A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications

Abstract:

For a root system in $\mathbb {R}^d$ furnished with its Coxeter-Weyl group $W$ and a multiplicity nonnegative function $k$, we consider the associated commuting system of Dunkl operators $D_1,\ldots ,D_d$ and the Dunkl-Laplacian $\Delta _k=D^{2}_{1}+\ldots +D^{2}_{d}$. This paper studies the properties of the functions $u$ defined on an open $W$-invariant set $\Omega \subset \mathbb {R}^d$ and satisfying $\Delta _k u=0$ on $\Omega$ (D-harmonicity). In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack’s type theorem and a Bôcher’s theorem for D-harmonic functions.