Functions with ultradifferentiable powers

TL;DR

This paper investigates the regularity of smooth functions whose powers belong to a Denjoy-Carleman class, establishing conditions under which the functions themselves inherit this regularity, extending classical results to ultradifferentiable contexts.

Contribution

It generalizes a classical theorem of H. Joris to ultradifferentiable functions, showing that if powers of a function are in a Denjoy-Carleman class and certain conditions hold, then the function itself is in that class.

Findings

Functions with powers in a Denjoy-Carleman class can inherit regularity under moderate growth conditions.

The main theorem extends classical smooth function results to ultradifferentiable classes.

Various examples illustrate the applicability of the main results.

Abstract

We study the regularity of smooth functions $f$ defined on an open set of $\mathbb{R}^n$ and such that, for certain integers $p\geq 2$, 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.