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

SIGMA 5 (2009), 037, 17 pages      arXiv:0903.4369      https://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.

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