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Composite quasianalytic functions

Published online by Cambridge University Press:  17 August 2018

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

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.

Type
Research Article
Copyright
© The Authors 2018 

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References

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