Hausdorff operators on weighted Bergman and Hardy spaces

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.