Extension zero of $W^{k,p}(\Omega)$ function

Question

Suppose $\Omega$ is an open set in $\mathbb{R}^N$ and $\omega$ compactly contained in $\Omega$. Suppose $u \in W^{k,p}(\omega)$ and supp $u$ compactly contained in $\Omega$.

I want to ask about the zero extension of $u$ on $\Omega$. Is it actually in $W^{k,p}(\Omega)$?

Thanks for your helps.