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

Abstract

We prove a Mikhlin--Hörmander multiplier theorem for the partial harmonic oscillator $H_{\textup{par}}=-\pa_ρ^2-Δ_x+|x|^2$ 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 $\mathbb{R}\times \mathbb{N}$.

14 pages, no figure. All comments are welcome