A Characterization of the Hardy Space Associated with the Dunkl Transform

Abstract

For $p\ge p_0:=2\lambda /(2\lambda +1)$ with $\lambda >0$, the Hardy space $H_\lambda ^p({\mathbb {R}}^2_+)$ associated with the Dunkl transform ${\mathscr {F}}_{\lambda }$ and the Dunkl operator D on the line ${\mathbb {R}}$, where $(D_xf)(x)=f'(x)+\frac{\lambda }{x} (f(x)-f(-x))$, is the set of functions $F=u+iv$ on the half plane ${\mathbb {R}}^2_+=\{(x,y):\,x\in {\mathbb {R}}, y>0\}$, satisfying the generalized Cauchy–Riemann equations $D_xu-\partial _yv=0$, $\partial _yu+D_xv=0$, and $\sup _{y>0}\int _{{\mathbb {R}}}| F(x,y)|^p|x|^{2\lambda }dx<+\infty$; and the real Hardy space $H_{\lambda }^p({\mathbb {R}})$ on the line ${\mathbb {R}}$ is the collection of boundary functions of the real parts of functions $F\in H_{\lambda }^p({\mathbb {R}}^2_+)$. In this paper, we establish the Hardy-Littlewood-Sobolev type theorem on the Hardy spaces for the Riesz potential $I_{\lambda }^{\alpha }$ associated to the Dunkl transform; and as the main result, we prove the equality $D(I_{\lambda }^{1}f)=- {\mathscr {H}}_{\lambda }f$ for $f\in H^1_{\lambda }({\mathbb {R}})$ in a weak sense, where ${\mathscr {H}}_{\lambda }$ is the generalized Hilbert transform related to the Dunkl transform, which gives a characterization for $f\in H_\lambda ^1({\mathbb {R}})$.