*([The question](https://math.stackexchange.com/q/1351154) was originally posted on Math StackExchange.)*


**Preliminaries:** Given ${s \geq 0}$, let ${\mathrm{H}\_P^s}$ denote the ${\mathrm{L}^2}$-based fractional-order Sobolev space of ${P}$-periodic functions on the line with norm
\begin{equation\*}
\| u \|\_{\mathrm{H}\_P^s} = \left( \sum\_{n \in \mathbb{Z}} \bigg(1 + \frac{4 \pi^2 n^2}{P^2}\bigg)^{s} |\widehat{u}\_n|^2 \right)^\frac{1}{2},
\end{equation\*}
where ${\widehat{u}}$ is the ${n}$th Fourier coefficient. We set ${\mathrm{L}\_P^2 \equiv \mathrm{H}\_P^0}$ and recall Parseval's theorem giving the equivalent $\mathrm{H}^s(-\frac{P}{2},\frac{P}{2})$ norm when $s$ is a nonnegative integer. Moreover, let ${\| \cdot \|\_{\infty}}$ express the supremum norm on an interval of length ${P}$.



> 
> **Question:** Fix ${s > \frac{1}{2}}$ and ${p > 2}$ and consider functions ${u}$ in an open ball of ${\mathrm{H}\_P^s}$, that is, ${\| u
> \|\_{\mathrm{H}\_P^s} < R}$. Is it possible to estimate
>  \begin{equation\*} \| u \|\_{\mathrm{L}\_P^p}^{p} := \int\_{-\frac{P}{2}}^{\frac{P}{2}} u^{p} \,
>  \mathrm{d}x \lesssim \| u \|\_{\mathrm{L}\_P^2}^{2 + \epsilon}
> \end{equation\*} for some ${\epsilon > 0}$ (likely depending on ${p}$)?
> 
> 
> 


**Ideas:** We have
\begin{equation\*}
\| u \|\_{\mathrm{L}\_P^p}^{p} \leq \| u \|\_{\infty}^{p - 2} \| u \|\_{\mathrm{L}\_P^2}^{2},
\end{equation\*}
so it is perhaps easiest to establish ${\| u \|\_{\infty}^{p - 2} \lesssim \| u \|\_{\mathrm{L}\_P^2}^{\epsilon}}$.



> 
> That is, do we have
>  $${\| u \|\_{\infty} \lesssim \| u \|\_{\mathrm{L}\_P^2}^{\tilde{\epsilon}}}$$
>  for some $\tilde{\epsilon} > 0$ given that ${\| u
> \|\_{\mathrm{H}\_P^s} < R}$?
> 
> 
> 


When ${s = 1}$ this is true based on the estimate
\begin{equation}
\| u \|\_{\infty} \lesssim \| u' \|\_{\mathrm{L}\_P^2}^{\frac{1}{2}} \| u \|\_{\mathrm{L}\_P^2}^{\frac{1}{2}} \lesssim \| u \|\_{\mathrm{L}\_P^2}^{\frac{1}{2}}
\end{equation}
(since ${\| u' \|\_{\mathrm{L}\_P^2} \leq \| u \|\_{\mathrm{H}\_P^1} < R}$). This estimate is found in these [[lecture notes, page 21]](http://math.cts.nthu.edu.tw/Mathematics/2012Summer-AIFD/Lecture%201.pdf) when the domain is the real line ${\mathbb{R}}$, but it would be nice to see an argument/reference to the periodic case. And does it extend to the general setting with $s > \frac{1}{2}$?



A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces
Let me expand my comment into an answer.
Let $s>\frac12$ and pick any $r\in(\frac12,s)$.
Interpolating between Sobolev spaces gives $\|u\|\_{H^r\_P}\lesssim\|u\|\_{H^0\_P}^\delta\|u\|\_{H^s\_P}^{1-\delta}$ for some $\delta>0$.
As was shown to you in [the MSE answer](https://math.stackexchange.com/a/1354391/166535) to the question, $\|u\|\_{L^\infty}\lesssim\|u\|\_{H^r\_P}$.
Putting these estimates together gives $\|u\|\_{L^\infty}\lesssim\|u\|\_{L^2\_P}^\delta\|u\|\_{H^s\_P}^{1-\delta}$.
Assuming $\|u\|\_{H^s\_P}\lesssim1$, this homogeneous estimate gives you the inhomogeneous one you need: $\|u\|\_{L^\infty}\lesssim\|u\|\_{L^2\_P}^\delta$.


