Pre-Compact Sets in the Generalized Morrey Spaces in Terms of the Averaging Function

Abstract

In this paper, sufficient conditions for compactness of sets in generalized Morrey spaces are given in terms of an averaging function. This result is analogous to the well-known Fréchet-Kolmogorov theorem for the pre-compacting of sets in Lebesgue spaces. Our main result consisted of four conditions on the behavior of the function norm and the norm of its averages in generalised Morrey spaces that are sufficient for a set to be pre-compact in these spaces. An example is provided to demonstrate that not all conditions obtained in the main result are necessary for a set to be pre-compacted in generalised Morrey spaces.