Almansi decomposition for Dunkl operators  

作　　者：REN Guangbin 

机构地区：[1]Department of Mathematics,University of Science and Technology of China,Hefei,230026,China

出　　处：《Science China Mathematics》2005年第z1期333-342,共10页中国科学：数学（英文版）

基　　金：supported by the National Natural Science Foundation of China(Grant No.10471134).

摘　　要：Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+|x|2f1(x)+…+|x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf= 0 for some integer n. Here△h= ∑Nj=1D2j is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G,Djf(x)=(δ)/(δ)xjf(x)+∑v∈R+kvf(x)-f(бvx)/<x,v>vj,where kv is a multiplicity function on R and σv is the reflection with respect to the root v.We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+│x│2f1(x)+…+│x│2(n-1)fn-1(x),(A)x∈Ω,where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

关 键 词：Dunkl operators  Almansi decomposition  Dunkl polyharmonic. 

分 类 号：N[自然科学总论]