On $C^1$ Whitney extension theorem in Banach spaces

Abstract

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jiménez-Sevilla and L. Sánchez-González which generalise (the basic statement of) the classical Whitney extension theorem for $C^1$-smooth real functions on ${\mathbb{R}}^n$ to the case of real functions on $X$ (\cite{JS1}) and to the case of mappings from $X$ to $Y$ (\cite{JS2}) for some Banach spaces $X$ and $Y$. Since the proof from \cite{JS2} 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 $F\in C^1(X;Y)$ which extends $f$ given on a closed set $A\subset X$ can be, in some cases, $C^{\infty }$-smooth (or $C^k$-smooth with $k>1$) on $X\setminus A$. Of course, also this improved result is weaker than Whitney's result (for $X={\mathbb{R}}^n$, $Y=\mathbb{R}$) which asserts that $F$ is even analytic on $X\setminus A$. Further, following another Whitney's article and using the above results, we prove results on extensions of $C^1$-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.

There has been some progress concerning the (non)validity of the $C^k$ Whitney extension theorem for $k\ge 3$, which makes Section 6 from the previous version obsolete. The section was removed and the still relevant parts will appear in another article