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Hausdorff operators on holomorphic Hardy spaces and applications

Published online by Cambridge University Press:  30 January 2019

Ha Duy Hung
High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam (
Luong Dang Ky
Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam (;
Thai Thuan Quang
Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam (;


The aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator

$${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$
is bounded on the Hardy spaces of the upper half-plane ${\rm {\cal H}}_a^p ({\open C}_ + )$, $p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.

Research Article
Copyright © 2019 The Royal Society of Edinburgh

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