Suppose to have a sufficiently regular domain $\Omega\subseteq\mathbb{R}^d$. I know that, for $s\in\mathbb{R}_+$, the space $H^{-s}(\Omega)$ is defined as the dual of $H^s_0(\Omega)$, endowed with the norm $$\|f\|_{H^{-s}(\Omega)}:=\sup\{\langle f,g\rangle : g \in H^s_0(\Omega), \;\|g\|_{H^{s}(\Omega)}\leq 1\}.$$ Is there a more practical characterization of the $H^{-s}(\Omega)$-norm?
Roughly speaking, suppose $\Omega=(0,1)$ and $f(x)=x^2$. How do I compute, e.g., $\|f\|_{H^{-1/2}(\Omega)}$?
I would like to exploit the characterization based on the Fourier Transform or on the Fourier Series, but how to deal with the fact that we are not in the case $\Omega=\mathbb{R}^d$ or $\Omega=(\mathbb{R}/\mathbb{Z})^d$?
I think a way to define $H^{s}$ for negative $s$ is as follows:
$$ \mu\in H^{s}\leftrightarrow (1+|\xi|^{2})^{s}(F\mu)(\xi)\in L^{2} $$ and you define the norm on the domain $\Omega$ by restriction map. For detail see the Wikipedia article. But it does not seem to me that this norm is explicitly computable in most cases. For example, the Fourier transform for most non-Schwartz functions has to be carried out in the framework of tempered distributions, and even some relatively simple tempered distributions' Fourier transform are often tricky to compute, not to say evaluating its norm.
I think there is another way to define $H^{s}$ by $$ \mu\in H^{s}\leftrightarrow A\mu\in L^{2},\forall A\in \Psi^{s}_{X} $$ where $\Psi^{s}_{X}$ is a class of pseudo-differential operator depend on your choice for the regularity of $\Omega$. This is used in index theory on manifolds with boundary a lot. Of course in the case $\Omega=\mathbb{R}^{N}$ the two definitions are identical, but I think for abstract Riemannian manifolds the second definition might be more natural, though of course even harder to evaluate.