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

Abstract

Given an arbitrary set $E \subset \mathbb R^n$, $n\ge 2$, and a function $f: E \rightarrow \mathbb R$, consider the problem of extending $f$ to a $C^1$ function defined on the entire $\mathbb R^n$. 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.