The $p$-harmonic boundary and $D_p$-massive subsets of a graph of bounded degree

Abstract

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.

arXiv commentExpanded the introduction to include more background material. Added an alternative definition for a function to be p-harmonic on a graph