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

TL;DR

This paper establishes a H{"o}rmander spectral multiplier theorem for operators with Poisson kernel estimates, extending previous results by weakening assumptions and applying to spaces of homogeneous type.

Contribution

It introduces a H{"o}rmander theorem under Poisson bounds instead of Gaussian bounds, using $ ext{H}^ ext{infinity}$ calculus and proving $R$-boundedness of the functional calculus.

Findings

The theorem applies to operators with Poisson estimates on complex semigroup kernels.

The derived functional calculus is $R$-bounded.

The results are demonstrated through specific examples.

Abstract

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 $z.$ 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 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 $\HI$ 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 $\Omega.$ 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.