| 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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|
