POSITIVE p-HARMONIC FUNCTIONS ON GRAPHS

Abstract

Suppose that an inflnite graph G of bounded degree has flnite number of ends, each of which is p-regular, where 1 < p < 1. Then we can identify all the positive (bounded, respectively) p-harmonic functions on G. In this paper, we study the Liouville type property on a graph. By the Liouville property on a graph G, we mean that every bounded har- monic function on G is constant. It immediately follows that the set of all bounded harmonic functions on G having Liouville property is in one to one correspondence with the real line R. It seems natural to regard the case that the set of all bounded harmonic functions on G is in one to one correspondence with R l for some positive integer l as a general- ized version of the Liouville property. In line with this view point, we consider the case of the p-Laplacian operator (1 < p < 1) and the posi- tive (bounded, respectively) p-harmonic functions on graphs of bounded degree. In the case that a graph G has a flnite number of ends, each of which is p-regular, we identify all the positive (bounded, respectively) p-harmonic functions on G in section 4. On the other hand, Kanai(5) proved that if G and H are roughly isometric graphs of bounded degree, then H is parabolic whenever G is. In (9), Soardi proved that if G and H are roughly isometric graphs of bounded degree, and if G has no nonconstant harmonic functions with flnite energy, then neither has H. Later, the second author (8) proved that the dimension of the space of harmonic functions with flnite energy is preserved under rough isometries between graphs of bounded degree. In this paper, we also discuss its rough isometric invariance in section 3