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

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References

Belotto da Silva, A., Biborski, I. and Bierstone, E., Solutions of quasianalytic equations , Selecta Math. (N.S.) 23 (2017), 2523–2552.Google Scholar
Bierstone, E. and Milman, P. D., Semianalytic and subanalytic sets , Publ. Math. Inst. Hautes Études Sci. 67 (1988), 5–42.Google Scholar
Bierstone, E. and Milman, P. D., Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant , Invent. Math. 128 (1997), 207–302.Google Scholar
Bierstone, E. and Milman, P. D., Resolution of singularities in Denjoy–Carleman classes , Selecta Math. (N.S.) 10 (2004), 1–28.Google Scholar
Bierstone, E., Milman, P. D. and Valette, G., Arc-quasianalytic functions , Proc. Amer. Math. Soc. 143 (2015), 3915–3925.Google Scholar
Borel, E., Sur la généralisation du prolongement analytique , C. R. Math. Acad. Sci. Paris 130 (1900), 1115–1118.Google Scholar
Carleman, T., Les fonctions quasi-analytiques, Collection Borel (Gauthier-Villars, Paris, 1926).Google Scholar
Chaumat, J. and Chollet, A., Sur la division et la composition dans des classes ultradifférentiables , Studia Math. 136 (1999), 49–70.Google Scholar
Denjoy, A., Sur les fonctions quasi-analytiques de variable reélle , C. R. Math. Acad. Sci. Paris 173 (1921), 1329–1331.Google Scholar
Glaeser, G., Fonctions composées différentiables , Ann. of Math. (2) 77 (1963), 193–209.Google Scholar
Hadamard, J., Lectures on Cauchy’s problem in linear partial differential equations (Yale University Press, New Haven, 1923).Google Scholar
Hörmander, L., The analysis of linear partial differential operators I (Springer, Berlin–Heidelberg–New York, 1983).Google Scholar
Jaffe, E., Pathological phenomena in Denjoy–Carleman classes , Canad. J. Math. 68 (2016), 88–108.Google Scholar
Komatsu, H., The implicit function theorem for ultradifferentiable mappings , Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), 69–72.Google Scholar
Malgrange, B., Frobenius avec singularités, 2. Le cas général , Invent. Math. 39 (1977), 67–89.Google Scholar
Mandelbrojt, S., Séries Adhérentes, Régularisation des Suites, Applications, Collection Borel (Gauthiers-Villars, Paris, 1952).Google Scholar
Miller, C., Infinite differentiability in polynomially bounded o-minimal structures , Proc. Amer. Math. Soc. 123 (1995), 2551–2555.Google Scholar
Nowak, K. J., A note on Bierstone-Milman-Pawłucki’s paper ‘Composite differentiable functions’ , Ann. Polon. Math. 102 (2011), 293–299.Google Scholar
Nowak, K. J., On division of quasianalytic function germs , Int. J. Math. 13 (2013), 1–5.Google Scholar
Nowak, K. J., Quantifier elimination in quasianalytic structures via non-standard analysis , Ann. Polon. Math. 114 (2015), 235–267.Google Scholar
Rolin, J.-P., Speissegger, P. and Wilkie, A. J., Quasianalytic Denjoy–Carleman classes and o-minimality , J. Amer. Math. Soc. 16 (2003), 751–777.Google Scholar
Roumieu, C., Ultradistributions définies sur ℝ n et sur certaines classes de variétés différentiables , J. Anal. Math. 10 (1962–63), 153–192.Google Scholar
Thilliez, V., On quasianalytic local rings , Expo. Math. 26 (2008), 1–23.Google Scholar