A Mikhlin--Hörmander multiplier theorem for the partial harmonic oscillator

We prove a Mikhlin–Hörmander multiplier theorem for the partial harmonic oscillator $H_{\text{par}}=-{\text{pa}}_{\rho }^2-{\mathrm{\Delta }}_x+\mathrm{\mid }x{\mathrm{\mid }}^2$ for $(\rho ,x)\in \mathbb{R}\times {\mathbb{R}}^d$ by using the Littlewood–Paley $g$ and $g^{\ast }$ functions and the associated heat kernel estimate. The multiplier we have investigated is defined on $\mathbb{R}\times \mathbb{N}$.