| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.ClosedLinear |
| import OAI.Geometry.CAT0Fillings.Compactness.BV |
| import OAI.Geometry.CAT0Fillings.Compactness.IntrinsicBV |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| 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 gradientFunction_mul {u v : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hv : LipschitzWith J v) : |
| q.gradientFunction (fun x => u x*v x) =ᵐ[q.atlasMeasure] |
| fun w => u (q.atlasParam w) • q.gradientFunction v w+ |
| v (q.atlasParam w) • q.gradientFunction u w := by |
| apply Measure.ae_sum_iff.mpr |
| intro i |
| apply (measurableEmbedding_prodMk_left i).ae_map_iff.mpr |
| apply (withDensity_absolutelyContinuous _ _).ae_le |
| filter_upwards [(q.chart i).ae_fderivWithin_scalar_mul hu hv, |
| ae_restrict_mem (q.chart i).borel] with z hz hzi |
| change q.normalizedCovector i (fun x => u x*v x) z = _ |
| unfold normalizedCovector covector |
| rw [hz] |
| have hr (L M : Euc (k+1) →L[ℝ] ℝ) (a b : ℝ) : |
| differentialRow (a • L+b • M) = a • differentialRow L+b • differentialRow M := rfl |
| rw [hr,Matrix.mulVec_add,Matrix.mulVec_smul,Matrix.mulVec_smul,WithLp.toLp_add, |
| WithLp.toLp_smul,WithLp.toLp_smul] |
| simp only [gradientFunction,normalizedCovector,covector,atlasParam, |
| IntegerChart.paramExtended,IntegerChart.scalar,dite_eq_left hzi] |
|
|
| lemma integral_gradientFunction_norm {u : X → ℝ} {K : ℝ≥0} (hu : LipschitzWith K u) : |
| (∫ w, ‖q.gradientFunction u w‖ ∂q.atlasMeasure) ≤ |
| ‖q.gradient hu‖*Real.sqrt (q.atlasMeasure.real univ) := by |
| have hh := integral_norm_le_lpNorm_two (q.memLp_gradientFunction hu) |
| have hn : lpNorm (q.gradientFunction u) 2 q.atlasMeasure = ‖q.gradient hu‖ := by |
| rw [gradient,Lp.norm_toLp,toReal_eLpNorm] |
| rwa [hn] at hh |
|
|
| lemma integral_gradient_mul_bound {u χ : X → ℝ} {K J : ℝ≥0} |
| (hu : LipschitzWith K u) (hχ : LipschitzWith J χ) {B : ℝ} |
| (hB : 0 ≤ B) (hχB : ∀ x, |χ x| ≤ B) : |
| (∫ w, ‖q.gradientFunction (fun x => χ x*u x) w‖ ∂q.atlasMeasure) ≤ |
| B*‖q.gradient hu‖*Real.sqrt (q.atlasMeasure.real univ)+ |
| J*‖value (hT := hT) hu‖*Real.sqrt ((MassMeasure.currentMassMeasure hT).real univ) := by |
| let μ := MassMeasure.currentMassMeasure hT |
| have hχb := Foundations.boundedLip_of_lipschitz hχ |
| have hub := Foundations.boundedLip_of_lipschitz hu |
| obtain ⟨M,hm⟩ := (hχb.mul hub).1 |
| have hia := ((q.memLp_gradientFunction hu).integrable (by norm_num)).norm.const_mul B |
| have hiu : Integrable (fun w => |u (q.atlasParam w)|) q.atlasMeasure := by |
| have hh : Integrable (fun x => |u x|) (q.atlasMeasure.map q.atlasParam) := by |
| rw [q.map_atlasMeasure] |
| exact ((hub.memLp 2).integrable (by norm_num)).abs |
| exact (integrable_map_measure (hu.continuous.abs.aestronglyMeasurable) |
| q.measurable_atlasParam.aemeasurable).mp hh |
| have hib := hiu.const_mul (J:ℝ) |
| have hpoint : ∀ᵐ w ∂q.atlasMeasure, |
| ‖q.gradientFunction (fun x => χ x*u x) w‖ ≤ |
| B*‖q.gradientFunction u w‖+J*|u (q.atlasParam w)| := by |
| filter_upwards [q.gradientFunction_mul hχ hu,q.ae_gradientFunction_bound hχ] with w hw hwb |
| rw [hw] |
