| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear |
| import OAI.Geometry.CAT0Fillings.Gradient.Multiplier |
| import OAI.Geometry.CAT0Fillings.Calculus.ClosedChain |
|
|
| 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 sobolev_tendsto {P : ℕ → q.Sobolev} {Q : q.Sobolev} |
| (hv : Tendsto (fun j => q.inclusion (P j)) atTop (𝓝 (q.inclusion Q))) |
| (hg : Tendsto (fun j => q.closedGradient (P j)) atTop (𝓝 (q.closedGradient Q))) : |
| Tendsto P atTop (𝓝 Q) := by |
| apply tendsto_subtype_rng.mpr |
| exact ((WithLp.prodContinuousLinearEquiv 2 ℝ |
| (Lp ℝ 2 (MassMeasure.currentMassMeasure hT)) (Lp (Euc (k+1)) 2 q.atlasMeasure)).symm.continuous.tendsto |
| (q.inclusion Q,q.closedGradient Q)).comp (hv.prodMk_nhds hg) |
|
|
| lemma closed_chain_in_closed {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) |
| {S : Set q.Sobolev} (hS : IsClosed S) |
| (hmem : ∀ (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev (hF.comp hu) ∈ S) : |
| ∃ Q : q.Sobolev, Q ∈ S ∧ |
| (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 point => hbF (u (s j) (q.atlasParam point)) |
| have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := |
| Eventually.of_forall fun point => hbF ((q.inclusion P) (q.atlasParam point)) |
| 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,?_,?_,?_⟩ |
| · apply hS.mem_of_tendsto (q.sobolev_tendsto (P := fun j => q.lipSobolev (hF.comp (hu (s j)))) (by simpa only [q.inclusion_lipSobolev,hQ] using hvl) (by simpa only [q.closedGradient_lipSobolev,hG] using hgl)) |
| exact Eventually.of_forall fun j => hmem _ _ (hu (s j)) |
| · rw [hQ] |
| exact q.composeValue_ae hF (q.inclusion P) |
| · rw [hG] |
| exact ClosedCalculus.mulLp_ae hc hcC _ |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| 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 closed_chain_uniform_in_closed {F D : ℝ → ℝ} {J : ℝ≥0} (hF : LipschitzWith J F) |
| (hD : Measurable D) {C : ℝ} (hC : 0 ≤ C) (hbD : ∀ t, |D t| ≤ C) |
| (Fj Dj : ℕ → ℝ → ℝ) (K : ℕ → ℝ≥0) |
| (hFj : ∀ j, LipschitzWith (K j) (Fj j)) |
| (hdj : ∀ j t, HasDerivAt (Fj j) (Dj j t) t) |
| (hcj : ∀ j, Continuous (Dj j)) (hbj : ∀ j t, |Dj j t| ≤ C) |
| (hdlim : ∀ t, Tendsto (fun j => Dj j t) atTop (𝓝 (D t))) |
| (a : ℕ → ℝ) (ha : Tendsto a atTop (𝓝 0)) |
| (he : ∀ j t, |Fj j t-F t| ≤ a j) (P : q.Sobolev) |
| {S : Set q.Sobolev} (hclosed : IsClosed S) |
| (hmem : ∀ j (u : X → ℝ) (K : ℝ≥0) (hu : LipschitzWith K u), q.lipSobolev ((hFj j).comp hu) ∈ S) : |
| ∃ Q : q.Sobolev, Q ∈ S ∧ |
| (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 => D ((q.inclusion P) (q.atlasParam w)) • (q.closedGradient P) w) := by |
| choose Q hQS hQ hG using (fun j => q.closed_chain_in_closed (hFj j) (hdj j) (hcj j) hC (hbj j) P hclosed (hmem j)) |
| let b (j : ℕ) (w : ℕ × Euc (k+1)) := Dj j ((q.inclusion P) (q.atlasParam w)) |
| let c (w : ℕ × Euc (k+1)) := D ((q.inclusion P) (q.atlasParam w)) |
| have hpm := (Lp.aestronglyMeasurable (q.inclusion P)).comp_measurePreserving q.atlas_preserving |
| have hb (j : ℕ) : AEStronglyMeasurable (b j) q.atlasMeasure := (hcj j).comp_aestronglyMeasurable hpm |
| have hc : AEStronglyMeasurable c q.atlasMeasure := (hD.comp_aemeasurable hpm.aemeasurable).aestronglyMeasurable |
| have hbC (j : ℕ) : ∀ᵐ w ∂q.atlasMeasure, |b j w| ≤ C := Eventually.of_forall fun _ => hbj j _ |
| have hcC : ∀ᵐ w ∂q.atlasMeasure, |c w| ≤ C := Eventually.of_forall fun _ => hbD _ |
| have hgl := ClosedCalculus.mulLp_tendsto hb hc hC hbC hcC |
| (Eventually.of_forall fun point => hdlim ((q.inclusion P) (q.atlasParam point))) |
| (tendsto_const_nhds (x := q.closedGradient P)) |
| have heG (j : ℕ) : ClosedCalculus.mulLp (hb j) (hbC j) (q.closedGradient P) = q.closedGradient (Q j) := by |
| apply Lp.ext |
| exact (ClosedCalculus.mulLp_ae (hb j) (hbC j) _).trans (hG j).symm |
| simp only [heG] at hgl |
| let V := q.composeValue hF (q.inclusion P) |
| have hv : Tendsto (fun j => q.inclusion (Q j)) atTop (𝓝 V) := by |
| apply tendsto_iff_norm_sub_tendsto_zero.mpr |
| apply squeeze_zero (fun _ => norm_nonneg _) |
| (g := fun j => a j*‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖) |
| · intro j |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| filter_upwards [Lp.coeFn_sub (q.inclusion (Q j)) V,hQ j,q.composeValue_ae hF (q.inclusion P), |
| ((Foundations.boundedLip_of_lipschitz (LipschitzWith.const (1:ℝ))).memLp |
| (μ := MassMeasure.currentMassMeasure hT) 2).coeFn_toLp] with x hs hq hv h1 |
| change (value (hT := hT) (LipschitzWith.const (1:ℝ)) : X → ℝ) x = 1 at h1 |
| rw [hs,Pi.sub_apply,hq,hv,h1,Real.norm_eq_abs,norm_one,mul_one] |
| exact he j _ |
| · simpa only [zero_mul] using ha.mul_const ‖value (hT := hT) (LipschitzWith.const (1:ℝ))‖ |
| obtain ⟨R,hR,hRG⟩ := q.closedGradient_closed V (ClosedCalculus.mulLp hc hcC (q.closedGradient P)) Q hv hgl |
| refine ⟨R,?_,?_,?_⟩ |
| · apply hclosed.mem_of_tendsto (q.sobolev_tendsto (P := Q) (by simpa only [hR] using hv) (by simpa only [hRG] using hgl)) |
| exact Eventually.of_forall hQS |
| · rw [hR] |
| exact q.composeValue_ae hF (q.inclusion P) |
| · rw [hRG] |
| exact ClosedCalculus.mulLp_ae hc hcC _ |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
