### THE HARDY SPACE H 1 IN THE RATIONAL DUNKL SETTING

"... Abstract. This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1. These results are proved here in the one ..."

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Abstract. This paper consists in a first study of the Hardy space H 1 in the rational Dunkl setting. Following Uchiyama’s approach, we characterizee H 1 atomically and by means of the heat maximal operator. We also obtain a Fourier multiplier theorem for H 1. These results are proved here in the one-dimensional case and in the product case. hal-00864457, version 1- 21 Sep 2013 1.

### 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.

### Symmetry, Integrability and Geometry: Methods and Applications External Ellipsoidal Harmonics for the Dunkl–Laplacian ⋆

"... Abstract. The paper introduces external ellipsoidal and external sphero-conal h-harmonics for the Dunkl–Laplacian. These external h-harmonics admit integral representations, and they are connected by a formula of Niven’s type. External h-harmonics in the plane are expressed in terms of Jacobi polyno ..."

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Abstract. The paper introduces external ellipsoidal and external sphero-conal h-harmonics for the Dunkl–Laplacian. These external h-harmonics admit integral representations, and they are connected by a formula of Niven’s type. External h-harmonics in the plane are expressed in terms of Jacobi polynomials P α,β n kind. and Jacobi’s functions Q α,β n of the second Key words: external ellipsoidal harmonics; Stieltjes polynomials; Dunkl–Laplacian; fundamental solution; Niven’s formula; Jacobi’s function of the second kind 2000 Mathematics Subject Classification: 33C52; 35C10 1

### Symmetry, Integrability and Geometry: Methods and Applications Generalized Bessel function of Type D ⋆

"... Abstract. We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. Key words: radial Dunkl processes; Brownian motions in Weyl ..."

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Abstract. We write down the generalized Bessel function associated with the root system of type D by means of multivariate hypergeometric series. Our hint comes from the particular case of the Brownian motion in the Weyl chamber of type D. Key words: radial Dunkl processes; Brownian motions in Weyl chambers; generalized Bessel function; multivariate hypergeometric series 2000 Mathematics Subject Classification: 33C20; 33C52; 60J60; 60J65 1 Root systems and related processes We refer the reader to [11] for facts on root systems. Let (V, 〈·〉) be an Euclidean space of finite dimension m ≥ 1. A reduced root system R is a finite set of non zero vectors in V such that 1) R ∩ Rα = {α, −α} for all α ∈ R, 2) σα(R) = R, where σα is the reflection with respect to the hyperplane Hα orthogonal to α 〈α, x〉 σα(x) = x − 2 α, x ∈ V. 〈α, α〉

### Liouville Theorem for Dunkl Polyharmonic Functions ⋆

, 2008

"... Original article is available at ..."

### 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.

### DUNKL OPERATORS AS COVARIANT DERIVATIVES IN A QUANTUM PRINCIPAL BUNDLE

"... Abstract. A quantum principal bundle is constructed for every Coxeter group acting on a finite-dimensional 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 finite-dimensional 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 commutivity of these operators among themselves as a consequence of a geometric property, namely, that the connection has curvature zero. 1.

### Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

"... Abstract. Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line. ..."

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Abstract. Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line.