Abstract
After reviewing Grunsky operator and Faber operator acting on Dirichlet space, we discuss the boundedness of Faber operator on BMOA, a new subject which turns out to be closely related to the BMO theory of the universal Teichmüller space. In particular, we show that the Faber operator acts as a bounded operator on BMOA if the symbol conformal map stays nearly to the base point in the BMO-Teichmüller space. Meanwhile, we obtain several results on quasiconformal mappings, BMO-Teichmüller space and chord-arc curves as well. As by-products, this provides a complex analysis approach to the boundedness of the Cauchy integral acting on BMO functions on a chord-arc curve near to the unit circle in the BMO-Teichmüller space.
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Supported by NSFC (Grant No. 12171346)
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Liu, T.L., Shen, Y.L. The Faber Operator Acting on BMOA, BMO-Teichmüller Space and Chord-arc Curves. Acta. Math. Sin.-English Ser. (2024). https://doi.org/10.1007/s10114-024-2184-4
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DOI: https://doi.org/10.1007/s10114-024-2184-4
Keywords
- Faber operator
- quasiconformal mapping
- quasicircle
- BMO-Teichmüller space
- chord-arc curve
- Grunsky operator
- BMO
- BMOA
- Carleson measure
- Cauchy integral