\({\mathcal C}^m\)-norms on finite sets and \({\mathcal C}^m\) extension. (English) Zbl 1116.58005

Bierstone, Edward; Milman, Pierre D.

Duke Math. J. 137, No. 1, 1-18 (2007).

In recent papers [Ann. Math. (2) 161, No. 1, 509–577 (2005; Zbl 1102.58005); Ann. Math. (2) 164, No. 1, 313–359 (2006; Zbl 1109.58016)] C. L. Fefferman has given criteria for a function defined on a compact subset \( E \) of \( {\mathbb R}^n \) to extend to a \( C^m \) or \( C^{m,\omega} \) function of \( {\mathbb R}^n \) (here, \( \omega \) denotes a regular modulus of continuity). These criteria rely on uniform bounds for the \( C^m \) or \( C^{m,\omega} \) norms of extensions from finite subsets \( S \) of \( E \) of cardinality at most a large integer \( k^\sharp \) depending only on \( m \) and \( n \). If \( d \) denotes the dimension of the vector space of real polynomials of degree at most \( m \) on \( {\mathbb R}^n \), Fefferman shows that one can take \( k^\sharp=(d+1)^{3\cdot 2^d} \) in the \( C^{m,\omega} \) case, and he gives a higher bound in the \( C^m \) case.
In the present article, the authors show that one can take \( k^\sharp=2^d \) in both cases. They also show that the geometric \( C^m \) paratangent bundle of \( E \) that plays a key role in the dual formulation of \( C^m \) extension criteria [E. Bierstone, P. D. Milman, and W. Pawłucki, Ann. Math. (2) 164, No. 1, 361–370 (2006; Zbl 1109.58015)] can be defined using limits of distributions supported on \( 2^{d-1} \) points. The proofs involve the combinatorics of clusterings of finite subsets \( S \), based on partitions \( S=\bigcup S_i \) where the distances between points of each \( S_i \) are comparable to the diameter of \( S_i \), and the distances between the subsets \( S_i \) are relatively large in comparison with their diameters, both properties being uniform with respect to \( S \).

Reviewer: Vincent Thilliez (Villeneuve d’Ascq)

MSC:

58C25	Differentiable maps on manifolds
26B05	Continuity and differentiation questions
26B35	Special properties of functions of several variables, Hölder conditions, etc.
58A20	Jets in global analysis

Keywords:

extension problems; differentiable functions; Whitney jets; paratangent spaces

Citations:

Zbl 1102.58005; Zbl 1109.58016; Zbl 1109.58015

References:

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