Bochner-Riesz means for the Hermite and special Hermite expansions

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.