## 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.

### MSC:

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

### Citations:

Zbl 1031.58002; Zbl 1109.58016
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