| import Mathlib |
| import OAI.Geometry.CAT0Fillings.Compactness.ChartDensity |
|
|
| namespace OAI |
|
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| section |
|
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| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.Slicing |
| open Foundations MassMeasure BorelCoefficients |
|
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [CompactSpace X] [Nonempty X] {k : ℕ} {T : Functional X k} |
|
|
| lemma fullSlice_borel_sub (h : NormalApprox k T) (π : Fin k → X → ℝ) (z : Euc k) |
| {b c : X → ℝ} (hb : Measurable b) (hc : Measurable c) |
| {B C : ℝ} (hB : ∀ x, |b x| ≤ B) (hC : ∀ x, |c x| ≤ C) : |
| currentBorelAction (fullSlice h π z) (fun x => b x-c x) (fun j => Fin.elim0 j) = |
| currentBorelAction (fullSlice h π z) b (fun j => Fin.elim0 j)- |
| currentBorelAction (fullSlice h π z) c (fun j => Fin.elim0 j) := by |
| simp only [currentBorelAction_eq (fullSlice_approx h π z).metric] |
| exact borelAction_sub _ (fullSlice_approx h π z).metric |
| (integrable_bounded_measurable _ hb hB) (integrable_bounded_measurable _ hc hC) _ |
| (fun j => Fin.elim0 j) |
|
|
| lemma fullSlice_borel_difference_bound (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| {b c : X → ℝ} (hb : Measurable b) (hc : Measurable c) |
| {B C : ℝ≥0} (hB : ∀ x, |b x| ≤ B) (hC : ∀ x, |c x| ≤ C) : |
| (∫ z, |currentBorelAction (fullSlice h π z) b (fun j => Fin.elim0 j)- |
| currentBorelAction (fullSlice h π z) c (fun j => Fin.elim0 j)|) ≤ |
| (K:ℝ)^k*∫ x, |b x-c x| ∂currentMassMeasure h.metric := by |
| have hd x : |b x-c x| ≤ (B+C:ℝ≥0) := |
| (abs_sub _ _).trans (add_le_add (hB x) (hC x)) |
| have hh := fullSlice_borel_L1_bound h hX π hπ (hb.sub hc) (B+C) hd |
| change (∫ z, |currentBorelAction (fullSlice h π z) (fun x => b x-c x) (fun j => Fin.elim0 j)|) ≤ |
| (K:ℝ)^k*∫ x, |b x-c x| ∂currentMassMeasure h.metric at hh |
| simp only [fullSlice_borel_sub h π _ hb hc hB hC] at hh |
| exact hh |
|
|
| lemma fullSlice_cutoff_difference_bound (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| {b c u : X → ℝ} (hb : Measurable b) (hc : Measurable c) (hu : Measurable u) |
| {B C U : ℝ≥0} (hB : ∀ x, |b x| ≤ B) (hC : ∀ x, |c x| ≤ C) (hU : ∀ x, |u x| ≤ U) : |
| (∫ z, |currentBorelAction (fullSlice h π z) (fun x => b x*u x) (fun j => Fin.elim0 j)- |
| currentBorelAction (fullSlice h π z) (fun x => c x*u x) (fun j => Fin.elim0 j)|) ≤ |
| (K:ℝ)^k*lpNorm (fun x => b x-c x) 2 (currentMassMeasure h.metric)* |
| lpNorm u 2 (currentMassMeasure h.metric) := by |
| have hbu 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 |
| have hcu x : |c x*u x| ≤ (C*U:ℝ≥0) := by |
| rw [abs_mul]; exact mul_le_mul (hC x) (hU x) (abs_nonneg _) C.coe_nonneg |
| apply (fullSlice_borel_difference_bound h hX π hπ (hb.mul hu) (hc.mul hu) hbu hcu).trans |
| rw [mul_assoc] |
| apply mul_le_mul_of_nonneg_left _ (pow_nonneg K.coe_nonneg _) |
| have hb2 : MemLp b 2 (currentMassMeasure h.metric) := |
| MemLp.of_bound hb.aestronglyMeasurable B (Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hB x)) |
| have hc2 : MemLp c 2 (currentMassMeasure h.metric) := |
| MemLp.of_bound hc.aestronglyMeasurable C (Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hC x)) |
