# Bounded geometry and $p$-harmonic functions under uniformization and hyperbolization

## TL;DR

This paper explores how geometric transformations called uniformization and hyperbolization affect measures and harmonic functions on Gromov hyperbolic spaces, establishing conditions for the existence of finite energy p-harmonic functions.

## Contribution

It introduces a measure transformation framework under uniformization and hyperbolization, linking local and global properties, and characterizes when nonconstant p-harmonic functions exist on hyperbolic spaces.

## Key findings

- Transformed measures become globally doubling under certain conditions.
- Global properties on uniform spaces imply local properties on hyperbolized spaces.
- Constructs an indirect product of Gromov hyperbolic spaces that remains hyperbolic.

## Abstract

The uniformization and hyperbolization transformations formulated by Bonk, Heinonen and Koskela in \emph{"Uniformizing Gromov Hyperbolic Spaces"}, Ast\'erisque {\bf 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 $p$-Poincar\'e inequality, then the transformed measure is globally doubling and supports a global $p$-Poincar\'e 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 $p$-Poincar\'e 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.