Weak boundedness of operator-valued Bochner–Riesz means for the Dunkl transform

Abstract

We consider operator-valued Bochner–Riesz means with weight function $h_{\kappa }^2$ under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means $B_R^{\delta }(h_{\kappa }^2;f)\left(x\right)$ for $\delta >{\lambda }_{\kappa }:=\frac{d-1}{2}+{\sum }_{j=1}^d{\kappa }_j$. Furthermore, we also obtain the corresponding pointwise convergence theorem.