| have hu2 : MemLp u 2 (currentMassMeasure h.metric) := |
| MemLp.of_bound hu.aestronglyMeasurable U (Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hU x)) |
| have he : (fun x => |b x*u x-c x*u x|) = (fun x => |(b-c) x| * |u x|) := by |
| funext x |
| rw [←sub_mul,abs_mul] |
| rfl |
| change (∫ x, |b x*u x-c x*u x| ∂currentMassMeasure h.metric) ≤ _ |
| rw [he] |
| exact integral_abs_mul_le_lpNorm_two (hb2.sub hc2) hu2 |
|
|
| noncomputable def projectedL1 (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| {b : X → ℝ} (hb : Measurable b) (B : ℝ≥0) (hB : ∀ x, |b x| ≤ B) : Lp ℝ 1 (volume : Measure (Euc k)) := |
| (fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb B hB).1.toL1 _ |
|
|
| lemma dist_projectedL1 (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| {b c : X → ℝ} (hb : Measurable b) (hc : Measurable c) |
| {B C : ℝ≥0} (hB : ∀ x, |b x| ≤ B) (hC : ∀ x, |c x| ≤ C) : |
| dist (projectedL1 h hX π hπ hb B hB) (projectedL1 h hX π hπ hc C hC) = |
| ∫ z, |currentBorelAction (fullSlice h π z) b (fun j => Fin.elim0 j)- |
| currentBorelAction (fullSlice h π z) c (fun j => Fin.elim0 j)| := by |
| rw [dist_eq_norm,L1.norm_eq_integral_norm] |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (projectedL1 h hX π hπ hb B hB) (projectedL1 h hX π hπ hc C hC), |
| (fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb B hB).1.coeFn_toL1, |
| (fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hc C hC).1.coeFn_toL1] |
| with z hz hzb hzc |
| rw [hz] |
| change ‖((fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb B hB).1.toL1 _) z- |
| ((fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hc C hC).1.toL1 _) z‖ = _ |
| rw [hzb,hzc,Real.norm_eq_abs] |
|
|
| lemma projectedL1_lipschitz (h : NormalApprox k T) (hX : IsCAT0 X) |
| (π : Fin k → X → ℝ) {K : ℝ≥0} (hπ : ∀ i, LipschitzWith K (π i)) |
| {b : X → ℝ} (hb : BoundedLip b) {B : ℝ≥0} (hB : ∀ x, |b x| ≤ B) : |
| projectedL1 h hX π hπ hb.continuous.measurable B hB = |
| (fullSlice_eval_integrable h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb |
| (fun j => Fin.elim0 j)).toL1 (fun z => fullSlice h π z b (fun j => Fin.elim0 j)) := by |
| apply Lp.ext |
| filter_upwards [(fullSlice_borel_integral h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb.continuous.measurable B hB).1.coeFn_toL1, |
| (fullSlice_eval_integrable h hX π (fun i => boundedLip_of_lipschitz (hπ i)) hb (fun j => Fin.elim0 j)).coeFn_toL1] |
| with z hz hz' |
| exact hz.trans ((currentBorelAction_eq_action (fullSlice_approx h π z).metric ⟨hb,fun j => Fin.elim0 j⟩).trans hz'.symm) |
|
|
| end CAT0Fillings.Slicing |
| end |
|
|
| section |
|
|
| open Set Filter MeasureTheory |
| open scoped Topology NNReal ENNReal |
|
|
| namespace CAT0Fillings.ChartGeometry |
| open Slicing Foundations MassMeasure BorelCoefficients |
|
|
| 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 coordinate_memLp (i : ℕ) {u : X → ℝ} (hu : Measurable u) (U : ℝ≥0) |
| (hU : ∀ x, |u x| ≤ U) (p : ℝ≥0∞) : |
| MemLp (fun z => u ((q.chart i).paramExtended z)) p (q.coordinateMeasure i) := |
| MemLp.of_bound (hu.comp (q.chart i).measurable_paramExtended).aestronglyMeasurable U |
