Inverse closedness and localization in extended Gevrey regularity

Abstract

We consider classes \( \mathcal {E}_{\tau ,\sigma }(U)\) of ultradifferentiable functions which are extension of Gevrey classes, and prove that such classes are inverse closed. This result is used to construct an element from \( \mathcal {E}_{\tau ,\sigma }(U)\) which is not a Gevrey regular function. Furthermore, we show that the singular support of a distribution \(u\in \mathcal {D}'(U)\) related to local regularity in \( \mathcal {E}_{\tau ,\sigma }(U)\) coincides with the standard projection of the corresponding wave front set \( {\text {WF}}_{\tau ,\sigma }(u)\).