| import OAI.Geometry.RCD.Heat.LocalLogarithmicLaplacian |
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| namespace OAI |
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| /-! # Bounded Lipschitz logarithmic extensions with their actual source -/ |
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| open Set Filter MeasureTheory |
| open scoped ENNReal NNReal |
| noncomputable section |
| namespace RegularPointCoordinates |
| variable {X : Type*} [MetricSpace X] [MeasurableSpace X] [BorelSpace X] |
| [ProperSpace X] (μ : Measure X) [IsFiniteMeasureOnCompacts μ] (Q : QuadraticCheegerEnergy μ) |
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| theorem bounded_local_logarithm (u : SobolevSpace μ Q) (d : Lp ℝ 2 μ) |
| (hd : IsWeakLaplacian μ Q u d) (F : X → ℝ) {L A : ℝ≥0} |
| (hF : LipschitzWith L F) (hA : ∀ x, |F x|≤A) |
| (hu : SobolevSpace.toLp μ Q u =ᵐ[μ] F) |
| {Ω : Set X} {a b : ℝ} (ha : 0<a) (hab : a≤b) |
| (hv : ∀ᵐ x ∂μ.restrict Ω, F x∈Icc a b) : |
| ∃ (v : SobolevSpace μ Q) (q : Lp ℝ 2 μ) (V : X → ℝ) (M B : ℝ≥0), |
| IsWeakLaplacian μ Q v q ∧ LipschitzWith M V ∧ |
| (∀ x, |V x|≤B) ∧ (SobolevSpace.toLp μ Q v =ᵐ[μ] V) ∧ |
| (SobolevSpace.toLp μ Q v =ᵐ[μ.restrict Ω] fun x => Real.log (F x)) := by |
| obtain ⟨φ,ψ,χ,C,D,B,E,hφ,hψ,hχ,hφ0,hψ0,hdφ,hdψ,hB,hE,he⟩ := |
| exists_local_smooth_log ha hab |
| let U := SobolevSpace.toLp μ Q u |
| have hp : MemLp (fun x => ψ (U x)) ∞ μ := memLp_top_of_bound |
| (hψ.continuous.comp_aestronglyMeasurable (Lp.aestronglyMeasurable U)) B |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hB (U x)) |
| have hc : MemLp (fun x => χ (U x)) ∞ μ := memLp_top_of_bound |
| (hχ.comp_aestronglyMeasurable (Lp.aestronglyMeasurable U)) E |
| (Eventually.of_forall fun x => by simpa only [Real.norm_eq_abs] using hE (U x)) |
| have hL := SobolevSpace.minimalSlope_le_lipschitz μ Q u hF hu |
| have hq : MemLp (fun x => ψ (U x)*d x+χ (U x)*localGamma μ Q u u x) 2 μ := |
| (hp.mul (Lp.memLp d)).add (hc.mul (localGamma_memLp_of_bounded_gradient μ Q u u L hL)) |
| let q := hq.toLp _ |
| let v := SobolevSpace.comp μ Q hφ hφ0 u |
| have hrep : SobolevSpace.toLp μ Q v =ᵐ[μ] fun x => φ (F x) := by |
| filter_upwards [SobolevSpace.comp_coe μ Q hφ hφ0 u,hu] with x hx hy |
| exact hx.trans (congrArg φ hy) |
| refine ⟨v,q,φ ∘ F,C*L,C*A, |
| weakLaplacian_comp_C2 μ Q u d q hd hφ hφ0 hψ hψ0 hχ hdφ hdψ B E hB hE hq.coeFn_toLp, |
| hφ.comp hF,?_,hrep,?_⟩ |
| · intro x |
| have hh := hφ.dist_le_mul (F x) 0 |
| simp only [hφ0,Real.dist_eq,sub_zero] at hh |
| exact hh.trans (by simpa only [NNReal.coe_mul] using |
| mul_le_mul_of_nonneg_left (hA x) C.coe_nonneg) |
| · filter_upwards [ae_restrict_of_ae hrep,hv] with x hx hy |
| exact hx.trans (he (F x) hy).1 |
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| end RegularPointCoordinates |
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| end |
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| end OAI |
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