Functions with ultradifferentiable powers

Abstract

We study the regularity of smooth functions $f$ defined on an open set of ${\mathbb{R}}^n$ and such that, for certain integers $p\ge 2$, the powers $f^p:x\mapsto (f(x){)}^p$ belong to a Denjoy-Carleman class ${\mathcal{C}}_M$ associated with a suitable weight sequence $M$. Our main result is a statement analogous to a classic theorem of H. Joris on ${\mathcal{C}}^{\infty }$ functions: if a function $f:\mathbb{R}\to \mathbb{R}$ is such that both functions $f^p$ and $f^q$ with $\gcd (p,q)=1$ are of class ${\mathcal{C}}_M$ on $\mathbb{R}$, and if the weight sequence $M$ satisfies the so-called moderate growth assumption, then $f$ itself is of class ${\mathcal{C}}_M$. Various ancillary results, corollaries and examples are presented.

15 pages