Division by Flat Ultradifferentiable Functions and Sectorial Extensions

Abstract

We consider classes ${\mathcal{A}}_M(S)$ of functions holomorphic in an open plane sector $S$ and belonging to a strongly non-quasianalytic class on the closure of $S$. In ${\mathcal{A}}_M(S)$, we construct functions which are flat at the vertex of $S$ with a sharp rate of vanishing. This allows us to obtain a Borel-Ritt type theorem for ${\mathcal{A}}_M(S)$ extending previous results by Schmets and Valdivia. We also derive a division property for ideals of flat ultradifferentiable functions, in the spirit of a classical $C^{\infty }$ result of Tougeron.

Slight update of the published version. The definition of closedness in subsections 4.1 and 4.2 is less restrictive. One minor typo corrected