Green's function for fractional operators

Question

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?

Answer 1

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.