Fractional Sobolev Embedding Theorem

Definitions
Let $S'$ denote the space of tempered distributions.

Let $\mathcal F : S' \to S'$ denote the Fourier transform.

Let $W^{s,p}(\R^n) = \{ u \in S' \ \mid {\mathcal F}^{-1} \langle \xi \rangle ^ s \mathcal F u \in L^p(\R^n) \}$ where $\langle \xi \rangle = ( 1 + |\xi|^2 ) ^ \frac 1 2$.

Theorem
1. If $s > t$ then $W^{s,p}(\R^n)$ embeds continuously into $W^{t,q}(\R^n)$ where $q$ is given by $\frac 1 q = \frac 1 p - \frac {s-t} n$.