What exactly is the $H^{-1/2}$ space? What exactly is the $H^{-1/2}$ space?
Definition for $H^{1/2}$ given e.g. here.

$Y = \widehat{H^{1/2}(\mathbb{R})}$ is the Hilbert space of functions $f \in L^2(\mathbb{R})$ with the norm $$\|f\|_Y^2 = \int_{-\infty}^\infty (1+|x|) |f(x)|^2dx$$ ie. the inner product $$\langle f, g \rangle_Y = \int_{-\infty}^\infty (1+|x|) f(x)\overline{g(x)}dx$$ 
Its strong dual is $Y^*=\widehat{H^{1/2}(\mathbb{R})}^*$, the Hilbert space of functions with the norm $$\|g\|_{Y^*}^2 =\sup_{f \in Y} \frac{|\langle f, g \rangle_Y|^2}{\|f\|_Y^2}= \int_{-\infty}^\infty \frac{1}{1+|x|} |g(x)|^2dx$$ and inner product $$\langle g_1,g_2 \rangle_{Y^*} = \int_{-\infty}^\infty \frac{1}{1+|x|} g_1(x)\overline{g_2(x)}dx $$
Then $X=H^{1/2}(\mathbb{R})$ is the inverse Fourier transform of $Y$ and $X^*=H^{-1/2}(\mathbb{R})=H^{1/2}(\mathbb{R})^*$ is its strong dual, the inverse Fourier transform of $Y^*$.
We can also define those things in term of fractional derivatives (some self-adjoint and closed unbounded operators). 
Restricting the norm to $\Omega$ we obtain $H^{\pm 1/2} (\Omega)$.
