Resumen de Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces

Loukas Grafakos , Stefanie Petermichl , Oliver Dragicevic, María Cristina Pereyra

• 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.