| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric ContinuousLinearMap |
| open scoped Topology NNReal ENNReal Convolution Pointwise |
|
|
| 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_scalar_bound {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E} |
| [μ.IsAddHaarMeasure] {f ρ : E → ℝ} (hf : Integrable f μ) {C M : ℝ} |
| (hC : 0 ≤ C) (hρ : ∀ x, |ρ x| ≤ C) (hM : (∫ x, |f x| ∂μ) ≤ M) (x : E) : |
| |(f ⋆[lsmul ℝ ℝ, μ] ρ) x| ≤ C*M := by |
| rw [convolution_def] |
| have hb := norm_integral_le_of_norm_le (hf.abs.mul_const C) |
| (ae_of_all μ fun y => show ‖(lsmul ℝ ℝ (f y)) (ρ (x-y))‖ ≤ |f y| *C by |
| simpa only [lsmul_apply,smul_eq_mul,Real.norm_eq_abs,abs_mul] using |
| mul_le_mul_of_nonneg_left (hρ (x-y)) (abs_nonneg (f y))) |
| rw [integral_mul_const] at hb |
| exact hb.trans ((mul_le_mul_of_nonneg_right hM hC).trans_eq (mul_comm _ _)) |
|
|
| lemma convolution_scalar_lipschitz {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E} |
| [μ.IsAddHaarMeasure] {f ρ : E → ℝ} (hf : Integrable f μ) |
| (hρc : HasCompactSupport ρ) {K M : ℝ≥0} (hρ : LipschitzWith K ρ) |
| (hM : (∫ x, |f x| ∂μ) ≤ M) : |
| LipschitzWith (K*M) (f ⋆[lsmul ℝ ℝ, μ] ρ) := by |
| have hi (x : E) : Integrable (fun y => f y*ρ (x-y)) μ := |
| hρc.convolutionExists_right (lsmul ℝ ℝ) hf.locallyIntegrable hρ.continuous x |
| apply LipschitzWith.of_dist_le_mul |
| intro x y |
| simp only [Real.dist_eq,convolution_def,lsmul_apply,smul_eq_mul] |
| rw [←integral_sub (hi x) (hi y)] |
| have hh := norm_integral_le_of_norm_le (hf.abs.mul_const ((K:ℝ)*dist x y)) |
| (ae_of_all μ fun z => show ‖f z*ρ (x-z)-f z*ρ (y-z)‖ ≤ |f z| *((K:ℝ)*dist x y) by |
| rw [←mul_sub,Real.norm_eq_abs,abs_mul] |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) |
| simpa only [Real.dist_eq,dist_sub_right] using hρ.dist_le_mul (x-z) (y-z)) |
| rw [integral_mul_const] at hh |
| apply hh.trans |
| calc (∫ z, |f z| ∂μ)*((K:ℝ)*dist x y) ≤ (M:ℝ)*((K:ℝ)*dist x y) := |
| mul_le_mul_of_nonneg_right hM (mul_nonneg K.coe_nonneg dist_nonneg) |
| _ = ((K*M:ℝ≥0):ℝ)*dist x y := by push_cast; ring |
|
|
| lemma convolution_scalar_support {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] {μ : Measure E} |
| [μ.IsAddHaarMeasure] {f ρ : E → ℝ} {R ε : ℝ} |
| (hf : Function.support f ⊆ closedBall (0:E) R) |
| (hρ : Function.support ρ ⊆ closedBall (0:E) ε) : |
| Function.support (f ⋆[lsmul ℝ ℝ, μ] ρ) ⊆ closedBall (0:E) (R+ε) := by |
| intro x hx |
| obtain ⟨y,hy,z,hz,rfl⟩ := support_convolution_subset_swap (lsmul ℝ ℝ) hx |
| have hy' : ‖y‖ ≤ ε := by simpa only [mem_closedBall,dist_zero_right] using hρ hy |
| have hz' : ‖z‖ ≤ R := by simpa only [mem_closedBall,dist_zero_right] using hf hz |
| simp only [mem_closedBall,dist_zero_right] |
| exact (norm_add_le y z).trans ((add_le_add hy' hz').trans_eq (add_comm _ _)) |
