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
Note on Bessel functions of type AN−1
TL;DR: In this paper, an iterative method for integral formula of Bessel function of type AN−1 and a partial product formula for it was given through the theory of Jack polynomials.
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The Hardy space H1 in the rational Dunkl setting
TL;DR: In this article, the authors studied the Hardy space H1 in the rational Dunkl setting and characterized it atomically and by means of the heat maximal operator, and obtained a Fourier multiplier theorem for H1.
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Note on Bessel functions of type $A_{N-1}$
TL;DR: In this paper, an iterative method for integral formula of Dunkl-Bessel functions of type $A_{N-1}$ and a partial product formula for it was given.
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Riesz Transforms Characterizations of Hardy Spaces $$H^1$$ for the Rational Dunkl Setting and Multidimensional Bessel Operators
TL;DR: In this paper, the authors characterized the Hardy space in the rational Dunkl setting associated with the reflection group by means of special Riesz transforms for product of Bessel operators.
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Dunkl kernel associated with dihedral groups
TL;DR: In this article, it was shown that the Dunkl intertwining operator can be computed on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function.
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