On interpolation of cocompact imbeddings
2013 (English)In: REVISTA MATEMATICA COMPLUTENSE, ISSN 1139-1138, Vol. 26, no 1, 33-55 p.Article in journal (Refereed) Published
Cocompactness is a useful weaker counterpart of compactness in the study of imbeddings between function spaces. In this paper we prove that, under quite general conditions, cocompactness of imbeddings of Banach spaces persists under both real and complex interpolation. As an application, we obtain that subcritical continuous imbeddings of fractional Sobolev spaces and Besov spaces are cocompact relative to lattice shifts. We deduce this by interpolating the known cocompact imbeddings for classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We also apply cocompactness to prove compactness of imbeddings of some radial subspaces and to show the existence of minimizers in some isoperimetric problems. Our research complements a range of previous results, and recalls that there is a natural conceptual framework for unifying them.
Place, publisher, year, edition, pages
2013. Vol. 26, no 1, 33-55 p.
Besov spaces, Cocompact imbeddings, Concentration compactness, Fractional Sobolev spaces, Interpolation spaces, Minimizers, Mollifiers, Sobolev imbeddings
IdentifiersURN: urn:nbn:se:uu:diva-309906DOI: 10.1007/s13163-011-0087-2ISI: 000313128100002OAI: oai:DiVA.org:uu-309906DiVA: diva2:1056717