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

TL;DR

This paper introduces a new decomposition of measurable functions using local mean oscillations, leading to improved weighted inequalities for one-sided singular integrals and maximal operators.

Contribution

It presents a novel decomposition method inspired by Lerner's ideas, enabling more precise estimates for one-sided operators and broader classes of weights.

Findings

New decomposition of measurable functions based on local mean oscillations

Derived sharper weighted inequalities for one-sided singular integrals

Extended the class of weights for which inequalities hold

Abstract

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.