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

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