On a Local Mean Oscillation Decomposition

TL;DR

This paper introduces two local median oscillation decompositions for measurable functions and explores their applications to Calderón-Zygmund operators, emphasizing a key inequality involving sharp maximal functions and local estimates.

Contribution

It presents novel local median oscillation decompositions and extends their application to singular integral operators, including a local version of a fundamental inequality.

Findings

Established two local median oscillation decompositions.

Proved a key inequality relating sharp maximal functions and local maximal functions.

Discussed applications to Calderón-Zygmund singular integral operators.

Abstract

In this note we generate two local median oscillation decompositions of an arbitrary measurable function and discuss some applications to Calder\'{o}n-Zygmund singular integral operators $T$. These applications rely on the inequality $M^{\sharp}_{0,s}(Tf)(x) \leq c\,Mf(x)$, and we complete the results given here with a discussion of a local version of this estimate.