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Factorization of solutions of linear differential equations

Janne Gröhn

TL;DR

This work shows that, for analytic coefficients $A$ in the unit disk satisfying $|A(z)|^2(1-|z|^2)^3\,dm(z)$ being a Carleson measure, every nontrivial solution of $f''+Af=0$ admits a precise factorization $f=Be^g$ where $B$ is a Blaschke product with a uniformly separated zero set and $g\in{\rm BMOA}$ satisfies a specific interpolation at the zeros. The authors prove both the existence of such a factorization and a converse: any pair $(B,g)$ with these properties yields a solution for some $A$ of the same coefficient class, and zero-free solutions correspond to $f=e^g$ with $g\in{\rm BMOA}$. They also develop an intermediate case with $A\in H^\infty_2$ where $g$ lies in the Bloch space $\mathcal{B}$, and they connect these factorization results to Riccati equations via admissible linear subspaces, providing a unified framework that ties growth in Hardy spaces, zero-sets, and differential equation structure. The paper further identifies and analyzes admissible subspaces, including $\mathcal{B}$ and $\rm BMOA$, and gives concrete examples beyond the standard spaces, enriching the toolbox for studying linear differential equations in the unit disk.

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 $Λ$ is uniformly separated and where $g\in{\rm BMOA}$ satisfies the interpolation property $$g'(z_n) = -\frac{1}{2} \, \frac{B''(z_n)}{B'(z_n)}, \quad z_n\inΛ.$$ 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.

Factorization of solutions of linear differential equations

TL;DR

This work shows that, for analytic coefficients in the unit disk satisfying being a Carleson measure, every nontrivial solution of admits a precise factorization where is a Blaschke product with a uniformly separated zero set and satisfies a specific interpolation at the zeros. The authors prove both the existence of such a factorization and a converse: any pair with these properties yields a solution for some of the same coefficient class, and zero-free solutions correspond to with . They also develop an intermediate case with where lies in the Bloch space , and they connect these factorization results to Riccati equations via admissible linear subspaces, providing a unified framework that ties growth in Hardy spaces, zero-sets, and differential equation structure. The paper further identifies and analyzes admissible subspaces, including and , and gives concrete examples beyond the standard spaces, enriching the toolbox for studying linear differential equations in the unit disk.

Abstract

This paper supplements recents results on linear differential equations , where the coefficient is analytic in the unit disc of the complex plane . It is shown that, if is analytic and is a Carleson measure, then all non-trivial solutions of can be factorized as , where is a Blaschke product whose zero-sequence is uniformly separated and where satisfies the interpolation property Among other things, this factorization implies that all solutions of 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.
Paper Structure (7 sections, 8 theorems, 48 equations)

This paper contains 7 sections, 8 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Results
  3. The main results
  4. The intermediate case
  5. Zero-free solutions
  6. Proofs of the results
  7. Examples of admissible linear subspaces of $\mathcal{H}(\mathbb{D})$

Key Result

Theorem 1

If $A\in\mathcal{H}(\mathbb{D})$ and $|A(z)|^2(1-|z|^2)^3\, dm(z)$ is a Carleson measure, then all non-trivial solutions $f$ of eq:de2 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 interpol Conversely, if $B$ is any Blaschke product whose zero-sequence $\Lambda$ is uniformly separated and

Theorems & Definitions (16)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Corollary 4
  • Theorem 5
  • Proposition 6
  • Proposition 7
  • Theorem 8
  • Example 1
  • Example 2
  • ...and 6 more