| import OAI.Geometry.RCD.Sobolev.ScaledCompactFactorLipschitz |
| import OAI.Geometry.RCD.Metric.CenteredBallCompactness |
|
|
| namespace OAI |
|
|
| /-! # Spatial compactness of the actual normalized entropic potentials |
|
|
| Endpoint scaling functions and interior times may vary along the sequence. |
| Only the common support, noise range and interior-time margin enter the |
| uniform Lipschitz bound. This is spatial compactness; no identification |
| with a transport potential or time convergence is asserted here. |
| -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal ENNReal Topology |
| noncomputable section |
| namespace RegularPointCoordinates.NoncollapsedRCD |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) |
| (hn : 2≤n) (p : X) [SeparableSpace X] |
| [IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular] |
|
|
| def scaledCompactHeatPotential {S : Set X} [CompactSpace S] |
| (ν : Measure S) [IsProbabilityMeasure ν] (F : C(S, ℝ)) (ε s : ℝ) |
| (ht : 0<ε*s/2) (x : X) : ℝ := |
| ε*Real.log (hX.compactHeatFactor hn p ν F (ε*s/2) ht x) |
|
|
| theorem compact_heat_factor_centered_compactness {S : Set X} [CompactSpace S] |
| (ν : Measure S) [IsProbabilityMeasure ν] (ρ : X → ℝ) |
| (hρ : Measurable ρ) (hρ0 : ∀ x, 0≤ρ x) {B : ℝ≥0} |
| (hB : ∀ᵐ x ∂hausdorff n X, |ρ x|≤B) |
| (hν : ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x))) |
| (F : ℕ → C(S, ℝ)) (hF : ∀ k a, 0<F k a) |
| (ε s : ℕ → ℝ) (hε : ∀ k, 0<ε k) (hε1 : ∀ k, ε k≤1) |
| {δ : ℝ} (hδ : 0<δ) (hδs : ∀ k, δ≤ s k) (hs1 : ∀ k, s k≤1) |
| (z : X) {R r : ℝ} (hR : 0≤R) (hdist : ∀ a : S, dist z a≤R) (hr : 0<r) : |
| ∃ (L : ℝ≥0) (g : closedBall z r → ℝ) (a : ℕ → ℕ), StrictMono a ∧ |
| LipschitzWith L g ∧ g ⟨z,mem_closedBall_self hr.le⟩=0 ∧ |
| TendstoUniformly (fun k (x : closedBall z r) => |
| hX.scaledCompactHeatPotential hn p ν (F (a k)) (ε (a k)) (s (a k)) |
| (div_pos (mul_pos (hε _) (hδ.trans_le (hδs _))) (by norm_num)) x- |
| hX.scaledCompactHeatPotential hn p ν (F (a k)) (ε (a k)) (s (a k)) |
| (div_pos (mul_pos (hε _) (hδ.trans_le (hδs _))) (by norm_num)) z) g atTop := by |
| obtain ⟨C,D,_hC,_hD,hdata⟩ := hX.compact_heat_factor_scaled_lipschitz hn p |
| let L : ℝ≥0 := ⟨Real.sqrt (Real.exp |K| * (4*C/δ+16*D*(R+8*(r+1))^2/δ^2)),Real.sqrt_nonneg _⟩ |
| let u := fun k => hX.scaledCompactHeatPotential hn p ν (F k) (ε k) (s k) |
| (div_pos (mul_pos (hε k) (hδ.trans_le (hδs k))) (by norm_num)) |
| have hu (k : ℕ) : LipschitzOnWith L (u k) (ball z (r+1)) := |
| hdata ν ρ hρ hρ0 hB hν (F k) (hF k) (hε k) (hε1 k) hδ (hδs k) (hs1 k) |
| z R hR hdist (by linarith) |
| obtain ⟨g,a,ha,hg,hg0,hlim⟩ := centered_ball_subsequence z hr u L hu |
| exact ⟨L,g,a,ha,hg,hg0,hlim⟩ |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
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|
