The Hilbert Transform for Dunkl Differential Operators Associated to the Reflection Group ℤ2

The aim of this paper is to introduce the Dunkl—Hilbert transform Hk, 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 Hk on Lp(wk) for 1 < p < ∞, where wk(x) = ∣x∣2k. Finally, we introduce the maximal Hilbert operator H*k and establish an analogue of Cotlar’s theorem.