\({\mathcal C}^m\)-norms on finite sets and \({\mathcal C}^m\) extension
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)] \textit{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 [\textit{E. Bierstone, P. D. Milman}, and \textit{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 \( S \).
scientific article; zbMATH DE number 5145497
scientific article