On interpolation of cocompact imbeddnings

TL;DR

This paper demonstrates that certain fractional Sobolev and Besov space embeddings are cocompact relative to lattice shifts, extending known results and applying interpolation techniques to broader function space contexts.

Contribution

It establishes cocompactness of subcritical fractional Sobolev and Besov space embeddings using interpolation, unifying previous results and providing new applications.

Findings

Cocompactness holds for subcritical fractional Sobolev and Besov spaces.

Applications to radial subspace embeddings and isoperimetric minimizers.

Extension of cocompactness results via interpolation techniques.

Abstract

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.