A converse to the mean value property on homogeneous trees

Abstract:

The homogeneous tree ${\mathbf {T}}$ of degree $q + 1\quad (q \geq 2)$ may be considered as a discrete analogue of the open unit disc ${\mathbf {D}}$. On ${\mathbf {D}}$, every harmonic function satisfies the mean value property (MVP) at every point. Conversely, positive functions on ${\mathbf {D}}$ having the MVP with respect to a ball with specified radius at each point of ${\mathbf {D}}$ are harmonic under certain assumptions concerning the radius function: results of this type are due to J. R. Baxter, W. Veech and others. Here we consider harmonic functions on ${\mathbf {T}}$ with respect to a natural choice of a discrete Laplacian: the analogous MVP is true in this setting. We present a Lipschitz-type condition on the radius function (which now has integer values and refers to the discrete metric of ${\mathbf {T}}$) under which harmonicity holds for positive functions whose value at each point is the mean of its values over the ball of the radius assigned to this point. The method is based upon our previous results concerning the geometrical realization of Martin boundaries of certain transition operators as the space of ends of the underlying graph.