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Can the Green function for the fractional Laplacian operator be estimated from above and below. $$ \left\{\begin{aligned} (-\Delta_x)^{s} G(x, y)+ G(x, y)&= \delta_{y}(x) &&\text{in } \Omega \\ G(x,y) & =0 &&\text{ in } \mathbb{R}^N\setminus \Omega \end{aligned} \right.$$ when $N\geq 2s$ with $s\in (0, 1).$

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The usual Green function would rather satisfy $(-\Delta)^s G(\cdot, y) = \delta_y(\cdot)$, and bounds for this one and bounded $C^{1,1}$ open sets have been obtained independently in:

Z.-Q. Chen, R. Song, Estimates on Green functions and Poisson kernels for symmetric stable process, Math. Ann. 312(3) (1998), pp. 465–501

and:

T. Kulczycki, Properties of Green function of symmetric stable process, Probab. Math. Stat. 17(2) (1997), 339–364.

Similar (less explicit) estimates for Lipschitz domains are given in:

T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22(2) (2002), 419–441.

Arbitrary open sets can be studied in a similar way using the boundary Harnack inequality from my very first article:

K. Bogdan, T. Kulczycki, M. Kwaśnicki, Estimates and structure of $\alpha$-harmonic functions, Prob. Theory Rel. Fields 140(3–4) (2008), 345–381.

The Green function for $(-\Delta)^s + 1$ in bounded sets is comparable to that for $(-\Delta)^s$, discussed above. This property is certainly written somewhere, but I do not know a reference. Alternatively, estimates for both Green functions follow directly from the similar estimates for the heat kernel, found for $C^{1,1}$ open sets in:

Z.-Q. Chen, P. Kim, R. Song, Heat kernel estimates for Dirichlet fractional Laplacian, J. Eur. Math. Soc. 12(5) (2010), 1307–1329,

and for Lipschitz domains in:

K. Bogdan, T. Grzywny, M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, Ann. Probab. 38(5) (2010), 1901–1923.

(I apologize if the above list is incomplete.)

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  • $\begingroup$ I have a feeling these results should be true even in one dimension. Most of the results are for $N\geq 2.$ $\endgroup$ – GabS Apr 2 at 16:27
  • $\begingroup$ As long as $N > 2 s$, there should be no difference between $N = 1$ and $N \geqslant 2$. When $N < 2 s$, things get different: Green's function will be bounded. The case $N = 2 s$ is typically the most problematic one due to logarithmic singularity near the diagonal. $\endgroup$ – Mateusz Kwaśnicki Apr 2 at 19:09
  • $\begingroup$ I agree, the log makes life difficult. The case $N=2s$ is interesting. But the result should hold even in this case as well. $\endgroup$ – GabS Apr 3 at 10:18

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