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.
Bibliographical noteFunding Information:
* Sponsored by the Edmund Landau Center for research in Mathematical supported by the Minerva Foundation (Germany).
- Harmonic functions; mean value property; symmetric spaces of rank one
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