Real, complex and functional analysis

On the weighted generalized Hölder continuity of a hypersingular integral over a metric measure space

Yu. E. Drobotov^abc, B. G. Vakulov<sup>a

a</sup> Institute of Mathematics, Mechanics and Computer Sciences I.I. Vorovich, Milchakova str., 8a, 344090, Rostov-on-Don, Russia
^b North-Caucasus Center for Mathematical Research of the Vladikavkaz Scientific Centre of the Russian Academy of Sciences, ul. Vil'yamsa, 1, 363110, RSO-Alaniya, Prigorodnyj rajon, s.Mihajlovskoe, Russia
^c Don State Technical University, Gagarina squ., 1, 344000, Rostov-on-Don, Russia

Abstract: Weighted Zygmund-type estimates are obtained for a hypersingular integral over an almost homogeneous metric measure space. Based on these estimates, theorems on the action of this integral operator in weighted variable generalized Hölder spaces are proved. As the weight function, a representative from the power function class is considered, the exponent of which does not exceed 1. It is shown that the hypersingular integral "degrades" the characteristic of the generalized Hölder space by an order of the hypersingular integral. The conditions of the presented theorems are formulated in terms of the Bary–Stechkin class and a special analog of the Dini continuity condition.

Keywords: bounded operator, continuity, generalized Hölder spaces, hypersingular integral, integral equations, local modulus of continuity, modulus of submetry, Zygmund-type estimates.

UDC: 517.5

MSC: 31B10

Received January 10, 2023, published December 11, 2024

DOI: 10.33048/semi.2024.21.085