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Harmonic Quasiconformal Mappings in the BMO-Teichmüller space

  • Liu Tailiang
  • Shen Yuliang
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Abstract

Let HQS(ℝ) denote the class of increasing homeomorphisms of the real line ℝ that admit harmonic quasiconformal extensions to the upper half-plane ℍ. Kalaj and Pavlović proved that a homeomorphism h belongs to HQS(ℝ) if and only if h is bi-Lipschitz and the Hilbert transform of h' belongs to L∞(ℝ), and asked whether HQS(ℝ) is a group under composition. We give a negative answer to this question. Further, we show that the Beltrami coefficients of harmonic quasiconformal mappings of ℍ can be identified with the open unit ball of H∞(ℍ). Finally, we study the topology of HQS(ℝ) and prove that it is contractible in the topology inherited from the BMO-Teichmüller space.

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Cite

BibTeXRIS
BibTeX
@misc{tailiang2026harmonic,
  title = {{Harmonic Quasiconformal Mappings in the BMO-Teichmüller space}},
  author = {Liu Tailiang and Shen Yuliang},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00860},
  eprint = {2611.00860},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00860},
}
RIS
TY  - UNPB
TI  - Harmonic Quasiconformal Mappings in the BMO-Teichmüller space
AU  - Liu Tailiang
AU  - Shen Yuliang
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00860
UR  - https://arxiv.org/abs/2611.00860
AB  - Let HQS(ℝ) denote the class of increasing homeomorphisms of the real line ℝ that admit harmonic quasiconformal extensions to the upper half-plane ℍ. Kalaj and Pavlović proved that a homeomorphism h belongs to HQS(ℝ) if and only if h is bi-Lipschitz and the Hilbert transform of h' belongs to L^∞(ℝ), and asked whether HQS(ℝ) is a group under composition. We give a negative answer to this question. Further, we show that the Beltrami coefficients of harmonic quasiconformal mappings of ℍ can be identified with the open unit ball of H^∞(ℍ). Finally, we study the topology of HQS(ℝ) and prove that it is contractible in the topology inherited from the BMO-Teichmüller space.
ER  -