H{\"o}rmander Functional Calculus for Poisson Estimates

The aim of the article is to show a H{"o}rmander spectral multiplier theorem for an operator $A$ whose kernel of the semigroup $\exp (-zA)$ satisfies certain Poisson estimates for complex times $z\mathrm{.}$ Here $\exp (-zA)$ acts on $L^p(\mathrm{\Omega }),1<p<\mathrm{\infty },$ where $\mathrm{\Omega }$ is a space of homogeneous type with the additional condition that the measure of annuli is controlled. In most of the known H{"o}rmander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extend weaker Poisson bounds, and $\text{HI}$ calculus in place of self-adjointness. The order of derivation in our H{"o}rmander multiplier result is typically $\frac{d}{2},$ $d$ being the dimension of the space $\mathrm{\Omega }\mathrm{.}$ Moreover the functional calculus resulting from our H{"o}rmander theorem is shown to be $R$-bounded. Finally, the result is applied to some examples.