WEIGHTED NORM INEQUALITIES AND VECTOR-VALUED INEQUALITIES FOR CERTAIN ROUGH OPERATORS

We prove two weighted norm inequalities of the form integral \TF\p u less-than-or-equal-to C(p) integral Absolute value of f(p) Nu, where N is a maximal operator which may be chosen to be bounded on any given L(r), r > 1. The operators which we consider include Hilbert transforms and maximal functions along homogeneous curves, as well as other examples whose kernels fail to satisfy standard regularity conditions. A key element in our approach is the use of weighted non-isotropic Littlewood-Paley theory.