$C^1$ extensions of functions and stabilization of Glaeser refinements

Abstract

Abstract. Given an arbitrary set E ⊂ R n , n ≥ 2, and a function f : E → R, consider the problem of extending f to a C 1 function defined on the entire R n . A procedure for determining whether such an extension exists was suggested in 1958 by G. Glaeser. In 2004 C. Fefferman proposed a related procedure for dealing with the much more difficult cases of higher smoothness. The procedures in question require iterated computations of some bundles until the bundles stabilize. How many iterations are needed? We give a sharp estimate for the number of iterations that could be required in the C 1 case. Some related questions are discussed.

Whitney’s extension problem in o-minimal structures

“…This can be seen as providing an answer to the C 1 -case of a question posed by Bierstone and Milman (see [50]). Our proof follows the argument for WEP n,1 given by Klartag and Zobin [26], which in turn rests on a use of Michael's Selection Theorem from general topology. Therefore, a study of properties of definable set-valued maps and a definable version of this selection theorem occupy most of this paper.…”

“…In this section we treat a definable version of the well-known Michael Selection Theorem [32] for set-valued maps. The classical version of this theorem plays a crucial role in the approach to solving WEP n,1 by Klartag and Zobin [26]. Classically, this theorem is shown by a non-constructive iterative procedure; see [3, Section 9.1] or [23,24] for expositions.…”

“…In this section, following [18], we introduce affine bundles and Glaeser refinements and establish basic facts about them used later. The proofs are routine adaptations of those given in [26] and are only included for the convenience of the reader. Throughout this section we let X denote a subset of R n .…”

“…In this section, we follow the idea given in [26] to solve WEP n,1 . Throughout this section, we fix a definable closed subset X of R n and a definable continuous function f : X → R. Definition 7.1.…”

Whitney’s extension problems and interpolation of data

“…This notion is defined in terms of iterated limits. Examples given by Glaeser [23] and Klartag-Zobin [24] suggest strongly that any complete answer to Question 1 must bring in such iterated limits.…”