The well-known criterion for the precompactness of sets in a Banach function space states that a subset K of the absolutely continuous part Xa of a Banach function space X is precompact in X if and only if K is locally precompact in measure and K has uniformly absolutely continuous norm. There is a natural question whether this criterion characterizing precompact subsets in Banach function spaces can be extended to the setting of quasi-Banach function spaces when the elements of these spaces are not necessarily locally integrable (as for example is the case with the space Lp(Rn) with 0<p<1). We give a positive answer to this question. We also establish an extension of the well-known criterion characterizing precompact sets in the Lebesgue space Lp(Rn), 1\leq p < \infty, to the case when the space Lp(Rn) is replaced by a so-called power quasi-Banach function space over Rn. If time permits we show how to apply these criteria in order to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces over bounded measurable subsets W of Rn and illustrate such results on embeddings of Besov spaces Bsp,q( Rn), 0<s<1, 0<p,q\leq\infty into Lorentz-type spaces over bounded measurable subsets of Rn. This is joint work with Amiran Gogatishvili and Bohumír Opic.
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