Converse mean value theorems on trees and symmetric spaces

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

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