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

TL;DR

This paper establishes necessary and sufficient conditions for extending vector-valued functions defined on closed subsets of Banach spaces to globally smooth functions, covering various specific Banach space configurations.

Contribution

It provides a comprehensive characterization of when such smooth extensions exist across different Banach space settings.

Findings

Characterization for Hilbert spaces with separable domain

Conditions for Banach spaces with separable duals and Lipschitz retract targets

Extension criteria for $L_2$ to $L_p$ and vice versa

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 $2<p<\infty$.