Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron Kerman ¹   ; Luboš Pick ²

¹ Department of Mathematics Brock University 500 Glenridge Avenue St. Catharines, Ontario Canada L2S 3A1
² Department of Mathematical Analysis Faculty of Mathematics and Physics Charles University Sokolovská 83 186 75 Praha 8, Czech Republic

Studia Mathematica, Tome 186 (2008) no. 2, pp. 127-160

Résumé

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.

Détail

DOI : 10.4064/sm186-2-2

Keywords: necessary sufficient conditions pair rearrangement invariant norms varrho sigma order sobolev space varrho mit omega compactly imbedded rearrangement invariant space sigma mit omega where mit omega nbsp bounded domain mathbb lipschitz boundary leq leq n particular establish equivalence compactness nbsp sobolev imbedding compactness nbsp certain hardy operator varrho mit omega sigma mit omega results illustrated examples which varrho sigma orlicz norms lorentz gamma norms