On the $L^2$ boundedness of pseudo-multipliers for Hermite expansions

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.