is there an intermediate functional space $Y$ such that the Sobolev space $W^{1,N}$ compactly embeds in $Y$ and $Y$ continuously embeds in $L^N$, where $N$ is the ambient space dimension? Thank you
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2$\begingroup$ I am not quite sure to understand the question. Are you talking about the Sobolev/Lebesgue spaces on the whole $R^N$ or simply on some bounded domain? $\endgroup$– Delio MugnoloCommented Oct 14, 2013 at 12:54
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$\begingroup$ Oh sorry, on some bounded domain. $\endgroup$– FluxCommented Oct 14, 2013 at 13:15
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$\begingroup$ The question is too vague. Why can't you just take $Y=L^{N}(\Omega)$ ? $\endgroup$– Piero D'AnconaCommented Oct 15, 2013 at 8:17
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1 Answer
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well, if you are interested in open bounded domains, then it is trivial: $W^{1,2}(\Omega)$ is compactly embedded in any Sobolev space of real order $W^{s,2}(\Omega)$, which is continuously (in fact, even compactly) embedded in $L^2(\Omega)$, whenever $s\in (0,1)$ (for $\Omega\subset \mathbb R^2$, but not only).
if you need a reference, see e.g. thm. 1.4.3.2 in grisvard's elliptic problems in nonsmooth domains, where only lipschitz boundary of $\Omega$ is assumed.
