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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 | import OAI.Geometry.RCD.Heat.KernelHeatTimeMonotone
import OAI.Geometry.RCD.Metric.CompactFactorNonnegativeData
namespace OAI
/-! # A time comparison independent of the compact endpoint factor -/
open Set Filter MeasureTheory 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_time_lower (hK : K≤0)
{S : Set X} [CompactSpace S] (ν : Measure S) [IsProbabilityMeasure ν]
(ρ : X → ℝ) (hρ : Measurable ρ) (hρ0 : ∀ x, 0≤ρ x) {C : ℝ≥0}
(hC : ∀ᵐ x ∂hausdorff n X, |ρ x|≤C)
(hν : ν.map Subtype.val=(hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x)))
(F : C(S, ℝ)) (hF : ∀ a, 0<F a) {a b : ℝ}
(ha : 0<a) (hab : a≤b) (x : X) :
Real.log (hX.compactHeatFactor hn p ν F a ha x)-
Real.log (hX.compactHeatFactor hn p ν F b (ha.trans_le hab) x) ≤
(n:ℝ)*(Real.log b-Real.log a+|K| *(b-a)) := by
obtain ⟨f,hf0,⟨A,hA⟩,he⟩ :=
hX.compact_heat_factor_nonnegative_data hn p ν ρ hρ hρ0 hC hν F hF
have hh := hX.kernelHeat_time_monotone hn p hK
(hX.canonicalHeatKernel_isContinuousHeatKernel hn p) f hf0 A hA ha hab x
rw [←he a ha x,←he b (ha.trans_le hab) x] at hh
have ha0 := hX.compactHeatFactor_positive hn p ν F hF a ha x
have hb0 := hX.compactHeatFactor_positive hn p ν F hF b (ha.trans_le hab) x
have hl := Real.log_le_log (mul_pos (Real.exp_pos _) ha0) hh
rw [Real.log_mul (Real.exp_ne_zero _) ha0.ne',
Real.log_mul (Real.exp_ne_zero _) hb0.ne',Real.log_exp,Real.log_exp] at hl
nlinarith only [hl]
end RegularPointCoordinates.NoncollapsedRCD
end
end OAI
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