Factorization of solutions of linear differential equations

TL;DR

This paper demonstrates that solutions to certain linear differential equations with analytic coefficients in the unit disk can be factorized into a Blaschke product and an exponential of a BMOA function, revealing their Hardy space nature and zero-free solutions.

Contribution

It introduces a new factorization of solutions to $f''+Af=0$ under Carleson measure conditions, linking solutions to Hardy spaces and BMOA functions.

Findings

Solutions are in Hardy spaces

Solutions have no singular inner factors

Zero-free solutions relate to Riccati equations

Abstract

This paper supplements recents results on linear differential equations $f''+Af=0$, where the coefficient $A$ is analytic in the unit disc of the complex plane $\mathbb{C}$. It is shown that, if $A$ is analytic and $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure, then all non-trivial solutions of $f''+Af=0$ can be factorized as $f=Be^g$, where $B$ is a Blaschke product whose zero-sequence $\Lambda$ is uniformly separated and where $g\in{\rm BMOA}$ satisfies the interpolation property $$g'(z_n) = -\frac{1}{2} \, \frac{B''(z_n)}{B'(z_n)}, \quad z_n\in\Lambda.$$ Among other things, this factorization implies that all solutions of $f''+Af=0$ are functions in a Hardy space and have no singular inner factors. Zero-free solutions play an important role as their maximal growth is similar to the general case. The study of zero-free solutions produces a new result on Riccati differential equations.