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Hausdorff operators on holomorphic Hardy spaces and applications
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics / Volume 150 / Issue 3 / June 2020
- Published online by Cambridge University Press:
- 30 January 2019, pp. 1095-1112
- Print publication:
- June 2020
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- Article
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The aim of this paper is to characterize the non-negative functions φ defined on (0,∞) for which the Hausdorff operator
is bounded on the Hardy spaces of the upper half-plane
$${\rm {\cal H}}_\varphi f(z) = \int_0^\infty f \left( {\displaystyle{z \over t}} \right)\displaystyle{{\varphi (t)} \over t}{\rm d}t$$
${\rm {\cal H}}_a^p ({\open C}_ + )$, 
$p\in [1,\infty ]$. The corresponding operator norms and their applications are also given.
