Median-type John-Nirenberg space in metric measure spaces

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.