Sobolev embedding theorems for spaces Wk,p(x)(Ω)

This paper gives a Sobolev-type embedding theorem for the generalized Lebesgue-Sobolev space W-k.p(x)(Omega), where Omega is an open domain in R-N(N greater than or equal to 2) with cone property, and p(x) is a Lipschitz continuous function defined on fl satisfying 1 < p(-) less than or equal to p(+) < p(+) < N/k. The main result can be stated as follows: for any measurable function q(x)(x epsilon <()over bar>) with p(x) less than or equal to q(x) less than or equal to p(*)(x) := Np(x)/Np(x)/N - kp(x), there exists a continuous embedding from W-k,W-p(x)(Omega) to L-q(x)(Omega). (C) 2001 Academic Press.