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

TL;DR

This paper investigates the trace properties of Sobolev-type functions in metric spaces with boundaries of certain codimension, establishing existence, compactness, and surjectivity conditions for trace operators, with sharpness and counterexamples.

Contribution

It introduces new trace theorems for Sobolev functions in metric spaces, including conditions for existence, compactness, and surjectivity of trace operators, extending classical results to more general settings.

Findings

Existence of measurable trace under integrability conditions

Trace operator is compact from Sobolev to Besov spaces

Counterexamples show surjectivity can fail under certain conditions

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.