Graphs of bounded degree and the<i>p</i>-harmonic boundary

Abstract

Let p be a real number greater than one and let G be a connected graph of bounded degree.We introduce the p-harmonic boundary of G and use it to characterize the graphs G for which the constant functions are the only pharmonic functions on G.We show that any continuous function on the pharmonic boundary of G can be extended to a function that is p-harmonic on G.We also give some properties of this boundary that are preserved under rough-isometries.Now let be a finitely generated group.As an application of our results, we characterize the vanishing of the first reduced p -cohomology of 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 , the p-harmonic boundary of , and the first reduced p -cohomology of .