Hörmander Functional Calculus for Poisson Estimates

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 Lp(Ω), 1 < p < ∞, where Ωis 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 Hörmander 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 H ∞calculus in place of self-adjointness. The order of derivation in our Hormander multiplier result is typically d 2, d being the dimension of the space Ω. Moreover the functional calculus resulting from our H¨ormander theorem is shown to be R-bounded. Finally, the result is applied to some examples.