齐型空间上的有界平均振动函数与拟对称映射
Functions of bounded mean oscillation and quasisymmetric mappings on spaces of homogeneous type
摘要：
建立了函数空间BMO与$\widetilde{X} :=(X,ρ,μ)$上的拟对称映射理论之间的联系。联系是$η$ -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$\rm{BMO}(\widetilde{X})$中.在证明这一结果的过程中，我们首先证明了在$\widetilde{X}$上，一个逆Hölder权$w$的对数在$\rm{BMO}(\widetilde{X})$中，并且上述联系在度量测度空间$\widehat{X} :=(X,d,μ)$上成立.此外，我们构造了一个大类的空间$(X,ρ,μ)$，我们的结果适用。在证明的关键成分是适当的推广到$(X,ρ,μ)$从欧几里德或度量测度空间设置的Calderon-Zygmund分解，维塔利覆盖定理，Radon-Nikodym定理，一个引理，控制失真的集合下的$η$ -拟对称映射，Heinonen和Kotcha的一个结果表明，$η$ -拟对称映射的体积导数是一个逆Hölder权.
英文摘要：
We establish a connection between the function space BMO and the theory of quasisymmetric mappings on \emph{spaces of homogeneous type} $\widetilde{X} :=(X,ρ,μ)$. The connection is that the logarithm of the generalised Jacobian of an $η$-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-Hölder weight $w$ is in $\rm{BMO}(\widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $\widehat{X} :=(X,d,μ)$. 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.
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