I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve:

$$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. Here, the support of $f$ is contained in $B_1 \cup B_2$, and $r_{B_1 \cup B_2}$ restricts a function to $B_1 \cup B_2$. How would I go about solving this?

(I think the solution is the Green's function of the fractional Laplacian on two intervals, rather than just one interval, but I have been unable to find any literature on this matter).

Fractional Integrals and Potentials, 1996) and by Stefan Samko (Hypersingular Integrals and Their Applications, 2001). $\endgroup$3more comments