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.

arXiv comment21 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

References in corpus (8)

• Composition in ultradifferentiable classes
• Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis
• Smooth solutions of quasianalytic or ultraholomorphic equations
• Equivalence of stability properties for ultradifferentiable function classes
• On the extension of Whitney ultrajets, II
• On the extension of Whitney ultrajets
• Pathological phenomena in Denjoy-Carleman classes
• Arc-smooth functions on closed sets

Cited by in corpus (3)

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