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arXiv · 0811.0962

Liouville Theorem for Dunkl Polyharmonic Functions

Guangbin Ren·Liang Liu
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Abstract

Assume that $f$ is Dunkl polyharmonic in $\mathbb{R}^n$ (i.e. $(Δ_h)^p f=0$ for some integer $p$, where $Δ_h$ 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 $\le s$ 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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Guangbin Ren, Liang Liu. 2008-11-06. Liouville Theorem for Dunkl Polyharmonic Functions. https://doi.org/10.3842/sigma.2008.076

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