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arXiv · 2012.04110

BMO and the John-Nirenberg Inequality on Measure Spaces

Galia Dafni·Ryan Gibara·Andrew Lavigne
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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 The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones.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 $\mathscr{G}$.

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Galia Dafni, Ryan Gibara, Andrew Lavigne. 2020-12-07. BMO and the John-Nirenberg Inequality on Measure Spaces. https://arxiv.org/abs/2012.04110

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