⟩ |
| have ha := measurePreserving_prodAssoc μ angularMeasure parityMeasure |
| have hfi : Integrable (fun p : ℂ × (Circle × ZMod 2) => f ((p.1,p.2.1),p.2.2)) |
| (μ.prod (angularMeasure.prod parityMeasure)) := by |
| exact (ha.integrable_comp_emb MeasurableEquiv.prodAssoc.measurableEmbedding).mp hf |
| have hs : ∀ᵐ c ∂angularMeasure.prod parityMeasure, |
| Integrable (fun y => f ((y,c.1),c.2)) (volume.restrict Q.table) := by |
| filter_upwards [hfi.prod_left_ae] with c hc |
| exact (integrable_smul_measure (by simpa using Q.area_lt_top.ne) |
| (by simpa using Q.area_pos.ne')).mp hc |
| exact ha.quasiMeasurePreserving.ae |
| ((measurePreserving_snd (μ := μ) (ν := angularMeasure.prod parityMeasure)).quasiMeasurePreserving.ae hs) |
| |
| lemma equal_slice_ae (Q : Triangle) {f g : DoublePhase → ℂ} |
| (h : f =ᵐ[doubleMeasure Q] g) : |
| ∀ᵐ z ∂doubleMeasure Q, (fun y => f ((y,z.1.2),z.2)) =ᵐ[volume.restrict Q.table] |
| (fun y => g ((y,z.1.2),z.2)) := by |
| let μ : Measure ℂ := (volume Q.table)⁻¹ • volume.restrict Q.table |
| have hμ : IsProbabilityMeasure μ := ⟨Q.normalized_area_univ⟩ |
| have ha := measurePreserving_prodAssoc μ angularMeasure parityMeasure |
| have hi := ha.symm MeasurableEquiv.prodAssoc |
| have hc := (Measure.measurePreserving_swap (μ := angularMeasure.prod parityMeasure) (ν := μ)).quasiMeasurePreserving.ae |
| (hi.quasiMeasurePreserving.ae h) |
| have hs : ∀ᵐ c ∂angularMeasure.prod parityMeasure, |
| (fun y => f ((y,c.1),c.2)) =ᵐ[volume.restrict Q.table] (fun y => g ((y,c.1),c.2)) := by |
| filter_upwards [Measure.ae_ae_of_ae_prod hc] with c hc |
| exact (Measure.ae_ennreal_smul_measure_iff (by simpa using Q.area_lt_top.ne)).mp hc |
| exact ha.quasiMeasurePreserving.ae |
| ((measurePreserving_snd (μ := μ) (ν := angularMeasure.prod parityMeasure)).quasiMeasurePreserving.ae hs) |
| |
| lemma directGradS_congr_ae (Q : Triangle) (ε : ℝ) {f g : DoublePhase → ℂ} |
| (h : f =ᵐ[doubleMeasure Q] g) : directGradS Q ε f =ᵐ[doubleMeasure Q] directGradS Q ε g := by |
| filter_upwards [equal_slice_ae Q h] with z hz |
| apply integral_congr_ae |
| filter_upwards [hz] with y hy |
| rw [hy] |
| |
| lemma reflectedGradS_congr_ae (Q : Triangle) (i : Fin 3) (ε : ℝ) {f g : DoublePhase → ℂ} |
| (h : f =ᵐ[doubleMeasure Q] g) : |
| reflectedGradS Q i ε f =ᵐ[doubleMeasure Q] reflectedGradS Q i ε g := by |
| filter_upwards [(measurePreserving_directionParityReflection Q i).quasiMeasurePreserving.ae |
| (equal_slice_ae Q h)] with z hz |
| apply integral_congr_ae |
| filter_upwards [hz] with y hy |
| rw [hy] |
| |
| lemma reflectedSmoothingGradient_congr_ae (Q : Triangle) (ε : ℝ) {f g : DoublePhase → ℂ} |
| (h : f =ᵐ[doubleMeasure Q] g) : |
