Restricted Mean Values and Harmonic Functions
John R. Baxter · Transactions of the American Mathematical Society · 1972
A function h defined on a region R in ${{\mathbf {R}}^n}$ will be said to possess a restricted mean value property if the value of the function at each point is equal to the mean value of the function over one open ball in R, with centre at that point. It is proved here that this restricted mean value property implies h is harmonic under certain conditions.