Liouville Theorem for Dunkl Polyharmonic Functions

Abstract

Assume that f is Dunkl polyharmonic in Rn (i.e. (Δh)p f = 0 for some integer p, where Δh is the Dunkl Laplacian associated to a root system R and to a multiplicity function κ, defined on R and invariant with respect to the finite Coxeter group). Necessary and successful condition that f is a polynomial of degree ≤ s for s ≥ 2p – 2 is proved. As a direct corollary, a Dunkl harmonic function bounded above or below is constant.