| reflectedSmoothingGradient Q ε f =ᵐ[doubleMeasure Q] reflectedSmoothingGradient Q ε g := by |
| filter_upwards [directGradS_congr_ae Q ε h, |
| Filter.eventually_all.mpr (fun i => reflectedGradS_congr_ae Q i ε h)] with z hd hr |
| change directGradS Q ε f z + ∑ i, reflectedGradS Q i ε f z = |
| directGradS Q ε g z + ∑ i, reflectedGradS Q i ε g z |
| rw [hd] |
| congr 1 |
| exact Finset.sum_congr rfl (fun i _ => hr i) |
| |
| lemma smulRight_sub (k : ℂ →L[ℝ] ℝ) (a b : ℂ) : |
| k.smulRight (a-b) = k.smulRight a - k.smulRight b := by |
| ext v |
| change k v • (a-b) = k v • a - k v • b |
| exact smul_sub _ _ _ |
| |
| lemma smulRight_integrable {k : ℂ → (ℂ →L[ℝ] ℝ)} {f : ℂ → ℂ} |
| {ν : Measure ℂ} (hk : StronglyMeasurable k) (hf : StronglyMeasurable f) |
| (hi : Integrable f ν) {C : ℝ} (hb : ∀ y, ‖k y‖ ≤ C) : |
| Integrable (fun y => (k y).smulRight (f y)) ν := by |
| apply (hi.norm.const_mul C).mono' (smulRight_stronglyMeasurable hk hf).aestronglyMeasurable |
| exact Eventually.of_forall fun y => by |
| rw [ContinuousLinearMap.norm_smulRight_apply] |
| exact mul_le_mul_of_nonneg_right (hb y) (norm_nonneg _) |
|
|
| lemma directGradS_integrable_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hi : Integrable f (doubleMeasure Q)) : |
| ∀ᵐ z ∂doubleMeasure Q, Integrable (fun y => (fderiv ℝ (kernel ε) (z.1.1-y)).smulRight |
| (f ((y,z.1.2),z.2))) (volume.restrict Q.table) := by |
| obtain ⟨C,hC⟩ := ((kernel_hasCompactSupport hε).fderiv ℝ).exists_bound_of_continuous |
| ((kernel_contDiff ε).continuous_fderiv (by simp)) |
| filter_upwards [integrable_slice_ae Q hi] with z hz |
| exact smulRight_integrable |
| (((kernel_contDiff ε).continuous_fderiv (by simp)).stronglyMeasurable.comp_measurable |
| (measurable_const.sub measurable_id)) |
| (hf.comp_measurable ((measurable_id.prodMk measurable_const).prodMk measurable_const)) |
| hz (fun y => hC (z.1.1-y)) |
|
|
| lemma reflectedGradS_integrable_ae (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hi : Integrable f (doubleMeasure Q)) : |
| ∀ᵐ z ∂doubleMeasure Q, Integrable (fun y => |
| (fderiv ℝ (kernel ε) (z.1.1-wallReflection Q i y)).smulRight |
| (f ((y,reflectedDirection Q i z.1.2),z.2+1))) (volume.restrict Q.table) := by |
| obtain ⟨C,hC⟩ := ((kernel_hasCompactSupport hε).fderiv ℝ).exists_bound_of_continuous |
| ((kernel_contDiff ε).continuous_fderiv (by simp)) |
| filter_upwards [(measurePreserving_directionParityReflection Q i).quasiMeasurePreserving.ae |
| (integrable_slice_ae Q hi)] with z hz |
| exact smulRight_integrable |
| (((kernel_contDiff ε).continuous_fderiv (by simp)).stronglyMeasurable.comp_measurable |
| (measurable_const.sub (wallReflection_contDiff Q i).continuous.measurable)) |
