A “LOCAL MEAN OSCILLATION” DECOMPOSITION AND SOME ITS APPLICATIONS

1. Introduction A recent result by the author [39] establishes a pointwise control of an arbitrary measurable function in terms of its local mean oscillations. Soon after that, in a surprising work [12], D. Cruz-Uribe, J. Martell and C. P´erez showed that this result can be effectively applied in a variety of questions, including sharp weighted inequalities for classical singular integrals and the dyadic square function. In turn, based on [12] and on a recent concept of the intrinsic square function by M. Wilson [54], the author [40] obtained sharp weighted estimates for essentially any Littlewood-Paley operator. The aim of these notes is to present a unified, extended and almost self-contained exposition of the above-mentioned works [39, 12, 40].