| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Currents.AffineCalculus |
| import OAI.Geometry.CAT0Fillings.Calculus.Smoothing |
|
|
| namespace OAI |
|
|
| section |
| section |
| 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 MeasureTheory Filter Set Metric |
| open scoped Topology Pointwise NNReal |
| open Set MeasureTheory Measure Filter Module |
| open Set Filter MeasureTheory Measure Metric |
| open scoped Topology ContDiff |
| open Set Filter Metric |
| open scoped Topology NNReal |
| open Set MeasureTheory Filter |
| open scoped Topology ENNReal NNReal |
| open Set Filter MeasureTheory Measure ContinuousLinearMap |
| open scoped Topology Convolution NNReal |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
|
|
| lemma LipschitzWith.integral_deriv_eq_sub {f : ℝ → ℝ} {K : ℝ≥0} |
| (hf : LipschitzWith K f) (a b : ℝ) : |
| ∫ t in a..b, deriv f t = f b - f a := by |
| obtain ⟨g,hg,hpt,hderiv⟩ := exists_smooth_lipschitz_approximation volume hf |
| have hbound (j : ℕ) (t : ℝ) : ‖deriv (g j) t‖ ≤ K := |
| norm_deriv_le_of_lipschitz (hg j).2 |
| have hlim : ∀ᵐ t ∂volume, |
| Tendsto (fun j => deriv (g j) t) atTop (𝓝 (deriv f t)) := by |
| filter_upwards [hderiv] with t ht |
| have H := ((ContinuousLinearMap.apply ℝ ℝ (1:ℝ)).continuous.tendsto |
| (fderiv ℝ f t)).comp ht |
| exact H |
| have hi (j : ℕ) : IntervalIntegrable (deriv (g j)) volume a b := |
| ((hg j).1.continuous_deriv (by norm_num)).intervalIntegrable a b |
| have hlimI : Tendsto (fun j => ∫ t in a..b, deriv (g j) t) atTop |
| (𝓝 (∫ t in a..b, deriv f t)) := by |
| apply intervalIntegral.tendsto_integral_filter_of_dominated_convergence (bound := fun _ => (K:ℝ)) |
| · exact Filter.Eventually.of_forall (fun j => ((hg j).1.continuous_deriv (by norm_num)).aestronglyMeasurable) |
| · exact Filter.Eventually.of_forall (fun j => Filter.Eventually.of_forall (fun t _ => hbound j t)) |
| · exact intervalIntegrable_const |
| · exact hlim.mono (fun _ ht _ => ht) |
| have heq (j : ℕ) : ∫ t in a..b, deriv (g j) t = g j b - g j a := |
| intervalIntegral.integral_eq_sub_of_hasDerivAt |
| (fun t _ => ((hg j).1.differentiable (by norm_num) t).hasDerivAt) (hi j) |
| simp only [heq] at hlimI |
| exact tendsto_nhds_unique hlimI ((hpt b).sub (hpt a)) |
|
|
| end CAT0Fillings |
|
|
| namespace CAT0Fillings |
| open Set MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] |
| local notation "BL" => boundedLipSubmodule (X := X) |
|
|
| omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in |
| lemma lipschitz_slice {f : ℝ × X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (t : ℝ) : |
| LipschitzWith K (fun x => f (t,x)) := by |
| simpa [Function.comp_def] using hf.comp ((LipschitzWith.const t).prodMk |
| (LipschitzWith.id : LipschitzWith 1 (id : X → X))) |
|
|
| omit [MeasurableSpace X] [BorelSpace X] [CompactSpace X] in |
| lemma lipschitz_time {f : ℝ × X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (x : X) : |
| LipschitzWith K (fun t => f (t,x)) := by |
| simpa [Function.comp_def] using hf.comp ((LipschitzWith.id : LipschitzWith 1 (id : ℝ → ℝ)).prodMk |
| (LipschitzWith.const x)) |
|
|
| omit [MeasurableSpace X] [BorelSpace X] in |
| lemma boundedLip_of_lipschitz_compact {f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) : |
| BoundedLip f := by |
| refine ⟨⟨K,hf⟩,?_⟩ |
| obtain ⟨b,hb⟩ := (isCompact_range hf.continuous.abs).bddAbove |
| exact ⟨b,fun x => hb (mem_range_self x)⟩ |
|
|
| omit [CompactSpace X] in |
| lemma measurable_timeDeriv {f : ℝ × X → ℝ} (hf : Continuous f) : |
| Measurable (fun p : ℝ × X => deriv (fun t => f (t,p.2)) p.1) := by |
| exact (measurable_deriv_with_param (f := fun x t => f (t,x)) |
| (hf.comp continuous_swap)).comp measurable_swap |
|
|
| omit [CompactSpace X] in |
| lemma integrable_timeDeriv {f : ℝ × X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) |
| (μ : Measure X) [IsFiniteMeasure μ] (t : ℝ) : |
| Integrable (fun x => deriv (fun s => f (s,x)) t) μ := by |
| apply (integrable_const (K:ℝ)).mono' |
| ((measurable_timeDeriv hf.continuous).comp (measurable_const.prodMk measurable_id)).aestronglyMeasurable |
| filter_upwards with x |
| exact norm_deriv_le_of_lipschitz (lipschitz_time hf x) |
| |
| lemma ae_time_hasDeriv {f : ℝ × X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) |
| (μ : Measure X) [IsFiniteMeasure μ] : |
| ∀ᵐ t ∂volume, ∀ᵐ x ∂μ, HasDerivAt (fun s => f (s,x)) |
| (deriv (fun s => f (s,x)) t) t := by |
| have hmeas := measurableSet_of_differentiableAt_with_param ℝ |
| (f := fun x t => f (t,x)) (hf.continuous.comp continuous_swap) |
| have hp : ∀ᵐ p ∂μ.prod volume, DifferentiableAt ℝ (fun t => f (t,p.1)) p.2 := by |
| apply (ae_prod_iff_ae_ae hmeas).mpr |
| filter_upwards with x |
| exact (lipschitz_time hf x).ae_differentiableAt |
| have hq := (Measure.measurePreserving_swap (μ := volume) (ν := μ)).quasiMeasurePreserving.ae hp |
| filter_upwards [ae_ae_of_ae_prod hq] with t ht |
| filter_upwards [ht] with x hx |
| exact hx.hasDerivAt |
| |
| end CAT0Fillings |
| end |
| end |
| |
| end OAI |
| |
