**Journal of Lie Theory
**

Vol. 8, No. 2, pp. 229-254 (1998)

#
Converse mean value theorems on trees and symmetric spaces

##
E. Casadio Tarabusi, J. M. Cohen, A. Koranyi, and M. A. Picardello

Dipartimento di Matematica

"G. Castelnuovo"

Universita di Roma "La Sapienza"

Piazzale A. Moro 2

00185 Roma

Italy

casadio@alpha.science.unitn.it
Department of Mathematics

University of Maryland

College Park, MD 20742

USA

jmc@math.umd.edu

Department of Mathematics

H. H. Lehman College

CUNY

Bronx, NY 10468

USA

adam@alpha.lehman.cuny.edu

Dipartimento di Matematica

Universita di Roma "Tor Vergata"

Via della Ricerca Scientifica

00133 Roma

Italy

picard@mat.uniroma2.it

**Abstract:** Harmonic functions satisfy the mean value property with respect to all integrable radial weights: if $\scriptstyle f$ is harmonic then $\scriptstyle h*f=f\int h$ for any such weight $\scriptstyle h$. But need a function $\scriptstyle f$ that satisfies this relation with a given (non-negative) $\scriptstyle h$ be harmonic? By a classical result of Furstenberg the answer is positive for every bounded $\scriptstyle f$ on a Riemannian symmetric space, but if the boundedness condition is relaxed then the answer turns out to depend on the weight $\scriptstyle h$. \endgraf In this paper various types of weights are investigated on Euclidean and hyperbolic spaces as well as on homogeneous and semi-homogeneous trees. If $\scriptstyle h$ decays faster than exponentially then the mean value property $\scriptstyle h*f=f\int h$ does not imply harmonicity of $\scriptstyle f$. For weights decaying slower than exponentially, at least a weak converse mean value property holds: the eigenfunctions of the Laplace operator which satisfy $\scriptstyle h*f=f\int h$ are harmonic. The critical case is that of exponential decay. In this class we exhibit weights that characterize harmonicity and others that do not.\endgraf

**Keywords:** Mean value property, harmonic functions, Laplace operator, trees, symmetric spaces, hyperbolic spaces, convolution operators, exponential decay

**Classification (MSC91):** 43A85; 31B05, 05C05, 53C35

**Full text of the article:**

[Previous Article] [Next Article] [Contents of this Number]