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.
Download