Integrable harmonic functions on symmetric spaces of rank one

If f is an element of L-1(d mu) is harmonic in the space G/K, where mu is a radial measure with mu(G/K)= 1, we have, by the mean value property f = f * mu. Conversely, does this mean value property imply that f is harmonic? In this paper we give a new and natural proof of a result obtained by P. Ahern, A. Flores, W. Rudin (J. Funct. Anal. 11 (1993), 380-397) and A. Koranyi (Contemp. Math. 191 (1995), 107-116) and generalize their result by providing sufficient conditions for a finite set of radial measures mu(i) on a symmetric space of rank one for which f * mu(i) = f imply that f is harmonic. (C) 1998 Academic Press.