Hilbert Transforms Associated with Dunkl-Hermite Polynomials - Ben Salem, Nejib; Samaali, Taha (Symmetry, Integrability and Geometry: Methods and Applications, 2009)
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.