A pointwise estimate for the local sharp maximal function with applications to singular integrals

Following the ideas of Carleson, Garnett–Jones and Fujii, we obtain a decomposition of an arbitrary measurable function f in terms of local mean oscillations. As a main application, in the case p > n we prove a conjecture by Cruz‐Uribe and Pérez about two‐weight norm inequalities for singular integrals. Also we extend an inequality, due to the author, relating f and the local sharp maximal functionM λ # f . This allows us to get new estimates involving singular integrals and maximal functions.