Bounded Geometry and p-Harmonic Functions Under Uniformization and Hyperbolization

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.

Rough isometry between Gromov hyperbolic spaces and uniformization

“…Should Y be another Gromov hyperbolic metric space equipped with a uniformly locally doubling measure supporting a uniformly local Poincaré inequality, and X is roughly isometric to Y , then we know from our main theorem, Theorem 3.8, that Y ε is also a uniform space. It then follows from the results in [BBS2] that the induced measure on Y ε is doubling and supports a Poincaré inequality; the results of [BBS2] show that the trace of the Newton-Sobolev class of functions on Y ε is a Besov class on ∂Y ε , and that a subset of ∂Y ε is null for this Besov class if and only if it is null for the Newton-Sobolev class. Null sets for Newton-Sobolev classes are reasonably well understood, and this understanding translates to a reasonable understanding of Besov-null sets in ∂Y ε .…”

“…When Z does not support such a Poincaré inequality, for example if Z does not have sufficiently many rectifiable curves, then the Newton-Sobolev class is the wrong class for potential theory on Z; in this case, the more appropriate function class is a Besov class of functions on Z. See for example [GKS,BBS,BBS2] for more on Besov classes. There are Gromov hyperbolic spaces X for which X ε is a uniform domain but ∂X ε may not even be connected; hence the potential theory on ∂X ε should be via Besov classes.…”

Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces

“…We will consider how measures transform under the generalization of this uniformization procedure that we introduced in [9]. As in [2], we will show that uniformizing these measures upgrades uniformly local doubling properties and uniformly local Poincaré inequalities to global doubling and global Poincaré inequalities for the uniformized space. Our results allow us to construct a number of interesting new unbounded metric measure spaces supporting Poincaré inequalities.…”

“…Broadly speaking our work in this paper has three closely related objectives. The primary objective is to generalize the recent results of Björn-Björn-Shanmugalingam [2] concerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces [4]. We will consider how measures transform under the generalization of this uniformization procedure that we introduced in [9].…”

Trace and Density Results on Regular Trees

“…Partial motivation for this paper comes from boundary value problems for the p-Laplace equation. For the case of manifolds see [12,13] and for the setting of metric spaces see [2,4,15]. Classical trace results on the Euclidean spaces can be found in [1,6,9,14,16,22,25,29,30] and studies of parabolicity on infinite networks in [26,32].…”