Abstract

Hörmander’s famous Fourier multiplier theorem ensures the $L_p$-boundedness of $F$ whenever $F\in \mathcal{H}$ for some $s>\frac{D}{2}$, where we denote by $\mathcal{H} $ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A \ge 0$ and yield the $L_p$-boundedness of $F$ provided $F\in \mathcal{H}$ for some $s$ sufficiently large. The harmonic oscillator $A=-\Delta _{\mathbb{R}}+x^2$ shows that in general $s> \frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper, we prove the $L_p$-boundedness of $F$ whenever $F\in \mathcal{H}$ for some $s>\frac{D+1}{2}$, provided $A$ satisfies generalized gaussian estimates. This assumption allows to treat even operators $A$ without heat kernel which was impossible for all known spectral multiplier results