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b37e936 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | 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
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