Harmonic Analysis Associated with the One-Dimensional Dunkl Transform

Abstract

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D x u−∂ y v=0,∂ y u+D x v=0, where D x is the Dunkl operator (D x f)(x)=f′(x)+(λ/x)(f(x)−f(−x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥p 0=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$ ; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$ .