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Let $\Omega \subseteq \mathbb{R}^d$ be open and $D\subseteq \partial\Omega$be a closed part of its boundary. Under very mild assumptions on $\Omega$, weconstruct a bounded Sobolev extension operator for the Sobolev space$\mathrm{W}^{k , p}_D (\Omega)$, $1 \leq p < \infty$, which consists of allfunctions in $\mathrm{W}^{k , p} (\Omega)$ that vanish in a suitable sense on$D$. In contrast to earlier work, this construction is global and \emph{not}using a localization argument, which allows to work with a boundary regularitythat is sharp at the interface dividing $D$ and $\partial \Omega \setminus D$.Moreover, we provide homogeneous and local estimates for the extensionoperator. Also, we treat the case of Lipschitz function spaces with a vanishingtrace condition on $D$.

Extendability of functions with partially vanishing trace