Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces
Dragičević, Oliver ...
We obtain sharp weighted ▫$L^p$▫ estimates in the Rubio de Francia extrapolation theorem in terms of the ▫$A_p$▫ characteristic constant of the weight. 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 $w$, 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, we show that they are sharp for the Hilbert, Beurling, and martingale transforms.
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