The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets
type: Item
instance of: scholarly article
title: The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets
zbMATH Open document ID: 1309.46011
DOI: 10.4171/RMI/801
author: Garving K. Luli
author: Charles Fefferman
published in: Revista Matemática Iberoamericana
publication date: 2014-10-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.2449
review text: The so-called Brenner-Hochster-Kollár problem about vector-valued continuous solutions of finite systems of linear equations was solved by \textit{C. Fefferman} and \textit{J. Kollár} [Dev. Math. 28, 233--282 (2013; Zbl 1263.15003)]. In the paper under review, the analytic method to settle this problem is extended to solve it for the space \(C^m(\mathbb{R}^n)\) of real-valued functions whose derivatives up to order \(m\) are continuous and bounded in \(\mathbb{R}^n\), and for the space \(C^{m,\omega}(\mathbb{R}^n)\) of real-valued functions whose \(m\)-th derivatives have modulus of continuity \(\omega\). These cases had been left open. For \(C^m(\mathbb{R}^n)\), the authors solve a more general problem that includes also a general Whitney extension problem for vector-valued functions and jets. The more general problem is formulated in terms of sections for bundles. A variant of this general problem for \(C^{m,\omega}(\mathbb{R}^n,\mathbb{R}^d)\) is also solved by means of a theorem which is a type of ``finiteness principle''. The dependence of a certain ``finiteness constant'' with respect to \(m\), \(n\) and \(d\) is also discussed. This paper continues recent, deep work by Bierstone-Milman-Pawlucki, Brudnyi-Shvartsman, Fefferman-Kollár, among others.
reviewed by: José Bonet
Mathematics Subject Classification ID: 46E10
Mathematics Subject Classification ID: 26B99
Mathematics Subject Classification ID: 14Q99
zbMATH DE Number: 6353611
zbMATH Keywords: generalized Whitney problems
zbMATH Keywords: extending vector-valued functions
zbMATH Keywords: continuous closures
zbMATH Keywords: differentiable closures
zbMATH Keywords: Brenner-Hochster-Kollár problem
MaRDI profile type: Publication
arXiv ID: 1209.2449
cites work: \({\mathcal C}^m\)-norms on finite sets and \({\mathcal C}^m\) extension
cites work: Differentiable functions defined in closed sets. A problem of Whitney
cites work: Q4317270
cites work: The Whitney problem of existence of a linear extension operator
cites work: Whitney's extension problem for multivariate \(C^{1,\omega}\)-functions
cites work: Continuous closure, axes closure, and natural closure
OpenAlex ID: W2963563717
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description: scientific article; zbMATH DE number 6353611