Graphs of bounded degree and the $p$-harmonic boundary

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 $\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced ${\ell }^p$-cohomology of $\Gamma$ 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 $\Gamma$, the $p$-harmonic boundary of $\Gamma$ with the first reduced ${\ell }^p$-cohomology of $\Gamma$.

Give a new proof for theorem 4.7. Change the style of the text in the first two sections

References in corpus (1)

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

Cited by in corpus (2)

• Parabolic theory of the discrete p-Laplace operator
• Mixing, malnormal subgroups and cohomology in degree one