zbMATH — the first resource for mathematics

A linear extension operator for Whitney fields on closed o-minimal sets. (English) Zbl 1168.14040
K. Kurdyka and W. Pawłucki [Stud. Math. 124, No. 3, 269–280 (1997; Zbl 0955.32006)] proved a subanalytic version of Whitney’s seminal extension theorem. That is, given a subanalytic Whitney field \(f\) on a closed subanalytic set \(A\), then there is a subanalytic Whitney field on the ambient space extending \(f\). However, this construction is not a linear extension operator.
In the present article the author studies extensions of continuous and continuously differentiable o-minimal functions. He introduces an explicit extension operator for functions defined on sets \(A\) which are definable in an o-minimal expansion of the real field, see L. van den Dries and C. Miller [Duke Math. J. 84, No. 2, 497–540 (1996; Zbl 0889.03025)]. This operator decomposes into suboperators of particularly simple form which either involve integration with respect to one variable or preserve definability.
The extension operator takes moduli of continuity into account such that the modulus of continuity of the extended Whitney field is a scalar multiple of the modulus of continuity of the original Whitney field where the scalar depends only on the set \(A\). According to the construction, such scalar is implied by the finiteness of stratifications of o-minimal sets. For general subanalytic or locally definable sets this construction does not work, and new ideas seem to be needed.
Definability of the extension is also analyzed. If no derivatives are involved then the extension is definable in the original structure. Otherwise, integration is needed. However, one can control the loss of definability in case of the o-minimal structure \(\mathbb{R}_{an}\) – the structure consisting of all globally subanalytic sets; in this case, the extension is definable in the o-minimal expansion \(\mathbb{R}_{an,\exp}\) which expands \(\mathbb{R}_{an}\) by the (real) exponential function.

14P10 Semialgebraic sets and related spaces
26B05 Continuity and differentiation questions
32B20 Semi-analytic sets, subanalytic sets, and generalizations
03C64 Model theory of ordered structures; o-minimality
Full Text: DOI Numdam
[1] Coste, M., An Introduction to O-minimal Geometry, (2000), Istituti Editoriali e Poligrafici Internazionali, Pisa-Roma
[2] van den Dries, L., Tame Topology and O-minimal Structures, (1998), Cambridge University Press · Zbl 0953.03045
[3] van den Dries, L.; Miller, C., Geometric categories and o-minimal structures, Duke Math. J., 84, 497-540, (1996) · Zbl 0889.03025
[4] Glaeser, G., Étude de quelques algèbres tayloriennes, J. Anal. Math., 6, 1-124, (1958) · Zbl 0091.28103
[5] Kurdyka, K., On a subanalytic stratification satisfying a Whitney property with exponent 1, Proc. Conference Real Algebraic Geometry, 316-322, (1991), Springer, Rennes · Zbl 0779.32006
[6] Kurdyka, K.; Pawłucki, W., Subanalytic version of whitney’s extension theorem, Studia Math., 124 (3), 269-280, (1997) · Zbl 0955.32006
[7] Lion, J.-M.; Rolin, J.-P., Intégration des fonctions sous-analytiques et volumes des sous-ensembles sous-analytiques, Ann. Inst. Fourier, Grenoble, 68,3, 755-767, (1998) · Zbl 0912.32007
[8] Malgrange, B., Ideals of Differentiable Functions, (1966), Oxford University Press · Zbl 0177.17902
[9] Parusiński, A., Lipschitz stratification of subanalytic sets, Ann. Scient. Ec. Norm. Sup., 27, 661-696, (1994) · Zbl 0819.32007
[10] Pawłucki, W., A decomposition of a set definable in an o-minimal structure into perfectly situated sets, Ann. Polon. Math., LXXIX.2, 171-184, (2002) · Zbl 1024.03036
[11] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, (1970), Princeton University Press · Zbl 0207.13501
[12] Tougeron, J. Cl, Idéaux des Fonctions Différentiables, (1972), Springer · Zbl 0251.58001
[13] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Am. Math. Soc., 36, 63-89, (1934) · 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.