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Hausdorff operators on holomorphic Hardy spaces and applications

Part of: Special classes of linear operators Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  30 January 2019

Ha Duy Hung ,
Luong Dang Ky  and
Thai Thuan Quang
Ha Duy Hung
Affiliation:
High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam (hunghaduy@gmail.com)
Luong Dang Ky
Affiliation:
Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam (luongdangky@qnu.edu.vn; thaithuanquang@qnu.edu.vn)
Thai Thuan Quang
Affiliation:
Department of Mathematics, Quy Nhon University, 170 An Duong Vuong, Quy Nhon, Binh Dinh, Viet Nam (luongdangky@qnu.edu.vn; thaithuanquang@qnu.edu.vn)
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Abstract

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.

Keywords

Hausdorff operatorHardy spaceholomorphic functionHilbert transform

MSC classification

Primary: 47B38: Operators on function spaces (general)
Secondary: 42B30: $H^p$-spaces 46E15: Banach spaces of continuous, differentiable or analytic functions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 3 , June 2020 , pp. 1095 - 1112
Copyright
Copyright © 2019 The Royal Society of Edinburgh

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