We find necessary and sufficient conditions on a pair of rearrangement-invariant norms, ϱ and σ, in order that the Sobolev space be compactly imbedded into the rearrangement-invariant space , 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 into . The results are illustrated with examples in which ϱ and σ are both Orlicz norms or both Lorentz Gamma norms.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-2-2,
author = {Ron Kerman and Lubo\v s Pick},
title = {Compactness of Sobolev imbeddings involving rearrangement-invariant norms},
journal = {Studia Mathematica},
volume = {187},
year = {2008},
pages = {127-160},
zbl = {1158.46024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-2-2}
}
Ron Kerman; Luboš Pick. Compactness of Sobolev imbeddings involving rearrangement-invariant norms. Studia Mathematica, Tome 187 (2008) pp. 127-160. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm186-2-2/