I am studying Sobolev spaces and have a couple of question to those.
It is known that the space
$H^s(\mathbb{R}^n):=\lbrace f\in L^2(\mathbb{R}^n) : (1+|\xi|^2)^s |(\mathcal{F}f)(\xi)|^2 \in L^2(\mathbb{R}^d)\rbrace$
coincides with $W^{k,p}(\mathbb{R}^d)$ $(k\in \mathbb{N})$ when ever $s=k$ and $p=2$.
Here $W^{k,p}(\mathbb{R}^d)$ is the space of functions in $L^2(\mathbb{R})$ for which all $m^{\text{th}}$ weak derivative exists for $m=1,\ldots, k$ and are in $L^p(\mathbb{R}^n)$.
$\mathcal{F}$ is the Fourriertransform on $L^2(\mathbb{R}^n)$, which one can define by Plancherels theorem.
Now assume $\Omega \subseteq \mathbb{R}^n$ $(n\in \mathbb{N})$ is open and $k\in \mathbb{N}$, then
${W}^{k,p}_0(\Omega)$ is defined as the closure of the space $C_0^\infty(\Omega)$ wrt. to the known Sobolev norm
$\|f\|_{k,p,\Omega}^p:=\sum\limits_{|\alpha|\leq k} \|\partial^k f\|^p_{L^p(\Omega)}$
Where
$C^\infty_0(\Omega):=\lbrace f\in C^\infty(\Omega) : f\text{ is compactly supported and }\mathrm{supp}(f)\subseteq \Omega\rbrace$.
But in order to define $H_0^s(\Omega)$ for $s>0$ and $s\not\in \mathbb{N}$ one defines $H_0^s(\Omega)$ as the closure of
$C^\infty_0(\Omega):=\lbrace f\in C^\infty(\mathbb{R}^n) : f\text{ is compactly supported and }\mathrm{supp}(f)\subseteq \Omega\rbrace$
wrt. to the norm $\|\cdot\|_s:=\int\limits_{\mathbb{R}^n} (1+|\xi|^2)^s |\mathcal{F}(\cdot)(\xi)|^2 d\xi$
because the Fourriertransform only eats functions whose domain is $\mathbb{R}^d$.
In which way does $H^k_0(\Omega)$ then still coincides with $W^{k,2}_0(\Omega)$ ($k\in \mathbb{N}$), especially are they equal?
The only way that comes into my mind, is identifying functions in $W^{k,2}_0(\Omega)$ with their zero extention on $\Omega$ ($k\in\mathbb{N}$) if they are equal. But that would imply that functions in $H^s_0(\Omega)$, at least for non-integer $s$, are zero outside $\Omega$ i.e. $u=\chi_\Omega u$ for all $u\in H_0^s(\Omega)$. Then naturally I question if this also hold for non integer $s$, so if $\chi_\Omega u=u$ for $u\in H_0^s(\Omega)$ $s>0$? Note that $H_0^s(\Omega)$ for $s>0$ non-integers have domain $\mathbb{R}^n$.