| import Mathlib | |
| import OAI.Geometry.CAT0Fillings.BV.JointPoincare | |
| import OAI.Geometry.CAT0Fillings.Compactness.Translation | |
| namespace OAI | |
| section | |
| open Set Filter MeasureTheory Metric ContinuousLinearMap | |
| open scoped Topology NNReal ENNReal Convolution | |
| namespace CAT0Fillings.JointBV | |
| lemma scalar_precompR {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] : | |
| (lsmul ℝ ℝ).precompR E = (lsmul ℝ ℝ : ℝ →L[ℝ] (E →L[ℝ] ℝ) →L[ℝ] (E →L[ℝ] ℝ)) := by | |
| ext scalar linearMap | |
| simp only [precompR,coe_comp,Function.comp_apply,compL_apply,lsmul_apply,_root_.smul_apply] | |
| lemma convolution_bump_deriv_norm_integrable {k : ℕ} | |
| {f : EuclideanSpace ℝ (Fin k) → ℝ} | |
| {μ : Measure (EuclideanSpace ℝ (Fin k))} [μ.IsAddHaarMeasure] | |
| (hf : Integrable f μ) (η : ContDiffBump (0:EuclideanSpace ℝ (Fin k))) : | |
| Integrable (fun x => ‖(f ⋆[(lsmul ℝ ℝ).precompR (EuclideanSpace ℝ (Fin k)), μ] | |
| fderiv ℝ (η.normed μ)) x‖) μ := by | |
| have hc := η.hasCompactSupport_normed (μ := μ) | |
| have hd := (η.contDiff_normed (μ := μ) (n := (⊤ : ℕ∞))).continuous_fderiv (by simp) | |
| rw [scalar_precompR] | |
| exact (hf.integrable_convolution (lsmul ℝ ℝ : ℝ →L[ℝ] (EuclideanSpace ℝ (Fin k) →L[ℝ] ℝ) →L[ℝ] (EuclideanSpace ℝ (Fin k) →L[ℝ] ℝ)) | |
| (hd.integrable_of_hasCompactSupport (hc.fderiv ℝ))).norm | |
| end CAT0Fillings.JointBV | |
| end | |
| section | |
| open Set Filter MeasureTheory Metric ContinuousLinearMap | |
| open scoped Topology ENNReal NNReal Convolution | |
| namespace CAT0Fillings.JointBV | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] | |
| [MeasurableSpace E] [BorelSpace E] {μ : Measure E} [μ.IsAddHaarMeasure] | |
| lemma convolution_bump_hasFDerivAt {f : E → ℝ} (hf : Integrable f μ) | |
| (η : ContDiffBump (0:E)) (x : E) : | |
| HasFDerivAt (f ⋆[lsmul ℝ ℝ, μ] η.normed μ) | |
| ((f ⋆[(lsmul ℝ ℝ).precompR E, μ] fderiv ℝ (η.normed μ)) x) x := | |
| η.hasCompactSupport_normed.hasFDerivAt_convolution_right (lsmul ℝ ℝ) hf.locallyIntegrable η.contDiff_normed x | |
| lemma convolution_bump_deriv_continuous {f : E → ℝ} (hf : Integrable f μ) | |
| (η : ContDiffBump (0:E)) : | |
| Continuous (f ⋆[(lsmul ℝ ℝ).precompR E, μ] fderiv ℝ (η.normed μ)) := | |
| (η.hasCompactSupport_normed.fderiv ℝ).continuous_convolution_right | |
| ((lsmul ℝ ℝ).precompR E) hf.locallyIntegrable | |
| ((η.contDiff_normed (n := (⊤ : ℕ∞))).continuous_fderiv (by simp)) | |
| lemma convolution_bump_integrable {f : E → ℝ} (hf : Integrable f μ) | |
| (η : ContDiffBump (0:E)) : Integrable (f ⋆[lsmul ℝ ℝ, μ] η.normed μ) μ := | |
