On the Green Function and Poisson Integrals of the Dunkl Laplacian

Abstract

Abstract. We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian ∆ k in R d . As applications we derive the Poisson-Jensen formula for ∆ k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of ∆ k . We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in R d . These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.

The Z2n Dirac–Dunkl operator and a higher rank Bannai–Ito algebra

“…The result is proved by a direct calculation using the explicit expression (14) for the operators Γ A . On the one hand, one has…”

“…Upon expanding the second term i∈A µ i Γ A\{i} of the operator C A using (14), a straightforward calculation shows that one has indeed (29).…”

“…The proof of (16) follows by a direct calculation using (14). For simplicity, one can perform the calculation with A = [ ] and {j, k} = {1, + 1} and extend the result by symmetry.…”

“…Dunkl operators are families of differential-difference operators associated with finite reflection groups. Introduced in [6], these operators appear in many areas: they are at the core of the study of multivariate special functions associated with root systems [7,18], they arise in harmonic analysis and integral transforms [3,20], they are closely related to representations of double affine Hecke algebras [5], they are at the origin of Dunkl processes [14], and they play a central role in the analysis of Calogero-Sutherland systems [22]. Consider the Abelian reflection group Z n 2 = Z 2 × · · · × Z 2 ; the corresponding Dunkl operators T 1 , .…”

A Dirac–Dunkl Equation on S 2 and the Bannai–Ito Algebra

“…Following their introduction in [8,9,10], Dunkl operators have appeared in various areas. They enter the study of Calogero-Moser-Sutherland models [28], they play a central role in the theory of multivariate orthogonal polynomials associated to reflection groups [11], they give rise to families of stochastic processes [20,25], and they can be used to construct quantum superintegrable systems involving reflections [13,14]. Dunkl operators also find applications in harmonic analysis and integral transforms [7,24], as they naturally lead to the Laplace-Dunkl operators, which are second-order differential/difference operator that generalize the standard Laplace operator.…”