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

Abstract

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite $p$-energy $p$-harmonic and $p$-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local $p$-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We 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 $p$-energy as spaces having at least two well-separated $p$-hyperbolic sequences of sets towards infinity. We also show that every such space $X$ has a function $f\notin L^p(X)+\mathbf{R}$ with finite $p$-energy.