Hilbert Transforms Associated with Dunkl-Hermite Polynomials

We consider expansions of functions in $L^p(\mathbb{R},\mathrm{\mid }x{\mathrm{\mid }}^{2k}dx)$, $1\leq p<+\infty$ 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 Calder'on-Zygmund operators and hence their mapping properties follow from general results.