Czechoslovak Mathematical Journal

, Volume 69, Issue 1, pp 257–273 | Cite as

Higher Order Riesz Transforms for the Dunkl Ornstein-Uhlenbeck Operator

  • Walid NefziEmail author


The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. Nowak, L. Roncal and K. Stempak to higher order.


Dunkl Laplacian Dunkl Ornstein-Uhlenbeck operator generalized Hermite polynomial Riesz transform 

MSC 2010

26A33 42C10 42C20 43A15 47G40 


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  1. [1]
    T. S. Chihara: Generalized Hermite Polynomials. Thesis (Ph. D.). Purdue University, West Lafayette, 1955.Google Scholar
  2. [2]
    C. D. Dunkl: Differential-difference operators associated to reflection groups, Trans. Am. Math. Soc. 311 (1989), 167–183.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    P. Graczyk, J. J. Loeb, I. López, A. Nowak, W. Urbina: Higher order Riesz transforms, fractional derivatives, and Sobolev spaces for Laguerre expansions, J. Math. Pures Appl. 84 (2005), 375–405.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    N. N. Lebedev: Special Functions and Their Applications. Dover Publications, New York, 1972.zbMATHGoogle Scholar
  5. [5]
    B. Muckenhoupt: Conjugate functions for Laguerre expansions, Trans. Am. Math. Soc. 147 (1970), 403–418.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    W. Nefzi: Higher order Riesz transforms for the Dunkl harmonic oscillator, Taiwanese J. Math. 19 (2015), 567–583.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    A. Nowak, L. Roncal, K. Stempak: Riesz transforms for the Dunkl Ornstein-Uhlenbeck operator, Colloq. Math. 118 (2010), 669–684.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    A. Nowak, K. Stempak: Riesz transforms for the Dunkl harmonic oscillator, Math. Z. 262 (2009), 539–556.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    M. Rosenblum: Generalized Hermite polynomials and the Bose-like oscillator calculus. Nonselfadjoint Operators and Related Topics (A. Feintuch et al., eds.). Operator Theory: Advances and Applications 73, Birkhäuser, Basel, 1994, pp. 369–396.Google Scholar
  10. [10]
    M. Rösler: Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun. Math. Phys. 192 (1998), 519–542.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    M. Rösler: Dunkl operators: Theory and applications. Orthogonal Polynomials and Special Functions (E. Koelink et al., eds.). Lecture Notes in Mathematics 1817, Springer, Berlin, 2003, pp. 93–135.Google Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018

Authors and Affiliations

  1. 1.University of Tunis El Manar, Faculty of Sciences of Tunis, LR11ES11 Analyse Mathématiques et ApplicationsTunisTunisia

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