Revista Matemática Iberoamericana

Optimal Orlicz-Sobolev embeddings

Andrea Cianchi

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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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Rev. Mat. Iberoamericana, Volume 20, Number 2 (2004), 427-474.

First available in Project Euclid: 17 June 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Sobolev inequalities Orlicz spaces rearrangement invariant spaces interpolation


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

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