Functions with finite Dirichlet sum of orderpand 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.

Expansion Constants and Hyperbolic Embeddings of Finite Graphs

“…Let G H = (V H , E H ) be a 1-net of the hyperbolic space H n = (H n , d H ) of dimension n. From Proposition 9.3 (and its proof) in [15], we can deduce that, for p > n − 1,…”

“…Obviously, (R ( p) G ) 1/ p induces a distance on V and we note that for x, y ∈ V , R [14,15]). For p = 2, R…”

“…G (x, y) is called the effective resistance between x and y; the quantity R ( p) G (x, y) for p > 1 has been considered in [9,10,14,15]. We note that the reciprocal of R ( p) G (x, y) is the p-capacity of the pair (x, y).…”

“…For any δ and κ, a metric ball of G × Z admits no (δ, κ) quasimonomorphisms to H n if the radius is sufficiently large, since G × Z has no quasi-monomorphisms to H n . The existence of quasi-monomorphisms of infinite graphs to the hyperbolic space is relevant to the existence of non-trivial p-harmonic functions with finite Dirichlet sums of order p (see [15]). …”