So we have the whole set of theory for Sobolev spaces \begin{equation} H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\}, \end{equation} and we know that they are the same as \begin{equation} W^{s,2}=\{u:D^{\alpha}u\in\mathcal{L}^2(\mathbb{R}^d)\text{ for all} |\alpha|\le s\} \end{equation} when $s\in\mathbb{N}$.
We also know that $H_s$ is useful when $s$ is a negative integer since it can be identified as the dual space of $H_{-s}$.
But what is the use of $H_s$ when $s$ is not an integer?
Thanks!
The fractional Sobolev spaces are important when formulating elliptic boundary value problems in Sobolev spaces.
Consider the Dirichlet problem for the Poisson equation. That is, you are interested in a solution $u$ to $\Delta u = g$ on a bounded open domain $\Omega \subset \mathbb{R}^n$ that satisfies a given boundary condition $u | \partial \Omega = f$ for some $f$ that is defined on $\partial{\Omega}$.
How do you formulate "boundary" conditions when functions in Sobolev spaces aren't necessarily even continuous? The boundary $\partial \Omega$ of (a sufficiently nice open subset) $\Omega$ is of measure zero and functions in $H^k(\Omega)$ are defined a priori only a.e, and changing them on a measure zero subset doesn't affect them.
Still, if $\partial \Omega$ is nice enough (say, a $C^k$ manifold), and if $k \geq 1$ is an integer, one can show that there are trace operators $T : H^k(\Omega) \rightarrow L^2(\partial \Omega)$, that extend continuously the usual restriction map $f \mapsto f| \partial \Omega$ on $C^{\infty}(\Omega) \cap H^k(\Omega)$. What is the image of $T$? That is, what are all the possible "boundary values" of a function in $H^k(\Omega)$? It turns out to be precisely $H^{k-\frac{1}{2}}(\partial \Omega)$!
Using this, one can formulate and prove uniqueness and existence results for solutions of PDEs in Sobolev spaces. For example, one has that the map $$ u \mapsto (\Delta u, u|_{\partial \Omega}) = (\Delta u, Tu) $$ is an isomorphism of $H^1(\Omega) \rightarrow H^{-1}(\Omega) \times H^{\frac{1}{2}}(\Omega)$, and so, given $g \in H^{-1}(\Omega)$ and $f \in H^{\frac{1}{2}}(\Omega)$, the Dirichlet problem for the Poisson equation $\Delta u = g$, $u|_{\partial \Omega} = f$ has a unique solution in $H^1(\Omega)$.
There are also higher order trace operators, which correspond to normal derivatives of various orders, whose image again lies in fractional Sobolev spaces.