# Hörmander Functional Calculus for Poisson Estimates

Abstract : 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 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 Hörmander multiplier result is typically $\frac{d}{2},$ $d$ being the dimension of the space $\Omega.$ Moreover the functional calculus resulting from our Hörmander theorem is shown to be $R$-bounded. Finally, the result is applied to some examples.
Keywords :
Type de document :
Pré-publication, Document de travail
The manuscript has undergone several substantial improvements. The main result (Theorem 3.2 and C.. 2014
Domaine :

Littérature citée [19 références]

https://hal.archives-ouvertes.fr/hal-00794258
Contributeur : Christoph Kriegler <>
Soumis le : mercredi 9 avril 2014 - 13:02:37
Dernière modification le : jeudi 11 janvier 2018 - 06:20:33
Document(s) archivé(s) le : mercredi 9 juillet 2014 - 11:35:14

### Fichiers

Poisson.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• HAL Id : hal-00794258, version 2
• ARXIV : 1302.6104

### Citation

Christoph Kriegler. Hörmander Functional Calculus for Poisson Estimates. The manuscript has undergone several substantial improvements. The main result (Theorem 3.2 and C.. 2014. 〈hal-00794258v2〉

### Métriques

Consultations de la notice

## 195

Téléchargements de fichiers