Radial mollifiers, mean value operators and harmonic functions in Dunkl theory (Q342948)

The authors use mollifiers to regularize functions relative to a set of Dunkl operators in \(\mathbb R^d\) with Coxeter-Weyl group \(W,\) multiplicity function \(k\) and weight function NEWLINE\[NEWLINE \omega_k(x)=\prod_{\alpha\in R_+}|\langle\alpha,x \rangle|^{2k(\alpha)}. NEWLINE\]NEWLINE In particular for \(\Omega\) a \(W\)-invariant open subset of \(\mathbb R^d,\) for \(\phi\in {\mathfrak D}(\mathbb R^d)\) a radial function and \(u\in L^1_{\mathrm{loc}}(\Omega, \omega_k(x)dx)\) they study the Dunkl-convolution product \(u\ast_k\phi\) and the action of the Dunkl-Laplacian NEWLINE\[NEWLINE \Delta_k f(x)=\Delta f(x)+2\sum_{\alpha\in R_+} \left( \frac{\langle \nabla f(x),\alpha\rangle}{\langle \alpha,x\rangle}- \frac{f(x)-f(\sigma_\alpha(x))}{\langle\alpha,x\rangle^2} \right) NEWLINE\]NEWLINE and the volume mean operators on these functions. The results are then applied to obtain an analogue of the Weyl lemma for Dunkl-harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators.