A characterization of BMO self-maps of a metric measure space

TL;DR

This paper characterizes BMO self-maps on metric measure spaces with doubling measures, extending Euclidean results by employing generalized extremal functions and the John-Nirenberg lemma to broader metric contexts.

Contribution

It provides new characterizations of BMO-preserving maps in metric measure spaces, generalizing known Euclidean results to more abstract settings.

Findings

Characterizations of BMO self-maps in metric measure spaces.

Extension of Euclidean BMO results to metric spaces.

Use of generalized extremal functions and John-Nirenberg lemma.

Abstract

This paper studies functions of bounded mean oscillation (BMO) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve BMO. This extends the corresponding Euclidean results by Gotoh to metric measure spaces. The argument is based on a generalizations Uchiyama's construction of certain extremal BMO-functions and John-Nirenberg's lemma.