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

We establish a connection between the function space BMO and the theory of quasisymmetric mappings on spaces of homogeneous type X :=(X,ρ ,μ ) . The connection is that the logarithm of the generalised Jacobian of an η -quasisymmetric mapping f: X→X is always in BMO(X) . In the course of proving this result, we first show that on X , the logarithm of a reverse-Hölder weight w is in BMO(X) , and that the above-mentioned connection holds on metric measure spaces 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.