Polysuperharmonic functions on a harmonic space (Q2574435)

Solutions of the equation \(\Delta ^{m}u=0\) on \(\mathbb R^{n}\) are called polyharmonic functions of order \(m\). The purpose of this paper is to develop an axiomatic approach to the study of such functions that would, for example, cover more general second order differential operators in place of the Laplacian, and Riemannian manifolds in place of Euclidean space. Some time ago \textit{E. P. Smyrnélis} [Ann. Inst. Fourier 25, 35--97 (1975; Zbl 0295.31006)] developed an axiomatic theory of biharmonic functions on locally compact spaces. The authors now present an alternative approach which is well adapted to deal also with the case \(m>2\). They develop notions of polysuperharmonic functions and polypotentials, an analogue of the Riesz decomposition theorem, and so on.