A Characterization of the Hardy Space Associated with the Dunkl Transform

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