Multiparameter operators and sharp weighted inequalities

This article is concerned with the operators from harmonic analysis which are naturally associated to a multiple parameter family of dilations. We are especially interested here in dealing with questions from the theory of such operators whose answers cannot be obtained by a reduction to the case of product operators. We also introduce a new tool in order to carry out this multiparameter analysis which comes from the classical theory. This is the concept of "sharp weighted inequalities," where, for operators such as the Hilbert transform or classical square function, one asks for Hilbert space inequalities which imply the correct estimates of the norm on L^p of these operators as p becomes large.