Hörmander condition for pseudo-multipliers associated to the harmonic oscillator

In this paper we prove Hörmander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the L^p-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give L^p-boundedness and L^p-compactness properties for multipliers. (L^p, L^q)-estimates for spectral pseudo-multipliers also are investigated.