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