Trujillo-González, R.: New maximal functions and multiple weights for the multilinear CalderónZygmund theory Adv

Abstract. A multi(sub)linear maximal operator that acts on the product of m Lebesgue spaces and is smaller that the m-fold product of the Hardy-Littlewood maximal function is studied. The operator is used to obtain a precise control on mul-tilinear singular integral operators of Calderón-Zygmund type and to build a theory of weights adapted to the multilinear setting. A natural variant of the operator which is useful to control certain commutators of multilinear Calderón-Zygmund operators with BMO functions is then considered. The optimal range of strong type estimates, a sharp end-point estimate, and weighted norm inequalities involving both the clas-sical Muckenhoupt weights and the new multilinear ones are also obtained for the commutators. 1