  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
$p=+\infty$.
