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} $\tilde{X}:=(X,\rho ,\mu )$. The connection is that the logarithm of the generalised Jacobian of an $\eta$-quasisymmetric mapping $f:\tilde{X}\to \tilde{X}$ is always in $\mathrm{BMO}(\tilde{\mathrm{X}})$. In the course of proving this result, we first show that on $\tilde{X}$, the logarithm of a reverse-Hölder weight $w$ is in $\mathrm{BMO}(\tilde{\mathrm{X}})$, and that the above-mentioned connection holds on metric measure spaces $\hat{X}:=(X,d,\mu )$. 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 Calderón--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-Hölder weight.