Examples concerning Whitney's \({\mathcal C}^m\) extension problem
type: Item
instance of: scholarly article
title: Examples concerning Whitney's \({\mathcal C}^m\) extension problem
zbMATH Open document ID: 1109.58014
author: Edward Bierstone
author: Pierre Milman
author: Wiesław Pawłucki
author: Charles Fefferman
published in: Mathematical Research Letters
publication date: 2007-02-05T00:00:00Z
review text: The authors present several examples connected with their outstanding articles on Whitney's problem for \( C^m \) functions [\textit{E. Bierstone, P. D. Milman}, and \textit{W. Pawłucki}, Invent. Math. 151, No.~2, 329--352 (2003; Zbl 1031.58002), \textit{C. Fefferman}, Ann. Math. (2) 164, No. 1, 313--359 (2006; Zbl 1109.58016), \textit{E. Bierstone, P. D. Milman}, and \textit{W. Pawłucki}, Ann. Math. (2) 164, No. 1, 361--370 (2006; Zbl 1109.58015)]. The \( C^m \) extension criterion of \textit{C. Fefferman} [loc. cit.] is a variant of a condition of \textit{E. Bierstone, P. D. Milman}, and \textit{W. Pawłucki} [loc. cit.]. Both of them involve, in direct or dual form, iterated ``Glaeser operations'' on generalized finite difference operators. The first example of the present paper shows that Glaeser iterations cannot be avoided in Fefferman's criterion. The other ones are counterexamples to \( C^m \) extension under the weaker criterion that allowed Bierstone-Milman-Pawłucki to prove a \( C^\infty \) extension theorem (or a \( C^m \) extension theorem with a loss of differentiability) for closed subanalytic sets.
reviewed by: Vincent Thilliez
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 26B05
Mathematics Subject Classification ID: 26B99
zbMATH DE Number: 5121801
zbMATH Keywords: extension problems
zbMATH Keywords: differentiable functions
zbMATH Keywords: Whitney jets
zbMATH Keywords: paratangent spaces
MaRDI profile type: Publication
DOI: 10.4310/MRL.2006.V13.N6.A1
description: scientific article; zbMATH DE number 5121801