| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.AECertificates |
| import OAI.Analysis.BiholderTransport.Calculus.TaylorPositivity |
| import OAI.Analysis.BiholderTransport.Calculus.LowerTaylorComposition |
| import OAI.Analysis.BiholderTransport.Coordinates.Endpoint |
| import OAI.Analysis.BiholderTransport.Calculus.TaylorAlgebra |
| import OAI.Analysis.BiholderTransport.Coordinates.Stationary |
| import OAI.Analysis.BiholderTransport.Coordinates.Add |
| import OAI.Analysis.BiholderTransport.Regularity.FirstActiveLimit |
| import OAI.Analysis.BiholderTransport.Regularity.CenterPoleStationary |
| import OAI.Analysis.BiholderTransport.Coordinates.ExpNonconjugateLimit |
| import OAI.Analysis.BiholderTransport.Regularity.CenterSequenceRegular |
| import OAI.Analysis.BiholderTransport.Regularity.MaximumReindex |
| import OAI.Analysis.BiholderTransport.Convexity.ChartJensenFirstLimit |
| import OAI.Analysis.BiholderTransport.Oscillation.ClassPower |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open MeasureTheory Manifold Bundle |
| open scoped Topology ContDiff |
|
|
| namespace WeakMTWTransport |
|
|
| theorem uniform_biHolder_transport |
| {n : ℕ} (hn : 2 ≤ n) {M : Type*} |
| [MetricSpace M] [CompactSpace M] [ConnectedSpace M] |
| [ChartedSpace (Model n) M] [IsManifold 𝓘(ℝ, Model n) ∞ M] |
| [RiemannianBundle (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)] |
| [IsContMDiffRiemannianBundle 𝓘(ℝ, Model n) ∞ (Model n) |
| (fun x : M => TangentSpace 𝓘(ℝ, Model n) x)] |
| [IsRiemannianManifold 𝓘(ℝ, Model n) M] |
| [MeasurableSpace M] [BorelSpace M] |
| (hmtw : WeakMTW (n := n) (M := M)) |
| (lam cap : ℝ) (hlam : 0 < lam) (hcap : lam ≤ cap) |
| (hclass : ∃ rho : M → ℝ, AdmissibleDensity (metricVolume n) lam cap rho) : |
| ∃ alpha C : ℝ, 0 < alpha ∧ alpha ≤ 1 ∧ 0 ≤ C ∧ |
| ∀ rho0 rho1 : M → ℝ, |
| AdmissibleDensity (metricVolume n) lam cap rho0 → |
| AdmissibleDensity (metricVolume n) lam cap rho1 → |
| ∃ T : M ≃ₜ M, |
| IsOptimalMap (metricVolume n) rho0 rho1 T ∧ |
| (∀ S : M → M, IsOptimalMap (metricVolume n) rho0 rho1 S → |
| S =ᵐ[densityMeasure (metricVolume n) rho0] T) ∧ |
| BiHolderEstimate alpha C T := by |
| classical |
| let x0:M:=Classical.choice (inferInstance:Nonempty M) |
| exact hmtw.transport_endpoint_of_class_power hlam x0 |
| (hmtw.class_power (by omega) hlam hcap x0 (densityDualClass_nonempty hlam hclass x0)) |
|
|
| end WeakMTWTransport |
|
|
| end |
| end |
|
|
| end OAI |
|
|
