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

Abstract

We characterize the weights for the Stieltjes transform and the Calderon operator to be bounded on the weighted variable Lebesgue spaces $$L_w^{p(\cdot )}(0,\infty )$$ , assuming that the exponent function $${p(\cdot )}$$ is log-Holder 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 Duoandikoetxea et al. (Indiana Univ Math J 62(3):891–910, 2013) 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 Soria and Weiss (Indiana Univ Math J 43(1):187–204, 1994) for integral operators to the variable exponent setting.