Elsevier

Advances in Mathematics

Volume 212, Issue 1, 20 June 2007, Pages 225-276
Advances in Mathematics

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

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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 Lp spaces for 1<p<. Pointwise estimates are then replaced by appropriate localized LpLq estimates. We obtain weighted Lp 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.

MSC

42B20
42B25

Keywords

Good-λ inequalities
Calderón–Zygmund decomposition
Muckenhoupt weights
Vector-valued inequalities
Extrapolation
Singular non-integral operators
Commutators with bounded mean oscillation functions

Cited by (0)

This work was partially supported by the European Union (IHP Network “Harmonic Analysis and Related Problems” 2002–2006, Contract HPRN-CT-2001-00273-HARP). The second author was also supported by MEC “Programa Ramón y Cajal, 2005” and by MEC Grant MTM2004-00678.