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We provide necessary and sufficient conditions for a $1$-jet $(f,G):E\rightarrow \mathbb{R} \times X$ to admit an extension $(F, \nabla F)$ forsome $F\in C^{1, \omega}(X)$. Here $E$ stands for an arbitrary subset of aHilbert space $X$ and $\omega$ is a modulus of continuity. As a corollary, inthe particular case $X=\mathbb{R}^n$, we obtain an extension (nonlinear)operator whose norm does not depend on the dimension $n$. Furthermore, weconstruct extensions $(F, \nabla F)$ in such a way that: (1) the (nonlinear)operator $(f, G)\mapsto (F, \nabla F)$ is bounded with respect to a naturalseminorm arising from the constants in the given condition for extension (andthe bounds we obtain are almost sharp); (2) $F$ is given by an explicitformula; (3) $(F, \nabla F)$ depend continuously on the given data $(f, G)$;(4) if $f$ is bounded (resp. if $G$ is bounded) then so is $F$ (resp. $F$ isLipschitz). We also provide similar results on superreflexive Banach spaces.

$C^{1,\omega}$ extension formulas for $1$-jets on Hilbert spaces