If $\varphi \in C_0^\infty(\mathbb{R}^n)$ and $\mathscr{F}$ is the Fourier transform on $L^2(\mathbb{R}^n)$ then one has
$\mathscr{F} (\partial_{\alpha} \varphi)=i^{|\alpha|} x^\alpha \mathscr{F}\varphi$
where $x^\alpha:=x_1^{\alpha_1}\ldots x_n^{\alpha_n}$, $|\alpha|:=|\alpha_1|+\ldots+|\alpha_n|$, $\alpha:=(\alpha_1,\ldots,\alpha_n) \in \mathbb{N}^n_0$ and $\partial^\alpha=\partial_{x_1}^{\alpha_1}\ldots \partial_{x_n}^{\alpha_n}$.
The same applies for the weak derivatives up to order $m$ for functions in $H^{m}(\mathbb{R}^m)$. But how is it for functions in $H^{m}(\Omega)$ or $H^{m}_0(\Omega)$ for any domain $\Omega \subseteq \mathbb{R}^n$?
Regarding $H_0^m$ one could argue that we just need to pass to the limit and use the continuity of the Fourier transform however functions in $H_0^m(\Omega)$ have domain $\Omega$ and not $\mathbb{R}^n$, which is the problem I face.