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 $s\in (0, 1).$


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


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.)

  • $\begingroup$ 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 '19 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. $\endgroup$ Apr 2 '19 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 '19 at 10:18

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