| hf.integrable_convolution (lsmul ℝ ℝ) | |
| (η.continuous_normed.integrable_of_hasCompactSupport η.hasCompactSupport_normed) | |
| lemma mollified_gradient_total_bound {k : ℕ} {f : EuclideanSpace ℝ (Fin k) → ℝ} | |
| {μ : Measure (EuclideanSpace ℝ (Fin k))} [μ.IsAddHaarMeasure] | |
| {ν : Measure (EuclideanSpace ℝ (Fin k))} [IsFiniteMeasure ν] | |
| (hf : Integrable f μ) | |
| (hv : ∀ i, DirectionalVariation f (EuclideanSpace.basisFun (Fin k) ℝ i) μ ν) | |
| (η : ContDiffBump (0 : EuclideanSpace ℝ (Fin k))) : | |
| (∫ z, ‖(f ⋆[(lsmul ℝ ℝ).precompR (EuclideanSpace ℝ (Fin k)), μ] | |
| fderiv ℝ (η.normed μ)) z‖ ∂μ) ≤ k*ν.real univ := by | |
| apply integral_le_of_closedBall_bound | |
| (convolution_bump_deriv_norm_integrable hf η) | |
| intro r | |
| exact (mollified_gradient_integral_bound hf.locallyIntegrable hv η 0 r).trans | |
| (mul_le_mul_of_nonneg_left (measureReal_mono (subset_univ _)) (Nat.cast_nonneg k)) | |
| theorem translation_integral_bound {k : ℕ} {f : EuclideanSpace ℝ (Fin k) → ℝ} | |
| {μ : Measure (EuclideanSpace ℝ (Fin k))} [μ.IsAddHaarMeasure] | |
| {ν : Measure (EuclideanSpace ℝ (Fin k))} [IsFiniteMeasure ν] | |
| (hf : Integrable f μ) | |
| (hv : ∀ i, DirectionalVariation f (EuclideanSpace.basisFun (Fin k) ℝ i) μ ν) | |
| (w : EuclideanSpace ℝ (Fin k)) : | |
| (∫ x, |f (x+w)-f x| ∂μ) ≤ ‖w‖*(k*ν.real univ) := by | |
| let η (j : ℕ) := shrinkingBump (E := EuclideanSpace ℝ (Fin k)) j | |
| let F (j : ℕ) := f ⋆[lsmul ℝ ℝ, μ] (η j).normed μ | |
| let F' (j : ℕ) := f ⋆[(lsmul ℝ ℝ).precompR (EuclideanSpace ℝ (Fin k)), μ] | |
| fderiv ℝ ((η j).normed μ) | |
| have hF (j : ℕ) (x : EuclideanSpace ℝ (Fin k)) : HasFDerivAt (F j) (F' j x) x := | |
| convolution_bump_hasFDerivAt hf (η j) x | |
| have hF' (j : ℕ) : Continuous (F' j) := convolution_bump_deriv_continuous hf (η j) | |
| have hFi (j : ℕ) : Integrable (F j) μ := convolution_bump_integrable hf (η j) | |
| have hGi (j : ℕ) : Integrable (fun x => ‖F' j x‖) μ := | |
| convolution_bump_deriv_norm_integrable hf (η j) | |
| have hl : ∀ᵐ x ∂μ, Tendsto (fun j => F j x) atTop (𝓝 (f x)) := by | |
| have hh := ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable | |
| (φ := η) shrinkingBump_tendsto (K := 2) (Eventually.of_forall (fun j => le_rfl)) hf.locallyIntegrable | |
| filter_upwards [hh] with x hx | |
| simpa only [F,scalar_convolution_comm f] using hx | |
| have hlw := (measurePreserving_add_right μ w).quasiMeasurePreserving.ae hl | |
| apply integral_abs_le_of_ae_tendsto ((hf.comp_add_right w).sub hf) | |
| (fun j => ((hFi j).comp_add_right w).sub (hFi j)) ?_ (by positivity) | |
