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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | 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
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