Maximal operators with rough kernels on product domains

In this paper, we study the L^p boundedness of certain maximal operators on product domains with rough kernels in L(log L). We prove that our operators are bounded on L^p for all 2 ≤ p < ∞. Moreover, we show that our condition on the kernel is optimal in the sense that the space L(log L) cannot be replaced by L(log)^r for any r < 1. Our results resolve a problem left open in [Y. Ding, A note on a class of rough maximal operators on product domains, J. Math. Anal. Appl. 232 (1999) 222-228].