Examples concerning Whitney's \({\mathcal C}^m\) extension problem
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.
scientific article; zbMATH DE number 5121801