Maximal Operator and Weighted Norm Inequalities for Multilinear Singular Integrals

The analysis of multilinear singular integrals has much of its origins in several works by Coifman and Meyer in the 70’s; see for example [3]. More recently, in [4] and [5], an updated systematic treatment of multilinear singular integral operators of Calderón-Zygmund type was presented in light of some new developments. See also [6] and the references therein for a detailed description of previous work in the subject. In this article we prove the boundedness of a maximal operator associated to multilinear singular intergals and we use it to obtain multilinear weighted norm inequalities. We will consider multilinear operators T initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions, T : S(R) × · · · × S(R) → S ′(Rn). Every such operator is associated with a distributional kernel on (R). We will assume that this distributional kernel coincides with a function K defined away from the diagonal y0 = y1 = y2 = · · · = ym in (R) which satisfies the size estimate |K(y0, y1, . . . , ym)| ≤ A ( ∑m k,l=0 |yk − yl|)mn (1)