$$\sqrt{2}f(x) =\lim_{\delta \to 0^{+}}\left[x-i\delta-\int_{-1}^{1} \frac{|f(y)|^2}{y-i\delta-x}dy\right]$$
I'd like to know if there is a solution for $f\colon(-1,1) \to\mathbb{C}$. Of course if it doesn't cause any problems, the interval may also be the closed one $[-1,1]$.
I know,that if the $|f(y)|^2$ was replaced by $f(y)$ that would be a Fredholm-equation.
P.S.: I'm am new to stack exchange, so please tell me, if you'd rather deem this to be a question for mathoverflow.