Orlicz-Sobolev extensions and measure density condition

We study the extension properties of Orlicz-Sobolev functions both in Euclidean spaces and in metric measure spaces equipped with a doubling measure. We show that a set E subset of R satisfying a measure density condition admits a bounded linear extension operator from the trace space W-1,W-psi (R-n)vertical bar(E) to W-1,W-psi (R-n). Then we show that a domain, in which the Sobolev embedding theorem or a Poincare-type inequality holds, satisfies the measure density condition. It follows that the existence of a bounded, possibly non-linear extension operator or even the surjectivity of the trace operator implies the measure density condition and hence the existence of a bounded linear extension operator. (C) 2010 Elsevier Inc. All rights reserved.