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

TL;DR

This paper investigates the boundedness of the Cesàro operator on Hardy spaces associated with the Dunkl transform and operator, establishing key inequalities for functions in these spaces.

Contribution

It introduces and proves boundedness results for the Cesàro operator on Dunkl-associated Hardy spaces, extending classical analysis to Dunkl settings.

Findings

Boundedness of Cesàro operator on Dunkl Hardy spaces established

Inequality relating norms of Cesàro operator and original functions proved

Conditions on p, λ, and α for boundedness are identified

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 $\alpha>0$.