The Hilbert Transform for Dunkl Differential Operators Associated to the Reflection Group DOUBLE-STRUCK CAPITAL Z2

The aim of this paper is to introduce the Dunkl-Hilbert transform H k, with k = 0, induced by the Dunkl differential operator and associated with the reflection group Z2. For this end, we establish that the Dunkl-Poisson kernel and the conjugate Dunkl-Poisson kernel satisfy the Cauchy-Riemann equations in the Dunkl context. We prove the continuity of H k on Lp (wk) for 1 < p < 8, where wk(x) = |x|2k. Finally, we introduce the maximal Hilbert operator Hk * and establish an analogue of Cotlar's theorem.