发表机构
Jiangsu University of Technology; Soochow University（江苏理工学院; 苏州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文解决了通用Teichmüller空间VMO理论中的问题，证明单位圆周自同胚对应的拉回算子反解析分量紧当且仅当该同胚为强对称，完成了相关紧性刻画。
AI 中文摘要
单位圆周$\boldsymbol{\text{T}}$的自同胚$h$是强对称的，当且仅当它绝对连续且$\boldsymbol{\text{log}\thinspace h'\thinspace \text{∈}\thinspace \text{VMO}(\text{T})}$。设$\boldsymbol{P_h^-}$为单位圆盘$\boldsymbol{\text{D}}$上$\boldsymbol{\text{BMOA}(\text{D})}$的拉回算子$\boldsymbol{F\thinspace ↦\thinspace F\thinspace ∘\thinspace h}$的反解析分量，Fan、Hu和Shen证明$h$的强对称性可推出$\boldsymbol{P_h^-}$是紧算子，并提出其逆命题是否成立的问题。本文肯定回答该问题，证明$\boldsymbol{P_h^-}$紧当且仅当$\boldsymbol{\text{log}\thinspace h'\thinspace \text{∈}\thinspace \text{VMO}(\text{T})}$，完善了通用Teichmüller空间的VMO理论。
英文摘要
A self-homeomorphism $h$ of the unit circle $\mathbb{T}$ is strongly symmetric if it is absolutely continuous and $\log h'\in\text{VMO}(\mathbb{T})$. Let $P_h^-$ be the anti-analytic component of the pullback operator $P_h: F\mapsto F\circ h$ on $\text{BMOA}(\mathbb{D})$, where $\mathbb{D}$ is the unit disk. P. Jones proved $P_h$ is bounded on BMO if and only if $h$ is strongly quasisymmetric. Fan, Hu, and Shen showed that the strong symmetry of $h$ yields the compactness of $P_h^-$, and raised the question of whether the converse is true. We answer this question affirmatively, establishing that $P_h^-$ is compact if and only if $\log h'\in\text{VMO}(\mathbb{T}))$, which completes the VMO theory of the universal Teichmüller space.
