Functions of bounded mean oscillation and quasiconformal mappings on spaces of homogeneous type

We establish a connection between the function space BMO and the theory of quasiconformal mappings on 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 $\text{BMO}(\widetilde{X})$. 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 $\text{BMO}(\widetilde{X})$, and that the above-mentioned connection holds on a metric measure space $\widehat{X} :=(X,d,\mu)$. 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 and the Radon--Nikodym Theorem, and of the result of Heinonen and Koskela which shows that the volume derivative is a reverse-H\"{o}lder weight.

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