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.
Best Answer
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 $0<s<1$ $$ \sum_{k=1}^n \frac{1}{k^s}\le \frac{(2-s)n^{1-s}}{s-1}. $$