# Hausdorff operators on holomorphic Hardy spaces and applications

## TL;DR

This paper characterizes functions that ensure the Hausdorff operator is bounded on Hardy spaces of the upper half-plane, providing operator norms and exploring applications in complex analysis.

## Contribution

It offers a complete characterization of nonnegative functions for boundedness of Hausdorff operators on Hardy spaces, including explicit operator norms and applications.

## Key findings

- Characterization of functions for bounded Hausdorff operators on Hardy spaces
- Explicit formulas for operator norms
- Applications to complex analysis and operator theory

## Abstract

The aim of this paper is to characterize the nonnegative functions $\varphi$ defined on $(0,\infty)$ for which the Hausdorff operator $$\mathscr H_\varphi f(z)= \int_0^\infty f\left(\frac{z}{t}\right)\frac{\varphi(t)}{t}dt$$ 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.