Graphs of bounded degree and the p-harmonic boundary

2008/06/18 by Michael J. Puls, Puls, Michael J.
Computer Science · Mathematics · #31C20 #43A15 #60J50 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.0806.3073

openalex publication_date 2008/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let p be a real number greater than one and let G be a connected graph of bounded degree. In this paper we introduce the p-harmonic boundary of G. We use this boundary to characterize the graphs G for which the constant functions are the only p-harmonic functions on G. It is shown that any continuous function on the p-harmonic boundary of G can be extended to a function that is p-harmonic on G. Some properties of this boundary that are preserved under rough-isometries are also given. Now let Γ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced ℓ^p-cohomology of Γ in terms of the cardinality of its p-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on Γ, the p-harmonic boundary of Γ with the first reduced ℓ^p-cohomology of Γ.

Citations

• Vanishing of the first reduced cohomology with values in an L^p-representation

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