Multiple Weighted Norm Inequalities for Maximal Multilinear Singular Integrals with Non-Smooth Kernels Loukas Grafakos, Liguang Liu, and Dachun Yang Abstract Weighted norm inequalities for maximal truncated operators of multilinear singular integrals with non-smooth kernels in the sense of Duong, Grafakos, and Yan are obtained; this class of operators extends the class of multilinear Calder´on- Zygmund operators introduced by Coifman and Meyer and includes the higher order commutators of Calder´on. The weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of Lerner, Ombrosi, P´ erez, Torres, and Trujillo-Gonz´alez. The key ingredient in the proof is the introduction of a new multi-sublinear maximal operator that plays the role of the Hardy-Littlewood maximal function in a version of Cotlar’s inequality. As applications of these results, new weighted estimates for the m-th order Calder´on commutators and their maximal counterparts are deduced. 1 Introduction We consider multilinear operators T initially defined on the m-fold product of Schwartz spaces on R n and taking values into the space of tempered distributions, T : m times    S×···×S→S ′ . Every such operator is associated with a distribution kernel on (R n ) m+1 . Throughout the paper, we assume that the distribution kernel coincides with a function K defined away from the diagonal y 0 = y 1 = ··· = y m in (R n ) m+1 , and T is associated with the kernel K in the following way: T (f 1 , ··· ,f m )(x)=  (R n ) m K(x, y 1 , ··· ,y m )f 1 (y 1 ) ··· f m (y m ) dy 1 ··· dy m , (1.1) whenever x/ ∈∩ m j =1 supp f j and f 1 , ··· ,f m are C ∞ functions with compact support. 2000 Mathematics Subject Classification. Primary 42B20; Secondary 42B25. Key words and phrases. multilinear operator, maximal truncated operator, multiple weight, generalized Calder´on-Zygmund operator, commutator. The first author was supported by grant DMS 0900946 of the National Science Foundation of the USA and the University of Missouri Research Council. The third author was supported by National Natural Science Foundation (Grant No. 10871025) of China and Program for Changjiang Scholars and Innovative Research Team in University of China. 1