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

Abstract

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 L^p(w_k) for 1 < p < ∞, where w_k(x) = ∣x∣^2k. Finally, we introduce the maximal Hilbert operator H ^k_* and establish an analogue of Cotlar’s theorem.