Let $H^s(\mathbb T)$, where $s\in\mathbb R$,  be the space of $2\pi$-periodic functions, $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, such that 
$$
\|u\|_{H^s}^2=\sum_{k\in\mathbb Z}(1+k^2)^{s}\lvert \hat u_k\rvert^2<\infty.
$$
Assume now that $s\in \big(\frac{1}{2},\frac{3}{2}\big)$. I wish to find out whether there exists a constant $c=c_s$, such that
$$
\lvert u(x)-u(y)\rvert\le c \lvert u\rvert_{H^s}\lvert x-y\rvert^{s-\frac{1}{2}},
$$
where 
$$
\lvert u \rvert_{H^s}^2=\sum_{k\in\mathbb Z}\lvert k\rvert^{2s}\lvert \hat u_k\rvert^2.
$$

If $s=1$, then this is Morrey's inequality.

Any reference?

Note. I have asked this question in [math.stackexchange][1] without any luck.


  [1]: http://math.stackexchange.com/questions/1821243/morreys-inequality-for-sobolev-spaces-of-fractional-order