Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights

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-X inequality with two parameters and the other uses Calderon-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 repres,entation by a kernel with size estimates, even rough, so that they may not be bounded on all L-p spaces for I < p < infinity. Pointwise estimates are then replaced by appropriate localized L-p-L-q estimates. We obtain weighted L-p estimates for a range of p that is different from (1, infinity) and isolate the right class of weights. In particular, we prove an extrapolation theorem "a la Rubio de Francia" for such a class and thus vector-valued estimates. (C) 2006 Elsevier Inc. All rights reserved.