A decomposition of a measurable function f by a one-sided local sharp maximal function and applications to one-sided operators

Following the ideas of Andrei Lerner in [ A pointwise estimate for the local sharp maximal function with applications to singular integrals" Bull. London Math. Soc. 42 (2010) 843856], we obtain another decomposition of an arbitrary measurable function f in terms of local mean oscillations. This allows us to get new estimates involving one-sided singular integrals and one-sided maximal operator. As an application to this result we obtain two weighted inequality for one-sided singular integrals and a L(w) inequality relating a measurable function f and sharp one-sided operator. These estimates are more precise in sense that they are valid for a greater class of weights.