  Cocompactness is a useful weaker counterpart of compactness in the study of
imbeddings between function spaces. In this paper we show that subcritical
continuous imbeddings of fractional Sobolev spaces and Besov spaces over
\mathbb{R}^{N} are cocompact relative to lattice shifts. We use techniques of
interpolation spaces to deduce our results from known cocompact imbeddings for
classical Sobolev spaces ("vanishing" lemmas of Lieb and Lions). We give
examples of applications of cocompactness to compactness of imbeddings of some
radial subspaces and to 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.
