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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.

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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 $\mathbb{R}^n$.

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  • $\begingroup$ Thanks a lot, Mateusz Kwasnicki, that were helpful hints to me (though indeed Stos' article I couldn't find online). May I ask one mor thing, regarding my second question: You wrote in this post mathoverflow.net/questions/203578/… that it is known for the Green's function that $G(x)≈min(|x|−n−α,|x|−n+α)$ (where n is the dimension) - that is indeed what a first rule of thumb, starting from the Green's function of the usual Laplacian, would lead to. You also said that "there is no closed-form expression" - do you have any (elementary) reference? $\endgroup$ – Kira G. Jul 6 '18 at 15:41
  • $\begingroup$ @KiraG.: No, I do not. In fact I do not even think someone tried proving that a closed-form expression does not exist. As I understand my previous comment now, what I meant is: if you open a standard table of integrals (like Gradshteyn–Ryzhik) or ask Mathematica, you will not find nor get a closed-form answer. $\endgroup$ – Mateusz Kwaśnicki Jul 6 '18 at 17:58

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