Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian. (26th May 2010) - Ren, Guangbin
Abstract : LetΩ be aG -invariant convex domain inℂ N including0, whereG is a complex Coxeter group associated with reduced root systemR ⊂ ℝ N . We consider holomorphic functionsf defined inΩ which are Dunkl polyharmonic, that is, ( Δ h ) n f = 0 for some integern . HereΔ h = ∑ j = 1 N 𝒟 j 2 is the complex Dunkl Laplacian, and𝒟 j is the complex Dunkl operator attached to the Coxeter groupG, 𝒟 j f ( z ) = ( ∂ f / ∂ z j ) ( z ) + ∑ v ∈ R + κ v ( ( f ( z ) - f ( σ v z ) ) / 〈 z, v 〉 ) v j, whereκ v is a multiplicity function onR andσ v is the reflection with respect to the rootv . We prove that any complex Dunkl polyharmonic functionf has a decomposition of the formf ( z ) = f 0 ( z ) + ( ∑ n = 1 N z j 2 ) f 1 ( z ) + ⋯ + ( ∑ n = 1 N z j 2 ) n - 1 f n - 1 ( z ), for allz ∈ Ω, wheref j are complex Dunkl harmonic functions, that is, Δ h f j = 0 .