Malgrange division by quasianalytic functions

TL;DR

This paper generalizes Malgrange's division theorem to quasianalytic functions, expanding the understanding of division properties within these classes which include Denjoy-Carleman and o-minimal definable functions.

Contribution

It extends Malgrange's theorem from real-analytic functions to quasianalytic classes, broadening the scope of division properties in smooth function classes.

Findings

Proves division theorem for quasianalytic functions

Generalizes classical results to broader function classes

Enhances understanding of function division in analysis

Abstract

Quasianalytic classes are classes of infinitely differentiable functions that satisfy the analytic continuation property enjoyed by analytic functions. Two general examples are quasianalytic Denjoy-Carleman classes (of origin in the analysis of linear partial differential equations) and the class of infinitely differentiable functions that are definable in a polynomially bounded o-minimal structure (of origin in model theory). We prove a generalization to quasianalytic functions of Malgrange's celebrated theorem on the division of infinitely differentiable by real-analytic functions.