Hermite Multipliers on Modulation Spaces

We study multipliers associated to the Hermite operator $H=-\mathrm{\Delta }+\mathrm{\mid }x{\mathrm{\mid }}^2$ on modulation spaces $M^{p,q}({\mathbb{R}}^d)$. We prove that the operator $m(H)$ is bounded on $M^{p,q}({\mathbb{R}}^d)$ under standard conditions on $m,$ for suitable choice of $p$ and $q$. As an application, we point out that the solutions to the free wave and Schr"odinger equations associated to $H$ with initial data in a modulation space will remain in the same modulation space for all times. We also point out that Riesz transforms associated to $H$ are bounded on some modulation spaces.