Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Volume 186 / 2008

Ron Kerman, Luboš Pick Studia Mathematica 186 (2008), 127-160 MSC: Primary 46E35. DOI: 10.4064/sm186-2-2


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.


  • Ron KermanDepartment of Mathematics
    Brock University
    500 Glenridge Avenue
    St. Catharines, Ontario
    Canada L2S 3A1
  • Luboš PickDepartment of Mathematical Analysis
    Faculty of Mathematics and Physics
    Charles University
    Sokolovská 83
    186 75 Praha 8, Czech Republic

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image