Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces (Q558344)

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.