2024

Eriksson-Bique, Sylvester; Poggi-Corradini, Pietro

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 N1,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 inequali-ties. 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.