Hilbert Transforms Associated with Dunkl–Hermite Polynomials

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