Compactness of Sobolev imbeddings involving rearrangement-invariant norms

Ron Kerman, and Luboš Pick. "Compactness of Sobolev imbeddings involving rearrangement-invariant norms." Studia Mathematica 186.2 (2008): 127-160. <http://eudml.org/doc/286659>.

@article{RonKerman2008,
abstract = {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.},
author = {Ron Kerman, Luboš Pick},
journal = {Studia Mathematica},
keywords = {Sobolev imbedding; rearrangement invariant norm; interpolation; Orlicz spaces},
language = {eng},
number = {2},
pages = {127-160},
title = {Compactness of Sobolev imbeddings involving rearrangement-invariant norms},
url = {http://eudml.org/doc/286659},
volume = {186},
year = {2008},
}

TY - JOUR
AU - Ron Kerman
AU - Luboš Pick
TI - Compactness of Sobolev imbeddings involving rearrangement-invariant norms
JO - Studia Mathematica
PY - 2008
VL - 186
IS - 2
SP - 127
EP - 160
AB - 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.
LA - eng
KW - Sobolev imbedding; rearrangement invariant norm; interpolation; Orlicz spaces
UR - http://eudml.org/doc/286659
ER -