度量测度空间中的拟极值距离域

Quasi-extremal Distance Domains in Metric Measure Spaces

在度量测度空间中引进了拟极值距离域的概念 ,讨论了该区域的一些性质 .通过运用度量测度空间中曲线族的模、容器的容量以及测度论等工具 ,证明了所有的拟极值距离域都是拟凸和线性局部连通域 ,并且它的边界测度为零 ;得到了拟极值距离域上与测度和曲线的模有关的两个不等式 .从而得出结论 :在 Rn中 ,拟极值距离域许多性质的存在依赖于 Rn空间的正则性和连接任意两个不相交、非退化连续统的曲线族的模大于一个正常数这两个事实 .

This paper introduced the concept of quasi extremal distance (QED) domains in metric measure spaces and discussed the properties of these domains. By applying the tools such as the modulus of curves. the capacity of condensers and the theory of measure in metric measure spaces, it proved that all QED domains are quasi convex and linear locally connected domains and that the measure of the boundaries of QED domains is zero. These properties are well known in R n. Moreover, this paper obtained two inequalities which are related to the measure and the modulus of curves on QED domains. According to the above geometric properties of QED domains in metric measure spaces, it can conclude that the existence of many properties of QED domains in R n is based on the regularity of R n and the fact that the modulus of curves joining two disjoint and non degenerate continua in R n is larger than a positive constant, but not on its other properties.