A bound on the $\mathrm{L}^p$ norm in terms of the $\mathrm{L}^2$ norm in periodic Sobolev spaces

Question

(The question 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] 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 $s > \frac{1}{2}$?

Answer 1

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 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: $\|u\|_{L^\infty}\lesssim\|u\|_{L^2_P}^\delta$.