| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear |
| import OAI.Geometry.CAT0Fillings.Compactness.Ascoli |
| import OAI.Geometry.CAT0Fillings.Compactness.Coordinates |
| import OAI.Geometry.CAT0Fillings.Compactness.Projected |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open Foundations MassMeasure Slicing JointBV BorelCoefficients |
|
|
| 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) |
|
|
| omit [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [CompactSpace X] [Nonempty X] in |
| lemma mul_bounded {b u : X → ℝ} {B U : ℝ≥0} |
| (hb : ∀ x, |b x| ≤ B) (hu : ∀ x, |u x| ≤ U) (x : X) : |b x*u x| ≤ (B*U:ℝ≥0) := by |
| rw [abs_mul] |
| exact mul_le_mul (hb x) (hu x) (abs_nonneg _) B.coe_nonneg |
|
|
| theorem totallyBounded_borel_projected {ι : Type*} |
| (h : NormalApprox (k+1) T) (hz : boundarySucc T = 0) (hX : IsCAT0 X) |
| (π : Fin (k+1) → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| (u : ι → X → ℝ) (L U : ι → ℝ≥0) (hu : ∀ i, LipschitzWith (L i) (u i)) |
| (hU : ∀ i x, |u i x| ≤ U i) |
| {χ : X → ℝ} (hχ : Measurable χ) (B : ℝ≥0) (hB : ∀ x, |χ x| ≤ B) |
| (M : ℝ≥0) (hMv : ∀ i, ‖value (hT := hT) (hu i)‖ ≤ M) |
| (hMg : ∀ i, ‖q.gradient (hu i)‖ ≤ M) : |
| TotallyBounded (range fun i => projectedL1 h hX π hπ (hχ.mul (hu i).continuous.measurable) |
| (B*U i) (mul_bounded hB (hU i))) := by |
| have hχ2 : MemLp χ 2 (currentMassMeasure hT) := MemLp.of_bound hχ.aestronglyMeasurable B |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hB x) |
| obtain ⟨b,hb⟩ := exists_lipschitz_L2_approximation (currentMassMeasure hT) hχ2 |
| apply totallyBounded_range_of_uniform_approx |
| intro ε hε |
| have hlim : Tendsto (fun j => (K:ℝ)^(k+1)*lpNorm (fun x => (b j).val x-χ x) 2 (currentMassMeasure hT)*M) |
| atTop (𝓝 0) := by simpa using (hb.const_mul ((K:ℝ)^(k+1))).mul_const (M:ℝ) |
| obtain ⟨j,hj⟩ := (hlim.eventually (gt_mem_nhds hε)).exists |
| obtain ⟨C₀,hC₀⟩ := (b j).property.2 |
| let C : ℝ≥0 := ⟨max C₀ 0,le_max_right _ _⟩ |
| have hC x : |(b j).val x| ≤ C := (hC₀ x).trans (le_max_left _ _) |
| obtain ⟨J,hJ⟩ := (b j).property.1 |
| let g i := projectedL1 h hX π hπ ((b j).val.continuous.measurable.mul (hu i).continuous.measurable) |
| (C*U i) (mul_bounded hC (hU i)) |
| refine ⟨g,?_,?_⟩ |
| · have he : g = fun i => |
| (fullSlice_eval_integrable h hX π (fun a => boundedLip_of_lipschitz (hπ a)) |
| ((boundedLip_of_lipschitz hJ).mul (boundedLip_of_lipschitz (hu i))) (fun j => Fin.elim0 j)).toL1 |
| (fun z => fullSlice h π z (fun x => (b j).val x*u i x) (fun j => Fin.elim0 j)) := by |
| funext i |
| exact projectedL1_lipschitz h hX π hπ ((b j).property.mul (boundedLip_of_lipschitz (hu i))) _ |
| rw [he] |
| exact q.totallyBounded_whole_projected h hz hX π hπ u L hu hJ M hMv hMg |
| · intro i |
| rw [dist_comm] |
| change dist (projectedL1 h hX π hπ _ _ _) (projectedL1 h hX π hπ _ _ _) < ε |
| rw [dist_projectedL1] |
| apply (fullSlice_cutoff_difference_bound h hX π hπ (b j).val.continuous.measurable hχ |
| (hu i).continuous.measurable hC hB (hU i)).trans_lt |
| apply lt_of_le_of_lt _ hj |
| rw [←norm_value (hT := hT) (hu i)] |
| exact mul_le_mul_of_nonneg_left (hMv i) (mul_nonneg (pow_nonneg K.coe_nonneg _) lpNorm_nonneg) |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open Slicing Foundations MassMeasure BorelCoefficients JointBV |
|
|
| 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 coordinate_distance_le (h : NormalApprox (k+1) T) (hX : IsCAT0 X) (i : ℕ) |
| (F : Fin (k+1) → X → ℝ) {K J : ℝ≥0} (hF : ∀ a, LipschitzWith K (F a)) |
| (hJ : LipschitzWith J (q.chart i).param) |
| (hFinv : ∀ z ∈ (q.chart i).domain, coordinateMap F ((q.chart i).paramExtended z) = z) |
| (hjac : ∀ᵐ z ∂volume.restrict (q.chart i).domain, (q.chart i).jacobian F z = 1) |
| {u v : X → ℝ} {L P : ℝ≥0} (hu : LipschitzWith L u) (hv : LipschitzWith P v) |
