Integrable Harmonic Functions on Symmetric Spaces of Rank One

If f ∈L1(dμ) is harmonic in the spaceG/K, whereμis a radial measure withμ(G/K)=1, we have, by the mean value propertyf=f∗μ. Conversely, does this mean value property imply thatfis 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 measuresμion a symmetric space of rank one for whichf∗μi=fimply thatfis harmonic.