Density of continuous functions in Sobolev spaces with applications to capacity

2023/03/01 by Sylvester Eriksson‐Bique, Eriksson-Bique, Sylvester, Pietro Poggi‐Corradini +1 · 2 citations
Mathematics · #30L15 #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Approximation and Integration #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2303.00649

openalex publication_date 2023/03/01 · openalex created_date 2023/03/03 · openalex updated_date 2026/07/28

Abstract

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if (X,d,μ) is a locally complete and separable metric measure space, then continuous functions are dense in the Newtonian space N^1,p(X). Here the measure μ is Borel and is finite and positive on all metric balls. In particular, we don't assume properness of X, doubling of μ or any Poincaré inequalities. These resolve, partially or fully, questions posed by a number of authors, including J. Heinonen, A. Björn and J. Björn. In contrast to much of the past work, our results apply to locally complete spaces X and dispenses with the frequently used regularity assumptions: doubling, properness, Poincaré inequality, Loewner property or quasiconvexity.

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• Stability of local Riemannian Ricci curvature lower bounds
• Doubling measures and Poincaré inequalities for sphericalizations of metric spaces

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