I'm looking for some intuition for Sobolev spaces $H^{\frac{1}{2}}$ and $H^{-\frac{1}{2}}$. Any explanation I've seen is very technical, I'm looking for the most simple explanation possible that gives the motivation behind these spaces and why there are useful?
Once I understand that the technical explanations will be much easier to go through.
You probably know that the Fourier transform takes derivatives to polynomials, ie $F (\partial_i u) = \xi_i \cdot Fu$. So, if all $m$-th order derivatives are in $L^2$ you get that $(1 + |\xi|^2)^{m/2} Fu$ is in $L^2$. This provides an easy way to define Sobolev spaces for $\mathbb{R}^d$. One gets a natural generalization to arbitrary $m \in \mathbb{R}$. Applications of the $1/2$ Sobolev space is the trace theorem, which states that for any $s > \frac{1}{2}$ the restriction map $γ : \mathscr{S}(\mathbb{R^d}) \to \mathscr{S}(\mathbb{R^{d-1}})$ with $γ(u)(x') = u(x',0)$ extends to a map $γ : H^s(\mathbb{R^d}) \to H^{s-\frac 12}(\mathbb{R^{d-1}})$. Moreover this map is surjective. So, a function $v$ is in $H^{\frac 12}$ if there is a function $u \in H^1$ such that $γ u = v$.
Another example (not quite, but it gives intuition) is the delta distribution $δ(u) = u(0)$, which is in $H^{-d/2-ε}$ for all $ε > 0$.
Besides Sobolev spaces with real order are useful, because one can use interpolation theory.
Technically speaking we have to impose $u ∈ \mathscr{S}'$ so that the Fourier transform is defined.
A good reference is M. E. Taylor - Partial differential equations 1.