The compactness of sets in global Morrey-type spaces can be characterized by analyzing the difference of functions within the set. This approach provides a framework to understand compactness in terms of function behavior and properties, such as decay and smoothness, within the Morrey-type spaces. The main attention is paid to the study of the structural properties of these spaces and the analysis of the behavior of functions in them. In particular, it is proved that for the choice of the weight function w(r)=r^-2, where 0≤λ≤n/p, the formulated theorem implies a well-known result for the Morrey space. In case λ=0, the proven theorem reduces to the classical Frechet-Kolmogorov theorem, which is the cornerstone of the theory of functional analysis.

To achieve the main goal of the paper, several auxiliary lemmas are preliminarily proved concerning the estimation of the mean values ​​of functions in the global Morrey-type spaces. These lemmas are not only important tools for proving the main theorem, but are also of independent interest, since they can be applied to other problems of analysis.

In addition, the necessity and sufficiency of the obtained conditions are discussed, as well as their connection with the known results of the theory of functional spaces. Particular attention is paid to the discussion of applications of the obtained results to the study of compactness of operators and the theory of differential equations. Thus, the results of the work expand the existing ideas about the properties of global Morrey-type spaces and open up new possibilities for their application in various areas of analysis and operator theory.

MATIN D.T.

PhD Doctor, Associate Professor, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan.

Е-mail: d.matin@mail.kz, https://orcid.org/0000-0002-9784-9304

AKHAZHANOV T.B.

PhD Doctor, Associate Professor, L.N. Gumilyov Eurasian National University, Astana, Kazakhstan.

Е-mail: talgat_a2008@mail.ru, https://orcid.org/0000-0003-3346-4947

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