Median-Type John\u2013Nirenberg Space in Metric Measure Spaces

Kim Myyryläinen

doi:10.1007/s12220-022-00872-9

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 Calderón–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.

Highlights

The space of functions of bounded mean oscillation (BMO) was introduced by John and Nirenberg in [27]
The John– Nirenberg space contains BMO, and BMO is obtained as the limit of J N p as p → ∞
As a corollary of the global John–Nirenberg inequality, we show that the integral- and median-type John–Nirenberg spaces coincide in every open set under the assumption that balls are Boman sets with uniform parameters (Corollary 5.4)

Summary

Introduction

The space of functions of bounded mean oscillation (BMO) was introduced by John and Nirenberg in [27]. He proved the analogous John–Nirenberg inequality for the median-type BMO in a Euclidean space. The proof of the local John–Nirenberg inequality with medians (Theorem 4.4) in Sect. As a corollary of the global John–Nirenberg inequality, we show that the integral- and median-type John–Nirenberg spaces coincide in every open set under the assumption that balls are Boman sets with uniform parameters (Corollary 5.4).

Results

Conclusion