| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.Multiplier |
| import OAI.Geometry.CAT0Fillings.Sobolev.Constant |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology ENNReal NNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X (k+1)} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma atlas_preserving : MeasurePreserving q.atlasParam q.atlasMeasure |
| (MassMeasure.currentMassMeasure hT) := ⟨q.measurable_atlasParam,q.map_atlasMeasure⟩ |
|
|
| noncomputable def composeValue {F : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) |
| (f : Lp ℝ 2 (MassMeasure.currentMassMeasure hT)) : Lp ℝ 2 (MassMeasure.currentMassMeasure hT) := |
| (hF.sub (LipschitzWith.const (F 0))).compLp (by simp) f + q.inclusion (q.constant (F 0)) |
|
|
| lemma composeValue_ae {F : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) |
| (f : Lp ℝ 2 (MassMeasure.currentMassMeasure hT)) : |
| q.composeValue hF f =ᵐ[MassMeasure.currentMassMeasure hT] (fun x => F (f x)) := by |
| filter_upwards [Lp.coeFn_add ((hF.sub (LipschitzWith.const (F 0))).compLp (by simp) f) |
| (value (hT := hT) (LipschitzWith.const (F 0))), |
| (hF.sub (LipschitzWith.const (F 0))).coeFn_compLp (by simp) f, |
| ((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (F 0))).memLp |
| (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x ha hb hc |
| rw [composeValue,constant,q.inclusion_lipSobolev,ha] |
| change ((hF.sub (LipschitzWith.const (F 0))).compLp (by simp) f) x + (value (hT := hT) (LipschitzWith.const (F 0))) x = _ |
| change (value (hT := hT) (LipschitzWith.const (F 0))) x = F 0 at hc |
| rw [hb,hc] |
| simp only [Function.comp_apply,sub_add_cancel] |
|
|
| lemma continuous_composeValue {F : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) : |
| Continuous (q.composeValue hF) := |
| ((hF.sub (LipschitzWith.const (F 0))).continuous_compLp (by simp)).add continuous_const |
|
|
| lemma composeValue_value {F : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) |
| {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| q.composeValue hF (value hu) = value (hF.comp hu) := by |
| apply Lp.ext |
| filter_upwards [q.composeValue_ae hF (value hu), |
| ((Foundations.boundedLip_of_lipschitz hu).memLp (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp, |
| ((Foundations.boundedLip_of_lipschitz (hF.comp hu)).memLp |
| (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x ha hb hc |
| exact ha.trans ((congrArg F hb).trans hc.symm) |
|
|
| lemma closed_chain {F F' : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) |
| (hdF : ∀ t, HasDerivAt F (F' t) t) (hcF : Continuous F') |
| {C : ℝ} (hC : 0 ≤ C) (hbF : ∀ t, |F' t| ≤ C) (P : q.Sobolev) : |
| ∃ Q : q.Sobolev, |
| (q.inclusion Q : X → ℝ) =ᵐ[MassMeasure.currentMassMeasure hT] |
| (fun x => F ((q.inclusion P) x)) ∧ |
| (q.closedGradient Q : (ℕ × Euc (k+1)) → Euc (k+1)) =ᵐ[q.atlasMeasure] |
| (fun w => F' ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by |
| obtain ⟨u,K,hu,hl⟩ := q.exists_lipschitz_approximants P |
| have hv := (q.valueProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl |
| have hg := (q.gradientProjection.continuous.tendsto (P : q.GraphAmbient)).comp hl |
| change Tendsto (fun j => value (hu j)) atTop (𝓝 (q.inclusion P)) at hv |
| change Tendsto (fun j => q.gradient (hu j)) atTop (𝓝 (q.closedGradient P)) at hg |
| obtain ⟨s,hs,hae⟩ := (tendstoInMeasure_of_tendsto_Lp hv).exists_seq_tendsto_ae |
| have hnames : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, ∀ j, |
| (value (hT := hT) (hu (s j))) x = u (s j) x := by |
| apply ae_all_iff.mpr |
| intro j |
| exact ((Foundations.boundedLip_of_lipschitz (hu (s j))).memLp 2).coeFn_toLp |
| have hae' : ∀ᵐ x ∂MassMeasure.currentMassMeasure hT, |
| Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x)) := by |
| filter_upwards [hae,hnames] with x hx he |
| simpa only [he] using hx |
| let b (j : ℕ) (w : ℕ × Euc (k+1)) := F' (u (s j) (q.atlasParam w)) |
| let c (w : ℕ × Euc (k+1)) := F' ((q.inclusion P) (q.atlasParam w)) |
| have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := |
| (hcF.measurable.comp ((hu (s j)).continuous.measurable.comp q.measurable_atlasParam)).aestronglyMeasurable |
| have hc : AEStronglyMeasurable c q.atlasMeasure := |
| hcF.comp_aestronglyMeasurable ((Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving) |
| have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun w => hbF _ |
| have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun w => hbF _ |
| have hbc : ∀ᵐ w ∂q.atlasMeasure, Tendsto (fun j => b j w) atTop (𝓝 (c w)) := by |
| have ha : ∀ᵐ w ∂q.atlasMeasure, |
| Tendsto (fun j => u (s j) (q.atlasParam w)) atTop (𝓝 ((q.inclusion P) (q.atlasParam w))) := by |
| apply ae_of_ae_map (μ := q.atlasMeasure) (p := fun x => |
| Tendsto (fun j => u (s j) x) atTop (𝓝 ((q.inclusion P) x))) q.measurable_atlasParam.aemeasurable |
| rwa [q.map_atlasMeasure] |
| filter_upwards [ha] with w hw |
| exact (hcF.tendsto _).comp hw |
| have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC hbc (hg.comp hs.tendsto_atTop) |
| have he (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.gradient (hu (s j))) = |
| q.gradient (hF.comp (hu (s j))) := by |
| apply Lp.ext |
| exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (q.gradient_chain_ae (hu (s j)) hF hdF).symm |
| simp only [Function.comp_apply,he] at hgl |
| have hvl := (q.continuous_composeValue hF).tendsto (q.inclusion P) |>.comp (hv.comp hs.tendsto_atTop) |
| simp only [Function.comp_def,q.composeValue_value] at hvl |
| obtain ⟨Q,hQ,hG⟩ := q.closedGradient_closed (q.composeValue hF (q.inclusion P)) |
| (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) |
| (fun j => q.lipSobolev (hF.comp (hu (s j)))) hvl hgl |
| refine ⟨Q,?_,?_⟩ |
| · rw [hQ] |
| exact q.composeValue_ae hF (q.inclusion P) |
| · rw [hG] |
| exact ClosedCalculus.mulLp_ae hc hcC _ |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
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|
