Hilbert transform on the Dunkl-Hardy Spaces

Abstract

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 [7]. Then it is proved in [11] that the real Dunkl-Hardy Spaces $H_{\lambda}^p(\mathbb{R})$ for $\frac{1}{1+\gamma_\lambda}<p\leq1$ 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 $\frac{1}{1+\gamma_\lambda}<p\leq1$ with $\gamma_\lambda=1/(4\lambda+2)$ and extend the results of $\lambda$-Hilbert transform in [7].