On the Green Function and Poisson Integrals of the Dunkl Laplacian

@article{Graczyk2016OnTG,
  title={On the Green Function and Poisson Integrals of the Dunkl Laplacian},
  author={Piotr Graczyk and Tomasz Luks and Margit R{\"o}sler},
  journal={Potential Analysis},
  year={2016},
  volume={48},
  pages={337-360}
}
We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian △k in ℝd$\mathbb {R}^{d}$. 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 ℝd$\mathbb {R}^{d}$. These estimates contrast sharply with the well-known results in the potential… Expand
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References

SHOWING 1-10 OF 50 REFERENCES
Newtonian Potentials and Subharmonic Functions Associated to Root Systems
The purpose of this paper is to present a new theory of subharmonic functions for the Dunkl-Laplace operator Δk in ℝd$\mathbb {R}^{d}$ associated to a root system and a multiplicity function k ≥ 0.Expand
On the Dirichlet problem associated with the Dunkl Laplacian
This paper is devoted to the study of the Dirichlet problem associated with the Dunkl Laplaciank. We establish, under some condition on a bounded domain D of R d , the existence of a uniqueExpand
Estimates of Green functions for some perturbations of fractional Laplacian
Suppose that Y(t) is a d-dimensional Levy symmetric process for which its Levy measure differs from the Levy measure of the isotropic alpha-stable process (0 0, we prove that the Green functions areExpand
Properties of Green function of symmetric stable processes
Abstract: We study the Green function GD(x, y) of symmetricα-stable processes in R for an open set D(0 < α < 2, d ≥ 3). Our main result gives the upper and the lower bound estimates of GD(x, y) for aExpand
Estimates on Green functions and Schrodinger-type equations for non-symmetric diffusions with measure-valued drifts
Abstract In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of theseExpand
On the Green's function for second order parabolic differential equations with discontinuous coefficients
du d ( du) Lu s dt d%j \ dxj for (x, 0 G o r = i2X(0, r ] , where Q is a bounded open simply connected region in E (n^2), T an arbitrary positive number, and the da are assumed only to be bounded andExpand
A positive radial product formula for the Dunkl kernel
It is an open conjecture that generalized Bessel functions associated with root systems have a positive product formula for non-negative multiplicity parameters of the associated Dunkl operators. InExpand
Harmonic and subharmonic functions associated to root systems
In this thesis, we show that, for any root system $\mathcal{R}$ in the Euclidean space $\R^d$ and for any nonnegative multiplicity function $k$ on $\mathcal{R}$, we can develop in this geometricExpand
Dunkl Operators: Theory and Applications
These lecture notes are intended as an introduction to the theory of rational Dunkl operators and the associated special functions, with an emphasis on positivity and asymptotics. We start with anExpand
Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient
The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional openExpand
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