On the Green Function and Poisson Integrals of the Dunkl Laplacian

  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},
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
Green Function and Poisson Kernel Associated to Root Systems for Annular Regions
Let Δ k be the Dunkl Laplacian relative to a fixed root system R $\mathcal {R}$ in ℝ d $\mathbb {R}^{d}$ , d ≥ 2, and to a nonnegative multiplicity function k on R $\mathcal {R}$ . Our first purposeExpand
In this article, we consider flat and curved Riemannian symmetric spaces in the complex case and we study their basic integral kernels, in potential and spherical analysis: heat, Newton, PoissonExpand
Riesz Potential and Maximal Function for Dunkl transform
We study weighted $(L^p, L^q)$-boundedness properties of Riesz potentials and fractional maximal functions for the Dunkl transform. In particular, we obtain the weighted Hardy-Littlewood-Sobolev typeExpand
Potential kernels for radial Dunkl Laplacians
We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(\alpha)=1$, i.e. for flat complex symmetricExpand
Sobolev-type inequalities for Dunkl operators
Abstract In this paper we study the Sobolev inequality in the Dunkl setting using two new approaches which provide a simpler elementary proof of the classical case p = 2 , as well as an extension toExpand
We derive two-sided bounds for the Newton and Poisson kernels of the W -invariant Dunkl Laplacian in geometric complex case when the multiplicity k(α) = 1 i.e. for flat complex symmetric spaces. ForExpand
Li--Yau Inequalities for Dunkl Heat Equations
Motivated by recent works due to Yu–Zhao [J. Geom. Anal. 2020] and Weber– Zacher [arXiv:2012.12974], we study Li–Yau inequalities for the heat equation corresponding to the Dunkl Laplacian, which isExpand
An Introduction to Dunkl Theory and Its Analytic Aspects
The aims of these lecture notes are twofold. On the one hand, they are meant as an introduction to rational and trigonometric Dunkl theory, starting with the historical examples of special functionsExpand
Dimension-Free Square Function Estimates for Dunkl Operators
Dunkl operators may be regarded as differential-difference operators parameterized by finite reflection groups and multiplicity functions. In this paper, the Littlewood– Paley square function forExpand
Upper and lower bounds for Littlewood-Paley square functions in the Dunkl setting
The aim of this paper is to prove upper and lower $L^p$ estimates, $1<p<\infty$, for Littlewood-Paley square functions in the rational Dunkl setting.


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