Sobolev embeddings, extensions and measure density condition

There are two main results in the paper. In the first one, Theorem 1, we prove that if the Sobolev embedding theorem holds in Omega, in any of all the possible cases, then Omega satisfies the measure density condition. The second main result, Theorem 5, provides several characterizations of the W-m,W-p-extension domains for 1 < p < infinity. As a corollary we prove that the property of being a W-m,W-p-extension domain, 1 < p <= infinity, is invariant under bi-Lipschitz mappings, Theorem 8. (C) 2007 Elsevier Inc. All rights reserved.