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

Abstract

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}.

Mathematical Inequalities & Applications (to appear)