| import Mathlib |
| import OAI.Analysis.BiholderTransport.Coordinates.ChartCostJets |
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| namespace OAI |
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| noncomputable section |
| open Set Filter Manifold Bundle |
| open scoped Topology ContDiff |
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| namespace WeakMTWTransport |
| section MixedContinuity |
| variable {E F : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F] |
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| local instance genericMixedProdNormedGroup : NormedAddCommGroup (F×E) := Prod.normedAddCommGroup |
| local instance genericMixedProdNormedSpace : NormedSpace ℝ (F×E) := Prod.normedSpace |
| local instance genericMixedDualNormedGroup : NormedAddCommGroup ((F×E) →L[ℝ] ℝ) := inferInstance |
| local instance genericMixedDualNormedSpace : NormedSpace ℝ ((F×E) →L[ℝ] ℝ) := inferInstance |
| end MixedContinuity |
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| variable {n : ℕ} {M : Type*} [MetricSpace M] [CompactSpace 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] |
| local instance tangentFiniteTrivialization (x : M) : |
| FiniteDimensional ℝ (TangentSpace 𝓘(ℝ,Model n) x) := |
| inferInstanceAs (FiniteDimensional ℝ (Model n)) |
| local instance mixedTrivProductNormedGroup (x y : M) : |
| NormedAddCommGroup (TangentSpace 𝓘(ℝ,Model n) y × TangentSpace 𝓘(ℝ,Model n) x) := |
| Prod.normedAddCommGroup |
| local instance mixedTrivProductNormedSpace (x y : M) : |
| NormedSpace ℝ (TangentSpace 𝓘(ℝ,Model n) y × TangentSpace 𝓘(ℝ,Model n) x) := Prod.normedSpace |
| local instance mixedTrivDualNormedGroup (x y : M) : |
| NormedAddCommGroup ((TangentSpace 𝓘(ℝ,Model n) y × TangentSpace 𝓘(ℝ,Model n) x) →L[ℝ] ℝ) := inferInstance |
| local instance mixedTrivDualNormedSpace (x y : M) : |
| NormedSpace ℝ ((TangentSpace 𝓘(ℝ,Model n) y × TangentSpace 𝓘(ℝ,Model n) x) →L[ℝ] ℝ) := inferInstance |
| end WeakMTWTransport |
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| end |
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| noncomputable section |
| open Set Filter |
| open scoped Topology ContDiff |
|
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| namespace WeakMTWTransport |
| lemma compact_parameter_second_jet_on_graph {Q E : Type*} |
| [NormedAddCommGroup Q] [NormedSpace ℝ Q] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| {K : Set Q} (hK : IsCompact K) {v : Q → E} (hv : Continuous v) |
| {F : (ℝ×Q) → E → ℝ} {t : ℝ} |
| (hF : ∀ q∈K, ContDiffAt ℝ ∞ (Function.uncurry F) ((t,q),v q)) : |
| ∃ B>0, ∀ᶠ s in 𝓝 t, ∀ q∈K, |
| ContDiffAt ℝ 2 (F (s,q)) (v q) ∧ |
| ‖fderiv ℝ (fderiv ℝ (F (s,q))) (v q)‖≤B := by |
| let H := fun z : ℝ×Q => fderiv ℝ (fderiv ℝ (F z)) (v z.2) |
| have hc : Continuous (fun z : ℝ×Q => (z,v z.2)) := continuous_id.prodMk (hv.comp continuous_snd) |
| have hH : ∀ q∈K, ContinuousAt H (t,q) := by |
| intro q hq |
| exact (ContDiffAt.partial_snd_fderiv_two (hF q hq)).continuousAt.comp |
| (x := (t,q)) (f := fun z : ℝ×Q => (z,v z.2)) hc.continuousAt |
| obtain ⟨B,hB,HB⟩ := compact_eventually_fiberwise_bound hK |
| (f := fun _ : Q => ()) continuous_const (g := fun _ : ℝ => ()) |
| (continuousAt_const (x := t)) (h := H) (fun q hq _ => hH q hq) |
| have HS : ∀ᶠ s in 𝓝 t, ∀ q∈K, ()=() → ContDiffAt ℝ 2 (F (s,q)) (v q) := by |
| apply compact_eventually_fiberwise hK (f := fun _ : Q => ()) continuous_const |
| (g := fun _ : ℝ => ()) (continuousAt_const (x := t)) |
| intro q hq _ |
| have He := ((hF q hq).of_le |
| (m := 2) (ENat.natCast_le_of_coe_top_le_withTop le_rfl 2)).eventually (by simp) |
| filter_upwards [(hc.continuousAt (x := (t,q))).tendsto.eventually He] with z hz |
| exact hz.comp (v z.2) (contDiffAt_const.prodMk contDiffAt_id) |
| refine ⟨B,hB,?_⟩ |
| filter_upwards [HB,HS] with s hb hs |
| exact fun q hq => ⟨hs q hq rfl,hb q hq rfl⟩ |
| end WeakMTWTransport |
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| end |
|
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| end OAI |
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