TITLE: Morrey's inequality for Sobolev spaces of fractional order
QUESTION [5 upvotes]: Let $H^s(\mathbb T)$, where $s\in\mathbb R$,  be the space of $2\pi$-periodic functions (or distributions), $u(x)=\sum_{k\in\mathbb Z}\hat u_k\,\mathrm{e}^{ikx}$, 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 without any luck.

REPLY [2 votes]: The Sobolev space $H^s(\mathbb T)$ is $W^{s,2}(\mathbb T)$ and is also the Besov space
$B^{s}_{2,2}(\mathbb T)$. On the other hand the set of Hölder continuous functions with index $\alpha\in (0,1)$ is the Besov space
$B^{\alpha}_{\infty,\infty}(\mathbb T)$. For $s\in(1/2,3/2),$ you have the following continuous inclusion,
$$
B^{s}_{2,2}\subset B^{s-\frac12}_{\infty,\infty},
$$
which is a classical result, whose proof is immediate using the Littlewood-Paley decomposition: if $(\phi_\nu)_{\nu\in \mathbb N}$ is such a decomposition (here $\phi_\nu(\xi)=\phi(\xi 2^{-\nu})$ so that $1=\sum_\nu \phi_\nu(\xi)$, with $\phi\in C^\infty_c)$, the norm of a function  $u\in B^{s}_{2,2}(\mathbb T)$ is
$$
\bigl(\sum_{\nu\in \mathbb N}2^{2\nu s}\Vert{\phi_\nu(D)u}\Vert_{L^2}^2\bigr)^{1/2}
$$
whereas the norm in $B^{s-\frac12}_{\infty,\infty}$
is
$
\sup_{\nu\in \mathbb N}2^{\nu (s-\frac12)}\Vert{\phi_\nu(D)u}\Vert_{L^\infty}
$.
We have with $\psi\in C^\infty_c$ equal to 1 on the support of $\phi$,
$$
2^{\nu (s-\frac12)}\Vert{\phi_\nu(D)u}\Vert_{L^\infty}
=2^{\nu (s-\frac12)}\Vert{2^\nu\hat\psi(2^\nu \cdot)\ast \phi_\nu(D)u}\Vert_{L^\infty}\\
\lesssim
2^{\nu (s-\frac12)}\Vert2^\nu\hat\psi(2^\nu \cdot)\Vert_{L^2}
\Vert{\phi_\nu(D)u}\Vert_{L^2}\lesssim 2^{\nu s}\Vert{\phi_\nu(D)u}\Vert_{L^2}
$$
so that the following continuous injections hold
$$B^{s}_{2,2}\subset
B^{s}_{2,\infty}\subset B^{s-\frac12}_{\infty,\infty}.
$$