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

Talgat Akhazhanov; Mendybay  Turgumbayev; Dauren Matin; Gulmira  Kenzhebekova; Akpeiil  Imanbetova; Altynay  Bozhayeva

doi:10.37256/cm.6120255460

Authors

Talgat Akhazhanov Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan https://orcid.org/0000-0003-3346-4947
Mendybay Turgumbayev Faculty of Mathematics and Information Technology, Buketov Karaganda University, Karaganda, 100028, Kazakhstan https://orcid.org/0009-0007-1422-8160
Dauren Matin Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan https://orcid.org/0000-0002-9784-9304
Gulmira Kenzhebekova Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan https://orcid.org/0000-0001-8060-6376
Akpeiil Imanbetova Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan https://orcid.org/0009-0000-7784-9895
Altynay Bozhayeva Faculty of Mechanics and Mathematics, L. N. Gumilyov Eurasian National University, Astana, 010000, Kazakhstan https://orcid.org/0009-0009-6244-9683

DOI:

https://doi.org/10.37256/cm.6120255460

Keywords:

pre-compact, generalized Morrey spaces, averaging function, commutator, riesz potential

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.

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

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