Functions satisfying the mean value property in the limit
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.
scientific article; zbMATH DE number 4105325
scientific article