Nonlinear conditions for ultradifferentiability

TL;DR

This paper extends Joris's theorem on smooth functions to a broad class of ultradifferentiable functions, demonstrating division properties in non-quasianalytic and certain quasianalytic classes across various dimensions.

Contribution

It proves a division property analogous to Joris's theorem in diverse ultradifferentiable classes, including non-quasianalytic and some quasianalytic cases, in multiple dimensions.

Findings

Division property holds in all dimensions for non-quasianalytic classes.

Validity of the property in quasianalytic classes is established in dimension one.

Certain quasianalytic classes also satisfy the division property in higher dimensions.

Abstract

A remarkable theorem of Joris states that a function $f$ is $C^\infty$ if two relatively prime powers of $f$ are $C^\infty$. Recently, Thilliez showed that an analogous theorem holds in Denjoy--Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris's result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.