| import OAI.Geometry.RCD.Metric.CompactFactorL2 | |
| namespace OAI | |
| /-! # Nonnegative initial data for positive compact heat factors -/ | |
| 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_nonnegative_L2 {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) : | |
| ∃ (f : Lp ℝ 2 (hausdorff n X)) (A : ℝ≥0), 0≤f ∧ | |
| (∀ᵐ x ∂hausdorff n X, |f x|≤A) ∧ | |
| ∀ (t : ℝ) (ht : 0<t) (x : X), hX.compactHeatFactor hn p ν F t ht x = | |
| kernelHeat (hausdorff n X) hX.quadratic | |
| (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) t f x := by | |
| let m := hausdorff n X | |
| have hS : IsCompact S := isCompact_iff_compactSpace.mpr inferInstance | |
| let f₀ := fun x => ρ x*compactFactorExtension F x | |
| have hm : MemLp f₀ 2 m := compact_weighted_density_memLp hS F m hρ hC | |
| let f := hm.toLp f₀ | |
| have hf : f =ᵐ[m] f₀ := hm.coeFn_toLp | |
| have hf0 : 0≤f := by | |
| apply (Lp.coeFn_nonneg _).mp | |
| filter_upwards [hf] with x hx | |
| rw [hx] | |
| apply mul_nonneg (hρ0 x) | |
| by_cases hs : x∈S | |
| · exact (compactFactorExtension_coe F ⟨x,hs⟩) ▸ hF ⟨x,hs⟩ | |
| · rw [compactFactorExtension_off F hs] | |
| obtain ⟨A,hA⟩ := compact_weighted_density_bounded hS F m hC | |
| refine ⟨f,A,hf0,?_,?_⟩ | |
| · filter_upwards [hf,hA] with x hx hax | |
| exact hx ▸ hax | |
| · intro t ht x | |
| change (∫ a, hX.canonicalHeatKernel hn p t x a*F a ∂ν) = _ | |
| calc | |
| _ = ∫ y, hX.canonicalHeatKernel hn p t x y*f₀ y ∂m := | |
| compact_weighted_density_integral hS.isClosed.measurableSet m ν ρ hρ hρ0 hν F _ | |
| _ = ∫ y, hX.canonicalHeatKernel hn p t x y*f y ∂m := by | |
| apply integral_congr_ae | |
| filter_upwards [hf] with y hy | |
| rw [hy] | |
| _ = _ := (kernelHeat_eq_integral m hX.quadratic | |
| (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) ht f x).symm | |
| end RegularPointCoordinates.NoncollapsedRCD | |
| end | |
| end OAI | |