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

Abstract

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $\mu$ supporting a $p$-Poincaré 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 $HB^α_{p,p}(\partial\Om)$ 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 $T:D^{1,p}(\Om)\rightarrow HB^{1-θ/p}(\partial\Om)$ and a bounded linear extension operator $E:HB^{1-θ/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$ that is a right-inverse of $T$.