# The multi-parameter Hausdorff operators on $H^1$ and $L^p$

## TL;DR

This paper characterizes functions for which multi-parameter Hausdorff operators are bounded on Hardy and Lebesgue spaces, providing operator norms and improving previous results while answering an open question.

## Contribution

It offers a complete characterization of boundedness for multi-parameter Hausdorff operators on Hardy and Lebesgue spaces, advancing the theoretical understanding.

## Key findings

- Characterization of functions for boundedness of Hausdorff operators
- Explicit operator norms derived
- Improvement over previous results and resolution of an open question

## Abstract

In the present paper, we characterize the nonnegative functions $\varphi$ for which the multi-parameter Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on the multi-parameter Hardy space $H^1(\mathbb R\times\cdots\times\mathbb R)$ or $L^p(\mathbb R^n)$, $p\in [1,\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \cite{Li}.