Nonlinear conditions for ultradifferentiability

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.

21 pages; Version 2: The division property in the first version is equivalent to property (2) in the current version. This simplifies somewhat the proof of Lemma 6.1. 20 pages

Cited by in corpus (3)

• On generalized definitions of ultradifferentiable classes
• Ultradifferentiable extension theorems: a survey
• Nonlinear conditions for ultradifferentiability: a uniform approach