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In this paper we fix $1\le p<\infty$ and consider $(\Om,d,\mu)$ be anunbounded, locally compact, non-complete metric measure space equipped with adoubling measure $\mu$ supporting a $p$-Poincar\'e inequality such that $\Om$is a uniform domain in its completion $\bar\Om$. We realize the trace offunctions 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, $\partial\Om$ isequipped 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 linearextension operator $E:HB^{1-\theta/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$that is a right-inverse of $T$.

Trace and extension theorems for homogeneous Sobolev and Besov spaces for unbounded uniform domains in metric measure spaces