Rough Marcinkiewicz integral operators on product spaces

In this paper, we study the Marcinkiewicz integral operators $\mathcal{M}_{\Omega,h}$ on the product space $\mathbb{R}^n\times \mathbb{R}^m$. We prove that $\mathcal{M}_{\Omega,h}$ is bounded on $L^p(\mathbb{R}^n\times \mathbb{R}^m) (1 &lt p &lt \infty)$ provided that $h$ is a bounded radial function and $\Omega$ is a function in certain block space $B^{(0,0)}_q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$ for some $q > 1$. We also establish the optimality of our condition in the sense that the space $B^{(0,0)}_q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$ cannot be replaced by $B^{(0,r)}_q (\mathbb{S}^{n-1}\times \mathbb{S}^{m-1})$ for any $-1 &lt r &lt 0$. Our results improve some known results.