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 $D_{x}f(x)=f^{\prime }(x)+\frac{\lambda }{x}[f(x)-f(-x)]$, is the set of functions $F=u+iv$ on the upper half plane ${\mathbb{R}}_{+}^2=\left\{(x,y):x\in \mathbb{R},y>0\right\}$, satisfying $\lambda$-Cauchy-Riemann equations: $D_{x}u-{\partial }_{y}v=0$, ${\partial }_{y}u+D_{x}v=0$, and $\underset{y>0}{\sup }{\int }_{\mathbb{R}}|F(x+iy){|}^p|x{|}^{2\lambda }dx<\infty$ in [7]. Then it is proved in [11] that the real Dunkl-Hardy Spaces $H_{\lambda }^p(\mathbb{R})$ for $\frac{1}{1+{\gamma }_{\lambda }}<p\le 1$ 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 ${\gamma }_{\lambda }=1/(4\lambda +2)$ and extend the results of $\lambda$-Hilbert transform in [7].

arXiv commentarXiv admin note: substantial text overlap with arXiv:2106.05845; text overlap with arXiv:2106.08894