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59d1c80 | 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 | import Mathlib
import OAI.Geometry.IntegralFillings.Slicing.Profiles
namespace OAI
section
open Set MeasureTheory Measure Filter Module
open Set Filter MeasureTheory Measure ContinuousLinearMap
open scoped Topology Convolution NNReal
open Set Filter MeasureTheory Measure Metric
open scoped Topology ContDiff
open Set Filter Metric
open Filter Set
open Set Filter MeasureTheory TopologicalSpace
open scoped Topology ENNReal
open Set MeasureTheory
open scoped RealInnerProductSpace
open Matrix
open scoped RealInnerProductSpace MatrixOrder
open Set Filter MeasureTheory
open scoped Topology ENNReal NNReal
open MeasureTheory Filter Set Metric
open scoped Topology Pointwise NNReal
open scoped ENNReal NNReal Topology
open Set MeasureTheory Filter
open scoped Topology NNReal
namespace SharpIntegralFillings.SmoothCutoff
open Set MeasureTheory Filter Metric
open scoped Topology NNReal
lemma dprofile_integrable {a : ℝ≥0} (ha : 0 < a) (t : ℝ) :
Integrable (dprofile a t) := by
obtain ⟨C,hC⟩ := exists_profile_bounds
have hc := (hC a t).2.1.continuous
apply hc.integrable_of_hasCompactSupport
apply HasCompactSupport.intro isCompact_Icc
exact fun x hx => dprofile_zero ha hx
lemma integral_dprofile {a : ℝ≥0} (ha : 0 < a) (t : ℝ) :
(∫ x : ℝ, dprofile a t x) = 1 := by
have ha' : 0 < (a:ℝ) := ha
obtain ⟨C,hC⟩ := exists_profile_bounds
have hc := (hC a t).2.1.continuous
rw [←setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx => dprofile_zero ha hx),
integral_Icc_eq_integral_Ioc,
←intervalIntegral.integral_of_le (show t ≤ t+(a:ℝ)⁻¹ from le_add_of_nonneg_right (inv_nonneg.mpr ha'.le)),
intervalIntegral.integral_eq_sub_of_hasDerivAt (fun x _ => profile_hasDerivAt a t x)
(hc.intervalIntegrable _ _)]
simp [profile,ha'.ne',Real.smoothTransition.one_of_one_le]
lemma dprofile_mul_integrable {f : ℝ → ℝ} (hf : Integrable f) (a : ℝ≥0) (t : ℝ) :
Integrable (fun x => dprofile a t x * f x) := by
obtain ⟨C,hC⟩ := exists_profile_bounds
exact hf.bdd_mul (hC a t).2.1.continuous.aestronglyMeasurable
(Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using (hC a t).2.2 x)
theorem ae_tendsto_dprofile_integral {f : ℝ → ℝ} (hf : Integrable f) :
∀ᵐ t : ℝ, Tendsto (fun n : ℕ => ∫ x : ℝ,
dprofile ((n:ℝ≥0)+1) t x * f x) atTop (𝓝 (f t)) := by
obtain ⟨C,hC⟩ := exists_profile_bounds
let a : ℕ → ℝ≥0 := fun n => (n:ℝ≥0)+1
let r : ℕ → ℝ := fun n => ((a n:ℝ))⁻¹
have hap n : 0 < a n := by dsimp [a]; positivity
have harp n : 0 < (a n:ℝ) := hap n
have hrp n : 0 < r n := inv_pos.mpr (harp n)
have hrt : Tendsto r atTop (𝓝[>] 0) := by
apply tendsto_nhdsWithin_iff.mpr
refine ⟨?_,Eventually.of_forall fun n => hrp n⟩
exact tendsto_inv_atTop_zero.comp
(show Tendsto (fun n : ℕ => (a n:ℝ)) atTop atTop from by
simpa only [a,NNReal.coe_add,NNReal.coe_natCast,NNReal.coe_one] using
tendsto_atTop_add_const_right atTop 1 (tendsto_natCast_atTop_atTop (R := ℝ)))
filter_upwards [IsUnifLocDoublingMeasure.ae_tendsto_average_norm_sub volume hf.locallyIntegrable 0]
with t ht
have hav := ht (fun _ : ℕ => t) r hrt
(Eventually.of_forall fun n => by simp only [zero_mul]; exact mem_closedBall_self le_rfl)
have hbound n : ‖(∫ x : ℝ, dprofile (a n) t x * f x)-f t‖ ≤
(2*(C:ℝ)) * (⨍ x in closedBall t (r n), ‖f x-f t‖) := by
have hk := dprofile_integrable (hap n) t
have hprod := dprofile_mul_integrable hf (a n) t
have heq : (∫ x : ℝ, dprofile (a n) t x * f x)-f t =
∫ x : ℝ, dprofile (a n) t x * (f x-f t) := by
simp only [mul_sub,integral_sub hprod (hk.mul_const _),integral_mul_const,
integral_dprofile (hap n),one_mul]
rw [heq]
have herr : Integrable (fun x : ℝ => dprofile (a n) t x * (f x-f t)) := by
apply (hprod.sub (hk.mul_const (f t))).congr
filter_upwards [] with x
exact (mul_sub _ _ _).symm
have hnz (x : ℝ) (hx : x ∉ closedBall t (r n)) : dprofile (a n) t x = 0 := by
apply dprofile_zero (hap n)
intro hi
apply hx
rw [Real.closedBall_eq_Icc]
change t-r n ≤ x ∧ x ≤ t+r n
exact ⟨by linarith [hi.1,hrp n],hi.2⟩
have hloc : IntegrableOn (fun x : ℝ => ‖f x-f t‖) (closedBall t (r n)) :=
(hf.integrableOn.sub (integrableOn_const (measure_closedBall_lt_top.ne))).norm
calc
‖∫ x : ℝ, dprofile (a n) t x * (f x-f t)‖ ≤
∫ x : ℝ, ‖dprofile (a n) t x * (f x-f t)‖ := norm_integral_le_integral_norm _
_ = ∫ x in closedBall t (r n), ‖dprofile (a n) t x * (f x-f t)‖ :=
(setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx => by rw [hnz x hx,zero_mul,norm_zero])).symm
_ ≤ ∫ x in closedBall t (r n), ((C:ℝ)*(a n:ℝ)) * ‖f x-f t‖ := by
apply setIntegral_mono_on herr.norm.integrableOn (hloc.const_mul _) isClosed_closedBall.measurableSet
intro x _
rw [norm_mul]
apply mul_le_mul_of_nonneg_right _ (norm_nonneg _)
simpa only [Real.norm_eq_abs,NNReal.coe_mul] using (hC (a n) t).2.2 x
_ = (2*(C:ℝ)) * (⨍ x in closedBall t (r n), ‖f x-f t‖) := by
rw [integral_const_mul,setAverage_eq,Real.volume_real_closedBall (hrp n).le,
smul_eq_mul]
dsimp [r]
field_simp
have hzero := squeeze_zero_norm hbound (by simpa only [mul_zero] using hav.const_mul (2*(C:ℝ)))
simpa only [sub_add_cancel,zero_add] using hzero.add_const (f t)
end SharpIntegralFillings.SmoothCutoff
end
end OAI
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