论文标题
有界平均振荡和准对称映射的函数在均匀类型的空间上
Functions of bounded mean oscillation and quasisymmetric mappings on spaces of homogeneous type
论文作者
论文摘要
我们建立了函数空间BMO与\ emph {同质类型} $ \ wideTilde {x} {x}：=（x，x，ρ，μ）$的函数空间与甲合物映射理论之间的联系。连接是，$η$ - quasisymmetric映射$ f的广义雅各布的对数：\ widetilde {x} \ rightarrow \ wideTilde {x} $始终以$ \ rm {bmo}（bmo}（bmo}）（\ widetilde {x} {x}）$。在证明这一结果的过程中，我们首先表明，在$ \ wideTilde {x} $上，反向hölder重量$ w $的对数为$ \ rm {bmo}（\ widetilde {x}）$，上述连接上上述连接在公元中，$ \ wide}。此外，我们构建了一大批空间$（x，ρ，μ）$。 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表明，$η$ quasisymmmore映射的体积衍生物是反向 - 荷兰德的重量。
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.
