| import Mathlib | |
| import OAI.Geometry.IntegralFillings.Charts.Differentiability | |
| import OAI.Geometry.IntegralFillings.Charts.Jacobian | |
| namespace OAI | |
| section | |
| open Set Filter MeasureTheory | |
| open scoped Topology ENNReal NNReal | |
| open MeasureTheory Filter Set Metric | |
| open scoped Topology Pointwise NNReal | |
| namespace SharpIntegralFillings | |
| namespace IntegerChart | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [Nonempty X] | |
| {k : ℕ} (C : IntegerChart X k) | |
| noncomputable def paramExtended : Euc k → X := | |
| by classical exact fun z => if hz : z ∈ C.domain then C.param ⟨z,hz⟩ else Classical.arbitrary X | |
| lemma measurable_paramExtended : Measurable C.paramExtended := by | |
| classical | |
| obtain ⟨L,U,hL,_⟩ := C.bilipschitz | |
| exact hL.continuous.measurable.dite measurable_const C.borel | |
| omit [MeasurableSpace X] [BorelSpace X] in | |
| lemma scalar_ae_paramExtended (b : X → ℝ) : | |
| C.scalar b =ᵐ[volume.restrict C.domain] b ∘ C.paramExtended := by | |
| filter_upwards [ae_restrict_mem C.borel] with z hz | |
| simp only [scalar,paramExtended,Function.comp_apply,dite_eq_left hz] | |
| variable [CompactSpace X] | |
| omit [Nonempty X] in | |
| lemma measurableEmbedding_param : MeasurableEmbedding C.param := by | |
| let := C.borel.standardBorel | |
| obtain ⟨L,U,hL,hU⟩ := C.bilipschitz | |
| exact hL.continuous.measurable.measurableEmbedding hU.injective | |
| omit [MeasurableSpace X] [BorelSpace X] [Nonempty X] [CompactSpace X] in | |
| lemma integrable_weighted_jacobian {π : Fin k → X → ℝ} | |
| (hπ : ∀ i, ∃ K : ℝ≥0, LipschitzWith K (π i)) : | |
| Integrable (fun z => (C.multiplicity z : ℝ)*C.jacobian π z) | |
| (volume.restrict C.domain) := by | |
| obtain ⟨B,hB⟩ := C.exists_jacobian_bound hπ | |
| exact C.integrable.mul_bdd (C.jacobian_aestronglyMeasurable hπ) hB | |
| lemma weighted_jacobian_le_measure | |
| (hC : IsMetricCurrent C.action) {ν : Measure X} [IsFiniteMeasure ν] | |
| (hν : Controls C.action ν) (π : Fin k → X → ℝ) (K : Fin k → ℝ≥0) | |
| (hK : ∀ i, LipschitzWith (K i) (π i)) : | |
| densityPush (volume.restrict C.domain) C.paramExtended | |
| (fun z => |(C.multiplicity z : ℝ)*C.jacobian π z|) ≤ (∏ i, K i) • ν := by | |
| have he : MeasurableEmbedding (fun z : C.domain => C.paramExtended (z : Euc k)) := by | |
| convert C.measurableEmbedding_param using 1 | |
| funext z | |
| simp [paramExtended,z.property] | |
| apply signed_chart_control_dominates_on (volume.restrict C.domain) ((∏ i,K i) • ν) | |
| C.measurable_paramExtended (ae_restrict_mem C.borel) he | |
| (C.integrable_weighted_jacobian (fun i => ⟨K i,hK i⟩)) | |
| intro b hb | |
| have heq : (∫ z, (C.multiplicity z : ℝ)*C.jacobian π z*b (C.paramExtended z) | |
| ∂volume.restrict C.domain) = C.action b π := by | |
| rw [action,ite_eq_left ⟨hb,fun i => ⟨K i,hK i⟩⟩] | |
| apply integral_congr_ae | |
| filter_upwards [C.scalar_ae_paramExtended b] with z hz | |
| dsimp only [Function.comp_apply] at hz | |
| rw [hz] | |
| ring | |
