Trace and extension theorems for functions of bounded variation

Abstract

In this paper we show that every $L^1$-integrable function on $\partial \Omega$ can be obtained as the trace of a function of bounded variation in $\Omega$ whenever $\Omega$ is a domain with regular boundary $\partial \Omega$ in a doubling metric measure space. In particular, the trace class of $BV(\Omega )$ is $L^1(\partial \Omega )$ provided that $\Omega$ supports a 1-Poincaré inequality. We also construct a bounded linear extension from a Besov class of functions on $\partial \Omega$ to $BV(\Omega )$.