| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Gradient.Pairing |
| import OAI.Geometry.CAT0Fillings.Compactness.IntrinsicBV |
|
|
| namespace OAI |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open Foundations MassMeasure Slicing BorelCoefficients BorelRestriction |
|
|
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma exists_exact_inverse_coordinates (i : ℕ) : |
| ∃ (F : Fin n → X → ℝ) (K : ℝ≥0), (∀ a, LipschitzWith K (F a)) ∧ |
| (∀ z ∈ (q.chart i).domain, coordinateMap F ((q.chart i).paramExtended z) = z) ∧ |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, (q.chart i).jacobian F z = 1 := by |
| obtain ⟨K₁,K₂,hK₁,hK₂⟩ := (q.chart i).bilipschitz |
| have hrow : ∀ a y (hy : y ∈ (q.chart i).domain) z (hz : z ∈ (q.chart i).domain), |
| |EuclideanSpace.proj a y-EuclideanSpace.proj a z| ≤ |
| (K₂:ℝ)*dist ((q.chart i).param ⟨y,hy⟩) ((q.chart i).param ⟨z,hz⟩) := by |
| intro a y hy z hz |
| have hh := PiLp.norm_apply_le (p := (2:ℝ≥0∞)) (y-z) a |
| have he : |EuclideanSpace.proj a y-EuclideanSpace.proj a z| ≤ dist y z := by |
| change |y.ofLp a-z.ofLp a| ≤ ‖y-z‖ |
| simpa only [Real.norm_eq_abs,PiLp.sub_apply] using hh |
| exact he.trans (hK₂.le_mul_dist ⟨y,hy⟩ ⟨z,hz⟩) |
| obtain ⟨F,hF,hFeq,hjac⟩ := (q.chart i).exists_inverse_coordinate_tests |
| (q.chart i).borel subset_rfl (fun a => EuclideanSpace.proj a) K₂ hrow |
| refine ⟨F,K₂,hF,?_,?_⟩ |
| · intro z hz |
| ext a |
| have hh := hFeq a z hz |
| change F a ((q.chart i).paramExtended z) = (EuclideanSpace.proj a) z |
| simpa only [IntegerChart.scalar,IntegerChart.paramExtended,dite_eq_left hz] using hh |
| · have he : (Matrix.of fun i j : Fin n => EuclideanSpace.proj i (EuclideanSpace.single j (1:ℝ))) = 1 := by |
| ext i j |
| simp [Matrix.one_apply] |
| simpa only [he,Matrix.det_one] using hjac |
|
|
| lemma chart_isolated_action (i : ℕ) {b : X → ℝ} (hb : BoundedLip b) |
| (F : Fin n → X → ℝ) (hF : ∀ a, ∃ K : ℝ≥0, LipschitzWith K (F a)) : |
| currentBorelAction T ((q.chart i).image.indicator b) F = (q.chart i).action b F := by |
| rw [currentBorelAction_eq hT,←restrictCurrent_apply hT (q.chart i).image ⟨hb,hF⟩] |
| rw [chartFamily_restrict_eq hT q.chart q.disjoint q.current q.summable q.action i] |
|
|
| lemma ae_isolated_projected_density (h : NormalApprox n T) (hX : IsCAT0 X) |
| (i : ℕ) (F : Fin n → X → ℝ) {K : ℝ≥0} (hF : ∀ a, LipschitzWith K (F a)) |
| (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 : X → ℝ} {L : ℝ≥0} (hu : LipschitzWith L u) : |
| (fun z => currentBorelAction (fullSlice h F z) ((q.chart i).image.indicator u) (fun j => Fin.elim0 j)) =ᵐ[volume] |
| (q.chart i).domain.indicator (fun z => ((q.chart i).multiplicity z:ℝ)*u ((q.chart i).paramExtended z)) := by |
| let C := q.chart i |
| obtain ⟨U,hU⟩ := (boundedLip_of_lipschitz hu).2 |
| let B : ℝ≥0 := ⟨max U 0,le_max_right _ _⟩ |
| have huB x : |u x| ≤ B := (hU x).trans (le_max_left _ _) |
| have hbu : Measurable (C.image.indicator u) := hu.continuous.measurable.indicator C.measurableSet_image |
