Liouville Theorem for Dunkl Polyharmonic Functions

Abstract

Assume that $f$ is Dunkl polyharmonic in ${\mathbb{R}}^n$ (i.e. $({\Delta }_h{)}^{p}f=0$ for some integer $p$, where ${\Delta }_h$ is the Dunkl Laplacian associated to a root system $R$ and to a multiplicity function $\kappa$, defined on $R$ and invariant with respect to the finite Coxeter group). Necessary and successful condition that $f$ is a polynomial of degree $\le s$ for $s\ge 2p-2$ is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant.

This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/