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b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 | 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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