  Let $\theta$ be an inner function on the unit disk, and let
$K^p_\theta:=H^p\cap\theta\overline{H^p_0}$ be the associated star-invariant
subspace of the Hardy space $H^p$, with $p\ge1$. While a nontrivial function
$f\in K^p_\theta$ is never divisible by $\theta$, it may have a factor $h$
which is "not too different" from $\theta$ in the sense that the ratio
$h/\theta$ (or just the anti-analytic part thereof) is smooth on the circle. In
this case, $f$ is shown to have additional integrability and/or smoothness
properties, much in the spirit of the Hardy--Littlewood--Sobolev embedding
theorem. The appropriate norm estimates are established, and their sharpness is
discussed.
