Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 019, 15 pages      arXiv:0902.2765
Contribution to the Special Issue on Dunkl Operators and Related Topics

Besov-Type Spaces on Rd and Integrability for the Dunkl Transform

Chokri Abdelkefi a, Jean-Philippe Anker b, Feriel Sassi a and Mohamed Sifi c
a) Department of Mathematics, Preparatory Institute of Engineer Studies of Tunis, 1089 Monfleury Tunis, Tunisia
b) Department of Mathematics, University of Orleans & CNRS, Federation Denis Poisson (FR 2964), Laboratoire MAPMO (UMR 6628), B.P. 6759, 45067 Orleans cedex 2, France
c) Department of Mathematics, Faculty of Sciences of Tunis, 1060 Tunis, Tunisia

Received August 28, 2008, in final form February 05, 2009; Published online February 16, 2009

In this paper, we show the inclusion and the density of the Schwartz space in Besov-Dunkl spaces and we prove an interpolation formula for these spaces by the real method. We give another characterization for these spaces by convolution. Finally, we establish further results concerning integrability of the Dunkl transform of function in a suitable Besov-Dunkl space.

Key words: Dunkl operators; Dunkl transform; Dunkl translations; Dunkl convolution; Besov-Dunkl spaces.

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  1. Abdelkefi C., Sifi M., On the uniform convergence of partial Dunkl integrals in Besov-Dunkl spaces, Fract. Calc. Appl. Anal. 9 (2006), 43-56.
  2. Abdelkefi C., Sifi M., Further results of integrability for the Dunkl transform, Commun. Math. Anal. 2 (2007), 29-36.
  3. Abdelkefi C., Dunkl transform on Besov spaces and Herz spaces, Commun. Math. Anal. 2 (2007), 35-41.
  4. Abdelkefi C., Sifi M., Characterization of Besov spaces for the Dunkl operator on the real line, JIPAM. J. Inequal. Pure Appl. Math. 8 (2007), no. 3, Article 73, 11 pages.
  5. Bergh J., Löfström J., Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin - New York, 1976.
  6. Besov O.V., On a family of function spaces in connection with embeddings and extentions, Trudy. Mat. Inst. Steklov. 60 (1961), 42-81 (in Russian).
  7. Betancor J.J., Rodríguez-Mesa L., Lipschitz-Hankel spaces and partial Hankel integrals, Integral Transform. Spec. Funct. 7 (1998), 1-12.
  8. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  9. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  10. Dunkl C.F., Integral kernels with reflection group invariance, Canad. J. Math. 43, (1991), 1213-1227.
  11. Dunkl C.F., Hankel transforms associated to finite reflection groups, in Proc. of Special Session on Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, 1991), Contemp. Math. 138 (1992), 123-138.
  12. Dunkl C.F., Xu Y., Orthogonal polynomials of several variables, Encyclopedia of Mathematics and its Applications, Vol. 81, Cambridge University Press, Cambridge, 2001.
  13. Giang D.V., Mórciz F., On the uniform and the absolute convergence of Dirichlet integrals of functions in Besov-spaces, Acta Sci. Math. (Szeged) 59 (1994), 257-265.
  14. Kamoun L., Besov-type spaces for the Dunkl operator on the real line, J. Comput. Appl. Math. 199 (2007), 56-67.
  15. Pelczynski A., Wojciechowski M., Molecular decompositions and embbeding theorems for vector-valued Sobolev spaces with gradient norm, Studia Math. 107 (1993), 61-100.
  16. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  17. Rösler M., Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463, q-alg/9710029.
  18. Rösler M., Voit M., Markov processes with Dunkl operators, Adv. in Appl. Math. 21 (1998), 575-643.
  19. Thangavelyu S., Xu Y., Convolution operator and maximal function for Dunkl transform, J. Anal. Math. 97 (2005), 25-55.
  20. Titchmarsh E.C., Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1937.
  21. Triebel H., Theory of function spaces, Monographs in Mathematics, Vol. 78, Birkhäuser, Verlag, Basel, 1983.
  22. Trimèche K., The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms Spec. Funct. 12 (2001), 349-374.
  23. Trimèche K., Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms Spec. Funct. 13 (2002), 17-38.

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