Ultradifferentiable functions of class $M_p^{\t,\s}$ and microlocal regularity

TL;DR

This paper investigates ultradifferentiable function spaces extending Gevrey classes, establishing continuity properties of operators and developing a wave-front set concept tailored to these regularity conditions.

Contribution

It introduces a new framework for ultradifferentiable functions beyond Komatsu's condition, including localization procedures and wave-front set analysis.

Findings

Proved continuity of (ultra)differentiable operators on these spaces.

Developed a wave-front set concept for the new ultradifferentiable classes.

Connected wave-front set projections to singular supports of ultradifferentiable function spaces.

Abstract

We study spaces of ultradifferentiable functions which contain Gevrey classes. Although the corresponding defining sequences do not satisfy Komatsu's condition (M.2)', we prove appropriate continuity properties under the action of (ultra)differentiable operators. Furthermore, we study convenient localization procedure which leads to the concept of wave-front set with respect to our regularity conditions. As an application, we identify the standard projections of intersections/unions of wave-front sets as singular supports of suitable spaces of ultradifferentiable functions.