Hausdorff operators on weighted Bergman and Hardy spaces

TL;DR

This paper investigates the boundedness and properties of Hausdorff operators on weighted Bergman and Hardy spaces in the upper half-plane, providing new insights and applications in complex analysis.

Contribution

It introduces new results on the boundedness of Hausdorff operators on weighted Bergman and Hardy spaces, extending previous work to these specific function spaces.

Findings

Characterization of bounded Hausdorff operators on weighted Bergman spaces

Extension of results to weighted Hardy spaces

Applications to real-variable versions of the operator

Abstract

Let $1\leq p<\infty$, $\alpha>-1$, and let $\varphi$ be a measurable function on $(0,\infty)$. The main purpose of this paper is to study the Hausdorff operator \[ \mathscr H_\varphi f(z)=\int_0^\infty f\left(\frac{z}{t}\right) \frac{\varphi(t)}{t} dt, \quad z\in \mathbb C^+, \] on the weighted Bergman space $\mathcal A^p_\alpha(\mathbb C_+)$ and on the power weighted Hardy space $\mathcal H^p_{|\cdot|^\alpha}(\mathbb{C_+})$ of the upper half-plane. Some applications to the real version of $\mathscr H_\varphi$ are also given.