We prove a Mikhlin--H\"ormander multiplier theorem for the partial harmonic
oscillator H_{\textup{par}}=-\pa_\rho^2-\Delta_x+|x|^2 for (Ο,x)βRΓRd by using the Littlewood--Paley g and gβ functions
and the associated heat kernel estimate. The multiplier we have investigated is
defined on RΓN.Comment: 14 pages, no figure. All comments are welcom