Sobolev embedding theorems for spaces W-k,p(x)(Omega)

This paper gives a Sobolev-type embedding theorem for the generalized Lebes gue-Sobolev space W-k.p(x)(Omega), where Omega is an open domain in R-N(N g reater than or equal to 2) with cone property, and p(x) is a Lipschitz cont inuous 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 measurabl e function q(x)(x epsilon <(<Omega>)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.