Functions satisfying the mean value property in the limit (Q1122027)

Let \(\mu\) be a given positive radial measure on \(\underset \tilde{} R^ n\), of infinite mass, and let \(\mu_ r\) be its normalized restriction to the ball B(0,r). The authors consider the question of whether the harmonic functions on \(\underset \tilde{} R^ n\) can be characterized as those continuous functions for which \(\mu_ r*f\to f\) locally uniformly as \(r\to \infty\). The measure \(\mu\) is said to satisfy the comparison condition if, for any continuous radial function f on \(\underset \tilde{} R^ n\) such that \(\ell =\lim_{r\to \infty}\mu_ r*f(0)\) exists, the limit of \(\mu_ r*f(x)\) exists and is equal to \(\ell\). Standard techniques show that, if \(\mu\) satisfies the comparison condition and f is a continuous function such that \(\mu_ r*f\to f\) locally uniformly, then f is harmonic. A characterization of those measures, of the form \(d\mu (x)=\rho (| x|)dx\) for a non-negative differentiable function \(\rho\), that satisfy the comparison condition is proved. Several variants on the problem are also considered, and examples that illustrate the constraints on the hypotheses are given.