Functions with Ultradifferentiable Powers

Abstract

We study the regularity of smooth functions f defined on an open subset 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 $f:{\mathbb {R}}\rightarrow {\mathbb {C}}$ 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 a standard condition known as moderate growth, then f itself is of class ${\mathcal {C}}_M$. It is also shown that the result is no longer true without the moderate growth assumption. Various ancillary results, corollaries and examples are presented.