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

Abstract

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 p spaces for 1 < p < ∞ . 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 , ∞ ) 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.