# $W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$?

How can I prove that $W^{s_1 , 2} (\Bbb R^n ) \hookrightarrow W^{s_2,4}(\Bbb R^n )$ if $s_1 > s_2 + n/4$ ? $W^{s,p}$ denotes a general Sobolev space for $s =0,1,2,\cdots$. The hook means a continuous embedding.

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Theorem. If the domain $\Omega \subset \mathbb{R}^n$ has the cone property, the following embeddings are continuous: $$W^{j+m,p}(\Omega) \subset W^{j,q}(\Omega) \quad \text{provided that p \leq q \leq \frac{np}{n-mp}}.$$

If you choose $p=2$ and $q=4$, the condition reads $2 \leq 4 \leq \frac{2n}{n-2m}$, which is satisfied when $n \geq 2m$ and $m \geq \frac{n}{4}$.

Bibliography: Proposition 1 in these notes.

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