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

Anders Björn¹ ; Jana Björn¹ ; Nageswari Shanmugalingam²

¹ Linköping University, Department of Mathematics
² University of Cincinnati, Department of Mathematical Sciences

Annales Fennici Mathematici, Tome 47 (2022) no. 2, pp. 1025-1052.

Résumé

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.

Détail

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