Abstract
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.
Highlights
Studies of metric space geometry usually consider two types of synthetic negative curvature conditions: Alexandrov curvature (known as CAT(−1) spaces) and Gromov hyperbolicity
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, the transformed measure is globally doubling and supports a global p-Poincaré inequality on the corresponding uniformized space
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
Summary
Studies of metric space geometry usually consider two types of synthetic (i.e. axiomatic) negative curvature conditions: Alexandrov curvature (known as CAT(−1) spaces) and Gromov hyperbolicity. A homeomorphism : X → Y between two noncomplete metric spaces is quasisimilar if it is Cx -biLipschitz on every ball B(x, c0 dist(x, ∂ X )), for some 0 < c0 < 1 independent of x, and there exists a homeomorphism η : [0, ∞) → [0, ∞) such that for each distinct triple of points x, y, z ∈ X , dY ( (x), (y)) ≤ η dX (x, y) It was shown in [14, Theorem 4.36] that two roughly starlike Gromov hyperbolic spaces are biLipschitz equivalent if and only if any two of their uniformizations are quasisimilar. 10 we show that under certain natural conditions, the class of p-harmonic functions is preserved under the uniformization and hyperbolization procedures In this final section, we characterize which Gromov hyperbolic spaces with bounded geometry support the finite-energy Liouville theorem for p-harmonic functions. In the beginning of each section, we list the standing assumptions for that section in italicized text; in Sects. 2 and 4 these assumptions are given a little later
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