Geodesic
Classification of metric measure spaces and their ends using p-harmonic functions
Anders Björn1 ; Jana Björn1 ; Nageswari Shanmugalingam2
1Linköping University, Department of Mathematics
2University of Cincinnati, Department of Mathematical Sciences
Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 1025-1052

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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.
DOI : 10.54330/afm.120618
Keywords: Classification of metric measure spaces, doubling measure, end at infinity, finite p-energy, p-hyperbolic sequence, Liouville theorem, p-harmonic function, Poincaré inequality, p-parabolic, quasiharmonic function, quasiminimizer

Anders Björn  1   ; Jana Björn  1   ; Nageswari Shanmugalingam  2

1 Linköping University, Department of Mathematics
2 University of Cincinnati, Department of Mathematical Sciences 
Anders Björn; Jana Björn; Nageswari Shanmugalingam. Classification of metric measure spaces and their ends using p-harmonic functions. Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 1025-1052. doi: 10.54330/afm.120618
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     title = {Classification of metric measure spaces and their ends using p-harmonic functions},
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