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

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 $C_{\text{conv}}^{1,\alpha }(X)$.

A misprint in the proof of Lemma 4.14 has been corrected