Endpoint estimates for Rubio de Francia operators

Abstract

The extrapolation theory of Rubio de Francia provides a tool to obtain Ap weighted estimates on Lp spaces for every 1 <p< ∞, starting from information at a single 1 < p0 < ∞. However, the endpoint case p = 1 cannot be reached in general. Classical extrapolation arguments in the sense of Yano can be added to this setting to deduce results close to L1 without weights. In this paper, we present different approaches that produce endpoint estimates with respect to the whole A1 class. We give applications to the Carleson operator and maximally modulated singular integrals among others.