| calc _ ≤ ‖χ (q.atlasParam w) • q.gradientFunction u w‖+ |
| ‖u (q.atlasParam w) • q.gradientFunction χ w‖ := norm_add_le _ _ |
| _ ≤ B*‖q.gradientFunction u w‖+J*|u (q.atlasParam w)| := by |
| simp only [norm_smul,Real.norm_eq_abs] |
| exact add_le_add (mul_le_mul_of_nonneg_right (hχB _) (norm_nonneg _)) |
| ((mul_le_mul_of_nonneg_left hwb (abs_nonneg _)).trans_eq (mul_comm _ _)) |
| have hh := integral_mono_ae ((q.memLp_gradientFunction hm).integrable (by norm_num)).norm |
| (hia.add hib) hpoint |
| simp only [Pi.add_apply] at hh |
| rw [integral_add hia hib,integral_const_mul,integral_const_mul] at hh |
| have he : (∫ w, |u (q.atlasParam w)| ∂q.atlasMeasure) = ∫ x, |u x| ∂μ := by |
| dsimp only [μ] |
| rw [←q.map_atlasMeasure,integral_map q.measurable_atlasParam.aemeasurable |
| hu.continuous.abs.aestronglyMeasurable] |
| rw [he] at hh |
| have huL := integral_norm_le_lpNorm_two (hub.memLp (μ := μ) 2) |
| rw [←norm_value (hT := hT) hu] at huL |
| simp only [Real.norm_eq_abs] at huL |
| exact hh.trans (by |
| have h1 := mul_le_mul_of_nonneg_left (q.integral_gradientFunction_norm hu) hB |
| have h2 := mul_le_mul_of_nonneg_left huL J.coe_nonneg |
| nlinarith) |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| 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) |
|
|
| lemma intrinsicSliceControl_total (π : Fin (k+1) → X → ℝ) {K : ℝ≥0} |
| (hπ : ∀ i, LipschitzWith K (π i)) {u : X → ℝ} {L : ℝ≥0} |
| (hu : LipschitzWith L u) : |
| (q.intrinsicSliceControl π K u).real univ = |
| (K:ℝ)^k*∫ w, ‖q.gradientFunction u w‖ ∂q.atlasMeasure := by |
| simpa using q.intrinsicSliceControl_integral π hπ hu (φ := fun _ => (1:ℝ)) continuous_const |
|
|
| lemma fullSlice_L1_bound {u : X → ℝ} {L : ℝ≥0} (hu : LipschitzWith L u) |
| (h : NormalApprox (k+1) T) (hX : IsCAT0 X) |
| (π : Fin (k+1) → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) : |
| (∫ z, |fullSlice h π z u (fun j => Fin.elim0 j)|) ≤ |
| (K:ℝ)^(k+1)*∫ x, |u x| ∂currentMassMeasure hT := by |
| obtain ⟨B,hB⟩ := (boundedLip_of_lipschitz hu).2 |
| let B' : ℝ≥0 := ⟨max B 0,le_max_right _ _⟩ |
| have hh := fullSlice_borel_L1_bound h hX π hπ hu.continuous.measurable B' |
| (fun x => (hB x).trans (le_max_left _ _)) |
| simpa only [currentBorelAction_eq_action (fullSlice_approx h π _).metric |
| ⟨boundedLip_of_lipschitz hu,fun j => Fin.elim0 j⟩] using hh |
|
|
| theorem totallyBounded_whole_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 : ι → ℝ≥0) (hu : ∀ i, LipschitzWith (L i) (u i)) |
| {χ : X → ℝ} {J : ℝ≥0} (hχ : LipschitzWith J χ) |
| (M : ℝ≥0) (hMv : ∀ i, ‖value (hT := hT) (hu i)‖ ≤ M) |
| (hMg : ∀ i, ‖q.gradient (hu i)‖ ≤ M) : |
| TotallyBounded (range fun i => |
| (fullSlice_eval_integrable h hX π (fun a => boundedLip_of_lipschitz (hπ a)) |
| ((boundedLip_of_lipschitz hχ).mul (boundedLip_of_lipschitz (hu i))) |
| (fun j => Fin.elim0 j)).toL1 |
| (fun z => fullSlice h π z (fun x => χ x*u i x) (fun j => Fin.elim0 j))) := by |
| obtain ⟨B₀,hB₀⟩ := (boundedLip_of_lipschitz hχ).2 |
| let B : ℝ≥0 := ⟨max B₀ 0,le_max_right _ _⟩ |
| have hB x : |χ x| ≤ B := (hB₀ x).trans (le_max_left _ _) |
| let v i := fun x => χ x*u i x |
