  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 $D_xf(x)=f'(x)+\frac{\lambda}{x}[f(x)-f(-x)]$, is the set of functions
$F=u+iv$ on the upper half plane $\mathbb R^2_+=\left\{(x, y): x\in\mathbb R,
y>0\right\}$, satisfying $\lambda$-Cauchy-Riemann equations: $ D_xu-\partial_y
v=0$, $\partial_y u +D_xv=0$, and $\sup\limits_{y>0}\int_{\mathbb
R}|F(x+iy)|^p|x|^{2\lambda}dx<\infty$. In this paper, we will give a further
characterization of $H_{\lambda}^p(\mathbb R_+^2)$ in \cite{ZhongKai Li 3}. We
prove the inequality $\|F\|_{H_{\lambda}^p(\mathbb R_+^2)}\leq
c\|u_{\nabla}^*\|_{L^p_{\lambda}}$, which gives a Real Characterization of the
class $H_{\lambda}^p(\mathbb R_+^2)$ for $\frac{2\lambda}{2\lambda+1}<p\leq1$
as a main result.
