New maximal functions and multiple weights for the multilinear Calderón-Zygmund theory

A. LERNER, S. OMBROSI, C. PÉREZ, R. TORRES, R. TRUJILLO

ADVANCES IN MATHEMATICS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Año: 2009 p. 1222 - 1264

0001-8708

A multi(sub)linear maximal operator that acts on the product of mLebesgue spaces and is smaller that the m-fold product of the Hardy-Littlewoodmaximal function is studied. The operator is used to obtain a precise control on multilinearsingular integral operators of Calder´on-Zygmund type and to build a theoryof weights adapted to the multilinear setting. A natural variant of the operator whichis useful to control certain commutators of multilinear Calder´on-Zygmund operatorswith BMO functions is then considered. The optimal range of strong type estimates,a sharp end-point estimate, and weighted norm inequalities involving both the classicalMuckenhoupt weights and the new multilinear ones are also obtained for the commutators