Compactness of Sobolev imbeddings involving rearrangement-invariant norms

• Autorzy: Ron Kerman , Luboš Pick
• Języki publikacji: EN

Abstrakty

EN

We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space $W^{m,ϱ}(Ω)$ be compactly imbedded into the rearrangement-invariant space $L_{σ}(Ω)$, where Ω is a bounded domain in ℝⁿ with Lipschitz boundary and 1 ≤ m ≤ n-1. In particular, we establish the equivalence of the compactness of the Sobolev imbedding with the compactness of a certain Hardy operator from $L_{ϱ}(0,|Ω|)$ into $L_{σ}(0,|Ω|)$. The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.

Kategorie tematyczne

• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2008
• Tom: 186
• Numer: 2
• Strony: 127-160

Daty

wydano

2008

Twórcy

autor

Ron Kerman

• Department of Mathematics, Brock University, 500 Glenridge Avenue, St. Catharines, Ontario, Canada L2S 3A1

autor

Luboš Pick

• Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

Identyfikatory

DOI

10.4064/sm186-2-2