The Brenner-Hochster-Kollár and Whitney problems for vector-valued functions and jets
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.
scientific article; zbMATH DE number 6353611
scientific article