Compactness of Sobolev imbeddings involving rearrangement-invariant norms
Volume 186 / 2008
Studia Mathematica 186 (2008), 127-160
MSC: Primary 46E35.
DOI: 10.4064/sm186-2-2
Abstract
We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, $\varrho$ and $\sigma$, in order that the Sobolev space $W^{m,\varrho}({\mit\Omega})$ be compactly imbedded into the rearrangement-invariant space $L_\sigma({\mit\Omega})$, where ${\mit\Omega}$ is a bounded domain in ${\mathbb R}^n$ with Lipschitz boundary and $1\leq m\leq n-1$. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{\varrho}(0,|{\mit\Omega}|)$ into $L_{\sigma}(0,|{\mit\Omega}|)$. The results are illustrated with examples in which $\varrho$ and $\sigma$ are both Orlicz norms or both Lorentz Gamma norms.
