= Solution
If $\phi\in\mathcal D(\omega)$, let $\widetilde\phi$ be its extension by zero to $\Omega$. Its <compact support> is contained in $\omega$, so this extension remains a <test function>. The <restriction of a distribution> is
$$
\boxed{\langle u|_\omega,\phi\rangle=\langle u,\widetilde\phi\rangle.}
$$
Extension by zero preserves convergence in the <space of test functions>, so this defines a <distribution> on $\omega$.
Choose a <partition of unity> $(\chi_j)$ subordinate to the neighborhoods $\omega_x$ from the hypothesis. For an arbitrary $\phi\in\mathcal D(\Omega)$, local finiteness and <compactness> of $\operatorname{supp}\phi$ give a finite decomposition $\phi=\sum_j\chi_j\phi$. Each term is a <test function> supported in one $\omega_x$. Thus
$$
\boxed{\langle v,\phi\rangle=\sum_j\langle v|_{\omega_{x(j)}},\chi_j\phi\rangle=0.}
$$
Since this holds for every <test function>, $v=0$. This is the <locality of distributions>.
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