Higher-order tangents and Fefferman’s paper on Whitney’s extension problem. (English) Zbl 1109.58015

Bierstone, Edward; Milman, Pierre D.; Pawłucki, Wiesław

Ann. Math. (2) 164, No. 1, 361-370 (2006).

According to a classical result of Whitney, a function defined on a closed subset of \( {\mathbb R} \) is the restriction of a \( C^m \) function if the limiting values of all its \( m^{\text{th}} \) divided differences with supports converging to points form a continuous function. In their earlier work [Invent. Math. 151, No. 2, 329–352 (2003; Zbl 1031.58002)], the authors conjectured that a real-valued function \( \varphi \) defined on a closed subset \( E \) of \( {\mathbb R}^n \) is the restriction of a \( C^m \) function provided that \( \varphi \) extends to a function on a “paratangent bundle” defined using iterated limits of finite difference operators. In the very interesting present paper, they prove that their conjectures hold with the paratangent bundle replaced by a natural geometric variant which, in turn, coincides with the so-called “Zariski paratangent bundle” \( {\mathcal T}^m(E) \) defined as follows: let \( {\mathcal P} \) denote the vector space of real \( m^{\text{th}} \) degree polynomials on \( {\mathbb R}^n \) and let \( {\mathcal P}^* \) be its dual space; let \( T^m_yF \) denote the \( m^{\text{th}}\) Taylor polynomial of a function \( F\in C^m({\mathbb R}^m) \) at a point \( y \in E \). Then \( {\mathcal T}^m(E)=\{(y,\xi)\in E\times{\mathcal P}^* : \xi(T^m_yF)=0\;\text{ for\;all}\;F\in C^m({\mathbb R}^n)\;\text{ such\;that}\;F_{| E}=0\} \). The statements obtained by the authors are, in fact, equivalent to the solution of Whitney’s problem given by C. Fefferman [Ann. Math. (2) 164, No. 1, 313–359 (2006; Zbl 1109.58016)]. A key step consists in showing that the “Glaeser refinements” of Fefferman are dual to certain “Glaeser operations” in the sense of the authors loc. cit.

Reviewer: Vincent Thilliez (Villeneuve d’Ascq)

MSC:

58C25	Differentiable maps on manifolds
26B05	Continuity and differentiation questions
26B99	Functions of several variables

Keywords:

extension problems; differentiable functions; Whitney jets; paratangent spaces

Citations:

Zbl 1031.58002; Zbl 1109.58016