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

Michal Johanis, Luděk Zajíček

Published 2024-03-21Version 1

Our note is a complement to recent articles \cite{JS1} (2011) and \cite{JS2} (2013) by M. Jim\'enez-Sevilla and L. S\'anchez-Gonz\'alez 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 $F$ is $f$ is a Lipschitz mapping. We also present a new observation on the limitation of the possible validity of Whitney's extension theorem for $C^2$-smooth functions.