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