A linear extension operator for Whitney fields on closed o-minimal sets
type: Item
instance of: scholarly article
title: A linear extension operator for Whitney fields on closed o-minimal sets
zbMATH Open document ID: 1168.14040
author: Wiesław Pawłucki
publication date: 2008-07-04T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: \textit{K. Kurdyka} and \textit{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 \textit{L. van den Dries} and \textit{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.
Mathematics Subject Classification ID: 14P10
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 32B20
Mathematics Subject Classification ID: 03C64
zbMATH DE Number: 5298307
zbMATH Keywords: Whitney field
zbMATH Keywords: extension operator
zbMATH Keywords: o-minimal structure
zbMATH Keywords: subanalytic set
MaRDI profile type: Publication
OpenAlex ID: W2331754506
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DOI: 10.5802/AIF.2355
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5298307