Optimal Orlicz-Sobolev embeddings

  • Andrea Cianchi

    Universita di Firenze, Italy


An embedding theorem for the Orlicz-Sobolev space W01,A(G)W^{1,A}_{0}(G), GRnG\subset \mathbb{R}^n, into a space of Orlicz-Lorentz type is established for any given Young function AA. 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 W01,p(G)W^{1,p}_{0}(G) by O'Neil and by Peetre (1p<n1\leq p< n), and by Brezis-Wainger and by Hansson (p=np=n).

Cite this article

Andrea Cianchi, Optimal Orlicz-Sobolev embeddings. Rev. Mat. Iberoam. 20 (2004), no. 2, pp. 427–474

DOI 10.4171/RMI/396