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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 60 | import OAI.Geometry.RCD.Heat.CanonicalHeatBackwardRatio
import OAI.Geometry.RCD.Heat.CompactHeatFactors
namespace OAI
/-! # Backward logarithmic time control for compact endpoint factors -/
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 compact_heat_factor_log_backward_time_ratio :
∃ C D : ℝ, 0≤C ∧ 0<D ∧
∀ {S : Set X} [CompactSpace S] (ν : Measure S) [IsProbabilityMeasure ν]
(F : C(S, ℝ)), (∀ a, 0<F a) → ∀ (s t : ℝ) (hs : 0<s) (hst : s≤t), t≤1 →
∀ (x : X) (R : ℝ), 0≤R → (∀ a : S, dist x a≤R) →
Real.log (hX.compactHeatFactor hn p ν F t (hs.trans_le hst) x)-
Real.log (hX.compactHeatFactor hn p ν F s hs x)≤C+D*R^2/s := by
obtain ⟨A,D,hA,hD,hkernel⟩ := hX.canonicalHeatKernel_backward_time_ratio hn p
refine ⟨|Real.log A|,D,abs_nonneg _,hD,?_⟩
intro S _ ν _ F hF s t hs hst ht1 x R hR hdist
have ht : 0<t := hs.trans_le hst
let B := A*Real.exp (D*R^2/s)
have hratio : hX.compactHeatFactor hn p ν F t ht x≤B*hX.compactHeatFactor hn p ν F s hs x := by
have hi := compact_kernel_factor_integrable ν (hX.compactHeatFactorKernel hn p S t ht) F x
have hj := compact_kernel_factor_integrable ν (hX.compactHeatFactorKernel hn p S s hs) F x
change (∫ a, hX.canonicalHeatKernel hn p t x a*F a ∂ν)≤
B*(∫ a, hX.canonicalHeatKernel hn p s x a*F a ∂ν)
rw [←integral_const_mul]
apply integral_mono hi (hj.const_mul B)
intro a
have he : A*Real.exp (D*(dist x a)^2/s)≤B := by
apply mul_le_mul_of_nonneg_left _ hA.le
apply Real.exp_le_exp.mpr
apply div_le_div_of_nonneg_right _ hs.le
exact mul_le_mul_of_nonneg_left ((sq_le_sq₀ dist_nonneg hR).mpr (hdist a)) hD.le
have hh := (hkernel s t hs hst ht1 x a).trans
(mul_le_mul_of_nonneg_right he (hX.canonicalHeatKernel_pos hn p s hs x a).le)
change hX.canonicalHeatKernel hn p t x a*F a≤
B*(hX.canonicalHeatKernel hn p s x a*F a)
simpa only [mul_assoc] using mul_le_mul_of_nonneg_right hh (hF a).le
have hp := hX.compactHeatFactor_positive hn p ν F hF t ht x
have hq := hX.compactHeatFactor_positive hn p ν F hF s hs x
have hl := Real.log_le_log hp hratio
dsimp only [B] at hl
rw [Real.log_mul (mul_pos hA (Real.exp_pos _)).ne' hq.ne',
Real.log_mul hA.ne' (Real.exp_ne_zero _),Real.log_exp] at hl
linarith only [hl,le_abs_self (Real.log A)]
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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