On smooth extensions of vector-valued functions defined on closed subsets of Banach spaces

Abstract

Let $X$ and $Z$ be Banach spaces, $A$ a closed subset of $X$ and a mapping $f:A \to Z$. We give necessary and sufficient conditions to obtain a $C^1$ smooth mapping $F:X \to Z$ such that $F_{\mid_A}=f$, when either (i) $X$ and $Z$ are Hilbert spaces and $X$ is separable, or (ii) $X^*$ is separable and $Z$ is an absolute Lipschitz retract, or (iii) $X=L_2$ and $Z=L_p$ with $1<p<2$, or (iv) $X=L_p$ and $Z=L_2$ with $2<p<\infty$.