Hausdorff operators on weighted Bergman and Hardy spaces

Abstract

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

24 pages, J. Math. Anal. Appl. (2025), You can (before July 5, 2025) download the paper from https://www.sciencedirect.com/science/article/pii/S0022247X25004421?dgcid=author