A Liouville theorem for matrix-valued harmonic functions on nilpotent groups

摘要

Let sigma be a non-degenerate positive M-n-valued measure on a locally compact group G with parallel tosigmaparallel to = 1. An M-n-valued Borel function f on G is called sigma-harmonic if f (x) = integral(G)f(xy(-1))dsigma(y) for all x is an element of G. Given such a function f which is bounded and left uniformly continuous on G, it is shown that every central element in G is a period of f. Further, it is shown that f is constant if G is nilpotent or central. [References: 17]