Liouville Theorem for Dunkl Polyharmonic Functions

Details Liouville Theorem for Dunkl Polyharmonic Functions Liouville Theorem for Dunkl Polyharmonic Functions

2008-11-06

arxiv.org/abs/0811.0962v1

SIGMA 4 (2008), 076, 6 pages

Mathematics

Classical Analysis and ODEs

This is a contribution to the Special Issue on Dunkl Operators and Related Topics, published in SIGMA (Symmetry, Integrability

Scientific paper

10.3842/SIGMA.2008.076

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.

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