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

We obtain sharp weighted L p estimates in the Rubio de Francia extrapolation theorem in terms of the Ap characteristic constant of the weight. Precisely, if for a given 1 r it is bounded on L p (v) by the same increasing function of the Ap characteristic constant of v, and for p < r it is bounded on L p (v) by the same increasing function of the r 1 p 1 power of the Ap 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.