Classification of metric measure spaces and their ends using $p$-harmonic functions

TL;DR

This paper classifies metric measure spaces based on the behavior of $p$-harmonic functions, extending earlier results from Riemannian geometry to more general spaces, and explores their hyperbolic properties and ends.

Contribution

It introduces a classification framework for metric measure spaces using Liouville theorems for $p$-harmonic functions, generalizing known results to broader settings.

Findings

Spaces with nonconstant finite energy $p$-harmonic functions have at least two $p$-hyperbolic sequences.

Characterization of spaces supporting nonconstant $p$-harmonic functions with finite energy.

Existence of functions outside $L^p + ext{constants}$ with finite $p$-energy in such spaces.

Abstract

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local $p$-Poincar\'e inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We also study the inclusions between these classes of metric measure spaces, and their relationship to the $p$-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant $p$-harmonic functions with finite energy as spaces having at least two well-separated $p$-hyperbolic sequences. We also show that every such space $X$ has a function $f \notin L^p(X) + \mathbb{R} $ with finite $p$-energy.