# Explicit formulas for $C^{1, 1}$ and $C^{1, \omega}_{\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces

## TL;DR

This paper provides explicit formulas for extending 1-jets to convex and $C^{1, ext{omega}}$ functions in Hilbert and superreflexive spaces, with optimal Lipschitz constants and necessary and sufficient conditions.

## Contribution

It introduces simple explicit formulas for $C^{1, ext{omega}}$ and $C^{1,1}$ extensions of 1-jets, including convex functions, in Hilbert and superreflexive spaces, with optimal bounds.

## Key findings

- Explicit formulas for $C^{1, ext{omega}}$ extensions in Hilbert spaces.
- Necessary and sufficient conditions for convex $C^{1, ext{omega}}$ extensions.
- Optimal Lipschitz constants for $C^{1,1}$ extensions of 1-jets.

## Abstract

Given $X$ a Hilbert space, $\omega$ a modulus of continuity, $E$ an arbitrary subset of $X$, and functions $f:E\to\mathbb{R}$, $G:E\to X$, we provide necessary and sufficient conditions for the jet $(f,G)$ to admit an extension $(F, \nabla F)$ with $F:X\to \mathbb{R}$ convex and of class $C^{1, \omega}(X)$, by means of a simple explicit formula. As a consequence of this result, if $\omega$ is linear, we show that a variant of this formula provides explicit $C^{1,1}$ extensions of general (not necessarily convex) $1$-jets satisfying the usual Whitney extension condition, with best possible Lipschitz constants of the gradients of the extensions. Finally, if $X$ is a superreflexive Banach space, we establish similar results for the classes $C^{1, \alpha}_{\textrm{conv}}(X)$.