Sharp maximal function estimates for multilinear singular integrals

Abstract

A new proof of a weighted norm inequality for multilinear singu- lar integrals of Calderon-Zygmund type is presented through a more general estimate involving a sharp maximal function. An application is given to the study of certain multilinear commutators. The study of multilinear singular integrals of Calderon-Zygmun type continues to attract many researchers. Many results obtained parallel the linear theory of classical Calderon-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such operators can be found in the article by Grafakos and Torres (GT1). See also the work of Kenig and Stein (KS) and the survey article (GT2) for further references and details. One aspect of the theory that still is being developed is the one related to the study of maximal operators associated to multilinear singular integrals and appro- priate versions of multilinear weighted norm inequalities. In a recent work Grafakos and Torres (GT3) have obtained multilinear weighted norm inequalities based on a version of Cotlar's inequality in the multilinear setting. Their approach provides multilinear analogous of the works by Coifman (C) and Coifman and Feerman (CF). Here we present a dierent approach based on the use of a modified version of the sharp maximal function of Feerman