| import OAI.Geometry.RCD.Coordinates.BoundedRegularBochnerMass |
| import OAI.Geometry.RCD.Heat.BoundedHeatSobolevData |
| import OAI.Geometry.RCD.Sobolev.BochnerMeasureFromSource |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] [ProperSpace X] |
|
|
| theorem NoncollapsedRCD.bounded_heat_full_source {K : ℝ} {n : ℕ} |
| (hX : NoncollapsedRCD K n X) [(hausdorff n X).Regular] (hn : 2 ≤ n) (hK : K ≤ 0) |
| (p : X) (h : Lp ℝ 2 (hausdorff n X)) (A : ℝ≥0) |
| (hA : ∀ᵐ x ∂hausdorff n X, |h x|≤A) {s r : ℝ} (hs : 0 < s) (hr : 0 < r) : |
| let μ := hausdorff n X |
| let Q := hX.quadratic |
| let w := heatSobolev μ Q s hs h |
| let d := heatSobolevMap μ Q (s/2) (half_pos hs) (-heatGeneratorAt μ Q (s-s/2) h) |
| ∃ (H : SobolevSpace μ Q) (τ : Measure X) (hτ : IsFiniteMeasure τ), |
| letI := hτ; |
| (SobolevSpace.toLp μ Q H =ᵐ[μ] localGamma μ Q w w) ∧ |
| IsMeasureLaplacianOn μ Q (ball p (2*r)) H (bochnerGramSource μ Q K w w d d τ.toSignedMeasure) := by |
| intro μ Q w d |
| let := hX.separable |
| let := locallyFinite_of_finiteOnBounded μ hX.finiteOnBounded |
| let hk := hX.canonicalHeatKernel_isContinuousHeatKernel hn p |
| obtain ⟨hLip,_⟩ := hX.kernelHeat_bounded_regularity hn p hk h A hA s hs |
| have hw : SobolevSpace.toLp μ Q w =ᵐ[μ] kernelHeat μ Q hk s h := by |
| simpa only [w,heatSobolev_toLp] using (kernelHeat_represents_L2 μ Q hk hs h).symm |
| have hWA := (hX.bounded_heat_sobolev_data hn p h A hA hs).1 |
| have hd : IsWeakLaplacian μ Q w (SobolevSpace.toLp μ Q d) := heatTail_weak_laplacian μ Q s hs h |
| obtain ⟨H,hH⟩ := hX.bounded_heat_gram hn p h A hA hs |
| obtain ⟨_M,_hM,hM⟩ := hX.bounded_regular_bochner_mass hn hK p w d H hd |
| (kernelHeat μ Q hk s h) hLip hw A hWA hH |
| obtain ⟨τ,hτ,hτid,_hτmass⟩ := hM r hr |
| let := hτ |
| refine ⟨H,τ,hτ,hH,?_⟩ |
| apply bochner_measure_laplacian_of_source μ Q K w d H hH τ |
| intro ψ v hv hvc hvs hψ |
| exact hτid ψ v hv hvc (hvs.trans (ball_subset_ball (by linarith))) hψ |
|
|
| end RegularPointCoordinates |
|
|
| end |
|
|
| end OAI |
|
|
