Converse mean value theorems on trees and symmetric spaces

Harmonic functions satisfy the mean value property with respect to all integrable radial weights if f is harmonic then hf f h for any such weight h But need a function f that satises this relation with a given nonnegative h b e harmonic By a classical result of Furstenb erg the answer is p ositive for every b ounded f on a Riemannian symmetric space but if the boundedness condition is relaxed then the answer turns out to depend on the weight h In this paper various types of weights are investigated on Euclidean and hyp erb olic spaces as well as on homogeneous and semihomogeneous trees IRf h decays faster than exponentially then the mean value property hf f h does not imply harmonicity of f For weights than exponentially at least a weak converse mean value the eigenfunctions of the Laplace operator which satisfy harmonic The critical case is that of exp onential decay exhibit weights that characterize harmonicity and others that do no