Two results on the Dunkl maximal operator

In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the case ${\mathbb{Z}}_2^d$ by establishing a result of exponential integrability corresponding to the case $p=+\mathrm{\infty }$.