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