Explicit formulas for $C^{1, 1}$ and $C^{1, \omega}_{\textrm{conv}}$ 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 $C^{1, \alpha}_{\textrm{conv}}(X)$.
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https://hal.archives-ouvertes.fr/hal-01579253
Contributor : Marie-Annick Guillemer <>
Submitted on : Wednesday, August 30, 2017 - 5:20:41 PM
Last modification on : Friday, November 16, 2018 - 1:34:23 AM

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Daniel Azagra, Erwan Le Gruyer, Carlos Mudarra. Explicit formulas for $C^{1, 1}$ and $C^{1, \omega}_{\textrm{conv}}$ extensions of $1$-jets in Hilbert and superreflexive spaces. Journal of Functional Analysis, Elsevier, 2018, 274 (10), pp.3003-3032. ⟨10.1016/j.jfa.2017.12.007⟩. ⟨hal-01579253⟩

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