Sobolev and Quasiconformal Extension Domains
Manouchehr Ghamsari · Proceedings of the American Mathematical Society · 1993
A domain $D \subset {\mathbb {R}^n}$ has the quasiconformal extension property if each quasiconformal self-map of $D$ extends to a quasiconformal self-map of ${\mathbb {R}^n};\;D$ has the Sobolev extension property if there is a bounded linear operator $\Lambda :{W^{1,n}}(D) \to {W^{1,n}}({\mathbb {R}^n})$. We consider the relation between the above extension properties for $n \geqslant 3$. We show that for domains quasiconformally equivalent to a ball the quasiconformal extension property implies the Sobolev extension property. We show that this is not true in general. Next the Sobolev extension property does not imply the extension property for quasiconformal maps which extend as homeomorphisms. Finally if $G \subset {\mathbb {R}^2}$ and if $D = G \times \mathbb {R}$ is quasiconformally equivalent to a ball, then $D$ has the quasiconformal extension property if and only if $D$ is a quasiball.