| (U V : ℝ≥0) (hU : ∀ x, |u x| ≤ U) (hV : ∀ x, |v x| ≤ V) : |
| dist (q.coordinateL1 i hu.continuous.measurable U hU) (q.coordinateL1 i hv.continuous.measurable V hV) ≤ |
| (J:ℝ)^(k+1)*dist (q.isolatedL1 h hX i F hF hu.continuous.measurable U hU) |
| (q.isolatedL1 h hX i F hF hv.continuous.measurable V hV) := by |
| rw [q.dist_coordinateL1,q.dist_isolatedL1 h hX i F hF hFinv hjac hu hv,←integral_const_mul] |
| have hb x : abs |u x-v x| ≤ (U+V:ℝ≥0) := by |
| rw [abs_abs]; exact (abs_sub _ _).trans (add_le_add (hU x) (hV x)) |
| have hm := (hu.continuous.measurable.sub hv.continuous.measurable).norm |
| have hleft := integrable_weighted_comp (volume.restrict (q.chart i).domain) |
| (q.chart i).measurable_paramExtended (q.density_integrable i) hm ⟨(U+V:ℝ≥0),hb⟩ |
| have hright := integrable_weighted_comp (volume.restrict (q.chart i).domain) |
| (q.chart i).measurable_paramExtended (q.chart i).integrable.norm hm ⟨(U+V:ℝ≥0),hb⟩ |
| apply integral_mono_ae (by simpa only [Real.norm_eq_abs,Pi.sub_apply] using hleft) |
| (by simpa only [Real.norm_eq_abs,Pi.sub_apply] using hright.const_mul ((J:ℝ)^(k+1))) |
| filter_upwards [q.ae_chart_jacobian_le i hJ] with z hz |
| dsimp only [density] |
| calc _ ≤ |((q.chart i).multiplicity z:ℝ)| * ((J:ℝ)^(k+1))*|u ((q.chart i).paramExtended z)-v ((q.chart i).paramExtended z)| := |
| mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hz (abs_nonneg _)) (abs_nonneg _) |
| _ = _ := by ring |
|
|
| theorem totallyBounded_coordinate {ι : Type*} |
| (h : NormalApprox (k+1) T) (hz : boundarySucc T = 0) (hX : IsCAT0 X) |
| (u : ι → X → ℝ) (L U : ι → ℝ≥0) (hu : ∀ j, LipschitzWith (L j) (u j)) |
| (hU : ∀ j x, |u j x| ≤ U j) |
| (M : ℝ≥0) (hMv : ∀ j, ‖value (hT := hT) (hu j)‖ ≤ M) |
| (hMg : ∀ j, ‖q.gradient (hu j)‖ ≤ M) (i : ℕ) : |
| TotallyBounded (range fun j => q.coordinateL1 i (hu j).continuous.measurable (U j) (hU j)) := by |
| obtain ⟨F,K,hF,hFinv,hjac⟩ := q.exists_exact_inverse_coordinates i |
| obtain ⟨J,J',hJ,hJ'⟩ := (q.chart i).bilipschitz |
| have hib : ∀ x, |(q.chart i).image.indicator (fun _ => (1:ℝ)) x| ≤ (1:ℝ≥0) := by |
| intro x; by_cases hx : x ∈ (q.chart i).image <;> simp [hx] |
| have ht := q.totallyBounded_borel_projected h hz hX F hF u L U hu hU |
| (measurable_const.indicator (q.chart i).measurableSet_image) 1 hib M hMv hMg |
| have he : (fun j => projectedL1 h hX F hF ((measurable_const.indicator (q.chart i).measurableSet_image).mul (hu j).continuous.measurable) (1*U j) (mul_bounded hib (hU j))) = |
| (fun j => q.isolatedL1 h hX i F hF (hu j).continuous.measurable (U j) (hU j)) := by |
| funext j |
| apply Lp.ext |
| filter_upwards [(fullSlice_borel_integral h hX F (fun a => boundedLip_of_lipschitz (hF a)) |
| ((measurable_const.indicator (q.chart i).measurableSet_image).mul (hu j).continuous.measurable) |
| (1*U j) (mul_bounded hib (hU j))).1.coeFn_toL1, |
| (fullSlice_borel_integral h hX F (fun a => boundedLip_of_lipschitz (hF a)) |
| ((hu j).continuous.measurable.indicator (q.chart i).measurableSet_image) |
| (U j) (q.indicator_bound i (hU j))).1.coeFn_toL1] with z hz hz' |
| change _ = _ at hz hz' |
| dsimp only [projectedL1,isolatedL1] |
| rw [hz,hz'] |
| congr 2 |
| ext x |
| by_cases hx : x ∈ (q.chart i).image <;> simp [hx] |
| rw [he] at ht |
| apply totallyBounded_range_of_control ht |
| intro ε hε |
| refine ⟨ε/((J:ℝ)^(k+1)+1),div_pos hε (by positivity),?_⟩ |
| intro a b hab |
| apply (q.coordinate_distance_le h hX i F hF hJ hFinv hjac (hu a) (hu b) (U a) (U b) (hU a) (hU b)).trans_lt |
| have hc := (lt_div_iff₀ (show 0 < (J:ℝ)^(k+1)+1 by positivity)).mp hab |
| have hd : 0 ≤ dist (q.isolatedL1 h hX i F hF (hu a).continuous.measurable (U a) (hU a)) |
| (q.isolatedL1 h hX i F hF (hu b).continuous.measurable (U b) (hU b)) := dist_nonneg |
| nlinarith [pow_nonneg J.coe_nonneg (k+1)] |
| |
| end CAT0Fillings.ChartGeometry |
| end |
| |
| end OAI |
| |
