| 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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