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

Authors

  • Anders Björn Linköping University, Department of Mathematics
  • Jana Björn Linköping University, Department of Mathematics
  • Nageswari Shanmugalingam University of Cincinnati, Department of Mathematical Sciences

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

Abstract

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.
Section
Articles

Published

2022-07-16

How to Cite

Björn, A., Björn, J., & Shanmugalingam, N. (2022). Classification of metric measure spaces and their ends using p-harmonic functions. Annales Fennici Mathematici, 47(2), 1025–1052. https://doi.org/10.54330/afm.120618