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In p. 508 of the paper : http://www.jstor.org/stable/2243484 , it is mentioned that if $2l \geq d$, the embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ is compact, where $W^{l,2}(\mathbb{R}^d)$ is the $(l,2)-$Sobolev space on $\mathbb{R}^d$ and $C_0(\mathbb{R}^d)$ is the space of continuous functions $\mathbb{R}^d \rightarrow \mathbb{R}$ vanishing at infinity. I have tried to look in many references but haven't found this. So is it true or not? |
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It is not true, I think that there are two errors. The first is that you do not have the embedding $$\tag{1}W^{l,p}(\mathbb{R}^d)\subset C_o(\mathbb{R}^d)$$ in the critical case $lp=d$. You can find more information about this in Evans' book on PDE, 2nd edition, pag. 280 "The borderline case $p=n$". The second error is that the embedding (1), which holds when $lp>d$, is not compact. To wit, fix a function $\phi\in C^{\infty}_c(\mathbb{R}^d)$ and a unit vector $u\in \mathbb{S}^{d-1}$. Define $$\phi_n(x)=\phi(x-nu).$$ This is a bounded sequence in $W^{l,p}(\mathbb{R}^d)$ which does not have any uniformly convergent subsequence, meaning that the embedding (1) is not compact. |
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