Let $K(k, u)$ be an integral transform kernel that transforms $$\tilde{p}(k) = \int du K(k,u) p(u),$$ where all the variables are real. For a given function $g(u_1, u_2)$, is there a way to find out whether there exists $K(k,u)$ and $f(k)$ satisfying $$\int du_1\int du_2 K(k_1,u_1) K(k_2,u_2)\delta(u - g(u_1,u_2)) = \delta(k_1-k_2) K(k_1,u) f(k_1).$$ I'm particularly interested in the case where $g(u_1, u_2) = \frac{u_1+u_2}{1+u_1u_2}$. But even if it's not possible for this particular $g$, it'd still be helpful to have some examples with other nontrivial $g$s.
P.S. I can only think of the Fourier transform as an example with $g(u_1,u_2) = u_1+u_2$.
Thanks! Yantao