*Submitted Paper*

**Inserted:** 25 jun 2021

**Last Updated:** 25 jun 2021

**Year:** 2021

**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.*

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