| · exact Eventually.of_forall fun j => | |
| (smooth_translation_integral_bound (hF j) (hF' j) (hFi j) (hGi j) w).trans | |
| (mul_le_mul_of_nonneg_left (mollified_gradient_total_bound hf hv (η j)) (norm_nonneg w)) | |
| · filter_upwards [hlw,hl] with x hx hw | |
| exact hx.sub hw | |
| end CAT0Fillings.JointBV | |
| end | |
| section | |
| open Set Filter MeasureTheory Metric | |
| open scoped Topology NNReal ENNReal | |
| namespace CAT0Fillings.JointBV | |
| lemma DirectionalVariation.congr {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [MeasurableSpace E] {μ ν : Measure E} {f g : E → ℝ} {v : E} | |
| (h : DirectionalVariation f v μ ν) (he : f =ᵐ[μ] g) : | |
| DirectionalVariation g v μ ν := by | |
| intro φ hφ hc | |
| have hh := h φ hφ hc | |
| convert hh using 2 | |
| apply integral_congr_ae | |
| filter_upwards [he] with x hx | |
| rw [hx] | |
| theorem totallyBounded_L1_of_directional_variation {k : ℕ} {ι : Type*} | |
| (f : ι → EuclideanSpace ℝ (Fin k) → ℝ) (hf : ∀ i, Integrable (f i)) | |
| (ν : ι → Measure (EuclideanSpace ℝ (Fin k))) [∀ i, IsFiniteMeasure (ν i)] | |
| (hv : ∀ i l, DirectionalVariation (f i) (EuclideanSpace.basisFun (Fin k) ℝ l) volume (ν i)) | |
| (B : ℝ≥0) (hB : ∀ i, (∫ x, |f i x|) ≤ B) | |
| (M : ℝ≥0) (hM : ∀ i, (ν i).real univ ≤ M) | |
| (R : ℝ) (hs : ∀ i, ∀ᵐ x, x ∉ closedBall 0 R → f i x = 0) : | |
| TotallyBounded (range fun i => (hf i).toL1 (f i)) := by | |
| let g (i) := (closedBall 0 R).indicator (f i) | |
| have he (i) : f i =ᵐ[volume] g i := by | |
| filter_upwards [hs i] with x hx | |
| by_cases hxR : x ∈ closedBall 0 R | |
| · simp only [g,indicator_of_mem hxR] | |
| · simp only [g,indicator_of_notMem hxR,hx hxR] | |
| have hg (i) : Integrable (g i) := (hf i).congr (he i) | |
| have hge (i) : (hf i).toL1 (f i) = (hg i).toL1 (g i) := | |
| (Integrable.toL1_eq_toL1_iff _ _ _ _).mpr (he i) | |
| simp only [hge] | |
| apply totallyBounded_L1_of_translation g hg B ?_ R ?_ (C := k*(M:ℝ)) (by positivity) | |
| · intro i w | |
| have hh := translation_integral_bound (hg i) (fun l => (hv i l).congr (he i)) w | |
| apply hh.trans | |
| calc ‖w‖*((k:ℝ)*(ν i).real univ) ≤ ‖w‖*((k:ℝ)*M) := | |
| mul_le_mul_of_nonneg_left (mul_le_mul_of_nonneg_left (hM i) (Nat.cast_nonneg k)) (norm_nonneg w) | |
| _ = _ := mul_comm _ _ | |
| · intro i | |
| have hi : (∫ x, |g i x|) = ∫ x, |f i x| := integral_congr_ae ((he i).symm.fun_comp abs) | |
| rw [hi] | |
| exact hB i | |
| · intro i x hx | |
| by_contra hn | |
| exact hx (indicator_of_notMem hn _) | |
| end CAT0Fillings.JointBV | |
| end | |
| end OAI | |