The problem I would like to solve is as follows: $$ \min_{\substack{\hat{f} \\ \hat{f} \geq 0\\ \int\hat{f} \;dw = 1}} \int_{\mathcal{Z}} \left( \int_{\mathcal{W}}\left[\hat{g}(z|w)\hat{f}(w) - g(z|w)f(w)\right]dw \right)^2h(z)\;dz $$
Basically, I want to find a density $\hat{f}$ such that the integral above is minimized, for given density functions $\hat{g}(.)$, $g(.)$, $f(.)$, $h(.)$. I have some intuition of what the solution is but I would like to check if it is correct. I was not able to find any reference where a similiar problem appears yet. Any help will be really appreaciated, whether in terms of the solution to the problem or related papers.