Propriétés de moyenne pour les fonctions harmoniques et polyharmoniques au sens de Dunkl.

For a root system in R d furnished with its Weyl–Coxeter group W and a multiplicity function k ≥ 0 , we consider the associated commuting system of Dunkl operators D 1 , … , D d and the Dunkl Laplacian Δ k = D 1 2 + … + D d 2 . This paper studies the properties of the functions u of class C 2 m on an open W -invariant set Ω ⊂ R d and satisfying Δ k m u = 0 on Ω ( D -polyharmonicity if m > 1 and D -harmonicity if m = 1 ). In particular, we introduce a new operator, which is a generalization of the classical volume mean value operator and which characterizes D -harmonicity (resp. D -polyharmonicity) and we give some applications. [ABSTRACT FROM AUTHOR]