| (Eventually.of_forall fun z => by simpa only [Real.norm_eq_abs] using hU ((q.chart i).paramExtended z)) |
|
|
| noncomputable def coordinateL1 (i : ℕ) {u : X → ℝ} (hu : Measurable u) (U : ℝ≥0) |
| (hU : ∀ x, |u x| ≤ U) : Lp ℝ 1 (q.coordinateMeasure i) := |
| (q.coordinate_memLp i hu U hU 1).toLp _ |
|
|
| lemma dist_coordinateL1 (i : ℕ) {u v : X → ℝ} (hu : Measurable u) (hv : Measurable v) |
| (U V : ℝ≥0) (hU : ∀ x, |u x| ≤ U) (hV : ∀ x, |v x| ≤ V) : |
| dist (q.coordinateL1 i hu U hU) (q.coordinateL1 i hv V hV) = |
| ∫ z, q.density i z*|u ((q.chart i).paramExtended z)-v ((q.chart i).paramExtended z)| |
| ∂volume.restrict (q.chart i).domain := by |
| rw [dist_eq_norm,L1.norm_eq_integral_norm] |
| have he : (∫ z, ‖(q.coordinateL1 i hu U hU-q.coordinateL1 i hv V hV) z‖ ∂q.coordinateMeasure i) = |
| ∫ z, |u ((q.chart i).paramExtended z)-v ((q.chart i).paramExtended z)| ∂q.coordinateMeasure i := by |
| apply integral_congr_ae |
| filter_upwards [Lp.coeFn_sub (q.coordinateL1 i hu U hU) (q.coordinateL1 i hv V hV), |
| (q.coordinate_memLp i hu U hU 1).coeFn_toLp,(q.coordinate_memLp i hv V hV 1).coeFn_toLp] with z hz huz hvz |
| rw [hz] |
| change ‖((q.coordinate_memLp i hu U hU 1).toLp _) z-((q.coordinate_memLp i hv V hV 1).toLp _) z‖ = _ |
| rw [huz,hvz,Real.norm_eq_abs] |
| rw [he,coordinateMeasure,integral_withDensity_eq_integral_toReal_smul₀ |
| (q.density_integrable i).aestronglyMeasurable.aemeasurable.ennreal_ofReal |
| (Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| simp only [ENNReal.toReal_ofReal (q.density_nonneg i _),smul_eq_mul] |
|
|
| lemma indicator_bound (i : ℕ) {u : X → ℝ} {U : ℝ≥0} (hU : ∀ x, |u x| ≤ U) : |
| ∀ x, |(q.chart i).image.indicator u x| ≤ U := by |
| intro x |
| by_cases hx : x ∈ (q.chart i).image |
| · simpa only [indicator_of_mem hx] using hU x |
| · simpa only [indicator_of_notMem hx,abs_zero] using U.coe_nonneg |
|
|
| noncomputable def isolatedL1 (h : NormalApprox n T) (hX : IsCAT0 X) (i : ℕ) |
| (F : Fin n → X → ℝ) {K : ℝ≥0} (hF : ∀ a, LipschitzWith K (F a)) |
| {u : X → ℝ} (hu : Measurable u) (U : ℝ≥0) (hU : ∀ x, |u x| ≤ U) : Lp ℝ 1 (volume : Measure (Euc n)) := |
| projectedL1 h hX F hF (hu.indicator (q.chart i).measurableSet_image) U (q.indicator_bound i hU) |
|
|
| lemma dist_isolatedL1 (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 v : X → ℝ} {L J : ℝ≥0} (hu : LipschitzWith L u) (hv : LipschitzWith J v) |
| (U V : ℝ≥0) (hU : ∀ x, |u x| ≤ U) (hV : ∀ x, |v x| ≤ V) : |
| 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) = |
| ∫ z, |((q.chart i).multiplicity z:ℝ)| * |u ((q.chart i).paramExtended z)-v ((q.chart i).paramExtended z)| |
| ∂volume.restrict (q.chart i).domain := by |
| rw [isolatedL1,isolatedL1,dist_projectedL1] |
| rw [←integral_indicator (q.chart i).borel] |
| apply integral_congr_ae |
| filter_upwards [q.ae_isolated_projected_density h hX i F hF hFinv hjac hu, |
| q.ae_isolated_projected_density h hX i F hF hFinv hjac hv] with z huz hvz |
| rw [huz,hvz] |
| by_cases hz : z ∈ (q.chart i).domain |
| · simp only [indicator_of_mem hz,←mul_sub,abs_mul] |
| · simp only [indicator_of_notMem hz,sub_self,abs_zero] |
|
|
| end CAT0Fillings.ChartGeometry |
| end |
|
|
| end OAI |
|
|
