An Introduction to Dunkl Theory and Its Analytic Aspects
TL;DR: In this paper, the authors introduce rational and trigonometric Dunkl theory, starting with the historical examples of special functions associated with radial Fourier analysis on rank one symmetric spaces.
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Abstract: 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.
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Citations
Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications
TL;DR: In this article, the spherical mean value of the Dunkl-type generalized translation operator is shown to be a positive ρ-factor of the Riesz potential for generalized translation.
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The Heckman-Opdam Markov processes
Bruno Schapira,Bruno Schapira +1 more
TL;DR: In this article, the authors introduced the natural counterpart of the Dunkl Markov processes in a negatively curved setting and gave a semimartingale decomposition of the radial part and some properties of the jumps.
Sobolev-Type Inequalities for Dunkl Operators
TL;DR: In this paper, the Sobolev inequality in the Dunkl setting was studied and two new approaches were proposed which provided a simpler elementary proof of the classical case p = 2, as well as an extension to the coefficient p = 1 that was previously unknown.
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The differential equations associated with Calogero-Moser-Sutherland particle models in the freezing regime
TL;DR: In this article, it was shown that for a start in arbitrary boundary points, the ODEs always admit unique solutions in their domains for $t>0, where t < 0.
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References
The Hardy Space H^1 in the Rational Dunkl Setting
TL;DR: In this paper, the first attempt at a study of the Hardy space in the rational Dunkl setting was made, where the heat maximal operator was used to characterize $$H^1$$ by Uchiyama's approach.
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Opdam's hypergeometric functions: product formula and convolution structure in dimension 1
TL;DR: In this paper, the Dunkl-Cherednik operator is expressed as an integral integral in terms of the product of the eigenfunctions of the operator with an explicit kernel.
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Spherical harmonic analysis on affine buildings
TL;DR: In this article, the algebra homomorphisms from a regular affine building to a commutative algebra A spanned by vertex set averaging operators are studied in terms of the Macdonald spherical functions and an integral over the boundary of the building.
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A uniform proof of the Macdonald-Mehta-Opdam identity for finite Coxeter groups
TL;DR: In this article, a new proof of the Macdonald-mehta-Opdam integral identity for finite Coxeter groups is given, which does not use multivariable Bessel functions.
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Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC
TL;DR: In this article, the authors presented explicit product formulas for a continuous two-parameter family of Heckman-Opdam hypergeometric functions of type BC on Weyl chambers.
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