Composite quasianalytic functions

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 ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (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 ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ 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.

Arc-smooth functions on closed sets

“…We will not pursue this in this paper. Instead, combining our results with a result of [15] and [3], we show in Theorem 8.4 that for fat closed subanalytic sets the loss of regularity can be controlled in a precise way.…”

“…By Theorem 8.2, where is a union of quadrants in . By [BBC18, Theorem 1.4], for each there is a positive integer such that is of class on . It follows that , where .◻…”

“…By the Bochnak-Siciak theorem 1.5 and Result 1. 3, all open subsets X ⊆ R d are A ω -admissible and A M -admissible, for each log-convex non-quasianalytic M .…”

“…where Y α is a union of quadrants in R d . By [3,Theorem 1.4], for each α there is a positive integer a α such that f is of class…”