| rw [heq,integral_smul_nnreal_measure] | |
| simpa only [NNReal.smul_def,NNReal.coe_prod,smul_eq_mul] using hC.mass_bound hν hb K hK | |
| end IntegerChart | |
| end SharpIntegralFillings | |
| namespace SharpIntegralFillings | |
| namespace IntegerChart | |
| variable {X : Type*} [MetricSpace X] {k : ℕ} (C : IntegerChart X k) | |
| lemma ae_fderivWithin_scalar_eq_linear {t : Set (Euc k)} | |
| (ht : MeasurableSet t) (hts : t ⊆ C.domain) | |
| {f : X → ℝ} {K : ℝ≥0} (hf : LipschitzWith K f) (ℓ : Euc k →L[ℝ] ℝ) | |
| (heq : ∀ z ∈ t, C.scalar f z = ℓ z) : | |
| ∀ᵐ z ∂volume.restrict t, fderivWithin ℝ (C.scalar f) C.domain z = ℓ := by | |
| have hd := (C.ae_differentiableWithinAt_scalar hf).filter_mono | |
| (ae_mono (Measure.restrict_mono hts le_rfl)) | |
| filter_upwards [hd,ae_uniqueDiffWithinAt volume t,ae_restrict_mem ht] with z hz hu hzt | |
| rw [←fderivWithin_subset hts hu hz,fderivWithin_congr' heq hzt,ℓ.fderivWithin hu] | |
| lemma exists_inverse_coordinate_tests {t : Set (Euc k)} | |
| (ht : MeasurableSet t) (hts : t ⊆ C.domain) | |
| (ℓ : Fin k → Euc k →L[ℝ] ℝ) (K : ℝ≥0) | |
| (hrow : ∀ i y (hy : y ∈ t) z (hz : z ∈ t), | |
| |ℓ i y - ℓ i z| ≤ (K : ℝ)*dist (C.param ⟨y,hts hy⟩) (C.param ⟨z,hts hz⟩)) : | |
| ∃ π : Fin k → X → ℝ, (∀ i, LipschitzWith K (π i)) ∧ | |
| (∀ i, ∀ z ∈ t, C.scalar (π i) z = ℓ i z) ∧ | |
| ∀ᵐ z ∂volume.restrict t, | |
| C.jacobian π z = (Matrix.of fun i j => ℓ i (EuclideanSpace.single j 1)).det := by | |
| classical | |
| by_cases hne : t.Nonempty | |
| · let := hne.to_subtype | |
| let φ : t → X := fun z => C.param ⟨z,hts z.property⟩ | |
| have hinj : Function.Injective φ := by | |
| obtain ⟨L,U,hL,hU⟩ := C.bilipschitz | |
| intro a b hab | |
| apply Subtype.ext | |
| exact congrArg (fun z : C.domain => (z : Euc k)) (hU.injective hab) | |
| let q : Fin k → X → ℝ := fun i x => ℓ i ((Function.invFun φ x : t) : Euc k) | |
| have hq (i : Fin k) : LipschitzOnWith K (q i) (Set.range φ) := by | |
| apply lipschitzOnWith_iff_dist_le_mul.mpr | |
| rintro _ ⟨a,rfl⟩ _ ⟨b,rfl⟩ | |
| simpa only [q,Function.leftInverse_invFun hinj a, | |
| Function.leftInverse_invFun hinj b,Real.dist_eq] using | |
| hrow i a a.property b b.property | |
| choose π hπ heq using fun i => (hq i).extend_real | |
| have hagree (i : Fin k) (z : Euc k) (hz : z ∈ t) : C.scalar (π i) z = ℓ i z := by | |
| rw [C.scalar_eq (hts hz)] | |
| change π i (φ ⟨z,hz⟩) = _ | |
| rw [←heq i (Set.mem_range_self (⟨z,hz⟩ : t))] | |
| dsimp only [q] | |
| rw [Function.leftInverse_invFun hinj] | |
| refine ⟨π,hπ,hagree,?_⟩ | |
| have hae : ∀ᵐ z ∂volume.restrict t, ∀ i, | |
| fderivWithin ℝ (C.scalar (π i)) C.domain z = ℓ i := | |
| ae_all_iff.mpr fun i => C.ae_fderivWithin_scalar_eq_linear ht hts (hπ i) (ℓ i) (hagree i) | |
| filter_upwards [hae] with z hz | |
| simp only [jacobian,hz] | |
| rfl | |
| · have ht0 : t = ∅ := Set.not_nonempty_iff_eq_empty.mp hne | |
| subst t | |
| exact ⟨fun _ _ => 0,fun _ => LipschitzWith.const' (0 : ℝ),by simp,by simp⟩ | |
| end IntegerChart | |
| end SharpIntegralFillings | |
| end | |
| end OAI | |