| (hf.comp_measurable ((measurable_id.prodMk measurable_const).prodMk measurable_const)) |
| hz (fun y => hC (z.1.1-wallReflection Q i y)) |
|
|
| lemma directGradS_sub_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) |
| (hfi : Integrable f (doubleMeasure Q)) (hgi : Integrable g (doubleMeasure Q)) : |
| directGradS Q ε (f-(g : DoublePhase → ℂ)) =ᵐ[doubleMeasure Q] directGradS Q ε f - directGradS Q ε g := by |
| filter_upwards [directGradS_integrable_ae Q hε hf hfi, |
| directGradS_integrable_ae Q hε hg hgi] with z hz hw |
| change (∫ y in Q.table, (fderiv ℝ (kernel ε) (z.1.1-y)).smulRight (f ((y,z.1.2),z.2) - g ((y,z.1.2),z.2))) = _ |
| simp_rw [smulRight_sub] |
| exact integral_sub hz hw |
|
|
| lemma reflectedGradS_sub_ae (Q : Triangle) (i : Fin 3) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) |
| (hfi : Integrable f (doubleMeasure Q)) (hgi : Integrable g (doubleMeasure Q)) : |
| reflectedGradS Q i ε (f-(g : DoublePhase → ℂ)) =ᵐ[doubleMeasure Q] reflectedGradS Q i ε f - reflectedGradS Q i ε g := by |
| filter_upwards [reflectedGradS_integrable_ae Q i hε hf hfi, |
| reflectedGradS_integrable_ae Q i hε hg hgi] with z hz hw |
| change (∫ y in Q.table, (fderiv ℝ (kernel ε) (z.1.1-wallReflection Q i y)).smulRight |
| (f ((y,reflectedDirection Q i z.1.2),z.2+1) - g ((y,reflectedDirection Q i z.1.2),z.2+1))) = _ |
| simp_rw [smulRight_sub] |
| exact integral_sub hz hw |
|
|
| lemma reflectedSmoothingGradient_sub_ae (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f g : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hg : StronglyMeasurable g) |
| (hfi : Integrable f (doubleMeasure Q)) (hgi : Integrable g (doubleMeasure Q)) : |
| reflectedSmoothingGradient Q ε (f-(g : DoublePhase → ℂ)) =ᵐ[doubleMeasure Q] |
| reflectedSmoothingGradient Q ε f - reflectedSmoothingGradient Q ε g := by |
| filter_upwards [directGradS_sub_ae Q hε hf hg hfi hgi, |
| Filter.eventually_all.mpr (fun i => reflectedGradS_sub_ae Q i hε hf hg hfi hgi)] with z hd hr |
| change directGradS Q ε (f-(g : DoublePhase → ℂ)) z + ∑ i, reflectedGradS Q i ε (f-(g : DoublePhase → ℂ)) z = |
| (directGradS Q ε f z + ∑ i, reflectedGradS Q i ε f z) - |
| (directGradS Q ε g z + ∑ i, reflectedGradS Q i ε g z) |
| simp only [Pi.sub_apply] at hd hr |
| rw [hd] |
| simp_rw [hr] |
| rw [Finset.sum_sub_distrib] |
| abel |
|
|
| lemma scaled_gradient_lpNorm_bound (Q : Triangle) {ε : ℝ} (hε : 0 < ε) |
| {f : DoublePhase → ℂ} (hf : StronglyMeasurable f) (hf₂ : MemLp f 2 (doubleMeasure Q)) : |
| lpNorm (ε • reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q) ≤ |
| (4 * derivativeMass) * lpNorm f 2 (doubleMeasure Q) := by |
| rw [← toReal_eLpNorm, ← toReal_eLpNorm] |
