Journal of the
Korean Mathematical Society JKMS

We establish appropriate $L^p$ estimates for a class of maximal operators $\mathcal{S}_\Omega ^{(\gamma )}$ on the product space $% \mathbf{R}^n\times \mathbf{R}^m$ when $\Omega $ lacks regularity and $1\leq \gamma \leq 2$. Also, when $\gamma =2,$ we prove the $L^p$ ($2 \leq p1)$. Moreover, we show that the condition $\Omega \in B_q^{(0,0)}\vspace{0.05in}(\mathbf{S}^{n-1}\times \mathbf{S}^{m-1})$ is nearly optimal in the sense that the operator $\mathcal{S}_\Omega ^{(2)}$ may fail to be bounded on $L^{2\mbox{ }}$ if the condition $\Omega \in B_q^{(0,0)}\vspace{0.05in}(\mathbf{S}^{n-1}\times \mathbf{S}^{m-1})$ is replaced by the weaker conditions $\Omega \in $ $B_q^{(0,\varepsilon )}\vspace{0.05in}(\mathbf{S}^{n-1}\times \mathbf{S}^{m-1})$ for any $-1<0$.

Keywords: rough kernel, block space, singular integral, maximal operator, product domains

MSC numbers: 42B20, 42B25, 42B30