The Real Characterization of $H_{\lambda }^p({\mathbb{R}}_{+}^2)$ for $\frac{2\lambda }{2\lambda +1}<p\le 1$

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 this paper, we will give a further characterization of $H_{\lambda }^p({\mathbb{R}}_{+}^2)$ in \cite{ZhongKai Li 3}. We prove the inequality $\Vert F{\Vert }_{H_{\lambda }^p({\mathbb{R}}_{+}^2)}\le c\Vert u_{\nabla }^{\ast }{\Vert }_{L_{\lambda }^p}$, which gives a Real Characterization of the class $H_{\lambda }^p({\mathbb{R}}_{+}^2)$ for $\frac{2\lambda }{2\lambda +1}<p\le 1$ as a main result.

arXiv commentarXiv admin note: text overlap with arXiv:2004.01777