| import OAI.Geometry.RCD.Sobolev.BoundedLpSquare | |
| namespace OAI | |
| /-! # Bounded compact Lipschitz approximation in L² -/ | |
| open Set Filter MeasureTheory | |
| open scoped NNReal ENNReal Topology | |
| noncomputable section | |
| namespace RegularPointCoordinates | |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] | |
| [ProperSpace X] (μ : Measure X) (hQ : QuadraticCheegerEnergy μ) | |
| /-- Clip the compact recovery sequence to preserve the amplitude of the | |
| original bounded Sobolev function. L² convergence suffices after heat flow. -/ | |
| theorem SobolevSpace.bounded_compact_L2_approximation (f : SobolevSpace μ hQ) | |
| (C : ℝ≥0) (hf : ∀ᵐ x ∂μ, |SobolevSpace.toLp μ hQ f x| ≤ C) (p : X) : | |
| ∃ (us : ℕ → Lp ℝ 2 μ) (gs : ℕ → X → ℝ), | |
| Tendsto us atTop (𝓝 (SobolevSpace.toLp μ hQ f)) ∧ | |
| (∀ i, ∃ L : ℝ≥0, LipschitzWith L (gs i)) ∧ | |
| (∀ i, HasCompactSupport (gs i)) ∧ | |
| (∀ i, us i =ᵐ[μ] gs i) ∧ (∀ i x, |gs i x| ≤ C) := by | |
| obtain ⟨vs,ks,Gs,hl,hs,he,_hG,hv,_hGs⟩ := | |
| (SobolevSpace.minimalSlope_isRelaxed μ hQ f).compact_recovery μ p | |
| let P : Lp ℝ 2 μ → Lp ℝ 2 μ := (clipReal_lipschitz C).compLp (clipReal_zero C) | |
| have hPf : P (SobolevSpace.toLp μ hQ f) = SobolevSpace.toLp μ hQ f := by | |
| apply Lp.ext | |
| filter_upwards [(clipReal_lipschitz C).coeFn_compLp (clipReal_zero C) | |
| (SobolevSpace.toLp μ hQ f),hf] with x hx hfx | |
| exact hx.trans (clipReal_eq_self C hfx) | |
| refine ⟨fun i => P (vs i),fun i x => clipReal C (ks i x),?_,?_,?_,?_,?_⟩ | |
| · have hPc : Continuous P := (clipReal_lipschitz C).continuous_compLp (clipReal_zero C) | |
| simpa only [hPf,Function.comp_apply] using! (hPc.tendsto (SobolevSpace.toLp μ hQ f)).comp hv | |
| · intro i | |
| obtain ⟨L,hL⟩ := hl i | |
| exact ⟨1*L,(clipReal_lipschitz C).comp hL⟩ | |
| · intro i | |
| exact (hs i).comp_left (clipReal_zero C) | |
| · intro i | |
| filter_upwards [(clipReal_lipschitz C).coeFn_compLp (clipReal_zero C) (vs i),he i] with x hx he | |
| exact hx.trans (congrArg (clipReal C) he) | |
| · intro i x | |
| exact clipReal_abs_le C (ks i x) | |
| end RegularPointCoordinates | |
| end | |
| end OAI | |