Optimal Orlicz-Sobolev embeddings

Abstract

An embedding theorem for the Orlicz-Sobolev space W^{1,A}_{0}(G) , G\subset \mathbb{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\leq p< n ), and by Brezis-Wainger and by Hansson ( p=n ).