有界度图与$p$ -调和边界
Graphs of bounded degree and the $p$-harmonic boundary
摘要：
设$p$是大于1的实数，$G$是有界度的连通图.本文引入了$G$的$p$ -调和边界.我们用这个边界来刻画图$G$，其中常数函数是$G$上唯一的$p$ -调和函数。证明了在$G$的$p$ -调和边界上的任意连续函数都可以推广为在$G$上的$p$ -调和函数.还给出了该边界在粗糙等距下保留的一些性质。现在让$Γ$成为一个有限生成群。作为我们的结果的一个应用，我们刻画了$Γ$的第一约化$\ell^p$ -上同调的消失在其$p$ -调和边界的基数。我们还研究了$Γ$上函数的某个差空间上平移不变线性泛函之间的关系，$Γ$的$p$ -调和边界与$Γ$的第一约化$\ell^p$ -上同调.
英文摘要：
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 $\ell^p$-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 $\ell^p$-cohomology of $Γ$.
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