Median-Type John–Nirenberg Space in Metric Measure Spaces

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 Calderón–Zygmund decomposition and a good-λ 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.