Differentiable functions defined in closed sets. A problem of Whitney

• Published in: Inventiones mathematicae Vol. 151; no. 2; pp. 329 - 352
• Main Authors: Bierstone, Edward, Milman, Pierre D., Pawłucki, Wiesław
• Format: Journal Article
• Language: English
• Published: Heidelberg Springer Nature B.V 01.02.2003
• Subjects: Bundles; Constrictions; Criteria; Functions (mathematics); Mathematical analysis; Operators; Studies; Tangents; Topology
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-002-0255-6

In 1934, Whitney raised the question of how to recognize whether a function f defined on a closed subset X of ^sup n^ is the restriction of a function of class ðoe'z^sup p^. A necessary and sufficient criterion was given in the case n=1 by Whitney, using limits of finite differences, and in the case p=1 by Glaeser (1958), using limits of secants. We introduce a necessary geometric criterion, for general n and p, involving limits of finite differences, that we conjecture is sufficient at least if X has a "tame topology". We prove that, if X is a compact subanalytic set, then there exists q=q^sub X^(p) such that the criterion of order q implies that f is ðoe'z^sup p^. The result gives a new approach to higher-order tangent bundles (or bundles of differential operators) on singular spaces.[PUBLICATION ABSTRACT]