chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2021-22-4-114-135cheb-1135Research ArticleСтатьиArticleПотенциал Рисса для (𝑘, 1)-обобщенного преобразования ФурьеRiesz potential for (𝑘, 1)-generalized Fourier transformИвановВалерий ИвановичIvanovValerii Ivanovich

доктор физико-математических наук, профессор

doctor of physical and mathematical sciences, professor

ivaleryi@mail.ruТульский государственный университетРоссияTula State UniversityRussian Federation202110012022224114135Copyright © Иванов В.И., 20212021Иванов В.И.Ivanov V.I.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/1135

В пространствах с весом |𝑥|−1𝑣𝑘(𝑥), где 𝑣𝑘(𝑥) — вес Данкля, действует (𝑘, 1)-обобщенное преобразование Фурье. Гармонический анализ в этих пространствах важен, в частности, в задачах квантовой механики. В работе для (𝑘, 1)-обобщенного преобразования Фурье определен потенциал Рисса. Для потенциала Рисса доказано (𝐿𝑞,𝐿𝑝)-неравенство с радиальными степенными весами, являющееся аналогом известного неравенства Стейна–Вейса для классического потенциала Рисса. Для потенциала Рисса получено точное значение 𝐿𝑝-нормы с радиальными степенными весами. Точное значение 𝐿𝑝-нормы с радиальными степенными весами для классического потенциала Рисса было получено независимо У. Бекнером и С. Самко.

In spaces with weight |𝑥|−1𝑣𝑘(𝑥), where 𝑣𝑘(𝑥) is the Dunkl weight, there is the (𝑘, 1)- generalized Fourier transform. Harmonic analysis in these spaces is important, in particular, inproblems of quantum mechanics. We define the Riesz potential for the (𝑘, 1)-generalized Fourier transform and prove for it, a (𝐿𝑞,𝐿𝑝)-inequality with radial power weights, which is an analogue of the well-known Stein–Weiss inequality for the classical Riesz potential. For the Riesz potential we calculate the sharp value of the 𝐿𝑝-norm with radial power weights. The sharp value of the 𝐿𝑝-norm with radial power weights for the classical Riesz potential was obtained independently by W. Beckner and S. Samko.

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The authors declare that there are no conflicts of interest present.