Factorization and non-factorization theorems for pseudocontinuable functions

Let $\theta$ be an inner function on the unit disk, and let $K_{\theta }^p:=H^p\cap \theta \overline{H_0^p}$ be the associated star-invariant subspace of the Hardy space $H^p$, with $p\ge 1$. While a nontrivial function $f\in K_{\theta }^p$ is never divisible by $\theta$, it may have a factor $h$ which is “not too different” from $\theta$ in the sense that the ratio $h\mathrm{/}\theta$ (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.