Hausdorff operators on weighted Bergman and Hardy spaces

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