  In this paper we fix $1\le p<\infty$ and consider $(\Om,d,\mu)$ be an
unbounded, locally compact, non-complete metric measure space equipped with a
doubling measure $\mu$ supporting a $p$-Poincar\'e inequality such that $\Om$
is a uniform domain in its completion $\bar\Om$. We realize the trace of
functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary
$\partial\Om$ as functions in the homogeneous Besov space
$HB^\alpha_{p,p}(\partial\Om)$ for suitable $\alpha$; here, $\partial\Om$ is
equipped with a non-atomic Borel regular measure $\nu$. We show that if $\nu$
satisfies a $\theta$-codimensional condition with respect to $\mu$ for some
$0<\theta<p$, then there is a bounded linear trace operator
$T:D^{1,p}(\Om)\rightarrow HB^{1-\theta/p}(\partial\Om)$ and a bounded linear
extension operator $E:HB^{1-\theta/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$
that is a right-inverse of $T$.
