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Real-analytic realization of universal Teichmüller space via complex-structures on H1/2
Abstract
Let H be the Hilbert transform, let h be a quasisymmetric homeomorphism of the unit circle S1, and set Vhu=u∘ h and Jh=VhHVh−1, defined on the Sobolev space H1/2(S1). We prove that h↦ Vh is nowhere continuous in operator norm, although it is continuous in the strong operator topology. By contrast, the induced map h↦ Jh is a real-analytic diffeomorphism onto its image in the operator-norm topology. Based on this, we further compute the differential at the identity and show that it is precisely the Calderón commutator. In graph coordinates, the tangent map admits a weighted Hankel matrix representation, whose Hilbert-Schmidt norm recovers the Weil-Petersson tangent quadratic form.
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BibTeX
@misc{tailiang2026realanalytic,
title = {{Real-analytic realization of universal Teichmüller space via complex-structures on $H^1/2$}},
author = {Liu Tailiang and Shen Yuliang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08539},
eprint = {2609.08539},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08539},
}
RIS
TY - UNPB
TI - Real-analytic realization of universal Teichmüller space via complex-structures on H^1/2
AU - Liu Tailiang
AU - Shen Yuliang
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08539
UR - https://arxiv.org/abs/2609.08539
AB - Let H be the Hilbert transform, let h be a quasisymmetric homeomorphism of the unit circle S^1, and set V_hu=u∘ h and J_h=V_hHV_h^−1, defined on the Sobolev space H^1/2(S^1). We prove that h↦ V_h is nowhere continuous in operator norm, although it is continuous in the strong operator topology. By contrast, the induced map h↦ J_h is a real-analytic diffeomorphism onto its image in the operator-norm topology. Based on this, we further compute the differential at the identity and show that it is precisely the Calderón commutator. In graph coordinates, the tangent map admits a weighted Hankel matrix representation, whose Hilbert-Schmidt norm recovers the Weil-Petersson tangent quadratic form.
ER -