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 \((\Omega,d,\mu)\) 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 \(H\kern-1pt B^\alpha_{p,p}(\partial\Omega)\) 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 \(T \colon\, D^{1,p}(\Omega)\to H\kern-1pt B^{1-\theta/p}(\partial\Omega)\) and a bounded linear extension operator \(E \colon\, H\kern-1pt B^{1-\theta/p}(\partial\Omega)\to D^{1,p}(\Omega)\) that is a right-inverse of \(T\).