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Composite quasianalytic functions

Part of: Model theory Singularities Entire and meromorphic functions, and related topics Real functions: Miscellaneous topics

Published online by Cambridge University Press:  17 August 2018

André Belotto da Silva ,
Edward Bierstone  and
Michael Chow
André Belotto da Silva
Affiliation:
Université Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse Cedex 9, France email andre.belotto_da_silva@math.univ-toulouse.fr
Edward Bierstone
Affiliation:
University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON, CanadaM5S 2E4 email bierston@math.toronto.edu
Michael Chow
Affiliation:
University of Toronto, Department of Mathematics, 40 St. George Street, Toronto, ON, CanadaM5S 2E4 email mikey.chow@mail.utoronto.ca
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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.

Keywords

quasianalyticDenjoy–Carleman classo-minimal structureresolution of singularitiescomposite function theoremquasianalytic continuation

MSC classification

Primary: 03C64: Model theory of ordered structures; o-minimality 26E10: $C^infty$-functions, quasi-analytic functions 32S45: Modifications; resolution of singularities
Secondary: 30D60: Quasi-analytic and other classes of functions 32B20: Semi-analytic sets and subanalytic sets

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 9 , September 2018 , pp. 1960 - 1973
Copyright
© The Authors 2018 

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