An embedding theorem for the Orlicz-Sobolev space , , into a space of Orlicz-Lorentz type is established for any given Young function . 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 by O'Neil and by Peetre (), and by Brezis-Wainger and by Hansson ().
Cite this article
Andrea Cianchi, Optimal Orlicz-Sobolev embeddings. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 427–474DOI 10.4171/RMI/396