Factorization and non-factorization theorems for pseudocontinuable functions

Let θ be an inner function on the unit disk, and let Kθp:=Hp∩θH0p‾ be the associated star-invariant subspace of the Hardy space Hp, with p≥1. While a nontrivial function f∈Kθp is never divisible by θ, it may have a factor h which is ''not too different” from θ in the sense that the ratio h/θ (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.