MULTIPARAMETER OPERATORS AND SHARP WEIGHTED INEQUALITIES

This article is concerned with the operators from harmonic analysis wh ich are naturally associated to a multiple parameter family of dilatio ns. We are especially interested here in dealing with questions from t he theory of such operators whose answers cannot be obtained by a redu ction to the case of product operators. We also introduce a new tool i n order to carry out this multiparameter analysis which comes from the classical theory. This is the concept of ''sharp weighted inequalitie s,'' where, for operators such as the Hilbert transform or classical s quare function, one asks for Hilbert space inequalities which imply th e correct estimates of the norm on L(p) of these operators as p become s large.