| import OAI.Geometry.RCD.Heat.CanonicalMovingLogLower |
|
|
| namespace OAI |
|
|
| /-! # Uniform sharp lower action on compact endpoint sets -/ |
|
|
| 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] |
|
|
| theorem canonicalHeatKernel_compact_log_lower (hK : K ≤ 0) |
| (ε : ℕ → ℝ) (hε : ∀ k, 0 < ε k) (hε1 : ∀ k, ε k ≤ 1) |
| (hε0 : Tendsto ε atTop (𝓝 0)) {A B : Set X} (hA : IsCompact A) (hB : IsCompact B) |
| {T η : ℝ} (hT : 0 < T) (hT1 : T < 1) (hη : 0 < η) : |
| ∀ᶠ k in atTop, ∀ x ∈ A, ∀ y ∈ B, |
| -(dist x y)^2/(2*T)-η ≤ ε k*Real.log (hX.canonicalHeatKernel hn p (ε k*T/2) x y) := by |
| classical |
| by_contra hbad |
| have hfreq : ∃ᶠ k in atTop, ∃ x ∈ A, ∃ y ∈ B, |
| ε k*Real.log (hX.canonicalHeatKernel hn p (ε k*T/2) x y) < -(dist x y)^2/(2*T)-η := by |
| rw [not_eventually] at hbad |
| apply hbad.mono |
| intro k hk |
| by_contra hn |
| apply hk |
| intro x hx y hy |
| exact le_of_not_gt (fun hh => hn ⟨x,hx,y,hy,hh⟩) |
| obtain ⟨a,ha,hbad⟩ := extraction_of_frequently_atTop hfreq |
| choose u hu v hv hf using hbad |
| obtain ⟨q,hq,b,hb,hlim⟩ := (hA.prod hB).tendsto_subseq (fun k => |
| show (u k,v k) ∈ A ×ˢ B from ⟨hu k,hv k⟩) |
| have hl := hX.canonicalHeatKernel_moving_log_lower hn p hK |
| (fun k => ε (a (b k))) (fun k => hε (a (b k))) (fun k => hε1 (a (b k))) |
| (hε0.comp (ha.comp hb).tendsto_atTop) q.1 q.2 |
| (fun k => u (b k)) (fun k => v (b k)) |
| (continuous_fst.tendsto q |>.comp hlim) (continuous_snd.tendsto q |>.comp hlim) |
| hT hT1 (half_pos hη) |
| have hc : Tendsto (fun k => -(dist (u (b k)) (v (b k)))^2/(2*T)-η) atTop |
| (𝓝 (-(dist q.1 q.2)^2/(2*T)-η)) := by |
| have hd := (continuous_fst.tendsto q |>.comp hlim).dist |
| (continuous_snd.tendsto q |>.comp hlim) |
| exact ((hd.pow 2).neg.div_const (2*T)).sub_const η |
| have he := hc.eventually (gt_mem_nhds (by linarith : |
| -(dist q.1 q.2)^2/(2*T)-η < -(dist q.1 q.2)^2/(2*T)-η/2)) |
| obtain ⟨k,hk,hk'⟩ := (hl.and he).exists |
| have hh := hf (b k) |
| linarith |
| |
| end RegularPointCoordinates.NoncollapsedRCD |
| |
| end |
| |
| end OAI |
| |
