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Classification of metric measure spaces and their ends using \(p\)-harmonic functions. (English) Zbl 1503.31021

Summary: 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.

MSC:

31E05 Potential theory on fractals and metric spaces
30L99 Analysis on metric spaces
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