| have hb : eLpNorm (ε • reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q) ≤ |
| ENNReal.ofReal (4 * derivativeMass) * eLpNorm f 2 (doubleMeasure Q) := by |
| calc |
| _ ≤ ‖ε‖ₑ * eLpNorm (reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q) := eLpNorm_const_smul_le |
| _ ≤ ‖ε‖ₑ * (4 * ENNReal.ofReal (derivativeMass / ε) * eLpNorm f 2 (doubleMeasure Q)) := by |
| gcongr |
| exact reflectedSmoothingGradient_eLpNorm Q hε hf |
| _ = _ := by |
| rw [Real.enorm_eq_ofReal hε.le, ENNReal.ofReal_div_of_pos hε, |
| ENNReal.ofReal_mul (by norm_num : (0:ℝ) ≤ 4)] |
| have he : ENNReal.ofReal ε ≠ 0 := ENNReal.ofReal_ne_zero_iff.mpr hε |
| rw [div_eq_mul_inv] |
| calc |
| _ = 4 * ENNReal.ofReal derivativeMass * eLpNorm f 2 (doubleMeasure Q) * |
| (ENNReal.ofReal ε * (ENNReal.ofReal ε)⁻¹) := by ac_rfl |
| _ = _ := by rw [ENNReal.mul_inv_cancel he ENNReal.ofReal_ne_top, mul_one]; norm_num |
| have ht : ENNReal.ofReal (4 * derivativeMass) * eLpNorm f 2 (doubleMeasure Q) ≠ ⊤ := |
| ENNReal.mul_ne_top ENNReal.ofReal_ne_top hf₂.eLpNorm_ne_top |
| have h := ENNReal.toReal_mono ht hb |
| have hC : 0 ≤ 4 * derivativeMass := mul_nonneg (by norm_num) derivativeMass_nonneg |
| simpa only [ENNReal.toReal_mul, ENNReal.toReal_ofReal hC] using h |
|
|
| lemma lpNorm_congr_ae_of_strong {α : Type uAlpha} {E : Type uE} [MeasurableSpace α] [NormedAddCommGroup E] |
| {μ : Measure α} {p : ℝ≥0∞} {f g : α → E} |
| (hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (h : f =ᵐ[μ] g) : |
| lpNorm f p μ = lpNorm g p μ := by |
| rw [← toReal_eLpNorm, ← toReal_eLpNorm] |
| exact congrArg ENNReal.toReal (eLpNorm_congr_enorm_ae hf hg (h.fun_comp enorm)) |
|
|
| lemma scaled_gradient_lpNorm_tendsto_bounded (Q : Triangle) (f : DoublePhase →ᵇ ℂ) : |
| Tendsto (fun ε : ℝ => lpNorm (ε • reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q)) |
| (𝓝[>] 0) (𝓝 0) := by |
| have he (ε : ℝ) : lpNorm (ε • reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q) = |
| Real.sqrt (∫ z, ‖ε • reflectedSmoothingGradient Q ε f z‖ ^ 2 ∂doubleMeasure Q) := by |
| rw [lpNorm_eq_integral_norm_rpow_toReal (by norm_num) (by norm_num) |
| ((reflectedSmoothingGradient_stronglyMeasurable Q ε f.continuous.stronglyMeasurable).const_smul ε).aestronglyMeasurable] |
| simp only [ENNReal.toReal_ofNat, Pi.smul_apply, Real.rpow_two, Real.sqrt_eq_rpow] |
| norm_num |
| simp_rw [he] |
| simpa only [Real.sqrt_zero, Function.comp_def] using Real.continuous_sqrt.continuousAt.tendsto.comp |
| (scaled_gradient_integral_tendsto Q f) |
|
|
| /-- The small-scaled-derivative estimate for every L² input |
| and the reflected triangular kernel, not merely a dense test class. -/ |
| lemma scaled_gradient_lpNorm_tendsto (Q : Triangle) {f : DoublePhase → ℂ} |
