where 1<1p < oo; m = 0, 1, 2, a; cx=(a1, an, c) is an n-tuple of nonnegative integers; j cx j = a,1 + +a,,; Da = D .. Dnn where Dj=a/axj. The Sobolev space Wo '(G) is the Banach space obtained by completing Co (G) with respect to the norm (2). It is well known that if G is bounded then (1) is also a norm on Wo '(G) and it is equivalent to the usual norm (2). In particular if p = 2 then the polyharmonic operator (-A)m with null Dirichlet boundary data induces a positive definite operator in Woj2(G). In this paper we show that (1) and (2) are equivalent norms on W0m'(G) for a suitable class of unbounded domains G having suitably regular (n-1)-dimensional boundaries. In addition we strengthen somewhat a theorem on the compactness of certain imbeddings of Sobolev spaces on G which was obtained by the writer in [2 ]. We begin by noting that if (1) and (2) are equivalent norms in Wo '(G) then G must be quasicylindrical, i.e. dist.(x, bdry. G) must remain bounded for x in G. Otherwise for k = 1, 2, * * * there would exist a ball BkCG having radius k, and a function Uk belonging to Co (Bk) C Co (G) such that Uk (x) =1 in a ball of radius k -1 and I Dauk(x) I is bounded independently of k and x. It follows that