Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian
Let Ω be a G-invariant convex domain in ℂN including 0, where G is a complex Coxeter group associated with reduced root system R⊂ℝN. We consider holomorphic functions f defined in Ω which are Dunkl polyharmonic, that is, (Δh)nf=0...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2010-01-01
|
| Series: | Journal of Inequalities and Applications |
| Online Access: | http://dx.doi.org/10.1155/2010/947518 |
| Summary: | Let Ω be a G-invariant convex domain in ℂN including 0, where G is a complex Coxeter group associated with reduced root system R⊂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)+∑v∈R+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 z∈Ω, where fj are complex Dunkl harmonic functions, that is, Δhfj=0. |
|---|---|
| ISSN: | 1025-5834 1029-242X |