Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 076, 6 pages      arXiv:0811.0962
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

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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