On some quasianalytic classes of $C^\infty$ functions

TL;DR

This paper explores properties of various quasianalytic classes of smooth functions, focusing on the Borel mapping's behavior and extending known results from Denjoy-Carleman classes to other classes, including those definable in o-minimal structures.

Contribution

It investigates whether properties like non-surjectivity and monotonicity of the Borel mapping hold across different quasianalytic classes, extending previous findings.

Findings

Non-surjectivity of the Borel mapping in certain classes

Monotonicity properties of quasianalytic classes

Existence of quasianalytic classes with surjective Borel mapping

Abstract

This expository article is devoted to the notion of quasianalytic classes and the Borel mapping. Although quasianalytic classes are well known in analysis since several decades. We are interested in certain properties of Denjoy-Carleman's quasianalytic classes, such as the non-surjectivity of the Borel mapping, the property of monotonicity. We try to see if it remains true for other quasianalytic classes, such as for example, the classes of indefinitely differentiable functions definable in a polynomially bounded o-minimal structures. What motivated this is the fact of having shown in a previous article the existence of quasianalytic classes where Borel mapping is surjective.