Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs

Abstract

In this paper, we study some potential theoretic properties of connected infinite networks and then investigate the space of $p$-Dirichlet finite functions on connected infinite graphs, via quasi-monomorphisms. A main result shows that if a connected infinite graph of bounded degrees possesses a quasi-monomorphism into the hyperbolic space form of dimension $n$ and it is not $p$-parabolic for $p>n-1$, then it admits a lot of $p$-harmonic functions with finite Dirichlet sum of order $p$.