Iterates and Hypoellipticity of Partial Differential Operators on Non-Quasianalytic Classes

Abstract

Let P be a linear partial differential operator with constant coefficients. For a weight function ω and an open subset Ω of \({\mathbb{R}^N}\) , the class \({\mathcal{E}_{P,\{\omega\}}(\Omega)}\) of Roumieu type involving the successive iterates of the operator P is considered. The completeness of this space is characterized in terms of the hypoellipticity of P. Results of Komatsu and Newberger-Zielezny are extended. Moreover, for weights ω satisfying a certain growth condition, this class coincides with a class of ultradifferentiable functions if and only if P is elliptic. These results remain true in the Beurling case \({\mathcal{E}_{P,(\omega)}(\Omega)}\).