| import OAI.Geometry.RCD.Heat.HeatRadonDensity |
| import OAI.Geometry.RCD.Heat.BoundedHeatLimit |
| import OAI.Geometry.RCD.Metric.BoundedContinuousL2Approximation |
|
|
| namespace OAI |
|
|
| /-! # Heat measure representation for all bounded L² data -/ |
|
|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal ENNReal Topology CompactlySupported |
|
|
| 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] |
|
|
| /-- Uniform compact approximation and absolute continuity extend the |
| constructed measure representation to every bounded L² datum. -/ |
| theorem heatRadonMeasure_integral_bounded (t : ℝ) (ht : 0 < t) (x : X) |
| (f : Lp ℝ 2 (hausdorff n X)) (C : ℝ≥0) |
| (hf : ∀ᵐ y ∂hausdorff n X, |f y| ≤ C) : |
| (∫ y, f y ∂hX.heatRadonMeasure hn p t ht x) = hX.boundedHeat hn p f ⟨C, hf⟩ t ht x := by |
| let μ := hausdorff n X |
| let ν := hX.heatRadonMeasure hn p t ht x |
| obtain ⟨F, hF, hlim⟩ := exists_bounded_continuous_L2_approximation μ f C hf |
| have hb (i : ℕ) : ∀ᵐ y ∂μ, |compactTestToL2 μ (F i) y| ≤ C := by |
| filter_upwards [compactTestToL2_coe μ (F i)] with y hy |
| rw [hy] |
| exact hF i y |
| obtain ⟨s, hs, hsae⟩ := (tendstoInMeasure_of_tendsto_Lp hlim).exists_seq_tendsto_ae |
| have hrep : ∀ᵐ y ∂μ, ∀ i, compactTestToL2 μ (F i) y = F i y := |
| ae_all_iff.mpr (fun i => compactTestToL2_coe μ (F i)) |
| have hFa : ∀ᵐ y ∂μ, Tendsto (fun i => F (s i) y) atTop (𝓝 (f y)) := by |
| filter_upwards [hsae, hrep] with y hy he |
| simpa only [he] using hy |
| have hFan : ∀ᵐ y ∂ν, Tendsto (fun i => F (s i) y) atTop (𝓝 (f y)) := |
| (hX.heatRadonMeasure_absolutelyContinuous hn p t ht x).ae_le hFa |
| have hleft : Tendsto (fun i => ∫ y, F (s i) y ∂ν) atTop (𝓝 (∫ y, f y ∂ν)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => (C : ℝ)) |
| · intro i |
| exact (F (s i)).continuous.aestronglyMeasurable |
| · exact integrable_const _ |
| · intro i |
| exact Eventually.of_forall fun y => hF (s i) y |
| · exact hFan |
| have hright := (hX.boundedHeat_tendsto hn p (fun i => compactTestToL2 μ (F i)) |
| f C hb hf hlim t ht x).comp hs.tendsto_atTop |
| have heq (i : ℕ) : (∫ y, F (s i) y ∂ν) = |
| hX.boundedHeat hn p (compactTestToL2 μ (F (s i))) ⟨C, hb (s i)⟩ t ht x := |
| hX.heatRadonMeasure_integral hn p t ht x (F (s i)) |
| have hright' : Tendsto (fun i => ∫ y, F (s i) y ∂ν) atTop |
| (𝓝 (hX.boundedHeat hn p f ⟨C, hf⟩ t ht x)) := by |
| simp_rw [heq] |
| exact hright |
| exact tendsto_nhds_unique hleft hright' |
|
|
| /-- The actual probability density represents all bounded L² heat data. -/ |
| theorem heatRadonDensity_integral_bounded (t : ℝ) (ht : 0 < t) (x : X) |
| (f : Lp ℝ 2 (hausdorff n X)) (C : ℝ≥0) |
| (hf : ∀ᵐ y ∂hausdorff n X, |f y| ≤ C) : |
| (∫ y, hX.heatRadonDensity hn p t ht x y * f y ∂hausdorff n X) = |
| hX.boundedHeat hn p f ⟨C, hf⟩ t ht x := by |
| change (∫ y, ((hX.heatRadonMeasure hn p t ht x).rnDeriv (hausdorff n X) y).toReal * f y |
| ∂hausdorff n X) = _ |
| rw [integral_toReal_rnDeriv_mul (hX.heatRadonMeasure_absolutelyContinuous hn p t ht x)] |
| exact hX.heatRadonMeasure_integral_bounded hn p t ht x f C hf |
|
|
| end RegularPointCoordinates.NoncollapsedRCD |
|
|
| end |
|
|
| end OAI |
|
|
