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

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.


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