Is there trace operator for periodic functions? I know that for a smooth domain $\Omega$ we can build a trace operator $\gamma : H^s(\Omega) \to \prod_{0\leq j \leq s}H^{s-j-\frac{1}{2}}(\partial \Omega)$. In particular it has a right inverse which implies that $\gamma$ is surjective. Moreover, one can characterize $H^s_0(\Omega)$ as being the kernel of $\gamma$.
Now I am wondering if a similar result exists for periodic functions. So if I define $\Omega = (-\pi,\pi)^m$ and $H^s_{per}(\Omega)$ to be the usual Sobolev space for periodic functions, can we build a surjective operator $\gamma_{per}$, such that $H^s_{0,per}$ (= periodic functions in $H^s_0(\Omega)$) can be identified as the kernel of $\gamma_{per}$ ?
The point is that $\Omega$ is not smooth anymore but only Lipschitz. However, I was hoping that since we restrict to periodic solutions, there might be a way to obtain a similar result anyway.
It feels like the natural way to do the proof would be to deal with coefficients of Fourier series, and turn the problem into finding an operator on coefficient spaces. I just could not find out the right inverse in that manner.
Edit :
There is a constrution of a right inverse on the half plane in ''Strongly Elliptic Systems and Boundary Integral Equations'' by William McLean (page 101, chapter of trace operator). Now since the boundary of a cube looks locally like a half plane (not in the vertices, but we can decompose the boundary into pieces to avoid this), I was hoping to obtain a right inverse in that manner.
Also if you define $e_n(x) = e^{in.x}$ for $x \in \mathbb{R}^m$ and $<n>^s = (1+n_1^2+...+n_m^2)^{\frac{s}{2}}$, then you define $H^s_{per}(\Omega) := \{ \sum_{n \in \mathbb{Z}^m}a_ne_n | \sum_{n\in \mathbb{Z}^m}|a_n|^2<n>^{2s} <\infty\} $. So $H^s_{per}$ is a Hilbert space, and you can notice that its construction is very similar to $H^s(\mathbb{R}^m)$ using Fourier transform (see McLean chapter 3). That is also why I am expecting a possible construction of a right inverse, as in the half plane case.
 A: Think of your space as not consisting of functions defined on $\Omega$, but consisting of functions defined in $\mathbb{R}^m$, with the periodicity condition built in. For example, one can define $H_{per}$ as the completion of smooth periodic functions in $\mathbb{R}^m$, with respect to the $H^1$-norm on $\Omega$. Then the trace in your sense would be the usual trace onto two lines, one horizontal and the other vertical. So the usual trace theory works.
A: I found a book that solves the problem for the 2D and 3D cases : ''Polynomials in the Sobolev World'' by Christine BERNARDI, Monique DAUGE and Yvon MADAY (chapters 5 and 6). https://perso.univ-rennes1.fr/monique.dauge/publis/BeDaMa07.pdf
They give the actual conditions that are needed in order to fully characterize the image of the trace, and so to be able to define a surjective operator.
The thing to be careful about (in 2D) is that there is a limiting case when $s$ is an integer and the norm needs to be adapted to handle this specific case. Otherwise, the compatibility conditions are pretty natural and they basically inforce that $\partial^k_x\partial^l_yu(P) = \partial^l_y\partial^k_xu(P)$ if $P$ is a corner of the square.
