Title: Can optimal rearrangement invariant Sobolev imbeddings be further extended?

Authors: Guillermo P. Curbera and Werner J. Ricker

Issue: Volume 56 (2007), Issue 3, 1479-1498

Abstract: Sobolev imbeddings (over suitable open subsets of $\mathbb{R}^n$) can be extended from the classical $L^p$-setting to that of more general norms (required to be \textit{rearrangement invariant}) on the underlying function spaces. This has been thoroughly studied in recent years and shown to be intimately connected to an associated kernel operator (of one variable). This kernel operator always has an optimal domain (being a Banach function space, but typically \textit{not} rearrangement invariant) to which it can be continuously extended. So, techniques aside, there is no a priori reason not to treat Sobolev imbeddings for \textit{non}-rearrangement invariant norms. This is the aim of the present paper.