Abstract
For a root system in Rd furnished with its Weyl–Coxeter group W and a multiplicity function k≥0, we consider the associated commuting system of Dunkl operators D1,…,Dd and the Dunkl Laplacian Δk=D12+…+Dd2. This paper studies the properties of the functions u of class C2m on an open W-invariant set Ω⊂Rd and satisfying Δkmu=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.
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