Is it possible to find $f,g\in L^2[(0,1)\times(0,1)]$ such that $$\log|x-y|=\int\_0^1f(x,t)g(t,y)dt\:\:\:\forall x,y\in(0,1).$$



Solvability of an integral equation
No. I'm elaborating on my comment: By CS, it would follow that $\left|\ln |x-y|\right| \le A(x)B(y)$, with $A,B\in L^2(0,1)$. So $\left| \ln |x-y|\right| \le C A(x)$ for a.e. $x$ and all $y$ from a set of almost full measure.


