Liouville Theorem for Dunkl Polyharmonic Functions

TL;DR

This paper establishes a Liouville-type theorem for Dunkl polyharmonic functions, showing under what conditions such functions must be polynomials or constants, extending classical harmonic analysis to Dunkl operators.

Contribution

It provides necessary and sufficient conditions for Dunkl polyharmonic functions to be polynomials of bounded degree, generalizing classical Liouville theorems to the Dunkl setting.

Findings

Dunkl polyharmonic functions are polynomials of degree ≤ s under certain conditions.

Bounded Dunkl harmonic functions are constant.

The paper extends classical harmonic analysis results to Dunkl operators.

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.