| import OAI.Geometry.RCD.Heat.HeatBoundedSmoothing |
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| namespace OAI |
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| /-! # Canonical bounded-data heat representatives |
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| The actual Cheeger semigroup and the proved reverse Poincare estimate give |
| continuous representatives on every RCD space in this application. Full |
| support makes the representative unique; positivity, linearity and the |
| Markov amplitude bound therefore hold at every point. No heat kernel is |
| assumed in this construction. |
| -/ |
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|
| open Set Filter MeasureTheory Metric TopologicalSpace |
| open scoped NNReal Topology |
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| noncomputable section |
| namespace RegularPointCoordinates |
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| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| {K : ℝ} {n : ℕ} (hX : NoncollapsedRCD K n X) (hn : 2 ≤ n) (p : X) |
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| / |
| def HasBoundedAmplitude (f : Lp ℝ 2 (hausdorff n X)) : Prop := |
| ∃ C : ℝ≥0, ∀ᵐ x ∂hausdorff n X, |f x| ≤ C |
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| namespace NoncollapsedRCD |
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| variable [ProperSpace X] [SeparableSpace X] |
| [IsLocallyFiniteMeasure (hausdorff n X)] [(hausdorff n X).Regular] |
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| / |
| def boundedHeat (f : Lp ℝ 2 (hausdorff n X)) (hf : HasBoundedAmplitude f) |
| (t : ℝ) (ht : 0 < t) : X → ℝ := |
| (hX.heat_bounded_lipschitz K n hn p f hf.choose hf.choose_spec t ht).choose |
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| omit [SeparableSpace X] [IsLocallyFiniteMeasure (hausdorff n X)] in |
| theorem boundedHeat_represents (f : Lp ℝ 2 (hausdorff n X)) |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) : |
| heatAt (hausdorff n X) hX.quadratic t f =ᵐ[hausdorff n X] |
| hX.boundedHeat hn p f hf t ht := |
| (hX.heat_bounded_lipschitz K n hn p f hf.choose hf.choose_spec t ht).choose_spec.2 |
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| omit [ProperSpace X] [SeparableSpace X] [IsLocallyFiniteMeasure (hausdorff n X)] |
| [(hausdorff n X).Regular] in |
| theorem boundedHeat_continuous (f : Lp ℝ 2 (hausdorff n X)) |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) : |
| Continuous (hX.boundedHeat hn p f hf t ht) := |
| (hX.heat_bounded_lipschitz K n hn p f hf.choose hf.choose_spec t ht).choose_spec.1.continuous |
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| omit [SeparableSpace X] [IsLocallyFiniteMeasure (hausdorff n X)] in |
| / |
| theorem boundedHeat_unique (f : Lp ℝ 2 (hausdorff n X)) |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) |
| {v : X → ℝ} (hv : Continuous v) |
| (hrep : heatAt (hausdorff n X) hX.quadratic t f =ᵐ[hausdorff n X] v) : |
| hX.boundedHeat hn p f hf t ht = v := by |
| let := hX.openPos K n |
| exact MeasureTheory.Measure.eq_of_ae_eq ((hX.boundedHeat_represents hn p f hf t ht).symm.trans hrep) |
| (hX.boundedHeat_continuous hn p f hf t ht) hv |
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| omit [ProperSpace X] [(hausdorff n X).Regular] in |
| theorem boundedHeat_lipschitz (f : Lp ℝ 2 (hausdorff n X)) |
| (hf : HasBoundedAmplitude f) (t : ℝ) (ht : 0 < t) |
| (C : ℝ≥0) (hC : ∀ᵐ x ∂hausdorff n X, |f x| ≤ C) : |
| LipschitzWith ⟨Real.sqrt ((C:ℝ)^2/(2*t*Real.exp (-2*|K| *t))),Real.sqrt_nonneg _⟩ |
| (hX.boundedHeat hn p f hf t ht) := by |
| let : ProperSpace X := hX.proper K n hn |
| let : (hausdorff n X).Regular := inferInstance |
| obtain ⟨v, hv, hrep⟩ := hX.heat_bounded_lipschitz K n hn p f C hC t ht |
| rw [hX.boundedHeat_unique hn p f hf t ht hv.continuous hrep] |
| exact hv |
|
|
| end NoncollapsedRCD |
| end RegularPointCoordinates |
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| end |
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| end OAI |
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|
