A characterization of BV and Sobolev functions via nonlocal functionals in metric spaces

TL;DR

This paper characterizes BV and Sobolev functions in metric spaces using generalized nonlocal functionals, revealing differences from Euclidean spaces and providing a counterexample for the case p=1.

Contribution

It extends the characterization of BV and Sobolev functions to more general nonlocal functionals in metric spaces with doubling measures and Poincaré inequalities.

Findings

More general nonlocal functionals are considered.

The limit of nonlocal functions is only comparable to the variation measure in metric spaces.

A counterexample for p=1 shows the limit is not necessarily equal to the variation measure.

Abstract

We study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case $p=1$ demonstrating that unlike in Euclidean spaces, in metric measure spaces the limit of the nonlocal functions is only comparable, not necessarily equal, to the variation measure $\| Df\|(\Omega)$.