Fractional Sobolev Embedding Theorem

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

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

For each $s \in \R$ and $p \in [1, \infty]$ let $W^{s,p} \left({\R^n}\right) = \left\{{u \in S': \langle \xi \rangle^s \hat{u} \in L^p \left({\R^n}\right)}\right\}$ where $\displaystyle \left \langle{\xi}\right \rangle = \left({1 + \left|{\xi}\right|^2}\right)^\frac 1 2$.

Then


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