Asymptotic mean value property for eigenfunctions of the Laplace-Beltrami operator on Damek-Ricci spaces

Abstract

Let $S$ be a Damek-Ricci space equipped with the Laplace-Beltrami operator $\Delta$. In this paper we characterize all eigenfunctions of $\Delta $ through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to infinity.