  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-Poincar\'e inequality. We also construct a bounded linear extension from a
Besov class of functions on $\partial\Omega$ to $BV(\Omega)$.
