The Range of the Spectral Projection Associated with the Dunkl Laplacian

For s∈ℝ, denote by Pksf the “projection” of a function f in Dℝd into the eigenspaces of the Dunkl Laplacian Δk corresponding to the eigenvalue −s2. The parameter k comes from Dunkl’s theory of differential-difference operators. We shall characterize the range of Pks on the space of functions f∈Dℝd supported inside the closed ball BO,R¯. As an application, we provide a spectral version of the Paley-Wiener theorem for the Dunkl transform.

Boundary value problems for the Dunkl Laplacian

Let Δk be the Dunkl Laplacian on ℜd associated with a reflection group W and a multiplicity function k. The purpose of this paper is to establish the existence and the uniqueness of a positive solution on the unit ball B of ℜd to the following boundary value problem:Δku = φu in B and u = ƒ on ∂BWe distinguish two cases of nonnegative perturbation φ: trivial and nontrivial.

Riesz potentials of Radon measures associated to reflection groups

For a root systemRon{\mathbb{R}^{d}}and a nonnegative multiplicity functionkonR, we consider the heat kernel{p_{k}(t,x,y)}associated to the Dunkl Laplacian operator{\Delta_{k}}. For{\beta\in{]0,d+2\gamma[}}, where{\gamma=\frac{1}{2}\sum_{\alpha\in R}k(\alpha)}, we study the{\Delta_{k}}-Riesz kernel of index β, defined by{R_{k,\beta}(x,y)=\frac{1}{\Gamma(\beta/2)}\int_{0}^{+\infty}t^{{\beta/2}-1}p_% {k}(t,x,y)\,dt}, and the corresponding{\Delta_{k}}-Riesz potential{I_{k,\beta}[\mu]}of a Radon measure μ on{\mathbb{R}^{d}}. According to the values of β, we study the{\Delta_{k}}-superharmonicity of these functions, and we give some applications like the{\Delta_{k}}-Riesz measure of{I_{k,\beta}[\mu]}, the uniqueness principle and a pointwise Hedberg inequality.

Potential theory associated with the Dunkl Laplacian

The main goal of this paper is to develop a potential theoretical approach to study the Dunkl Laplacian