In the context of reproducing some known expressions, I'm trying to simplify the integral $$ \int \dfrac{dx dy}{|x-y|} \delta(x-r_j)\delta(y-r_l) \;, $$ which I know it should give as result $$ \int \dfrac{dk}{k^2}e^{-i k r_j}e^{-i k r_l} \;. $$ [Here I reported the integrals I actually care about in a simplified form, so that the main point of the calculation is emphasised. More actual details are not so relevant.]
I think that the idea in going from the first to the second integral must involve the Fourier transform, but I was not able to find out how. I have the impression one has to FT the fraction, and then integrate out $x,y$, but by taking FT of $1/|x|$ I get stuck.
By definition of the Dirac delta, your integral is just $$ I = \int \frac{1}{|r_j-y|} \,\delta(y-r_l) \,\mathrm d y = \frac{1}{|r_j-r_l|} $$ Assuming that you works in dimension $3$, the Fourier transform of $1/|x|$ is $1/(2\pi^2|x|^2)$, so the above function is just the Fourier transform of $1/(2\pi^2|x|^2)$ evaluated at $r_j-r_l$, that is $$ I = \int e^{-i(r_j-r_l)k} \frac{1}{2\pi^2|k|^2} \,\mathrm d k. $$ In particular, you have a sign error in one of the exponentials.