Fractional Sobolev Embedding Theorem

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

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

For each $s \in \R$ and $p \in \closedint 1 \infty$, let:
 * $\map {W^{s, p} } {\R^n} = \set {u \in S': \sequence \xi^s \hat u \in \map {L^p} {\R^n} }$

where:
 * $\sequence \xi = \paren {1 + \size \xi^2}^\frac 1 2$

Then:
 * $(1): \quad$ If $s > t$ then $\map {W^{s, p} } {\R^n}$ embeds continuously into $\map {W^{t, q} } {\R^n}$ where $q$ is given by $\dfrac 1 q = \dfrac 1 p - \dfrac {s - t} n$.