Almansi decomposition for Dunkl operators

作者：Ren GB^*

来源：Science in China Series A-Mathematics, 2005, 48: 333-342.

DOI：10.1360/05za0027

• Dunkl operators
• Almansi decomposition
• Dunkl polyharmonic

Let Omega be a G-invariant convex domain in R-N including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Q which are Dunkl polyharmonic, i.e. (Delta(h))(n)f = 0 for some integer n. Here Delta(h) = Sigma(j=1)(N) D-j(2) is the Dunkl Laplacian, and D-j is the Dunkl operator attached to the Coxeter group G,
Dj f(x) = partial derivative/partial derivative x(j) f(x) + Sigma(v is an element of R+) kappa(v)f(x) - f(sigma(v)x)/< x, v)v(j)
where K, is a multiplicity function on R and sigma(v), 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) = f(0)(x) + vertical bar x vertical bar(2)f(1)(x) + center dot center dot center dot + vertical bar x vertical bar(2(n-1))f(n-1)(x), for all x is an element of Omega,
where f(j) are Dunkl harmonic functions, i.e. Delta(h)f(j) = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.

• 出版日期2005-5