Open Access
June, 2004 Optimal Orlicz-Sobolev embeddings
Andrea Cianchi
Rev. Mat. Iberoamericana 20(2): 427-474 (June, 2004).


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 ($p=n$).


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Andrea Cianchi. "Optimal Orlicz-Sobolev embeddings." Rev. Mat. Iberoamericana 20 (2) 427 - 474, June, 2004.


Published: June, 2004
First available in Project Euclid: 17 June 2004

zbMATH: 1061.46031
MathSciNet: MR2073127

Primary: 46E30 , 46E35

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

Rights: Copyright © 2004 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.20 • No. 2 • June, 2004
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