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Weighted norm inequalities, off-diagonal estimates and elliptic operators. I: General operator theory and weights. (English) Zbl 1213.42030
Summary: 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-$\lambda$ inequality with two parameters and the other uses the 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 $1<p<\infty$. 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 $(1,\infty )$ 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:
42B20Singular and oscillatory integrals, several variables
47F05Partial differential operators
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