Maximal operators with rough kernels on product domains

In this paper, we study the Lp boundedness of certain maximal operators on product domains with rough kernels in L(log L). We prove that our operators are bounded on Lp 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 L)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].  2005 Elsevier Inc. All rights reserved.