# Factorization and non-factorization theorems for pseudocontinuable functions

## TL;DR

This paper investigates the factorization properties of functions in star-invariant subspaces of Hardy spaces associated with inner functions, establishing new theorems that relate factorization, smoothness, and integrability, with sharp norm estimates.

## Contribution

It introduces factorization and non-factorization theorems for functions in star-invariant subspaces, linking their smoothness properties to their factorization behavior in Hardy spaces.

## Key findings

- Functions in $K^p_ heta$ can have factors close to $ heta$ with smooth ratios.
- Additional integrability and smoothness properties are established for such functions.
- Sharp norm estimates for these properties are provided.

## Abstract

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.