Almansi decomposition for Dunkl operators

Let Ωbe 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 Ωwhich 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.

Almansi decomposition for Dunkl operators

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)/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.