Inverse closedness and localization in extended Gevrey regularity

We consider classes E τ , σ ( 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 E τ , σ ( U ) which is not a Gevrey regular function. Furthermore, we show that the singular support of a distribution u ∈ D ′ ( U ) related to local regularity in E τ , σ ( U ) coincides with the standard projection of the corresponding wave front set WF τ , σ ( u ) .