| import OAI.Geometry.RCD.Heat.CompactHeatFactors | |
| import OAI.Geometry.RCD.Metric.CompactWeightedDensityIntegral | |
| import OAI.Geometry.RCD.Metric.CompactWeightedDensityL2 | |
| import OAI.Geometry.RCD.Heat.KernelHeatGeneratorDerivative | |
| namespace OAI | |
| /-! # The constructed compact heat factor is the actual L² heat flow | |
| The initial datum is the endpoint density multiplied by the zero-extended | |
| continuous scaling factor. Its L² class and bounded amplitude are proved. | |
| -/ | |
| 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_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)) : | |
| ∃ f : Lp ℝ 2 (hausdorff n X), HasBoundedAmplitude f ∧ | |
| ∀ (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 μ := 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 μ := compact_weighted_density_memLp hS F μ hρ hC | |
| let f := hm.toLp f₀ | |
| have hf : f =ᵐ[μ] f₀ := hm.coeFn_toLp | |
| obtain ⟨D,hD⟩ := compact_weighted_density_bounded hS F μ hC | |
| have hbound : HasBoundedAmplitude f := by | |
| refine ⟨D,?_⟩ | |
| filter_upwards [hf,hD] with x hx hDx | |
| rw [hx] | |
| exact hDx | |
| refine ⟨f,hbound,?_⟩ | |
| 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 ∂μ := | |
| compact_weighted_density_integral hS.isClosed.measurableSet μ ν ρ hρ hρ0 hν F _ | |
| _ = ∫ y, hX.canonicalHeatKernel hn p t x y*f y ∂μ := by | |
| apply integral_congr_ae | |
| filter_upwards [hf] with y hy | |
| rw [hy] | |
| _ = _ := (kernelHeat_eq_integral μ hX.quadratic | |
| (hX.canonicalHeatKernel_isContinuousHeatKernel hn p) ht f x).symm | |
| end RegularPointCoordinates.NoncollapsedRCD | |
| end | |
| end OAI | |