Median-type John-Nirenberg space in metric measure spaces

TL;DR

This paper extends the John-Nirenberg space concept to metric measure spaces using medians instead of averages, establishing inequalities and equivalences under certain conditions.

Contribution

It introduces a median-based John-Nirenberg space in metric measure spaces and proves key inequalities and boundary behavior results.

Findings

Established local and global John-Nirenberg inequalities.

Proved the equivalence of integral and median-type spaces under chaining assumptions.

Developed a Calderón-Zygmund decomposition and good-$\lambda$ inequality for medians.

Abstract

We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calder\'{o}n-Zygmund decomposition and a good-$\lambda$ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by using chaining arguments. As a consequence, the integral-type and the median-type John-Nirenberg spaces coincide under a Boman-type chaining assumption.