O-minimal version of Whitney’s extension theorem. (English) Zbl 1318.14052

Kurdyka, Krzysztof; Pawłucki, Wiesław

Stud. Math. 224, No. 1, 81-96 (2014).

In the paper under review the authors give a generalized and improved version of [K. Kurdyka and W. Pawłucki, Stud. Math. 124, No. 3, 269–280 (1997; Zbl 0955.32006)]. The main result is the following:
Given an o-minimal structure on a real closed field \(R\), let \(\Omega\) be an open definable subset of \(R^n\) and let \(E\) be a definable closed subset of \(\Omega\). Let \(p,q\) be natural numbers with \(p\leq q\) and let \[ F(x,X)=\sum_{|\kappa|\leq p}\frac{1}{\kappa !}F^\kappa(x)X^\kappa \] be a definable \(C^p\)-Whitney field on \(E\) (i.e. all \(F^\kappa\)’s are definable \(C^p\)-functions). Then there exists a definable \(C^p\)-function \(f:\Omega\to R\) that is \(C^q\) on \(\Omega\setminus E\) such that \(D^\kappa f=F^\kappa\) on \(E\) for all \(\kappa\) with \(|\kappa|\leq p\).

Reviewer: Tobias Kaiser (Passau)

MSC:

14P10	Semialgebraic sets and related spaces
32B20	Semi-analytic sets, subanalytic sets, and generalizations
03C64	Model theory of ordered structures; o-minimality
14P15	Real-analytic and semi-analytic sets

Keywords:

Whitney field; o-minimal structure

References:

[1]	[C]M. Coste, An Introduction to O-minimal Geometry, Dottorato di Ricerca in Matematica, Dipartimento di Matematica, Università di Pisa, Istituti Editoriali e Poligrafici Internazionali, Pisa, 2000.
[2]	[G]G. Glaeser, Étude de quelques algèbres tayloriennes, J. Anal. Math. 6 (1958), 1–124. 96K. Kurdyka and W. Pawłucki · Zbl 0091.28103
[3]	[K]K. Kurdyka, On a subanalytic stratification satisfying a Whitney property with exponent 1, in: Real Algebraic Geometry (Rennes, 1991), Lecture Notes in Math. 1524, Springer, 1992, 316–322. · Zbl 0779.32006
[4]	[KP]K. Kurdyka and A. Parusiński, Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture, in: Singularity Theory and its Applications, Adv. Stud. Pure Math. 43, Math. Soc. Japan, Tokyo, 2006, 137–177. [KPaw] K. Kurdyka and W. Pawłucki, Subanalytic version of Whitney’s extension theorem, Studia Math. 124 (1997), 269–280.
[5]	[M]B. Malgrange, Ideals of Differentiable Functions, Oxford Univ. Press, 1966. [Par]A. Parusiński, Lipschitz stratification of subanalytic sets, Ann. Sci. École Norm. Sup. 27 (1994), 661–696. [Paw1] W. Pawłucki, Lipschitz cell decomposition in o-minimal structures. I, Illinois J. Math. 52 (2008), 1045–1063. [Paw2] W. Pawłucki, A linear extension operator for Whitney fields on closed o-minimal sets, Ann. Inst. Fourier (Grenoble) 58 (2008), 383–404. [Paw3] W. Pawłucki, A decomposition of a set definable in an o-minimal structure into perfectly situated sets, Ann. Polon. Math. 79 (2002), 171–184. [Th]A. Thamrongthanyalak, Whitney’s extension theorem in o-minimal structures, MODNET Preprint 626.
[6]	[T]J.-Cl. Tougeron, Idéaux de fonctions différentiables, Springer, Berlin, 1972. [vdD]L. van den Dries, Tame Topology and O-minimal Structures, Cambridge Univ. Press, 1998.
[7]	[W]H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89. · Zbl 0008.24902