The uniformization and hyperbolization transformations formulated by Bonk et al. in “Uniformizing Gromov Hyperbolic Spaces”, Astérisque, vol 270 (2001), dealt with geometric properties of metric spaces. In this paper we consider metric measure spaces and construct a parallel transformation of measures under the uniformization and hyperbolization procedures. We show that if a locally compact roughly starlike Gromov hyperbolic space is equipped with a measure that is uniformly locally doubling and supports a uniformly local p-Poincaré inequality, then the transformed measure is globally doubling and supports a global p-Poincaré inequality on the corresponding uniformized space. In the opposite direction, we show that such global properties on bounded locally compact uniform spaces yield similar uniformly local properties for the transformed measures on the corresponding hyperbolized spaces. We use the above results on uniformization of measures to characterize when a Gromov hyperbolic space, equipped with a uniformly locally doubling measure supporting a uniformly local p-Poincaré inequality, carries nonconstant globally defined p-harmonic functions with finite p-energy. We also study some geometric properties of Gromov hyperbolic and uniform spaces. While the Cartesian product of two Gromov hyperbolic spaces need not be Gromov hyperbolic, we construct an indirect product of such spaces that does result in a Gromov hyperbolic space. This is done by first showing that the Cartesian product of two bounded uniform domains is a uniform domain.

Bonk等人制定的均一化和超代谢转变。在Astérisque，第270卷（2001年）的“ Universizing Gromov双曲空间”中，讨论了度量空间的几何特性。在本文中，我们考虑了度量测度空间，并在均匀化和超音化程序下构造了测度的并行转换。我们表明，如果局部紧凑的近似星形的格罗莫夫双曲空间配备了一个一致的局部加倍的度量，并且支持一致的局部p- Poincaré不等式，那么变换后的度量就是全局加倍的并且支持一个全局p-对应一致空间上的庞加莱不等式。在相反的方向上，我们表明，在有界局部紧致均匀空间上的此类全局属性对于相应超增空间上的变换度量产生了相似的均匀局部属性。我们将上述结果用于度量的均一化，以表征配备了支持局部p- Poincaré不等式的局部一致加倍度量的Gromov双曲空间具有有限p的非恒定全局定义p-谐函数-能源。我们还研究了Gromov双曲和均匀空间的一些几何性质。尽管两个Gromov双曲空间的笛卡尔积不一定是Gromov双曲，但我们构造了此类空间的间接乘积，但确实产生了Gromov双曲空间。首先显示两个有界均匀域的笛卡尔积是一个均匀域。