  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.
