2016/01/19 by Cyril Tintarev, Tintarev, Cyril
Mathematics · #46B20 #46B50 #46B99 (Primary) #46E15 #46E35 #47N20 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:46B20 #msc:46B50 #msc:46B99 #msc:46E15 #msc:46E35 #msc:47N20
paper · pdf · doi:10.48550/arxiv.1601.04873
arxiv created 2016/01/19 · arxiv updated 2016/01/20
Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured "bubble decomposition" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis.