| import OAI.Geometry.RCD.Heat.KernelHeatInitialUniqueness | |
| import OAI.Geometry.RCD.Metric.CompactFactorL2 | |
| namespace OAI | |
| /-! # Identification of the initial datum of a compact heat factor -/ | |
| open Set Filter MeasureTheory TopologicalSpace | |
| open scoped ENNReal NNReal | |
| 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_initial_value {S : Set X} [CompactSpace S] | |
| (ν : Measure S) (ρ : X → ℝ) (hρ : Measurable ρ) {C : ℝ≥0} | |
| (hC : ∀ᵐ x ∂hausdorff n X, |ρ x| ≤ C) | |
| (F : C(S,ℝ)) (f : Lp ℝ 2 (hausdorff n X)) | |
| (hrep : ∀ t : ℝ, 0 < t → ∀ x, | |
| kernelHeat (hausdorff n X) hX.quadratic | |
| (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) t f x = | |
| ∫ a, hX.canonicalHeatKernel hn p t x a*F a ∂ν) | |
| (hν : ν.map Subtype.val = (hausdorff n X).withDensity (fun x => ENNReal.ofReal (ρ x))) | |
| (hρ0 : ∀ x, 0 ≤ ρ x) : | |
| f =ᵐ[hausdorff n X] fun x => ρ x*compactFactorExtension F x := by | |
| let m := hausdorff n X | |
| let Q := hX.quadratic | |
| let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p | |
| have hS : IsCompact S := isCompact_iff_compactSpace.mpr inferInstance | |
| have hm := compact_weighted_density_memLp hS F m hρ hC | |
| let g := hm.toLp _ | |
| have hg : g =ᵐ[m] fun x => ρ x*compactFactorExtension F x := hm.coeFn_toLp | |
| have hfg : f = g := kernelHeat_initial_unique m Q hk f g (by | |
| intro t ht x | |
| rw [hrep t ht x,kernelHeat_eq_integral m Q hk ht g x] | |
| rw [compact_weighted_density_integral hS.measurableSet m ν ρ hρ hρ0 hν F] | |
| apply integral_congr_ae | |
| filter_upwards [hg] with y hy | |
| rw [hy]) | |
| exact hfg ▸ hg | |
| end RegularPointCoordinates.NoncollapsedRCD | |
| end | |
| end OAI | |