Extensions of the John-Nirenberg theorem and applications

TL;DR

This paper extends the John-Nirenberg theorem to relate maximal functions and weighted oscillations, with applications to inequalities, weight classes, and polynomial BMO spaces.

Contribution

It introduces two new extensions of the John-Nirenberg theorem, connecting maximal functions and weighted oscillations, and explores their applications.

Findings

Relation between dyadic maximal and sharp maximal functions.

Generalization of weighted mean oscillation results.

Applications to Poincaré inequalities and weight classes.

Abstract

The John-Nirenberg theorem states that functions of bounded mean oscillation are exponentially integrable. In this article we give two extensions of this theorem. The first one relates the dyadic maximal function to the sharp maximal function of Fefferman-Stein, while the second one concerns local weighted mean oscillations, generalizing a result of Muckenhoupt and Wheeden. Applications to the context of generalized Poincar\'e type inequalities and to the context of the $C_p$ class of weights are given. Extensions to the case of polynomial BMO type spaces are also given.