  We construct whole-space extensions of functions in a fractional Sobolev
space of order $s\in (0,1)$ and integrability $p\in (0,\infty)$ on an open set
$O$ which vanish in a suitable sense on a portion $D$ of the boundary $\partial
O$ of $O$. The set $O$ is supposed to satisfy the so-called interior thickness
condition in $\partial O \setminus D$, which is much weaker than the global
interior thickness condition. The proof works by means of a reduction to the
case $D=\emptyset$ using a geometric construction.

Functional Analysis