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

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 Uchiyamas construction of certain extremal BMO-functions and John-Nirenbergs lemma.