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

Department of Mathematics
University of Maryland
College Park, MD 20742

Department of Mathematics
H. H. Lehman College
Bronx, NY 10468

Dipartimento di Matematica
Universita di Roma "Tor Vergata"
Via della Ricerca Scientifica
00133 Roma

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]