Cesàro operator on Hardy spaces associated with the Dunkl setting ($\frac{2\lambda }{2\lambda +1}<p<\infty$)

Abstract

For $p>\frac{2\lambda }{2\lambda +1}$ with $\lambda >0$, the Hardy spaces $H_{\lambda }^p({\mathbb{R}}_{+}^2)$ associated with the Dunkl transform ${\mathcal{F}}_{\lambda }$ and the Dunkl operator $D_x$ on the line, where $D_{x}f(x)=f^{\prime }(x)+\frac{\lambda }{x}[f(x)-f(-x)]$, is the set of function $F=u+iv$ on the upper half plane ${\mathbb{R}}_{+}^2=\{(x,y):y>0\}$, satisfying the $\lambda$-Cauchy-Riemann equations: $D_{x}u-{\partial }_{y}v=0,{\partial }_{y}u+D_{x}v=0$, and ${\sup }_{y>0}{\int }_{\mathbb{R}}|F(x,y)||x{|}^{2\lambda }dx<0$. In this paper, we will study the boundedness of Cesàro operator on $H_{\lambda }^p({\mathbb{R}}_{+}^2)$. We will prove the following inequality $$\Vert C_{\alpha }f{\Vert }_{H_{\lambda }^p({\mathbb{R}}_{+}^2)}\le C\Vert f{\Vert }_{H_{\lambda }^p(v_{+}^2)},$$ for $\frac{2\lambda }{2\lambda +1}<p<\infty$, where C is dependent on $\alpha$, $p$, $\lambda$, and the average function for the Cesàro operator $C_{\alpha }$ is ${\varphi }_{\alpha }(t)=\alpha (1-t{)}^{\alpha -1}$ with $\alpha >0$.

arXiv commentarXiv admin note: text overlap with arXiv:2106.05845