Rough singular integrals on product spaces

where, p.v. denotes the principal value. It is known that if Φ is of finite type at 0 (see Definition 2.2) and Ω ∈ 1(Sn−1), then TΦ,Ω is bounded on Lp for 1 1. Subsequently, the Lp (1< p <∞) boundedness of TΦ,Ω was established under conditions much weaker than Ω ∈ Lq(Sn−1) [1, 6]. In particular, Al-Qassem et al. [1] established the Lp boundedness of TΦ,Ω under the condition that the function Ω belongs to the block space B 0,0 q (Sn−1) introduced by Jiang and Lu in (see [14]). In fact, they proved the following theorem.