Compactness in quasi-Banach function spaces and applications to compact embeddings of Besov-type spaces

António Caetano, Amiran Gogatishvili, Bohumír Opic

Published 2013-07-20, updated 2017-01-10Version 3

There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \infty$, into Lorentz-type spaces.