Hormander Functional Calculus for Poisson Estimates

The aim of the article is to show a Hormander 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 , where is a space of homogeneous type with the additional conditions that the volume of balls grows polynomially of exponent d and the measure of annuli is controlled by the corresponding euclidean term. In most of the known Hormander 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 extent weaker Poisson bounds, and calculus in place of self-adjointness. The order of derivation in our Hormander multiplier result is typically , d being the dimension of the space . Moreover the functional calculus resulting from our Hormander theorem is shown to be -bounded. Finally, the result is applied to some examples.