Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights |
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Authors: | Pascal Auscher |
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Institution: | a Université de Paris-Sud et CNRS UMR 8628, 91405 Orsay Cedex, France b Departamento de Matemáticas - IMAFF, Consejo Superior de Investigaciones Científicas, C/ Serrano 123, 28006 Madrid, Spain c Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain |
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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 Lp-Lq 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. |
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Keywords: | 42B20 42B25 |
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