Higher-order tangents and Fefferman’s paper on Whitney’s extension problem

Abstract

Whitney [W2] proved that a function defined on a closed subset of R is the restriction of a C m function if the limiting values of all m th divided differences form a continuous function. We show that Fefferman's solution of Whitney's problem for R n [F, Th. 1] is equivalent to a variant of our conjecture in [BMP2] giving a criterion for C m extension in terms of iterated limits of finite differences.

Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators

“…. ,9, 18, 19, 20], and Bierstone-Milman-Paw lucki [1,2]). Here, C m,ω (R n ) denotes the space of all C m functions on R n whose m th derivatives have modulus of continuity ω.…”

“…We write C m (E) for the Banach space of all restrictions to E of functions F ∈ C m (E). The norm on C m (E) is given by (2) f C m (E) = inf…”

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

“…Let us show that these operations commute (see also Theorem 3.2 in [2] for a closely related result).…”

“…This notion, introduced by Fefferman, is related to the notions of paratangent and iterated paratangent bundles, introduced by Glaeser [8] and Bierstone-Milman-Paw lucki [1,2].…”

“…They introduced the notion of iterated paratangent bundles which allowed them to formulate and prove the result. See also [2] for further developments.…”