Optimal Orlicz-Sobolev embeddings

An embedding theorem for the Orlicz-Sobolev space W^{1,A}_0(G), where G is an open subset of R^n, into a space of Orlicz-Lorentz type is established for any given Young function A. Such a space is shown to be the best possible among all rearrangement invariant spaces. A version of the theorem for anisotropic spaces is also exhibited. In particular, our results recover and provide a unified framework for various well-known Sobolev type embeddings, including the classical inequalities for the standard Sobolev space W^{1,p}_0(G) by O'Neil and by Peetre (1<p< n), and by Brezis-Wainger and by Hansson (p=n).