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Hausdorff operators on some classical spaces of analytic functions

Part of: Integral, integro-differential, and pseudodifferential operators Spaces and algebras of analytic functions

Published online by Cambridge University Press:  29 February 2024

Huayou Xie  and
Qingze Lin Open the ORCID record for Qingze Lin [Opens in a new window]
Huayou Xie
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China e-mail: xiehy33@mail2.sysu.edu.cn
Qingze Lin*
Affiliation:
Department of Mathematics, Shantou University, Shantou, 515063, People’s Republic of China
*
e-mail: gdlqz@e.gzhu.edu.cn
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Abstract

In this note, we start on the study of the sufficient conditions for the boundedness of Hausdorff operators

$$ \begin{align*}(\mathcal{H}_{K,\mu}f)(z):=\int_{\mathbb{D}}K(w)f(\sigma_w(z))d\mu(w)\end{align*} $$
on three important function spaces (i.e., derivative Hardy spaces, weighted Dirichlet spaces, and Bloch type spaces), which is a continuation of the previous works of Mirotin et al. Here, $\mu $ is a positive Radon measure, K is a $\mu $-measurable function on the open unit disk $\mathbb {D}$, and $\sigma _w(z)$ is the classical Möbius transform of $\mathbb {D}$.

Keywords

Hausdorff operatorweighted Dirichlet spacederivative Hardy spaceBloch type spaceboundedness

MSC classification

Primary: 47G10: Integral operators 30H05: Bounded analytic functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 3 , September 2024 , pp. 768 - 780
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Qingze Lin is supported by the Natural Science Foundation of Jiaxing (Grant No. 2023AY40003).

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