Hankel Operators on Bounded Analytic Functions

For $U$ a domain in the complex plane and $g$ a bounded measurable function on $U$, the generalized Hankel operator $S_g$ on $H^\infty(U)$ is the operator of multiplication by $g$ followed by projection into $L^\infty/H^\infty$. Under certain conditions on $U$ we show that either $S_g$ is compact or there is an embedded $\ell^\infty$ on which $S_g$ is bicontinuous. We characterize those $g$'s for which $S_g$ is compact in the case that $U$ is a Behrens roadrunner domain.