zbMATH — the first resource for mathematics

O-minimal version of Whitney’s extension theorem. (English) Zbl 1318.14052
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 \(|\kappa|\leq p\).

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
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.