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


SIGMA 5 (2009), 037, 17 pages      arXiv:0903.4369      http://dx.doi.org/10.3842/SIGMA.2009.037
Contribution to the Special Issue on Dunkl Operators and Related Topics

Hilbert Transforms Associated with Dunkl-Hermite Polynomials

Néjib Ben Salem and Taha Samaali
Department of Mathematics, Faculty of Sciences of Tunis, Campus Universitaire, 2092 Tunis, Tunisia

Received October 14, 2008, in final form March 12, 2009; Published online March 25, 2009

Abstract
We consider expansions of functions in Lp(R,|x|2kdx), 1 ≤ p < +∞ with respect to Dunkl-Hermite functions in the rank-one setting. We actually define the heat-diffusion and Poisson integrals in the one-dimensional Dunkl setting and study their properties. Next, we define and deal with Hilbert transforms and conjugate Poisson integrals in the same setting. The formers occur to be Calderón-Zygmund operators and hence their mapping properties follow from general results.

Key words: Dunkl operator; Dunkl-Hermite functions; Hilbert transforms; conjugate Poisson integrals; Calderón-Zygmund operators.

pdf (252 kb)   ps (180 kb)   tex (13 kb)

References

  1. de Jeu M.F.E., The Dunkl transform, Invent. Math. 113 (1993), 147-162.
  2. Dunkl C.F., Hankel transforms associated to finite reflection groups, in Hypergeometric Functions on Domains of Positivity, Jack Polynomials and Applications (Tampa, 1991), Contemp. Math. 138 (1992), 123-138.
  3. El Garna A., The left-definite spectral theory for the Dunkl-Hermite differential-difference equation, J. Math. Anal. Appl. 298 (2004), 463-486.
  4. Gosselin J., Stempak K., Conjugate expansions for Hermite functions, Illinois J. Math. 38 (1994), 177-197.
  5. Journé J.-L., Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón, Lecture Notes in Mathematics, Vol. 994, Springer-Verlag, Berlin, 1983.
  6. Nowak A., Stempak K., Riesz transforms for the Dunkl harmonic oscillator, Math. Z., to appear, arXiv:0802.0474.
  7. Rosenblum M., Generalized Hermite polynomials and the Bose-like oscillator calculus, in Nonselfadjoint Operators and Related Topics (Beer Sheva, 1992), Oper. Theory Adv. Appl., Vol. 73, Birkhäuser, Basel, 1994, 369-396.
  8. Rösler M., Generalized Hermite polynomials and the heat equation for Dunkl operators, Comm. Math. Phys. 192 (1998), 519-542, q-alg/9703006.
  9. Rösler M., Positivity of Dunkl's intertwining operator, Duke Math. J. 98 (1999), 445-463, q-alg/9710029.
  10. Stein E.M., Topics in harmonic analysis related to the Littlewood-Paley theory, Annals of Mathematics Studies, no. 63, Princeton University Press, Princeton, NJ, 1970.
  11. Stempak K., Torrea J.L., Poisson integrals and Riesz transforms for Hermite function expansions with weights, J. Funct. Anal. 202 (2003), 443-472.
  12. Thangavelu S., Lectures on Hermite and Laguerre expansions, Mathematical Notes, Vol. 42, Princeton University Press, Princeton, NJ, 1993.

Previous article   Next article   Contents of Volume 5 (2009)