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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.

MSC:
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
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References:
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