arXiv subjects
Publications and source records attributed to Michael Puls.
Let $p$ be a real number greater than one and let $X$ be a locally compact, noncompact metric measure space that satisfies certain conditions. The $p$-Royden and $p$-harmonic boundaries of $X$ are constructed by using the $p$-Royden algebra of functions on $X$ and a Dirichlet type problem is solved for the $p$-Royden boundary. We also characterize the metric measure spaces whose $p$-harmonic boundary is empty.
Let $p$ be a real number greater than one and let $\Gamma$ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma$ and the $D_p$-massive subsets of $\Gamma$. In particular, if there are $n$ pairwise disjoint $D_p$-massive subsets of $\Gamma$, then the $p$-harmonic boundary of $\Gamma$ consists of at least $n$ elements. We also show that the converse of this statement is also true.
Let $p$ be a real number greater than one and let $\Gamma$ be a connected graph of bounded degree. We show that the $p$-Royden boundary of $\Gamma$ with the $p$-harmonic boundary removed is a $F_{\sigma}$-set. We also characterize the $p$-harmonic boundary of $\Gamma$ in terms of the intersection of the extreme points of a certain subset of one-sided infinite paths in $\Gamma$.
Let $p$ be a real number greater than one and let $G$ be a finitely generated, infinite group. In this paper we introduce the $p$-harmonic boundary of $G$. We then characterize the vanishing of the first reduced $\ell^p$-cohomology of $G$ in terms of the cardinality of this boundary. Some properties of $p$-harmonic boundaries that are preserved under rough isometries are also given. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $G$, the $p$-harmonic boundary of $G$ and the first reduced $\ell^p$-cohomology of $G$.
Let $G$ be a finitely generated, infinite group, let $p>1$, and let $L^p(G)$ denote the Banach space $\{\sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of $G$ with coefficients in $L^p(G)$, and the first reduced $L^p$-cohomology space of $G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups.