Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 947518, 13 pages
Research Article

Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian

1Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China
2Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal

Received 28 December 2009; Accepted 26 March 2010

Academic Editor: Yuming Xing

Copyright © 2010 Guangbin Ren and Helmuth R. Malonek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let Ω be a G-invariant convex domain in N including 0, where G is a complex Coxeter group associated with reduced root system RN. We consider holomorphic functions f defined in Ω which are Dunkl polyharmonic, that is, (Δh)nf=0 for some integer n. Here Δh=j=1N𝒟j2 is the complex Dunkl Laplacian, and 𝒟j is the complex Dunkl operator attached to the Coxeter group G, 𝒟jf(z)=(f/zj)(z)+vR+κv((f(z)-f(σvz))/z,v)vj, where κv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any complex Dunkl polyharmonic function f has a decomposition of the form f(z)=f0(z)+(n=1Nzj2)f1(z)++(n=1Nzj2)n-1fn-1(z), for all zΩ, where fj are complex Dunkl harmonic functions, that is, Δhfj=0.