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It is known that there exists a compactness results involving fractional sobolev spaces in bounded domain. What about unbounded domain? More precisely, Under which conditions, we can extend the compact embedding between the fractional sobolev space $W^{s,2}(D)$ and $L^{2}(D)$, with $s\in (0,1)$ to an unbounded domain $D$. Could you provied me by some references contain such conditions and examples. Thanks.

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To the best of my knowledge, the more advanced results in this field are in the last sections of the book of Adams and Fournier. Don't seem to be very easy to apply tough. Anyway, they introduce a class of domains they call "quasi-bounded" and show that quasi-boundedness is a necessary condition for compactness of the imbedding.

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