Mean value characterizations of the Dunkl polyharmonic functions

Abstract

We give characterizations of the Dunkl polyharmonic functions, i.e., solutions to the iteration of the Dunkl-Laplace operator \(\Delta_\kappa\) which is a differential-reflection operator associated with a Coxeter–Weil group \(W\) generated by a finite set of reflections and an invariant multiplicity function \(\kappa\), in terms of integral means over Euclidean balls and spheres.