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

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 locally doubling measure supporting local $p$-Poincar\'e inequality. Similar classifications have earlier been obtained Riemann surfaces Riemannian manifolds. We also study the inclusions between these classes of spaces, their relationship to $p$-hyperbolicity space its ends. In particular, characterize that carry nonconstant functions as having at least two well-separated $p$-hyperbolic sequences. show every such $X$ has function $f \notin L^p(X) + \mathbb{R} $ $p$-energy.