Decomposition of Polyharmonic Functions with Respect to the Complex Dunkl Laplacian

Guangbin Ren,Helmuth R Malonek

doi:10.1155/2010/947518

Abstract

Let be a -invariant convex domain in including , where is a complex Coxeter group associated with reduced root system . We consider holomorphic functions defined in which are Dunkl polyharmonic, that is, for some integer . Here is the complex Dunkl Laplacian, and is the complex Dunkl operator attached to the Coxeter group , where is a multiplicity function on and is the reflection with respect to the root . We prove that any complex Dunkl polyharmonic function has a decomposition of the form , for all , where are complex Dunkl harmonic functions, that is, .

Highlights

A fundamental result in the theory of polyharmonic functions is the celebrated Almansi theorem 1–3, which shows that for any polyharmonic function f of degree n in a starlike domain D in RN with center 0, that is, ⎛ ⎞n ΔR nf : N ⎝ ∂2 ⎠ j 1 ∂xj2, f1.1 there exist uniquely harmonic functions f0, . . . , fn−1 such that f x f0 x |x|2f1 x · · · |x|2 n−1 fn−1 x, ∀x ∈ D.Journal of Inequalities and ApplicationsThe Almansi formula is a genuine analogy to the Taylor formula: ft f0 t f
We prove that any complex Dunkl polyharmonic function f has a decomposition of the form f z f0 z
The Almansi formula is a genuine analogy to the Taylor formula: ft f0 t f tn f n0 n!

Summary

Introduction

A fundamental result in the theory of polyharmonic functions is the celebrated Almansi theorem 1–3 , which shows that for any polyharmonic function f of degree n in a starlike domain D in RN with center 0, that is,. With Dunkl operators in place of the usual partial derivatives, one can define the Laplacian in the Dunkl setting, which is a parametrized operator and invariant under reflection groups These parametrized Laplacian suggests the theory of Dunkl harmonics. Let Dj be the Dunkl operator attached to the Coxeter group G and the multiplicity function κ, defined by see 16. The Dunkl operators enjoy the regularity property: if f ∈ H Ω , the space of holomorphic functions in Ω, Dif ∈ H Ω This follows immediately from the formula f z − f σvz z, v. Fn−1 Dunkl harmonic in Ω, defines a Dunkl polyharmonic function in Ω of degree n. These formulae are new even in the classical case κ 0

Preliminaries

Proof of the Main Theorem

Gauss Decomposition