A Hörmander-type spectral multiplier theorem for operators without heat kernel

Blunck, Sönke

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 449-459.

Hörmander’s famous Fourier multiplier theorem ensures the $L_p$-boundedness of $F(-{\Delta }_{ℝ}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ 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 $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta }_{ℝ}+x^2$ 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 $F\left(A\right)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s>\frac{D+1}{2}$, provided $A$ satisfies generalized gaussian estimates. This assumption allows to treat even operators $A$ without heat kernel (e.g. operators of higher order and operators with complex or unbounded coefficients) which was impossible for all known spectral multiplier results.

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