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SIGMA 4 (2008), 076, 6 pages arXiv:0811.0962
https://doi.org/10.3842/SIGMA.2008.076
Contribution to the Special Issue on Dunkl Operators and Related Topics
Liouville Theorem for Dunkl Polyharmonic Functions
Guangbin Ren a, b and Liang Liu a
a) Department of Mathematics,
University of Science and Technology of China, Hefei, Anhui 230026, P.R. China
b) Departamento de Matemática, Universidade de Aveiro, P-3810-193, Aveiro, Portugal
Received July 03, 2008, in final form October 30, 2008; Published online November 06, 2008
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.
Key words:
Liouville theorem; Dunkl Laplacian; polyharmonic functions.
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