Restricted mean value property for Balayage spaces with jumps (Q663508)

A continuous function \(f:D\to \mathbb R\) defined on an open set \(D\subset \mathbb R^d\) is known to be harmonic if and only if, for every \(x\in D\) and every \(r>0\) such that the closed ball \(\bar B(x,r)\) with center \(x\) and radius \(r\) is contained in \(D\), the volume mean of \(f\) over \(\bar B(x,r)\) is equal to \(f(x)\). A similar characterization holds if we replace volume means by surface means over the sphere \(\partial \bar B(x,r)\). However, the question whether an analogous characterization holds if the mean value property holds just for \textit{some} \(r=r(x)\) is more difficult. In the paper under review the author considers this restricted mean-value problem for functions harmonic with respect to Riesz potentials (potentials generated by \(\alpha\)-stable processes) or, more generally, in the setting of balayage spaces with jumps. Surprisingly, it is shown that the restricted mean-value problem in the setting of Riesz potentials is simpler than in the classical setting.