Let $\Omega$ be a bounded set and let $f:\Omega \to \mathbb{R}$. What are the weakest conditions of $f$ so that $$\int_{\Omega}uvf \leq C\lVert u \rVert\lVert v\rVert$$ holds for all $u, v\in L^2(\Omega)$, where the norms are on $L^2(\Omega)$?
Obviously $f \in L^\infty(\Omega)$ suffices. But what are the weakest it can be? Thanks.
The condition that $f\in L^\infty(\Omega)$ is the weakest possible. Suppose $f\notin L^\infty(\Omega)$, so for any $y$ we have some set $S_y$ of nonzero measure on which $|f(x)|>y$. Assume WLOG that $f(x)$ is positive on $S_y$, so $f(x)>y$. Then let $u_y$ be the indicator function of $S_y$ and $v\equiv 1$. This gives us $$\int_\Omega u_y vf\ge y\cdot m(S_y)=\frac{y}{m(\Omega)}\|u_y\|\|v\|$$ so letting $y\to \infty$ we see that no $C$ suffices to make the desired inequality hold.