| have hvb i : BoundedLip (v i) := (boundedLip_of_lipschitz hχ).mul (boundedLip_of_lipschitz (hu i)) |
| choose N hv using fun i => (hvb i).1 |
| let f i := fun z => fullSlice h π z (v i) (fun j => Fin.elim0 j) |
| have hfi i : Integrable (f i) := fullSlice_eval_integrable h hX π |
| (fun a => boundedLip_of_lipschitz (hπ a)) (hvb i) _ |
| let ν i := q.intrinsicSliceControl π K (v i) |
| have : ∀ index, IsFiniteMeasure (ν index) := |
| fun index => q.intrinsicSliceControl_finite π K (hv index) |
| let m := Real.sqrt ((currentMassMeasure hT).real univ) |
| let a := Real.sqrt (q.atlasMeasure.real univ) |
| let C : ℝ≥0 := ⟨(K:ℝ)^k*(B*M*a+J*M*m),by dsimp [a,m]; positivity⟩ |
| let D : ℝ≥0 := ⟨(K:ℝ)^(k+1)*B*M*m,by dsimp [m]; positivity⟩ |
| obtain ⟨K',hK'⟩ := coordinateMap_lipschitz π hπ |
| obtain ⟨R,hR⟩ := (isCompact_range hK'.continuous).isBounded.subset_closedBall (0:Euc (k+1)) |
| apply totallyBounded_L1_of_directional_variation f hfi ν |
| (fun i l => q.fullSlice_intrinsic_directional_variation h hz hX π hπ (hv i) l) D ?_ C ?_ R ?_ |
| · intro i |
| apply (fullSlice_L1_bound (hv i) h hX π hπ).trans |
| have hint : (∫ x, |v i x| ∂currentMassMeasure hT) ≤ |
| B*∫ x, |u i x| ∂currentMassMeasure hT := by |
| rw [←integral_const_mul] |
| apply integral_mono (integrable_boundedLip _ (hvb i)).abs |
| ((integrable_boundedLip _ (boundedLip_of_lipschitz (hu i))).abs.const_mul B) |
| intro x |
| exact (abs_mul _ _).le.trans (mul_le_mul_of_nonneg_right (hB x) (abs_nonneg _)) |
| have hil := integral_norm_le_lpNorm_two ((boundedLip_of_lipschitz (hu i)).memLp (μ := currentMassMeasure hT) 2) |
| rw [←norm_value (hT := hT) (hu i)] at hil |
| simp only [Real.norm_eq_abs] at hil |
| have him : (∫ x, |u i x| ∂currentMassMeasure hT) ≤ M*m := |
| hil.trans (mul_le_mul_of_nonneg_right (hMv i) (Real.sqrt_nonneg _)) |
| change _ ≤ (K:ℝ)^(k+1)*B*M*m |
| nlinarith [pow_nonneg K.coe_nonneg (k+1),mul_le_mul_of_nonneg_left hint (pow_nonneg K.coe_nonneg (k+1)), |
| mul_le_mul_of_nonneg_left him (mul_nonneg (pow_nonneg K.coe_nonneg (k+1)) B.coe_nonneg)] |
| · intro i |
| rw [intrinsicSliceControl_total q π hπ (hv i)] |
| apply (mul_le_mul_of_nonneg_left (q.integral_gradient_mul_bound (hu i) hχ B.coe_nonneg hB) |
| (pow_nonneg K.coe_nonneg k)).trans |
| change _ ≤ (K:ℝ)^k*(B*M*a+J*M*m) |
| apply mul_le_mul_of_nonneg_left _ (pow_nonneg K.coe_nonneg k) |
| exact add_le_add |
| (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hMg i) B.coe_nonneg) (Real.sqrt_nonneg _)) |
| (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left (hMv i) J.coe_nonneg) (Real.sqrt_nonneg _)) |
| · intro i |
| obtain ⟨B',hB'⟩ := (hvb i).2 |
| let B'' : ℝ≥0 := ⟨max B' 0,le_max_right _ _⟩ |
| have hs := ae_fullSlice_borel_support h hX π (fun a => boundedLip_of_lipschitz (hπ a)) |
| (hv i).continuous.measurable B'' (fun x => (hB' x).trans (le_max_left _ _)) |
| filter_upwards [hs] with z hz |
| intro hnot |
| have hh := hz (fun hh => hnot (hR hh)) |
| simpa only [currentBorelAction_eq_action (fullSlice_approx h π z).metric |
| ⟨hvb i,fun j => Fin.elim0 j⟩] using hh |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
