Dyadic John-Nirenberg space

TL;DR

This paper introduces the dyadic John-Nirenberg space, extending BMO functions, and establishes key properties like boundedness of the dyadic maximal operator and space completeness, using median-based inequalities.

Contribution

It generalizes the classical John-Nirenberg space by incorporating medians and dyadic structures, and proves fundamental properties including boundedness and completeness.

Findings

Boundedness of the dyadic maximal operator on the space

Construction method for nontrivial functions in the space

Proof of the space's completeness

Abstract

We discuss the dyadic John-Nirenberg space that is a generalization of functions of bounded mean oscillation. A John-Nirenberg inequality, which gives a weak type estimate for the oscillation of a function, is discussed in the setting of medians instead of integral averages. We show that the dyadic maximal operator is bounded on the dyadic John-Nirenberg space and provide a method to construct nontrivial functions in the dyadic John-Nirenberg space. Moreover, we prove that the John-Nirenberg space is complete. Several open problems are also discussed.