  We consider the Bochner-Riesz means for the Hermite and special Hermite
expansions and study their $L^p$ boundedness with the sharp summability index
in a local setting. In two dimensions we establish the boundedness on the
optimal range of $p$ and extend the previously known range in higher
dimensions. Furthermore, we prove a new lower bound on the $L^p$ summability
index for the Hermite Bochner-Riesz means in $\mathbb R^d$, $d\ge 2.$ This
invalidates the conventional conjecture which was expected to be true.
