BMO and the John-Nirenberg Inequality on Measure Spaces

TL;DR

This paper extends the theory of BMO spaces to general measure spaces, exploring their Banach space properties, relation to weights, and the John-Nirenberg inequality under measure-theoretic conditions.

Contribution

It provides necessary and sufficient conditions for BMO to be a Banach space and introduces Denjoy families to ensure the John-Nirenberg inequality holds in this setting.

Findings

BMO is a Banach space modulo constants under certain measure space conditions

Denjoy families guarantee the John-Nirenberg inequality in measure spaces

Conditions for BMO's properties are characterized in decomposable measure spaces

Abstract

We study the space BMO in the general setting of a measure space $\mathbb{X}$ with a fixed collection $\mathscr{G}$ of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in $\mathscr{G}$. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space $\mathbb{X}$ for BMO to be a Banach space modulo constants. We also develop the notion of a Denjoy family $\mathscr{G}$, which guarantees that functions in BMO satisfy the John-Nirenberg inequality on the elements of $\mathscr{G}$.