%0 Journal Article
%T Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian
%A Guangbin Ren
%A Helmuth R. Malonek
%J Journal of Inequalities and Applications
%D 2010
%I Springer
%R 10.1155/2010/947518
%X Let 次 be a G-invariant convex domain in  N including 0, where G is a complex Coxeter group associated with reduced root system R  N. We consider holomorphic functions f defined in 次 which are Dunkl polyharmonic, that is, (忖h)nf=0 for some integer n. Here 忖h=﹉j=1N  j2 is the complex Dunkl Laplacian, and   j is the complex Dunkl operator attached to the Coxeter group G,   jf(z)=( f/ zj)(z)+﹉v﹋R+百v((f(z)-f(考vz))/ z,v )vj, where 百v is a multiplicity function on R and 考v 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 z﹋次, where fj are complex Dunkl harmonic functions, that is, 忖hfj=0.
%U http://dx.doi.org/10.1155/2010/947518