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The Clifford Deformation of the Hermite Semigroup
"... Abstract. This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many res ..."
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Abstract. This paper is a continuation of the paper [De Bie H., Ørsted B., Somberg P., Souček V., Trans. Amer. Math. Soc. 364 (2012), 3875–3902], investigating a natural radial deformation of the Fourier transform in the setting of Clifford analysis. At the same time, it gives extensions of many results obtained in [Ben Saïd S., Kobayashi T., Ørsted B., Compos. Math. 148 (2012), 1265–1336]. We establish the analogues of Bochner’s formula and the Heisenberg uncertainty relation in the framework of the (holomorphic) Hermite semigroup, and also give a detailed analytic treatment of the series expansion of the associated integral transform.
TWO RESULTS ON THE DUNKL MAXIMAL OPERATOR
 STUDIA MATHEMATICA 203 (2011) 4768
, 2011
"... In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vectorvalued theorem for the Dunkltype FeffermanStein operator in the case Zd ..."
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In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vectorvalued theorem for the Dunkltype FeffermanStein operator in the case Zd 2 by establishing a result of exponential integrability corresponding to the case p = +∞.
On fundamental harmonic analysis operators in certain Dunkl and Bessel settings
 J. Math. Anal. Appl
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Symmetry, Integrability and Geometry: Methods and Applications Dunkl Operators as Covariant Derivatives in a Quantum Principal Bundle
"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a prog ..."
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Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finitedimensional Euclidean space E, and then a connection is also defined on this bundle. The covariant derivatives associated to this connection are the Dunkl operators, originally introduced as part of a program to generalize harmonic analysis in Euclidean spaces. This gives us a new, geometric way of viewing the Dunkl operators. In particular, we present a new proof of the commutativity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero.
THREE RESULTS IN DUNKL ANALYSIS
, 904
"... a distinguished polish mathematician, a guide and a friend, who has left many orphans in Wroclaw and around the world. We miss you. Abstract. 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 ..."
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a distinguished polish mathematician, a guide and a friend, who has left many orphans in Wroclaw and around the world. We miss you. Abstract. 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 → 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. 1.
Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra
, 2011
"... Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(1 ..."
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Abstract. A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl−1(2), this algebra encompasses the Lie superalgebra osp(12). It is obtained as a q = −1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible representations. It possesses a noncocommutative coproduct. The Clebsch–Gordan coefficients (CGC) of sl−1(2) are obtained and expressed in terms of the dual −1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization. Key words: parabosonic algebra; dual Hahn polynomials; Clebsch–Gordan coefficients 2010 Mathematics Subject Classification: 17B37; 17B80; 33C45