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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.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
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