My last question hasn't got any replies so I'll try another..
Is there a way to split the following integral ($g$ is arbitrary) $\int{f^2g}$ so that I instead have an expression involving the $L^2$ norm on $f$ and either $L^2$ or $L^\infty$ norm on $g$? In particular, I want something like $A\lVert f \rVert_{L^2}^2 + B\lVert g \rVert^c_{L^p} \leq \int{f^2g}$ where $A$ and $B$ and $c$ are constants. I tried to use Holder's inequality but that gives me an upper bound instead.