Hausdorff operators on holomorphic Hardy spaces and applications

The aim of this paper is to characterize the nonnegative functions $φ$ defined on $(0,\infty)$ for which the Hausdorff operator $\mathscr H_φf(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{φ(t)}{t}dt$ is bounded on the Hardy spaces of the upper half-plane $\mathcal H_a^p(\mathbb C_+)$, $p\in[1,\infty]$. The corresponding operator norms and their applications are also given.

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