# Is the Sobolev embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C_0(\mathbb{R}^d)$ compact?

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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The standard counterexample, in those things, is made up by translating a bump function in space. This should disprove compactness of that embedding. –  Giuseppe Negro Feb 28 '13 at 23:34
Either the author of that paper made a mistake, or used strange notation. Sobolev embedding yields a compact embedding $W^{l,2}(\mathbb{R}^d) \rightarrow C^m(\mathbb{R}^d)$, where $m < n - l/2$. Perhaps the author takes $m = 0$ and uses a subscript where there is normally a superscript? –  Christopher A. Wong Mar 1 '13 at 0:19
Thanks a lot for that answer, would you maybe have a reference about what you said? PS: no, it's not a strange notation because he says that $C_0$ is Banach, so he really doesn't mean $"C"$ –  user44670 Mar 1 '13 at 0:23
Yeah, I mean $d$. See any textbook on partial differential equations, such as Partial Differential Equations by Evans, or Elliptic partial differential equations of second order by Gilbarg and Trudinger. You should probably just be able to find the result by googling the Sobolev embedding theorem. –  Christopher A. Wong Mar 1 '13 at 0:27
@GiuseppeNegro: why not post your first comment as an answer? It answers the question explicitly asked by the OP. –  Willie Wong Mar 1 '13 at 0:56

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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