  We give various conditions for Hermite pseudo-multipliers to be bounded on
$L^2(\mathbb{R}^n)$. As a by-product we also give results on
$L^p(\mathbb{R}^n)$, as well as new results for pseudo-multipliers for the
Gaussian measure setting. One of our key tools is a new integration by-parts
formula for Hermite expansions.
