| import OAI.Geometry.RCD.Heat.CanonicalStrictPositivity |
| import OAI.Geometry.RCD.Heat.CanonicalGaussianUpper |
| import OAI.Geometry.RCD.Heat.BoundedHeatVolumePower |
| import OAI.Geometry.RCD.Heat.SmallNoiseLogScale |
| import OAI.Geometry.RCD.Heat.GaussianLogUpper |
| import OAI.Geometry.RCD.Heat.GaussianActionSlack |
|
|
| namespace OAI |
|
|
| /-! # Uniform sharp upper action on bounded 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_bounded_log_upper |
| (ε : ℕ → ℝ) (hε : ∀ k, 0 < ε k) (hε1 : ∀ k, ε k ≤ 1) |
| (hε0 : Tendsto ε atTop (𝓝 0)) {R T η : ℝ} |
| (hR : 0 ≤ R) (hT : 0 < T) (hT1 : T < 1) (hη : 0 < η) : |
| ∀ᶠ k in atTop, ∀ x ∈ closedBall p R, ∀ y ∈ closedBall p R, |
| ε k*Real.log (hX.canonicalHeatKernel hn p (ε k*T/2) x y) ≤ -(dist x y)^2/(2*T)+η := by |
| let a := η*T/(2*(R^2+1)) |
| have ha : 0 < a := by dsimp only [a]; positivity |
| obtain ⟨C,D,hC,_hD,hupper⟩ := hX.canonicalHeatKernel_gaussian_upper hn p ha |
| obtain ⟨c,hc,hvolume⟩ := hX.bounded_heat_volume_power hn p R |
| let E := fun k => ε k*Real.log (C/c) - (n:ℝ)*ε k/2*Real.log (ε k*T/2) + D*(ε k)^2*T/2 |
| have hlog : Tendsto (fun k => ε k*Real.log (ε k*T/2)) atTop (𝓝 0) := by |
| simpa only [mul_div_assoc] using small_noise_log_scale hε hε0 (half_pos hT) |
| have hElim : Tendsto E atTop (𝓝 0) := by |
| have hh := ((hε0.mul_const (Real.log (C/c))).sub (hlog.const_mul ((n:ℝ)/2))).add |
| (((hε0.pow 2).const_mul D).mul_const (T/2)) |
| convert hh using 1 |
| · funext k; dsimp only [E]; ring |
| · norm_num |
| filter_upwards [hElim.eventually (gt_mem_nhds (half_pos hη))] with k hk |
| intro x hx y hy |
| have ht : 0 < ε k*T/2 := div_pos (mul_pos (hε k) hT) (by norm_num) |
| have ht1 : ε k*T/2 ≤ 1 := by |
| have hh := mul_le_mul_of_nonneg_right (hε1 k) hT.le |
| nlinarith |
| have hp := hX.canonicalHeatKernel_pos hn p (ε k*T/2) ht x y |
| have hvol := hvolume y hy (ε k*T/2) ht ht1 |
| have hh := gaussian_log_upper hp hC hc ht (hε k).le hvol (hupper (ε k*T/2) ht x y) |
| have he : ε k*(-(dist x y)^2/((4+a)*(ε k*T/2))+D*(ε k*T/2)) = |
| -2*(dist x y)^2/((4+a)*T)+D*(ε k)^2*T/2 := by |
| field_simp [ne_of_gt (hε k),ne_of_gt hT,ne_of_gt (by linarith : 0 < 4+a)] |
| rw [he] at hh |
| simp only [neg_mul,neg_div] at hh |
| have hd : dist x y ≤ 2*R := by |
| have hx' : dist x p ≤ R := hx |
| have hy' : dist y p ≤ R := hy |
| have hd := dist_triangle x p y |
| rw [dist_comm p y] at hd |
| linarith |
| have hs := gaussian_action_slack hT hη hR (dist_nonneg (x := x) (y := y)) hd |
| change (dist x y)^2/(2*T)-2*(dist x y)^2/((4+a)*T) ≤ η/4 at hs |
| dsimp only [E] at hk |
| rw [neg_div] |
| nlinarith |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
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|
