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Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces. (English) Zbl 1081.42007
The authors obtain sharp weighted $$L^p$$ estimates in an extrapolation theorem which was first introduced by Rubio de Francia. They obtain the estimates in terms of the $$A_p$$ characteristic constant of the weight. More precisely, if for a given $$1<r<\infty$$ the norm of a sublinear operator on $$L^r(w)$$ is bounded by a function of the $$A_r$$ characteristic constant of the weight function $$w$$, then, for $$p>r$$, it is bounded on $$L^p(v)$$ by the same increasing function of the $$A_p$$ characteristic constant of $$v$$, and for $$p<r$$ it is bounded on $$L^p(v)$$ by the same increasing function of the $$\frac{r-1}{p-1}$$ power of the $$A_p$$ characteristic constant of $$v$$. For some operators these bounds are sharp, but not always. In particular, the authors show that they are sharp for the Hilbert, Beurling, and martingale transforms.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42A50 Conjugate functions, conjugate series, singular integrals 42B25 Maximal functions, Littlewood-Paley theory 46M35 Abstract interpolation of topological vector spaces 44A15 Special integral transforms (Legendre, Hilbert, etc.) 47B38 Linear operators on function spaces (general)
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