Density of Lipschitz functions in Energy

Abstract

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty )$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls. Emphatically, $p=1$ is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this is that we do not use any form of Poincaré inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of existing literature on the topic.

arXiv comment19 pages, comments welcome. This version is substantially revised and improved from the last version. The main change is an expanded section 2 with many details fleshed out, and some auxiliary terminology. A reader may also find the expanded discussion in the introduction interesting