Analysis On Symmetric Cones

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 functions associated with radial Fourier analysis on rank one symmetric spaces. On the other hand, we have tried to give an up-to-date account of the main results and problems, with an emphasis on analytic aspects.

An Introduction to Dunkl Theory and Its Analytic Aspects

We obtain several versions of the Hausdorff–Young and Hardy–Littlewood inequalities for the (k,a)-generalized Fourier transform recently investigated at length by Ben Said, Kobayashi, and Orsted. We also obtain a number of weighted inequalities — in particular Pitt’s inequality — that have application to uncertainty principles. Specifically we obtain several analogs of the Heisenberg–Pauli–Weyl principle for Lp-functions, local Cowling–Price-type inequalities, Donoho–Stark-type inequalities and qualitative extensions. We finally use the Hausdorff–Young inequality as a means to obtain entropic uncertainty inequalities.

Weighted inequalities and uncertainty principles for the (k,a)-generalized Fourier transform

Scuola normale superiore - pisa

In this work we extend the theory of the classical Hardy space $$H^1$$
 to the rational Dunkl setting. Specifically, let $$\Delta $$
 be the Dunkl Laplacian on a Euclidean space $$\mathbb {R}^N$$
 . On the half-space $$\mathbb {R}_+\times \mathbb {R}^N$$
 , we consider systems of conjugate $$(\partial _t^2+\Delta _{\mathbf {x}})$$
 -harmonic functions satisfying an appropriate uniform $$L^1$$
 condition. We prove that the boundary values of such harmonic functions, which constitute the real Hardy space $$H^1_{\Delta }$$
 , can be characterized in several different ways, namely by means of atoms, Riesz transforms, maximal functions or Littlewood–Paley square functions.

/pdf/harmonic-functions-conjugate-harmonic-functions-and-the-4a64tgqagt.pdf

Harmonic Functions, Conjugate Harmonic Functions and the Hardy Space $$H^1$$ H 1 in the Rational Dunkl Setting

In this paper we obtain the $L^p$-boundedness of Riesz transforms for Dunkl transform for all $1<p<\infty$.

/pdf/riesz-transforms-for-dunkl-transform-4521znbcps.pdf

Riesz transforms for Dunkl transform

Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. Rosler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$.

Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

In this article, we establish first a geometric Paley-Wiener theorem for the Dunkl transform in the crystallographic case. Next we obtain an optimal bound for the $L^p\to L^p$ norm of Dunkl translations in dimension 1. Finally we describe more precisely the support of the distribution associated to Dunkl translations in higher dimension.

https://hal.archives-ouvertes.fr/docs/00/42/12/80/PDF/AAS.pdf

Three results in Dunkl analysis

Let $G_{\lambda}^{(\alpha,\beta)}$ be the eigenfunctions of the Dunkl-Cherednik operator $T^{(\alpha,\beta)}$ on $\mathbb{R}$. In this paper we express the product $G_{\lambda}^{(\alpha,\beta)}(x)G_{\lambda}^{(\alpha,\beta)}(y)$ as an integral in terms of $G_{\lambda}^{(\alpha,\beta)}(z)$ with an explicit kernel. In general this kernel is not positive. Furthermore, by taking the so-called rational limit, we recover the product formula of M. R\"osler for the Dunkl kernel. We then define and study a convolution structure associated to $G_{\lambda}^{(\alpha,\beta)}$.

The main purpose of this article is to study the Lp-boundedness of linear and bilinear multiplier operators for the Dunkl transform in the one dimensional case.

Mohamed Sifi

Papers

Riesz transforms for Dunkl transform

Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Three results in Dunkl analysis

Opdam's hypergeometric functions: product formula and convolution structure in dimension 1

Linear and Bilinear Multiplier Operators for the Dunkl Transform