重新审视ODE的缓慢增长解决方案
Slowly growing solutions of ODEs revisited
摘要：
假设$A$在单位圆盘$\mathbb{D}$中解析并且满足\begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$} \end{equation}，则考虑微分方程$f''+Af=0$的解。根据文献中的最新结果，这种限制已与将所有解置于布洛赫空间中的系数条件相关联$\mathcal{B}$。在本文中，它表明，任何系数条件意味着\eqref{eq:dag}未能检测某些情况下，布洛赫解决方案确实出现。逆问题也得到解决：什么可以说的增长系数$A$，如果所有的解决方案$f''+Af=0$属于$\mathcal{B}$？对缓慢增长的解决方案进行了全面的修订，强调函数空间$\mathcal{B}$，$\rm{BMOA}$和$\rm{VMOA}$。
英文摘要：
Solutions of the differential equation $f''+Af=0$ are considered assuming that $A$ is analytic in the unit disc $\mathbb{D}$ and satisfies \begin{equation} \label{eq:dag} \sup_{z\in\mathbb{D}} \, |A(z)| (1-|z|^2)^2 \log\frac{e}{1-|z|} < \infty. \tag{$\star$} \end{equation} By recent results in the literature, such restriction has been associated to coefficient conditions which place all solutions in the Bloch space $\mathcal{B}$. In this paper it is shown that any coefficient condition implying \eqref{eq:dag} fails to detect certain cases when Bloch solutions do appear. The converse problem is also addressed: What can be said about the growth of the coefficient $A$ if all solutions of $f''+Af=0$ belong to $\mathcal{B}$? An overall revised look into slowly growing solutions is presented, emphasizing function spaces $\mathcal{B}$, $\rm{BMOA}$ and $\rm{VMOA}$.
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。
