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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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