ar X iv : m at h / 06 03 64 0 v 1 [ m at h . C A ] 2 8 M ar 2 00 6 WEIGHTED NORM INEQUALITIES , OFF - DIAGONAL ESTIMATES AND ELLIPTIC OPERATORS

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-λ inequality with two-parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular “non-integral” operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, “non-integral” that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all L spaces for 1 < p < ∞. Pointwise estimates are then replaced by appropriate localized L − L estimates. We obtain weighted L estimates for a range of p that is different from (1,∞) and isolate the right class of weights. In particular, we prove an extrapolation theorem “à la Rubio de Francia” for such a class and thus vector-valued estimates.