Assume is continuous and satisfies the following two conditions
in for some
and the function
Let us define the following space
On , we can use the following norm
In this note, we prove that is compactly embedded in for all . This is Lemma 2.4 in a paper by Y. Yang published in JFA this 2012 although the technique used is standard.
Indeed, by the first condition on , the standard Sobolev embedding theorem implies that the following embedding
is obviously continuous for any . It follows from the second condition on and the Holder inequality that
For any , there holds
Thus, we get continuous embedding
for all . To prove that the above embedding is also compact, we follow definition. Take a sequence of functions such that for all , we must prove that up to a subsequence, there exists some such that converges to strongly in for all .
Since is reflexive, without loss of generality, we may assume that
weakly in and strongly in for all .
In view of the second condition on , for any , there exists such that
Note that this depends on the norm of . On the other hand, since strongly in , we get that
Since is arbitrary, we obtain
For , it follows from the continuous embedding , , that