On C1 Whitney extension theorem in Banach spaces

Abstract

Our note is a complement to recent articles [17] (2011) and [18] (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 $R^n$ to the case of real functions on X ([17]) and to the case of mappings from X to Y ([18]) for some Banach spaces X and Y. Since the proof from [18] 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 $X=R^n$, $Y=R$) 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 [17], [18] we also consider the question concerning the Lipschitz constant of F if f is a Lipschitz mapping.