Functions of Bounded Mean Oscillation and Quasisymmetric Mappings on Spaces of Homogeneous Type

Abstract

We establish a connection between the function space BMO and the theory of quasisymmetric mappings on \emph{spaces of homogeneous type} $\widetilde{X} :=(X,\rho,\mu)$. The connection is that the logarithm of the generalised Jacobian of an $\eta$-quasisymmetric mapping $f: \widetilde{X} \rightarrow \widetilde{X}$ is always in $\rm{BMO}(\widetilde{X})$. In the course of proving this result, we first show that on $\widetilde{X}$, the logarithm of a reverse-H\"{o}lder weight $w$ is in $\rm{BMO}(\widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $\widehat{X} :=(X,d,\mu)$. Furthermore, we construct a large class of spaces $(X,\rho,\mu)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,\rho,\mu)$ from the Euclidean or metric measure space settings of the Calder\'{o}n--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $\eta$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $\eta$-quasisymmetric mapping is a reverse-H\"{o}lder weight.