线性微分方程解的因式分解
Factorization of solutions of linear differential equations
AI总结：
本文证明在Carleson测度条件下，线性微分方程的解可分解为Blaschke乘积与BMOA函数之积，并由此推出解属于Hardy空间且无奇异内因子，同时给出Riccati方程的新结果。
摘要：
本文补充了关于线性微分方程 $f''+Af=0$ 的最新结果，其中系数 $A$ 在复平面 $\mathbb{C}$ 的单位圆盘内解析。研究表明，若 $A$ 解析且 $|A(z)|^2(1-|z|^2)^3\\, dm(z)$ 是 Carleson 测度，则 $f''+Af=0$ 的所有非平凡解可分解为 $f=Be^g$，其中 $B$ 是零序列 $\Lambda$ 一致分离的 Blaschke 乘积，且 $g\in{\rm BMOA}$ 满足插值性质 $$g'(z_n) = -\frac{1}{2} \\, \frac{B''(z_n)}{B'(z_n)}, \quad z_n\in\Lambda.$$ 除其他性质外，该因式分解蕴含 $f''+Af=0$ 的所有解都属于 Hardy 空间且没有奇异内因子。无零解因其最大增长与一般情形相似而扮演重要角色。对无零解的研究产生了关于 Riccati 微分方程的一个新结果。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
