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• # Artykuł - szczegóły

## Studia Mathematica

2008 | 186 | 2 | 127-160

## Compactness of Sobolev imbeddings involving rearrangement-invariant norms

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.

127-160

wydano
2008

### Twórcy

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