Trace and extension theorems for Sobolev-type functions in metric spaces

Abstract

Trace classes of Sobolev-type functions in metric spaces are subject of this paper. In particular, functions on domains whose boundary has an upper codimension-$\theta$ bound are considered. Based on a Poincar\'e inequality, existence of a Borel measurable trace is proven whenever the power of integrability of the "gradient" exceeds $\theta$. The trace $T$ is shown to be a compact operator mapping a Sobolev-type space on a domain into a Besov space on the boundary. Sufficient conditions for $T$ to be surjective are found and counterexamples showing that surjectivity may fail are also provided. The case when the exponent of integrability of the "gradient" is equal to $\theta$, i.e., the codimension of the boundary, is also discussed. Under some additional assumptions, the trace lies in $L^\theta$ on the boundary then. Essential sharpness of these extra assumptions is illustrated by an example.