Ces\`aro 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 $\mathscr{F}_\lambda$ and the Dunkl operator $D_x$ on the line, where $D_xf(x)=f'(x)+\frac{\lambda}{x}[f(x)-f(-x)]$, is the set of function $F=u+iv$ on the upper half plane $\mathbb{R}_+^2=\big\{(x, y): y>0\big\}$, satisfying the $\lambda$-Cauchy-Riemann equations: $D_xu-\partial_y v=0, \partial_y u +D_xv=0$, and $\sup_{y>0}\int_{\mathbb{R}}|F(x, y)||x|^{2\lambda}dx<0$. In this paper, we will study the boundedness of Ces\`{a}ro operator on $H_{\lambda}^{p}(\mathbb{R}^{2}_+)$. We will prove the following inequality $$ \|C_{\alpha}f\|_{H_{\lambda}^p(\mathbb{R}_+^2)}\leq C\|f\|_{H_{\lambda}^p(v_+^2)},$$ for $\frac{2\lambda}{2\lambda+1}< p<\infty$, where C is dependent on $\alpha$, $p$, $\lambda$, and the average function for the Ces\`{a}ro operator $C_{\alpha}$ is $\phi_{\alpha}(t)=\alpha(1-t)^{\alpha-1}$ with $\alpha>0$.