It is well known that there are bounded domains Ω⊂ℝn whose boundaries ∂Ω are not smooth enough for there to exist a bounded linear extension for the Sobolev space W1p(Ω) into W1p(ℝn), but the embedding W1p(Ω)⊂ Lp(Ω) is nevertheless compact. For the Lipγ boundaries (0<γ<1) studied in [3, 4], there does not exist in general an extension operator of W1p(Ω) into W1p(ℝn) but there is a bounded linear extension of W1p(Ω) into Wγp(ℝn) and the smoothness retained by this extension is enough to ensure that the embedding W1p(Ω)⊂ Lp(Ω) is compact. It is natural to ask if this is typical for bounded domains which are such that W1p(Ω)⊂ Lp(Ω) is compact, that is, that there exists a bounded extension into a space of functions in ℝn which enjoy adequate smoothness. This is the question which originally motivated this paper. Specifically we study the ‘extension by zero’ operator on a space of functions with given ‘generalized’ smoothness defined on a domain with an irregular boundary, and determine the target space with respect to which it is bounded.