Hausdorff operators on weighted Bergman and Hardy spaces

Let $1\le p<\mathrm{\infty }$, $\alpha >-1$, and let $\phi$ be a measurable function on $(0,\mathrm{\infty })$. The main purpose of this paper is to study the Hausdorff operator $${\mathcal{H}}_{\phi }f(z)={\int }_0^{\mathrm{\infty }}f\left(\frac{z}{t}\right)\frac{\phi (t)}{t}dt,z\in {\mathbb{C}}^{+},$$ on the weighted Bergman space ${\mathcal{A}}_{\alpha }^p({\mathbb{C}}_{+})$ and on the power weighted Hardy space ${\mathcal{H}}_{\mathrm{\mid }\cdot {\mathrm{\mid }}^{\alpha }}^p({\mathbb{C}}_{+})$ of the upper half-plane. Some applications to the real version of ${\mathcal{H}}_{\phi }$ are also given.