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} $\widetilde{X} :=(X,ρ,μ)$. The connection is that the logarithm of the generalised Jacobian of an $η$-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-Hölder weight $w$ is in $\rm{BMO}(\widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $\widehat{X} :=(X,d,μ)$. Furthermore, we construct a large class of spaces $(X,ρ,μ)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,ρ,μ)$ from the Euclidean or metric measure space settings of the Calderón--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $η$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $η$-quasisymmetric mapping is a reverse-Hölder weight.