On Whitney-type extension theorems for $C^{1,+}$, $C^2$, $C^{2,+}$, and $C^3$-smooth mappings between Banach spaces

TL;DR

This paper investigates Whitney extension theorems for various smoothness classes of mappings between Banach spaces, showing that certain vector-valued extensions fail in infinite-dimensional settings, with implications for smooth function extension theory.

Contribution

The paper demonstrates the limitations of Whitney extension theorems for vector-valued functions in infinite-dimensional Banach spaces, extending and generalizing previous results.

Findings

Vector-valued Whitney extensions hold only in special cases like injective Banach spaces.

Extensions fail for mappings into 'somewhat Euclidean' infinite-dimensional spaces.

Negative results for $C^{2,+}$, $C^{2, ext{omega}}$, and $C^3$ smooth mappings.

Abstract

In 1973 J. C. Wells showed that a variant of the Whitney extension theorem holds for $C^{1,1}$-smooth real-valued functions on Hilbert spaces. In 2021 D. Azagra and C. Mudarra generalised this result to $C^{1,\omega}$-smooth functions on certain super-reflexive spaces. We show that while the vector-valued version of these results do hold in some rare cases (when the target space is an injective Banach space, e.g. $\ell_\infty$), it does not hold for mappings from infinite-dimensional spaces into "somewhat euclidean" spaces (e.g. infinite-dimensional spaces of a non-trivial type), and neither does the $C^2$-smooth variant. Further, we prove negative results concerning the real-valued $C^{2,+}$, $C^{2,\omega}$, and $C^3$-smooth versions generalising older results of J. C. Wells.