order of the singularity of a Green's function to the fractional Laplacian I was looking at a problem which involves the Green's function of a fractional Poisson equation. 
To fix notation, let $D\subset \mathbb{R}^n$ very nice, i.e. a hypercube, and
\begin{equation}
\begin{aligned}
(-\Delta)^{\alpha/2} u &= f \quad in \; D\\
\end{aligned}
\end{equation}
with periodic boundary conditions.
Define $(-\Delta)^{\alpha/2}$ in the spectral way, i.e.  given $\alpha>0$,
\begin{equation}
   (-\Delta)^{\alpha/2} v  = \sum_{k=1}^{\infty} \lambda^{\alpha/2}_k \langle v, e_k\rangle e_k \\
\end{equation}
My questions are very simple and for sure well-known in the expert community, so I hope I can get some good reference by you:
1) is it true that the green's function looks like
\begin{equation}
G(x,y) = \sum_{k=1}^{\infty} \lambda_k^{-\alpha/2} e_k(x) e_k(y)   ?
\end{equation}
2) Are there any results about the order of the singularity of a Green's function to the fractional Laplacian? For the usual Poisson equation, we know that it has a logarithmic singularity, so I guess it should be sith like G(x,y) = |x-y|^? + g(x,y) and the little g has something to do with the boundary?
I'd be grateful for any literature advice you may have. I read https://arxiv.org/abs/1507.07356 , but the topic was not adressed, and the articles by Bogdan, Chen, Song or others are already very specialized for domains with $C^{1,1}$ or Lipschitz boundary, and I wasn't able to strip them down to the easy question I have. 
 A: Oh, well, I'll try my best to help.
(0) Hypercube is not really super-nice, it is merely Lipschitz! This is of course a joke, because periodic boundary conditions indeed make it perfectly regular. But then...
(1) Even for the usual Laplace operator the Green function (in the strict sense: as the kernel of the inverse operator $(-\Delta_{\mathbb{T}})^{-1}$) is not defined, because $\Delta_{\mathbb{T}}$ has eigenvalue $0$ (corresponding to a constant eigenfunction $e_1$; I use ${\mathbb{T}}$ for the hypercube here and below). The same is clearly true for the fractional power $(-\Delta_{\mathbb{T}})^{\alpha/2}$, if it is defined via spectral theory (in this case: Fourier series), as in the second display in the statement of the question.
(2) Noteworthy, it does not matter whether we take the fractional power of $-\Delta$ first and then periodize (is this the right word?), or vice versa. The easiest way to see this is to note that Bochner's subordination formula for the heat kernel of a fractional power commutes with periodization, so to say.
(3) The inverse operator $G^{(\alpha)}_{\mathbb{T}} = (-\Delta_{\mathbb{T}})^{-\alpha/2}$ is well-defined on the orthogonal complement of constants. Of course it can be expressed in terms of spectral theory: $$G^{(\alpha)}_{\mathbb{T}} v = \sum_{k = 2}^\infty \lambda_k^{-\alpha/2} \langle v, e_k \rangle e_k ,$$ but there is a more useful expression: the kernel of $G$ is a "compensated" integral of the heat kernel: $$G^{(\alpha)}_{\mathbb{T}}(x, y) = \int_0^\infty (p^{(\alpha)}_{\mathbb{T}}(t, x, y) - 1) dt .$$ Here $p^{(\alpha)}_{\mathbb{T}}(t, x, y)$ is the heat kernel of $(-\Delta_{\mathbb{T}})^{\alpha/2}$. Indeed, in order to prove the above formula, one simply expands the heat kernel in terms of the eigenfunctions and uses Fubini.
(4) Since we have already observed that $p^{(\alpha)}_{\mathbb{T}}(t, x, y)$ is the periodization of the heat kernel $p^{(\alpha)}(t, x, y)$ in full space, it is not hard to see that the order of the singularity of $G^{(\alpha)}(x, y)$ at the diagonal is preserved. Namely,
$$G^{(\alpha)}_{\mathbb{T}}(x, y) - G^{(\alpha)}(x - y) = \int_0^\infty \left(\sum_{k \in \mathbb{Z}^d \setminus \{0\}} p^{(\alpha)}(t, x, y + k) - 1\right) dt$$
is uniformly bounded due to well-known bounds on the heat kernel.
(5) In fact the order of singularity of the Green function of $(-\Delta)^{\alpha/2}$ is the same in essentially every reasonable space. One only needs Gaussian (or even sub-Gaussian!) bounds for the heat kernel of $(-\Delta)^{\alpha/2}$ for small times, and the subordination formula. I suppose this can be found in Andrzej Stós's article Symmetric $\alpha$-stable processes on $d$-sets, which unfortunately does not seem to be easily available online.
Finally, let me stress that I do not really know the literature on fractional Laplace operators in spaces other than $\mathbb{R}^n$.
