Traces and Sobolev extension domains

Abstract:

Assume that $\Omega \subset {\mathbb {R}^n}$ is a bounded domain and its boundary $\partial \Omega$ is $m$-regular, $n-1 \le m <n$. We show that if there exists a bounded trace operator $T:W^{1,p}(\Omega ) \to B^{p}_{1-\alpha }(\partial \Omega )$, $1<p<\infty$ and $\alpha = \tfrac {n-m}{p}$, and $(1-\lambda )$-Hölder continuous functions are dense in $W^{1,p}(\Omega )$, $0\le \lambda < n-m$, then the domain $\Omega$ is a $W^{1,p}$-extension domain.