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Consider the fractional Sobolev space $H^{1/2}_{2\pi}$. This space consists of the functions $u$ in the space $L^2(0, 2\pi)$ whose coefficients of their Fourier expansion $$u(t)=a_0+\sum_{k=1}^{\infty}(a_k \cos kt+ b_k\sin kt)$$ satisfy $$\sum_{k=1}^{\infty}k(a_k^2+b_k^2)<\infty.$$ If we also consider $v\in H^{1/2}_{2\pi}$ with the Fourier expansion $$v=\tilde{a}_0+\sum_{k=1}^{\infty}(\tilde{a}_k\cos kt+\tilde{b}_k\sin kt)$$ and define the inner product $(\cdot, \cdot):H^{1/2}_{2\pi}\times H^{1/2}_{2\pi}\to \mathbb{R}$ by $$(u, v)_{H^{1/2}_{2\pi}}=2\pi a_0\tilde{a}_0+\sum_{k=1}^{\infty}k(a_k\tilde{a}_k+b_k\tilde{b}_k),$$ then this inner product defines a Hilbert space structure on $H^{1/2}_{2\pi}$.

The fractional Sobolev space $H^{1/2}_{2\pi}$ is frequently considered in the literature when looking for periodic solutions of Hamiltonian systems (to be precise, the space that one considers in this context is $E:=\left(H^{1/2}_{2\pi}\right)^{2n}$, but there is nothing special about this space from a functional analytic perspective, so this is why I am considering the space $H^{1/2}_{2\pi}$ in this question). I first encountered this space in the well known book Minimax Methods in Critical Point Theory with Applications to Differential Equations by Paul Rabinowitz when I started learning about the Mountain Pass Theorem and its generalisations. One particularly important result about this space that is frequently used in this context is the following:

For all $s\in [1, \infty)$, $H^1_{2\pi}$ is compactly embedded in $L^s(0, 2\pi)$. In particular, there is an $\alpha_s>0$ such that $$\left\lVert u \right\rVert_{H^{1/2}}\le \alpha_s \left\lVert u \right\rVert_{L^s}.$$

Rabinowitz does not prove this result in his book. He gives two references to which I unfortunately do not have access to:

A. Friedman, Partial differential equations, Holt, Rinehart, and Winston, Inc., New York, 1969.
P. Rabinowitz, A variational method for finding periodic Solutions of differential equation, Nonlinear Evolution Equations (M. G. Crandall, ed.), Academic Press, New York, 1978 , pp. 225-251 .

Does anyone know a modern reference for this result? The proof seems nontrivial, at least I do not know how to approach it. I could not locate a proof of this result in the literature even if it is widely used in the context of Hamiltonian systems that I mentioned before (I am not sure if the fact that the embedding is compact is ever used, but the fact that the embedding is continuous is ubiquitous).

P.S. I am sorry if this question is too easy for MO, but it did not seem fit for MS because to the best of my knowledge the space $H^{1/2}_{2\pi}$ is pretty niche (at least I have not encountered it outside of this research area, but I am only a third year undergraduate student, so my assumption may be wrong due to my limited knowledge). Also please let me know if I should include any further details.

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An excellent reference for this, with exercises, full details etc, is

Demengel, Françoise; Demengel, Gilbert, Functional spaces for the theory of elliptic partial differential equations. Transl. from the French by Reinie Erné, Universitext. Berlin: Springer (ISBN 978-1-4471-2806-9/pbk; 978-1-4471-2807-6/ebook). xviii, 465 p. (2012). ZBL1239.46001.

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  • $\begingroup$ thank you so much, this book is truly a gem especially for a beginner like me. $\endgroup$ Commented Apr 25, 2023 at 17:46
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I would say that the book Linear and Quasilinear Parabolic Problems, Volume II: Function Spaces by Herbert Amann qualifies as a modern reference (published in 2019).

You can find a version of a Rellich-Kondrachov compact embedding result in Theorem 7.3.6 for functions defined on a compact interval of $\mathbb{R}$, with fractional regularity, which includes the case $H^{1/2}(0,2\pi)$ compactly embedded in $L^p(0,2\pi)$ for $p < \infty$.

Amann's book is actually much, much, more general as he manipulates functions which are Banach-valued (but you can take this Banach to be merely $\mathbb{R}$).

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  • $\begingroup$ Your last sentence reminds me of the three-volume Analysis by Amann... $\endgroup$
    – Akira
    Commented Apr 25, 2023 at 10:38
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    $\begingroup$ @cs89 thank you for the reference! I decided to accept username's answer because the book he mentioned is more beginner friendly, but I will surely use the reference you gave me to further my knowledge of function spaces. $\endgroup$ Commented Apr 25, 2023 at 17:50
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    $\begingroup$ Sure, no problem. The other reference is indeed better suited for your question. $\endgroup$
    – cs89
    Commented Apr 25, 2023 at 19:02

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