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Here I am trying to prove the COMPACT embedding for the case $p>N$. The exercise shows that for $p>N$, then $W^{1,p}(\Omega)$ is compact embedded in $C^{0,\alpha}(\bar{\Omega})$ where $\alpha<1-N/p$. (Here $\Omega$ is bounded and has nice boundary). I tried it but it looks to me this is not a quick prove. (Yes, to show $W^{1,p}$ is compact embedded in $C^0(\bar{\Omega})$ is really quick.)

So I was wondering, maybe this is a theorem in some textbooks? If yes, could somebody provide me the name? Or if you know the prove, could you point out the key step? The difficulties I had in my prove is how to obtain strong convergence in Holder space...

Also, if you know a book who has some details treatment in Holder space, please let me know as well! The book I have, like Leoni, Evans, Adams, and Brezis does not treat Holder space in details, they rather just give the definition and leave some exercises..

Thx!

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First, embed into $C^{0,1-N/p}$ (Sobolev-Morrey); then use compact embedding between Hölder spaces.

Yes, Hölder spaces don't get a very detailed treatment in PDE books (or any book I know, for that matter). The properties are generally bad: nonseparable, nonreflexive... not much to work with. As a reference, I suggest Chapter 3 of the book Lectures on Elliptic and Parabolic Equations in Hölder Spaces by Krylov.

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