Morrey's inequality for Sobolev spaces of fractional order 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.
 A: I have produced an answer, not the most elegant one. 
Let $x\ne y$, then we have that
$$
\lvert u(x)-u(y)\rvert = \left|\sum_{k\in\mathbb Z}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|\le
\left|\sum_{\lvert k\rvert \le |x-y|^{-1}}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|+\left|
\sum_{\lvert k\rvert \ge |x-y|^{-1}}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|.
$$
We shall exploit the fact that 
$$
\big|\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big|\le
\min\big\{\lvert k\rvert\lvert x-y\rvert,2\big\}.
$$
For the first term we have two cases:
Case I. $s \le 1$,
$$
\left|\sum_{\lvert k\rvert \le |x-y|^{-1}}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right| \le
\sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert \hat u_k\rvert \lvert k\rvert \lvert x-y\rvert \\ =
\lvert x-y\rvert \sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert \hat u_k\rvert \lvert k\rvert^s \lvert k\rvert^{1-s}  \\ \le 
\lvert x-y\rvert \,
\left(\sum_{\lvert k\rvert \le |x-y|^{-1}}\lvert k\rvert^{2-2s}\right)^{1/2}
\left(\sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert k\rvert^{2s}\lvert \hat u_k\rvert^2 \right)^{1/2}\\
\lesssim \lvert x-y\rvert \, \|u\|_{H^s} 
\left(\frac{2}{\lvert x-y\rvert^{3-2s}}\right)^{1/2} \\
=2^{1/2}\|u\|_{H^s}\lvert x-y\rvert^{s-1/2}
$$
Case II. $1<s<3/2$. We have
$$
\left|\sum_{\lvert k\rvert \le |x-y|^{-1}}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right| \le
\sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert \hat u_k\rvert \lvert k\rvert \lvert x-y\rvert\\ =
\lvert x-y\rvert \sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert \hat u_k\rvert \lvert k\rvert^s \lvert k\rvert^{1-s}  \\ \le 
\lvert x-y\rvert \,
\left(\sum_{\lvert k\rvert \le |x-y|^{-1}}\lvert k\rvert^{2-2s}\right)^{1/2}
\left(\sum_{\lvert k\rvert \le |x-y|^{-1}}
\lvert k\rvert^{2s}\lvert \hat u_k\rvert^2 \right)^{1/2} \\
=\lvert x-y\rvert \, \|u\|_{H^s} 
\left(\frac{4s}{(2s-1)\lvert x-y\rvert^{3-2s}}\right)^{1/2} \\
=\left(\frac{4s}{2s-1}\right)^{1/2}\|u\|_{H^s}\lvert x-y\rvert^{s-1/2}
$$
For the second term we have
$$
\left|
\sum_{\lvert k\rvert \ge |x-y|^{-1}}
\hat u_k\big(\mathrm{e}^{ikx}-\mathrm{e}^{iky}\big)\right|\le 
2\sum_{\lvert k\rvert \ge |x-y|^{-1}}
\lvert \hat u_k\rvert=2\sum_{\lvert k\rvert \ge |x-y|^{-1}}
\lvert \hat u_k\rvert \lvert k\rvert^s \lvert k\rvert^{-s}\\ \le 2\,
\left(\sum_{\lvert k\rvert \ge |x-y|^{-1}}\frac{1}{\lvert k\rvert^{2s}}\right)^{1/2}
\left(\sum_{\lvert k\rvert \ge |x-y|^{-1}}
\lvert \hat u_k\rvert^2 \lvert k\rvert^{2s}
\right)^{1/2}\\ \le 2\cdot\left(\frac{2\lvert x-y\rvert^{2s-1}}{2s-1}\right)^{1/2}\|u\|_{H^s} \\
=\frac{2^{3/2}}{(2s-1)^{1/2}}\cdot\lvert x-y\rvert^{s-1/2}\|u\|_{H^s}
$$
Altogether, for every $s\in(1/2,3/2)$, there exists a $c_s>0$, such that
$$
\lvert u(x)-u(y)\rvert\le c_s\lvert x-y\rvert^{s-1/2}\|u\|_{H^s},
$$
for all $u\in H^s(\mathbb T)$.
Note. We have used the following rather crude inequalities
a. For $s>0$,
$$
\sum_{k=1}^n k^s\le n^{s+1}.
$$
b. For $s>1$, 
$$
\sum_{k=n}^\infty \frac{1}{k^s}\le \frac{s}{(s-1)n^{s-1}}.
$$
c. For $0<s<1$
$$
\sum_{k=1}^n \frac{1}{k^s}\le \frac{(2-s)n^{1-s}}{s-1}.
$$
A: This works for $s\ge 1$, by Sobolev embedding for fractional orders: if $u\in H^s=W^{s,2}$, then $u'\in L^p$ with $1/p=3/2-s$. Now your inequality follows by applying Hölder's inequality to $u(y)-u(x)=\int_x^y u'$.
A: 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
$$B^{s}_{2,2}\subset
B^{s}_{2,\infty}\subset B^{s-\frac12}_{\infty,\infty}.
$$
