JOURNAL OF INTEGRAL EQUATIONS AND APPLICATIONS Volume 17, Number 4, Winter 2005 A CLASS OF MAXIMAL OPERATORS RELATED TO ROUGH SINGULAR INTEGRALS ON PRODUCT SPACES H. AL-QASSEM AND Y. PAN ABSTRACT. This paper is concerned with studying the L p boundedness of a class of maximal operators S (γ) Ω related to rough singular integrals on product spaces. We obtain appro- priate L p bounds for such maximal operators and establish the optimality of our condition on the kernel for the L 2 bound- edness of S (2) Ω . Our results improve substantially the main result obtained by Ding in [8]. 1. Introduction and statement of results. Throughout this paper, we let ξ  denote ξ/ |ξ | for ξ ∈ R n \{0} and p  denote the exponent conjugate to p, that is, 1/p+1/p  = 1. Let n, m ≥ 2. Suppose that S d−1 (d = n or m) is the unit sphere of R d equipped with the normalized Lebesgue measure dσ = dσ (x  ). In [7], Chen and Lin studied the L p boundedness of a class of maximal operators M (γ) Ω defined by M (γ) Ω f (x) = sup h      R n f (x − y)h(|y|)Ω(y/ |y|) |y| −n dy     , where the supremum is taken over the set {h : h L γ (R + ,dr/r) ≤ 1}, γ> 1 and Ω ∈ L 1 (S n−1 ) is a function satisfying the cancelation condition (1.1)  S n−1 Ω(y  ) dσ(y  )=0. Chen and Lin in [7] proved the L p boundedness of the maximal operator M (γ) Ω under a smoothness condition on Ω as described in the following theorem: 2000 AMS Mathematics Subject Classification. Primary 42B20, Secondary 42B15, 42B25. Key words and phrases. Rough kernel, singular integral, product domains. Received by the editors in December 2004, and in revised form on November 1, 2005. Copyright c 2005 Rocky Mountain Mathematics Consortium 331