| have hbb x : |C.image.indicator u x| ≤ B := by |
| by_cases hx : x ∈ C.image |
| · simpa only [indicator_of_mem hx] using huB x |
| · simpa only [indicator_of_notMem hx,abs_zero] using B.coe_nonneg |
| have hi := (fullSlice_borel_integral h hX F (fun a => boundedLip_of_lipschitz (hF a)) hbu B hbb).1 |
| have hden := integrable_weighted_comp (volume.restrict C.domain) C.measurable_paramExtended |
| C.integrable hu.continuous.measurable ⟨B,huB⟩ |
| have hden' : Integrable (C.domain.indicator (fun z => (C.multiplicity z:ℝ)*u (C.paramExtended z))) := |
| (integrable_indicator_iff C.borel).mpr hden |
| apply ae_eq_of_integral_contDiff_smul_eq hi.locallyIntegrable hden'.locallyIntegrable |
| intro φ hφ hφs |
| obtain ⟨J,hJ⟩ := ContDiff.lipschitzWith_of_hasCompactSupport hφs hφ (by simp) |
| obtain ⟨M,hM⟩ := coordinateMap_lipschitz F hF |
| have hφb : BoundedLip φ := by |
| obtain ⟨V,hV⟩ := hφ.continuous.bounded_above_of_compact_support hφs |
| exact ⟨⟨J,hJ⟩,V,fun x => by simpa only [Real.norm_eq_abs] using hV x⟩ |
| obtain ⟨V,hV⟩ := hφb.2 |
| let D : ℝ≥0 := ⟨max V 0,le_max_right _ _⟩ |
| have hφD x : |φ x| ≤ D := (hV x).trans (le_max_left _ _) |
| have hw := fullSlice_borel_weighted_integral h hX F hF hbu B hbb hφ.continuous.measurable D hφD |
| have he : (fun x => φ (coordinateMap F x)*C.image.indicator u x) = |
| C.image.indicator (fun x => φ (coordinateMap F x)*u x) := by |
| ext x; by_cases hx : x ∈ C.image <;> simp [hx] |
| simp only [smul_eq_mul] |
| have hbφ : BoundedLip (fun x => φ (coordinateMap F x)) := boundedLip_of_lipschitz (hJ.comp hM) |
| have hprod := hbφ.mul (boundedLip_of_lipschitz hu) |
| rw [←hw.2,he,q.chart_isolated_action i hprod F (fun a => ⟨K,hF a⟩)] |
| rw [IntegerChart.action,ite_eq_left ⟨hprod,fun a => ⟨K,hF a⟩⟩] |
| have hei : (fun z => φ z*C.domain.indicator (fun z => (C.multiplicity z:ℝ)*u (C.paramExtended z)) z) = |
| C.domain.indicator (fun z => φ z*((C.multiplicity z:ℝ)*u (C.paramExtended z))) := by |
| ext z; by_cases hz : z ∈ C.domain <;> simp [hz] |
| rw [hei,integral_indicator C.borel] |
| apply integral_congr_ae |
| filter_upwards [ae_restrict_mem C.borel,hjac] with z hz hjz |
| have hh := hFinv z hz |
| simp only [IntegerChart.scalar,IntegerChart.paramExtended,dite_eq_left hz] at hh ⊢ |
| change (C.multiplicity z:ℝ)*(_)*C.jacobian F z = _ |
| rw [hjz,hh] |
| have hzC : z ∈ C.domain := hz |
| simp only [dite_eq_left hzC] |
| ring |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory Matrix Metric |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {n : ℕ} {T : Functional X n} |
| {hT : IsMetricCurrent T} (q : ChartGeometry hT) |
|
|
| lemma ae_chart_jacobian_le (i : ℕ) {K : ℝ≥0} |
| (hK : LipschitzWith K (q.chart i).param) : |
| ∀ᵐ z ∂volume.restrict (q.chart i).domain, |
| Real.sqrt (q.gram i z).det ≤ (K:ℝ)^n := by |
| filter_upwards [q.differential i,ae_restrict_mem (q.chart i).borel, |
| Besicovitch.ae_tendsto_measure_inter_div volume (q.chart i).domain] with z hz hzi hzd |
| have hb : ∀ v, q.field i z v ≤ (K:ℝ)*‖v‖ := by |
