Revista Matemática Iberoamericana

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Volume 20, Issue 2, 2004, pp. 427–474
DOI: 10.4171/RMI/396

Published online: 2004-08-31

Optimal Orlicz-Sobolev embeddings

Andrea Cianchi[1]

(1) Universita di Firenze, Italy

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 ($p=n$).

Keywords: Sobolev inequalities, Orlicz spaces, rearrangement invariant spaces, interpolation

Cianchi Andrea: Optimal Orlicz-Sobolev embeddings. Rev. Mat. Iberoam. 20 (2004), 427-474. doi: 10.4171/RMI/396