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

TL;DR

This paper establishes trace and extension theorems for Sobolev and Besov spaces on unbounded uniform domains within metric measure spaces, broadening understanding of boundary behavior in non-smooth, non-complete settings.

Contribution

It introduces new trace and extension operators linking Sobolev spaces to Besov spaces on irregular boundaries in metric measure spaces.

Findings

Existence of bounded trace operator from Sobolev to Besov spaces.

Existence of bounded extension operator from Besov to Sobolev spaces.

Trace operator is a right-inverse of the extension operator.

Abstract

In this paper we fix $1\le p<\infty$ and consider $(\Om,d,\mu)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $\mu$ supporting a $p$-Poincar\'e 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 $HB^\alpha_{p,p}(\partial\Om)$ 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 $T:D^{1,p}(\Om)\rightarrow HB^{1-\theta/p}(\partial\Om)$ and a bounded linear extension operator $E:HB^{1-\theta/p}(\partial\Om)\rightarrow D^{1,p}(\Om)$ that is a right-inverse of $T$.