Hilbert transform on the Dunkl-Hardy Spaces

For $p>p_0=\frac{2\lambda }{2\lambda +1}$ with $\lambda >0$, the Hardy space $H_{\lambda }^p({\mathbb{R}}_{+}^2)$ associated with the Dunkl transform ${\mathcal{F}}_{\lambda }$ and the Dunkl operator $D$ on the real line $\mathbb{R}$, where $D_{x}f(x)=f^{\mathrm{\prime }}(x)+\frac{\lambda }{x}[f(x)-f(-x)]$, is the set of functions $F=u+iv$ on the upper half plane ${\mathbb{R}}_{+}^2=\{(x,y):x\in \mathbb{R},y>0\}$, satisfying $\lambda$-Cauchy-Riemann equations: $D_{x}u-{\mathrm{\partial }}_{y}v=0$, ${\mathrm{\partial }}_{y}u+D_{x}v=0$, and $\underset{y>0}{\sup }{\int }_{\mathbb{R}}\mathrm{\mid }F(x+iy){\mathrm{\mid }}^p\mathrm{\mid }x{\mathrm{\mid }}^{2\lambda }dx<\mathrm{\infty }$ in [7]. Then it is proved in [11] that the real Dunkl-Hardy Spaces $H_{\lambda }^p(\mathbb{R})$ for $\frac{1}{1+{\gamma }_{\lambda }}<p\le 1$ are Homogeneous Hardy Spaces. In this paper, we will continue to investigate $\lambda$-Hilbert transform on the real Dunkl-Hardy Spaces $H_{\lambda }^p(\mathbb{R})$ for $\frac{1}{1+{\gamma }_{\lambda }}<p\le 1$ with ${\gamma }_{\lambda }=1\mathrm{/}(4\lambda +2)$ and extend the results of $\lambda$-Hilbert transform in [7].