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 $s\in (0, 1).$



Estimates for Green function for fractional Laplacian
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:



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



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> T. Kulczycki, *Properties of Green function of symmetric stable process*, Probab. Math. Stat. 17(2) (1997), 339–364.
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Similar (less explicit) estimates for Lipschitz domains are given in:



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> T. Jakubowski, *The estimates for the Green function in Lipschitz domains for the symmetric stable processes*. Probab. Math. Statist. 22(2) (2002), 419–441.
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Arbitrary open sets can be studied in a similar way using the boundary Harnack inequality from my very first article:



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



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



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> K. Bogdan, T. Grzywny, M. Ryznar, *Heat kernel estimates for the fractional Laplacian with Dirichlet conditions*, Ann. Probab. 38(5) (2010), 1901–1923.
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(I apologize if the above list is incomplete.)


