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 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 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.

arXiv comment27 pages

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• Prime ends for domains in metric spaces
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Cited by in corpus (1)

• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces