弦弧曲线的共形焊接
Conformal welding of chord-arc curves
- School of Mathematics and Physics, Jiangsu University of Technology（江苏科技大学数理学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究通过分析共形焊接相关算子的有界同构性，给出弦弧曲线的共形焊接刻画，解决了Semmes的公开问题，并建立了其与Faber积分算子的对应关系。
AI中文摘要：
我们研究弦弧曲线的几何性质与其共形焊接之间的关系。设$h$为闭若尔当曲线$\boldsymbol{\text{Γ}}$的共形焊接，根据琼斯定理，拉回算子$C_h$在BMO上有界当且仅当$h$对应于Bishop-Jones拟圆的焊接。设$A_h$为$C_h$的解析投影，我们证明$A_h$在BMOA上是有界同构当且仅当$\boldsymbol{\text{Γ}}$是弦弧曲线。这为弦弧曲线提供了完整的共形焊接刻画，解决了Semmes在20世纪80年代提出的一个公开问题。此外，我们建立了$A_h$的逆与经典Faber积分算子之间的精确对应，表明对于可求长曲线，Faber算子在BMOA上是有界同构当且仅当该曲线满足弦弧条件。
英文摘要：
We study the relation between the geometric properties of a chord-arc curve and its conformal welding. Let $h$ be the conformal welding of a closed quasicircle $Γ$. By Jones's theorem, the pull-back operator $C_h u=u\circ h$ is bounded on BMO if and only if $h$ corresponds to the welding of a Bishop-Jones quasicircle, equivalently, $h$ is strongly quasisymmetric. Let $A_h$ denote the analytic projection of $C_h$. We prove that $A_h$ is a bounded isomorphism on BMOA if and only if $Γ$ is a chord-arc curve. More strongly, the same characterization holds if invertibility is replaced by Fredholmness. This gives an intrinsic conformal-welding characterization of chord-arc curves and a complete geometric answer to the invertibility problem posed by Semmes in the 1980s. Furthermore, we establish an exact correspondence between the inverse of $A_h$ and the classical Faber integral operator, showing that for a rectifiable quasicircle, the Faber operator is a bounded isomorphism on BMOA if and only if the curve satisfies the chord-arc condition.
