Revista Matemática Complutense

, Volume 26, Issue 1, pp 33–55

On interpolation of cocompact imbeddings

Open AccessArticle

DOI: 10.1007/s13163-011-0087-2

Cite this article as:
Cwikel, M. & Tintarev, K. Rev Mat Complut (2013) 26: 33. doi:10.1007/s13163-011-0087-2

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 spacesCocompact imbeddingsConcentration compactnessFractional Sobolev spacesInterpolation spacesMinimizersMollifiersSobolev imbeddings

Mathematics Subject Classification (2000)

46B7046E3546B5030H2546N2049J45
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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