We establish a connection between the function space \(\mathrm{BMO}\) and the theory of quasisymmetric 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 \(\mathrm{BMO}({\widetilde{X}})\). In the course of proving this result, we first show that on \({\widetilde{X}}\), the logarithm of a reverse-Hölder weight w is in \(\mathrm{BMO}({\widetilde{X}})\), 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 Calderón–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-Hölder weight.

我们在函数空间\(\mathrm{BMO}\)和齐型空间上的拟对称映射理论 之间建立联系 \({\widetilde{X}} :=(X,\rho ,\mu )\) . 联系是\(\eta \) -拟对称映射 \(f: {\widetilde{X}} \rightarrow {\widetilde{X}}\)的广义雅可比矩阵的对数总是在 \(\mathrm {BMO}({\widetilde{X}})\)。在证明这个结果的过程中，我们首先证明在 \({\widetilde{X}}\) 上，反向 Hölder 权重w的对数 在 \(\mathrm{BMO}({\widetilde{X} })\)，并且上述连接在度量空间上成立 \({\widehat{X}} :=(X,d,\mu)\)。此外，我们构造了一大类空间 \((X,\rho ,\mu )\)来应用我们的结果。证明的关键要素包括从 Calderón-Zygmund 分解的欧几里得或度量空间设置、Vitali 覆盖定理、Radon-Nikodym 定理对\((X,\rho,\mu)\) 的适当推广，一个控制\(\eta \) -拟对称映射下集合失真的引理，以及 Heinonen 和 Koskela 的结果，表明\(\eta \) -拟对称映射的体积导数是一个反向 Hölder 权重.