of distributional solutions to partial differential equations by asking when such a solution is an honest function; this is the subject of the embedding theorems. We summarise the following useful properties of the $H^s$ spaces, the first of which is a relevant embedding theorem:

\[Sobolev prop\] The Sobolev spaces $H^s({\mathbb{R}}^n)$ have the following properties: $$\begin{array}{cl}
i) & \textrm{Let $k\in \{ 0\} \cup{\mathbb{N}}$ and $s\in{\mathbb{R}}$ satisfy $s>k+n/2$ then
} H^s({\mathbb{R}}^n) \subset C^k({\mathbb{R}}^n) \\ & \text{
is a continuous embedding} \\
ii) & H^s({\mathbb{R}}^n) \subset H^{s^\prime}({\mathbb{R}}^n) \, \forall s\geq s^\prime; \\
iii) & \textrm{if } u \in H^s({\mathbb{R}}^n), f \in C^k({\mathbb{R}}^n) \textit{ and } D^\alpha f  \in
L^\infty({\mathbb{R}}^n) \, \forall |\alpha|\leq k, \textit{ where $D$ is a} \\
& \textit{partial derivative operator and $\alpha$ is a multi-index, then for all } |s|\leq
k \\ &  
u \mapsto fu \textit{ is a bounded linear map of $H^s({\mathbb{R}}^n)$ into $H^s({\mathbb{R}}^n)$. In particular,} \\
& \textit{$H^s({\mathbb{R}}^n)$ is closed under multiplication by smooth functions.}
\end{array}$$

The concept of the Sobolev wave-front set will give a concise way of saying what it means for a distribution to microlocally fail to be an element of a Sobolev space. For convenience, we adopt the notation that $\dot{T}^*{\mathbb{R}}^n$ (similarly $\dot{T}{\mathbb{R}}^n$, 