Product Hardy Operators on Hardy Spaces

Abstract

We study the product Hausdorff operator $H_{\Phi}$ on the product Hardy spaces, and prove that, for a nonnegative valued function $\Phi$, $H_{\Phi}$ is bounded on the product Hardy space $H^{1}(\mathbb{R}\times \mathbb{R})$ if and only if $\Phi$ is a Lebesgue integrable function on $(0,\infty)\times (0,\infty)$. As an application, we know that the product Hardy operator $\mathcal{H}$ is not bounded on $H^{1}(\mathbb{R}\times \mathbb{R})$. On the other hand, we prove that $\left\Vert \mathcal{H}f\ \right\Vert_{H^{1}(\mathbb{R}\times \mathbb{R} )}$\ $\preceq \left\Vert f\ \right\Vert_{H^{1}(\mathbb{R}\times \mathbb{R} )}$ if $f$ is an even function. Furthermore, using the $H^{p}(\mathbb{R}\times \mathbb{R})$ boundedness criterion of Fefferman, we prove that the $k$-th order Hardy operator is bounded on $H^{p}(\mathbb{R}\times \mathbb{R})$ whenever $k>1/p-1$.