| (hf : StronglyMeasurable f) (hf₂ : MemLp f 2 (doubleMeasure Q)) : |
| Tendsto (fun ε : ℝ => lpNorm (ε • reflectedSmoothingGradient Q ε f) 2 (doubleMeasure Q)) |
| (𝓝[>] 0) (𝓝 0) := by |
| let C : ℝ := 4 * derivativeMass |
| have hC : 0 ≤ C := mul_nonneg (by norm_num) derivativeMass_nonneg |
| apply Metric.tendsto_nhds.mpr |
| intro δ hδ |
| let η : ℝ := δ / (2 * (C+1)) |
| have hη : 0 < η := div_pos hδ (by positivity) |
| obtain ⟨g,hg,hg₂⟩ := hf₂.exists_boundedContinuous_eLpNorm_sub_le |
| (by norm_num : (2 : ℝ≥0∞) ≠ ⊤) (ENNReal.ofReal_ne_zero_iff.mpr hη) |
| have hsmall : lpNorm (f - (g : DoublePhase → ℂ)) 2 (doubleMeasure Q) ≤ η := by |
| rw [← toReal_eLpNorm] |
| exact (ENNReal.toReal_mono ENNReal.ofReal_ne_top hg).trans_eq (ENNReal.toReal_ofReal hη.le) |
| have hgsmall := (Metric.tendsto_nhds.mp (scaled_gradient_lpNorm_tendsto_bounded Q g)) |
| (δ/2) (by positivity) |
| filter_upwards [self_mem_nhdsWithin, hgsmall] with ε hε he |
| have hgm := (reflectedSmoothingGradient_memLp Q hε g.continuous.stronglyMeasurable hg₂).const_smul ε |
| have hfm := (reflectedSmoothingGradient_memLp Q hε hf hf₂).const_smul ε |
| have hdiffm := (reflectedSmoothingGradient_memLp Q hε |
| (hf.sub g.continuous.stronglyMeasurable) (hf₂.sub hg₂)).const_smul ε |
| have hae : (ε • reflectedSmoothingGradient Q ε f - ε • reflectedSmoothingGradient Q ε g) =ᵐ[doubleMeasure Q] |
| ε • reflectedSmoothingGradient Q ε (f - (g : DoublePhase → ℂ)) := by |
| filter_upwards [reflectedSmoothingGradient_sub_ae Q hε hf g.continuous.stronglyMeasurable |
| (hf₂.integrable (by norm_num)) (hg₂.integrable (by norm_num))] with z hz |
| simp only [Pi.sub_apply, Pi.smul_apply] at hz ⊢ |
| rw [hz, smul_sub] |
| have hn := lpNorm_le_lpNorm_add_lpNorm_sub' hgm (f := ε • reflectedSmoothingGradient Q ε f) |
| (by norm_num : (1:ℝ≥0∞) ≤ 2) |
| rw [lpNorm_congr_ae_of_strong (hfm.sub hgm).aestronglyMeasurable |
| hdiffm.aestronglyMeasurable hae] at hn |
| have hb := scaled_gradient_lpNorm_bound Q hε (hf.sub g.continuous.stronglyMeasurable) (hf₂.sub hg₂) |
| have ht : C * η < δ/2 := by |
| dsimp [η] |
| have hc : (0:ℝ) < C+1 := by linarith |
| rw [← mul_div_assoc, div_lt_iff₀ (by positivity : 0 < 2*(C+1))] |
| nlinarith |
| have hnonneg : 0 ≤ lpNorm (ε • reflectedSmoothingGradient Q ε g) 2 (doubleMeasure Q) := lpNorm_nonneg |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg hnonneg] at he |
| rw [Real.dist_eq, sub_zero, abs_of_nonneg lpNorm_nonneg] |
| have hh : lpNorm (ε • reflectedSmoothingGradient Q ε (f - (g : DoublePhase → ℂ))) 2 |
| (doubleMeasure Q) ≤ C * η := hb.trans (mul_le_mul_of_nonneg_left hsmall hC) |
| linarith |
| |
| end TriangularBilliards |
| |
| end |
| end |
| end |
| |
| end OAI |
| |