|
|
| 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 mollifier_error_of_translation {f : E → ℝ} (hf : Integrable f μ) |
| {C : ℝ} (hC : 0 ≤ C) |
| (ht : ∀ w, (∫ x, |f (x+w)-f x| ∂μ) ≤ C*‖w‖) (η : ContDiffBump (0:E)) : |
| (∫ x, |f x-(η.normed μ ⋆[lsmul ℝ ℝ, μ] f) x| ∂μ) ≤ C*η.rOut := by |
| let ρ := η.normed μ |
| have hρ : Integrable ρ μ := η.continuous_normed.integrable_of_hasCompactSupport η.hasCompactSupport_normed |
| have hρ0 : ∀ x, 0 ≤ ρ x := η.nonneg_normed |
| have hfi : Integrable (ρ ⋆[lsmul ℝ ℝ, μ] f) μ := hρ.integrable_convolution (lsmul ℝ ℝ) hf |
| have hconv (x : E) : Integrable (fun y => ρ y*f (x-y)) μ := by |
| exact (η.hasCompactSupport_normed.convolutionExists_left (lsmul ℝ ℝ) |
| η.continuous_normed hf.locallyIntegrable x) |
| have hp : Integrable (fun p : E × E => ρ p.2*(f p.1-f (p.1-p.2))) (μ.prod μ) := by |
| have h1 := (hf.mul_prod hρ) |
| have h2 := hρ.convolution_integrand (lsmul ℝ ℝ) hf |
| convert h1.sub h2 using 1 |
| funext p |
| simp only [Pi.sub_apply,lsmul_apply,smul_eq_mul] |
| ring |
| have hpoint (x : E) : |f x-(ρ ⋆[lsmul ℝ ℝ, μ] f) x| ≤ |
| ∫ y, ρ y*|f x-f (x-y)| ∂μ := by |
| have he : f x-(ρ ⋆[lsmul ℝ ℝ, μ] f) x = ∫ y, ρ y*(f x-f (x-y)) ∂μ := by |
| simp only [mul_sub] |
| rw [integral_sub (hρ.mul_const _) (hconv x),integral_mul_const] |
| simp only [ρ,η.integral_normed,one_mul,convolution_def,lsmul_apply,smul_eq_mul] |
| rw [he] |
| have hn := norm_integral_le_integral_norm (f := fun y => ρ y*(f x-f (x-y))) (μ := μ) |
| simpa only [Real.norm_eq_abs,abs_mul,abs_of_nonneg (hρ0 _)] using hn |
| have hpn : Integrable (fun p : E × E => ρ p.2*|f p.1-f (p.1-p.2)|) (μ.prod μ) := by |
| simpa only [Real.norm_eq_abs,abs_mul,abs_of_nonneg (hρ0 _)] using hp.abs |
| have hh := integral_mono (hf.sub hfi).abs hpn.integral_prod_left hpoint |
| rw [integral_integral_swap hpn] at hh |
| have hb (y : E) : (∫ x, ρ y*|f x-f (x-y)| ∂μ) ≤ ρ y*(C*η.rOut) := by |
| rw [integral_const_mul] |
| by_cases hy : y ∈ Function.support ρ |
| · have hyn : ‖y‖ ≤ η.rOut := by |
| have hys : Function.support ρ ⊆ closedBall 0 η.rOut := by |
| dsimp [ρ] |
| rw [η.support_normed_eq] |
| exact ball_subset_closedBall |
| simpa only [mem_closedBall,dist_zero_right] using hys hy |
| have ht' : (∫ x, |f x-f (x-y)| ∂μ) ≤ C*‖y‖ := by |
| simpa only [←sub_eq_add_neg,norm_neg,abs_sub_comm] using ht (-y) |
| exact mul_le_mul_of_nonneg_left (ht'.trans (mul_le_mul_of_nonneg_left hyn hC)) (hρ0 y) |
| · have hz : ρ y = 0 := Function.notMem_support.mp hy |
| simp only [hz,zero_mul,le_refl] |
| have hm := integral_mono hpn.integral_prod_right (hρ.mul_const (C*η.rOut)) hb |
| rw [integral_mul_const] at hm |
| have hr : (∫ y, ρ y ∂μ) = 1 := η.integral_normed |
| rw [hr,one_mul] at hm |
| exact hh.trans hm |
|
|
| end CAT0Fillings.JointBV |
| end |
|
|
| end OAI |
|
|
