A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications (Q2790606)

For a root system \(R\) in \(\mathbb R^d\) endowed with its Coxeter-Weyl group \(W\) and a multiplicity nonnegative function \(k:R\to \mathbb C,\) the authors consider the associated commuting system of Dunkl operators \(D_1,\ldots,D_d\) NEWLINE\[NEWLINE D_jf(x)=\frac{\partial}{\partial x_j}f(x)+\sum_{\alpha\in R_+} k(\alpha)\alpha_j\frac{f(x)-f(\sigma_\alpha(x))}{\langle \alpha,x\rangle}, \quad \sigma_\alpha(x)=x-2\frac{\langle x,\alpha\rangle}{\|\alpha \|^2}\alpha, NEWLINE\]NEWLINE and the Dunkl-Laplacian \(\Delta_k=D_1^2+\ldots+D_d^2.\) They study the properties of the functions \(u\) defined on an open \(W\)-invariant set \(\Omega\subset\mathbb R^d\) and satisfying \(\Delta_ku=0\) on \(\Omega\) (\(D\)-harmonicity). In particular, the authors introduce and give a complete study of a new mean value operator which characterizes \(D\)-harmonicity. As applications they prove a strong maximum principle, a Harnack-type theorem and a Bôcher theorem for \(D\)-harmonic functions.