The Sobolev spaces with fractional order $H^s_0(\Omega)$ ($s>0,n\in\mathbb{N}, \Omega\subseteq \mathbb{R}^n$ open) can be defined as the closure of the test function space $C_0^\infty(\Omega)$ in $H^s(\Omega)$ with respect to the norm
$$\|u\|^2_{s}:= \|u\|_{H^m(\Omega)}^2+\sum\limits_{|\alpha|=m}\int\limits_{\Omega \times \Omega} \frac{|D^\alpha [u(x)-u(y)]|^2}{|x-y|^{n+2\sigma}} d(x,y)$$
where $m$ is an integer s.t. $\sigma=s-m\in(0,1)$.
If $\Omega=\mathbb{R}^n$ this norm is equivalent to this norm
$$\|u\|^2:= \int\limits_{\mathbb{R}^n} (1+|\xi|^{2s})|(\mathscr{F}u)(\xi)^2 |d\xi.$$
Where $\mathscr{F}$ is .... if you don't know what $\mathscr{F}$ is you can't help me anyway.
I think that if one would define $H^s_0(\Omega)$ as the closure of test functions $f:\mathbb{R}^n\rightarrow \mathbb{R}$ which are compactly supported with support in $\Omega$, wrt. the latter norm i.e. $\|u\|^2:= \int\limits_{\mathbb{R}^n} (1+|\xi|^{2s})|(\mathscr{F}u)(\xi)^2 |d\xi$. Then we would essentially obtain a space which is isometrically isomorphic to $\overline{C_0^\infty(\Omega)}^{\|\cdot\|_s}$, by obvious reasons. I think the norm via the Fourier transform allows one to relate these spaces easier to the ones with integer order, that's why I'm interested in this alternative definition.
But in the literature this is never mentioned. So I'm not sure if this is certainly true what I just said.
Im aware that this question is related to the one: comparing two Sobolev spaces:$W_0^{s,p}(\Omega)$ and $\overline{ C_c^{\infty}(\Omega)}^{~~W^{s,p}(\Bbb R^d)}$
But the answer given there does not answers my question. As the answer there says they are the same if $\Omega$ is Lipschitz and $0<s<1/2$. I think they are always the same in the above case as the extention by zero on $\mathbb{R}^d\setminus \Omega$ provides an isometric isomorphism between these spaces.
