About extension on Sobolev space

Question

Let $\Omega\subset\mathbb{R}^n$ be open connected with smooth boundary. Let $\zeta$ be a zero extension operator : $\forall u\in W^{k,p}(\Omega)$, $$\zeta u := \Big\{\begin{array}uu \quad\text{on $\Omega$},\\0 \quad\text{otherwise.}\end{array}$$ The question is, for which $k$, $\zeta(W^{k,p}(\Omega))\subset W^{k,p}(\mathbb{R}^n)$? Here $\Omega$ is not assumed to be bounded.