TITLE: Way to memorize relations between the Sobolev spaces?
QUESTION [46 upvotes]: Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel lost. Are there certain tricks to memorize the (continuous and compact) embeddings between the different $W^{k,p}(\Omega)$ or into $C^{r,\alpha}(\bar{\Omega})$ ?

REPLY [2 votes]: The way I remember is that the embedded space must have more smoothness, and the additional smoothness that is required gets larger when the difference in $1/p$ gets larger. Then you have to divide the smoothness by $n$, so there will be fraction lines everywhere. So a necessary condition for $W^{s,p}\hookrightarrow W^{r,q}$ is
$$
\frac{s-r}{n} \geq \frac1p - \frac1q.
$$