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On the boundedness of the Hardy operator in the weighted space BMO

Об ограниченности оператора Харди в весовом пространстве BMO

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Abstract

The result of Golubov [5, Theorem 2] on the boundedness of the Hardy-Littlewood operator

$$ \mathcal{B}f(x): = \frac{1} {x}\int_0^x {f(t)} dt $$

in the space BMO(ℝ) is well known. The author of the present paper solves the analogous problem in the weighted space BMO on the semi-axis ℝ+ for the operator

$$ T_w f(x): = \frac{1} {{W(x)}}\int_0^x {f(t)w(t)} dt $$

and also in the classical space BMO(ℝ+) for a class of integral operators involving, for example, the Riemann-Liouville fractional integral.

Реэюме

Хорошо известен результат Б.И. Голубова [5 Теорема 2] об ограниченности оператора Харди-Литтлвуда

$$ \mathcal{B}f(x): = \frac{1} {x}\int_0^x {f(t)} dt $$

в пространстве BMO(ℝ). Автором статьи решены аналогичные задачи в весовом пространстве BMO на полуоси ℝ+ для оператора

$$ T_w f(x): = \frac{1} {{W(x)}}\int_0^x {f(t)w(t)} dt $$

и в классическом пространстве BMO(ℝ+) для одного класса интегральных операторов, включаюьим, например, дробный интеграл Римана-Лиувилла.

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Authors and Affiliations

  1. Peoples’ Friendship University of Russia, ul. Miklukho Maklaya 6, Moscow, 117 198, Russia

    Pham Tien Dung

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  1. Pham Tien Dung
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Dung, P.T. On the boundedness of the Hardy operator in the weighted space BMO. Anal Math 35, 289–299 (2009). https://doi.org/10.1007/s10476-009-0404-x

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