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

In the present paper, we characterize the nonnegative functions $\phi$ for which the multi-parameter Hausdorff operator ${\mathcal{H}}_{\phi }$ generated by $\phi$ 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}.