Revista Matemática Complutense

, Volume 26, Issue 1, pp 33–55 | Cite as

On interpolation of cocompact imbeddings

Open Access
Article

Abstract

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.

Keywords

Besov spaces Cocompact imbeddings Concentration compactness Fractional Sobolev spaces Interpolation spaces Minimizers Mollifiers Sobolev imbeddings 

Mathematics Subject Classification (2000)

46B70 46E35 46B50 30H25 46N20 49J45 

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsTechnion–Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of MathematicsUppsala UniversityUppsalaSweden

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