Green's function for fractional operators

I am studying some papers about the fractional laplacian, and I am stuck on a formula that I do not understand. I would like to ask if anybody can give me some help. In this paper, on page 12, there is a formula for the Green function $G_\alpha$ of a problem in $1+1$ dimensions. However, in this paper by Xavier Cabré, it seems to me that the Green function should have a completely different form, as noticed in Remarks 3.8 and 3.10. In particular, I do not understand the presence of an exponential term in the paper by Kenig. So my question is: what is the relationship between the function $G_\alpha$ of the first paper and the functions appearing in Remarks 3.8 and 3.10 of the second paper?

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It looks like Kenig-Martel-Robbiano study a parabolic problem while Cabré-Sire study an elliptic problem. For ordinary Laplacian, the heat kernel $t^{-n/2}e^{-|x|^2/(4t)}$ does not look very much like the fundamental solution $|x|^{2-n}$ either.
After some more thoughts, I think they solve different problems. While Cabré solves $(-\Delta)^su=h$, Kenig et al. solve $(-\Delta)^s u+u=h$. It is not trivial to move from one equation to the other. Even for the ordinary laplacian, the Green function for $-\Delta+I$ decays exponentially fast at infinity, while the Green function for $-\Delta$ decays like a negative power. But there is still something I do not understand completely. –  Siminore Aug 18 '12 at 8:04