Composite quasianalytic functions

TL;DR

This paper establishes new theorems on the composition and regularity of quasianalytic functions within Denjoy-Carleman classes, including a local composite function theorem and regularity loss in o-minimal structures.

Contribution

It introduces a composite function theorem for quasianalytic Denjoy-Carleman classes and analyzes regularity loss in o-minimal structures involving these classes.

Findings

Formal composability propagates locally in quasianalytic classes.

Functions definable in the o-minimal structure exhibit regularity loss.

Estimates for regularity of solutions to functional equations are provided.

Abstract

We prove two main results on Denjoy-Carleman classes: (1) a composite function theorem which asserts that a function f(x) in a quasianalytic Denjoy-Carleman class Q, which is formally composite with a generically submersive mapping y=h(x) of class Q, at a single given point in the source (or in the target) of h, can be written locally as f(x) = g(h(x)), where g(y) belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar loss of regularity for functions definable in the o-minimal structure given by expansion of the real field by restricted functions of quasianalytic class Q. Both results depend on an estimate for the regularity of an infinitely differentiable solution g of the equation f(x) = g(h(x)), with f and h as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.