# Existence and almost uniqueness for $p$-harmonic Green functions on bounded domains in metric spaces

## TL;DR

This paper investigates the existence, uniqueness, and properties of $p$-harmonic Green functions in bounded metric measure spaces, establishing conditions for their existence and comparability, and providing new characterizations.

## Contribution

It demonstrates the existence of $p$-harmonic Green functions based on capacity conditions and shows that all such functions with the same singularity are comparable, extending understanding beyond Euclidean spaces.

## Key findings

- Green functions exist iff the complement has positive capacity
- All Green functions with the same singularity are comparable
- Provides characterizations of singular functions in metric spaces

## Abstract

We study ($p$-harmonic) singular functions, defined by means of upper gradients, in bounded domains in metric measure spaces. It is shown that singular functions exist if and only if the complement of the domain has positive capacity, and that they satisfy very precise capacitary identities for superlevel sets. Suitably normalized singular functions are called Green functions. Uniqueness of Green functions is largely an open problem beyond unweighted $\mathbf{R}^n$, but we show that all Green functions (in a given domain and with the same singularity) are comparable. As a consequence, for $p$-harmonic functions with a given pole we obtain a similar comparison result near the pole. Various characterizations of singular functions are also given. Our results hold in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, or under similar local assumptions.