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

In this paper we fix $1\le p<\mathrm{\infty }$ and consider $(\text{Om},d,\mu )$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $\mu$ supporting a $p$-Poincaré inequality such that $\text{Om}$ is a uniform domain in its completion $\overline{\text{Om}}$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\text{Om})$ on the boundary $\mathrm{\partial }\text{Om}$ as functions in the homogeneous Besov space $H{B}_{p,p}^{\alpha }(\mathrm{\partial }\text{Om})$ for suitable $\alpha$; here, $\mathrm{\partial }\text{Om}$ 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 $T:D^{1,p}(\text{Om})\to H{B}^{1-\theta \mathrm{/}p}(\mathrm{\partial }\text{Om})$ and a bounded linear extension operator $E:H{B}^{1-\theta \mathrm{/}p}(\mathrm{\partial }\text{Om})\to D^{1,p}(\text{Om})$ that is a right-inverse of $T$.