I am interested in a Gagliardo-Nirenberg type inequality for functions in the space $$H^1_T(\mathbb{R}^n)=\lbrace \psi=u+iv\in H^1_{loc}(\mathbb{R}^n,\mathbb{C}):\psi(x)=\psi(x_1+T,...,x_n+T)\rbrace$$ The usual norm of this space is $$ \Vert \psi \Vert^2=\int_Q|\psi|^2\ dx \ +\ \int_Q|\nabla\psi|^2\ dx $$ where $Q=[0,T]^n$. Please, let me know where can I find these inequalities in case they exist. Thanks in advance!

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    $\begingroup$ It seems to me that after a change of coordinates, this is equivalent to asking about functions on the cylinder $S^1 \times \mathbb{R}^{n-1}$. $\endgroup$ May 21 '19 at 17:07
  • $\begingroup$ Can you give me some references about that? $\endgroup$ May 21 '19 at 20:31
  • $\begingroup$ Have you tried adapting Gagliardo’s proof? $\endgroup$
    – Deane Yang
    May 21 '19 at 21:01
  • $\begingroup$ No, I haven't. I thought that this inequality I am looking for was known and easy to find on internet or books. But it seems that I'm wrong. Can you recommend me some book to read Gagliardo's proof? $\endgroup$ May 21 '19 at 21:30
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    $\begingroup$ Is your question about functions that are separately periodic in each of its coordinates, or functions that are only periodic if you add $T$ to all coordinates at the same time? The results are different in those cases. In the fully periodic case, the classical GN inequality is false due to constant functions. In the partially periodic case, I am not sure whether the correct scaling is the same as the flat case. $\endgroup$ May 21 at 15:10

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