A characterization of polyharmonic functions (Q343278)

The author generalizes the familiar mean value characterization of harmonic functions to the case of polyharmonic functions, as follows. Let \(m\in \mathbb{N}\cup \{0\}\) and \(u\) be a continuous function on an open set \(\Omega \) in \(\mathbb{R}^{n}\). Then \(\Delta ^{m+1}u=0\) on \(\Omega \) if and only if there are continuous functions \(u_{0},u_{1},\ldots,u_{2m}\) and \(\varepsilon :\Omega \rightarrow (0,\infty)\) such that the mean value of \(u\) over the ball \(B(x,R)\) is given by \(\sum_{k=0}^{2m}u_{k}(x)R^{k}\) whenever \(x\in \Omega \) and \(0<R<\varepsilon (x)\). This characterization is then used to define polyharmonic functions on metric measure spaces.