2024/03/21 by Michal Johanis, Johanis, Michal, Luděk Zaj́ıček +1 · 2 citations
Computer Science · Mathematics · #46G05 #46T20 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2403.14317
openalex publication_date 2024/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Our note is a complement to recent articles \citeJS1 (2011) and \citeJS2 (2013) by M. Jiménez-Sevilla and L. Sánchez-González which generalise (the basic statement of) the classical Whitney extension theorem for C1-smooth real functions on \mathbb Rn to the case of real functions on X (\citeJS1) and to the case of mappings from X to Y (\citeJS2) for some Banach spaces X and Y. Since the proof from \citeJS2 contains a serious flaw, we supply a different more transparent detailed proof under (probably) slightly stronger assumptions on X and Y. Our proof gives also extensions results from special sets (e.g. Lipschitz submanifolds or closed convex bodies) under substantially weaker assumptions on X and Y. Further, we observe that the mapping F∈ C1(X;Y) which extends f given on a closed set A⊂ X can be, in some cases, C^∞-smooth (or Ck-smooth with k>1) on X∖ A. Of course, also this improved result is weaker than Whitney's result (for X=\mathbb Rn, Y=\mathbb R) which asserts that F is even analytic on X∖ A. Further, following another Whitney's article and using the above results, we prove results on extensions of C1-smooth mappings from open ("weakly") quasiconvex subsets of X. Following the above mentioned articles we also consider the question concerning the Lipschitz constant of F if f is a Lipschitz mapping.