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

TL;DR

This paper extends precompactness criteria from Banach to quasi-Banach function spaces and applies these results to establish compact embeddings of Besov spaces into various quasi-Banach spaces, including Lorentz-type spaces.

Contribution

It generalizes precompactness criteria to quasi-Banach spaces and demonstrates their use in compactly embedding Besov spaces into Lorentz-type and other quasi-Banach spaces.

Findings

Extended precompactness criteria to quasi-Banach spaces.

Established compact embeddings of Besov spaces into Lorentz-type spaces.

Illustrated results with specific embeddings of $B^s_{p,q}$ spaces.

Abstract

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.