Uniform and Sobolev Extension Domains

Abstract

We prove that if a domain $D \subset {{\mathbf {R}}^n}$ is quasiconformally equivalent to a uniform domain, then $D$ is an extension domain for the Sobolev class $W_n^1$ if and only if $D$ is locally uniform. We provide examples which suggest that this result is best possible. We exhibit a list of equivalent conditions for domains quasiconformally equivalent to uniform domains, one of which characterizes the quasiconformal homeomorphisms between uniform and locally uniform domains.