| apply (hz.1 hzi).seminorm_le_on_piece volume subset_rfl hzd (q := normSeminorm ℝ (Euc n)) |
| intro y hy |
| change dist ((q.chart i).param ⟨y,hy⟩) ((q.chart i).param ⟨z,hzi⟩) ≤ (K:ℝ)*‖y-z‖ |
| simpa only [Subtype.dist_eq,dist_eq_norm] using hK.dist_le_mul ⟨y,hy⟩ ⟨z,hzi⟩ |
| apply sqrt_det_le_pow_of_quadratic_bound _ |
| (polarizationMatrix_posDef _ _ hz.2.2 hz.2.1).posSemidef K.coe_nonneg |
| intro v |
| let w : Euc n := WithLp.toLp 2 v |
| have hrepr : (EuclideanSpace.basisFun (Fin n) ℝ).toBasis.repr w = v := by ext j; simp [w] |
| have he : v ⬝ᵥ (q.gram i z).mulVec v = (q.field i z w)^2 := by |
| simpa only [gram,hrepr] using polarizationMatrix_quadratic |
| (EuclideanSpace.basisFun (Fin n) ℝ).toBasis (q.field i z) hz.2.2 hz.2.1 w |
| change Real.sqrt (v ⬝ᵥ (q.gram i z).mulVec v) ≤ _ |
| rw [he,Real.sqrt_sq (apply_nonneg _ _)] |
| exact hb w |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings |
| lemma integral_abs_mul_le_lpNorm_two {α : Type*} [MeasurableSpace α] |
| {μ : Measure α} {f g : α → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| (∫ x, |f x| * |g x| ∂μ) ≤ lpNorm f 2 μ*lpNorm g 2 μ := by |
| have hp : (2:ℝ).HolderConjugate 2 := by norm_num [Real.holderConjugate_iff] |
| have hh := integral_mul_norm_le_Lp_mul_Lq hp (by simpa using hf) (by simpa using hg) |
| rw [lpNorm_eq_integral_norm_rpow_toReal (by norm_num) (by norm_num) hf.aestronglyMeasurable, |
| lpNorm_eq_integral_norm_rpow_toReal (by norm_num) (by norm_num) hg.aestronglyMeasurable] |
| simpa only [Real.norm_eq_abs,ENNReal.toReal_ofNat,one_div] using hh |
|
|
| namespace BorelCoefficients |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
|
|
| noncomputable def lipToL2 : lipAlgebra (X := X) →ₗ[ℝ] Lp ℝ 2 μ := |
| (ContinuousMap.toLp 2 μ ℝ).toLinearMap.comp (lipAlgebra (X := X)).val.toLinearMap |
|
|
| lemma lipToL2_dense : DenseRange (lipToL2 μ) := by |
| apply (ContinuousMap.toLp_denseRange ℝ μ ℝ (by simp : (2:ℝ≥0∞) ≠ ∞)).comp |
| lipAlgebra_dense (ContinuousMap.toLp 2 μ ℝ).continuous |
|
|
| lemma exists_lipschitz_L2_approximation {b : X → ℝ} (hb : MemLp b 2 μ) : |
| ∃ f : ℕ → lipAlgebra (X := X), |
| Tendsto (fun j => lpNorm (fun x => (f j).val x-b x) 2 μ) atTop (𝓝 0) := by |
| obtain ⟨u,hu,hulim⟩ := mem_closure_iff_seq_limit.mp ((lipToL2_dense μ) (hb.toLp b)) |
| choose f hf using hu |
| have hfun : (fun j => lipToL2 μ (f j)) = u := funext hf |
| have hnorm := (tendsto_iff_norm_sub_tendsto_zero).mp (hfun ▸ hulim) |
| refine ⟨f,?_⟩ |
| convert hnorm using 1 |
| funext j |
| rw [←lpNorm_Lp] |
| rw [←toReal_eLpNorm,←toReal_eLpNorm] |
| congr 1 |
| apply eLpNorm_congr_ae |
| filter_upwards [Lp.coeFn_sub (lipToL2 μ (f j)) (hb.toLp b), |
| ContinuousMap.coeFn_toLp (p := 2) (𝕜 := ℝ) μ (f j).val,hb.coeFn_toLp] with x hx hfx hbx |
| change (f j).val x-b x = (lipToL2 μ (f j)-hb.toLp b) x |
| rw [hx] |
| change _ = (ContinuousMap.toLp 2 μ ℝ (f j).val) x-(hb.toLp b) x |
| rw [hfx,hbx] |
|
|
| end BorelCoefficients |
| end CAT0Fillings |
| end |
|
|
| end OAI |
|
|
