Вещественный, комплексный и и функциональный анализ

К исследованию весовой обобщенной гельдеровости гиперсингулярного интеграла на метрическом пространстве с мерой

Ю. Е. Дроботов^abc, Б. Г. Вакулов<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

Аннотация: 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.

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

УДК: 517.5

MSC: 31B10

Поступила 10 января 2023 г., опубликована 11 декабря 2024 г.

DOI: 10.33048/semi.2024.21.085