The Calderón operator and the Stieltjes transform on variable Lebesgue spaces with weights

Abstract

We characterize the weights for the Stieltjes transform and the Calder´on operator to be bounded on the weighted variable Lebesgue spaces $L_w^{p(cdot)}(0,infty)$, assuming that the exponent function $pp$ is log-H"older continuous at the origin and at infinity. We obtain a single Muckenhoupt-type condition by means of a maximal operator defined with respect to the basis of intervals ${ (0,b) : b>0}$ on $(0,infty)$. Our results extend those in cite{DMRO1} for the constant exponent $L^p$ spaces with weights. We also give two applications: the first is a weighted version of Hilbert´s inequality on variable Lebesgue spaces, and the second generalizes the results in cite{SW} for integral operators to the variable exponent setting.