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A Compactness Characterization of Strongly Symmetric Homeomorphisms
Abstract
A self-homeomorphism h of the unit circle 𝕋 is strongly symmetric if it is absolutely continuous and log h'∈VMO(𝕋). Let Ph- be the anti-analytic component of the pullback operator Ph: F↦ F∘ h on BMOA(𝔻), where 𝔻 is the unit disk. P. Jones proved Ph 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 Ph-, and raised the question of whether the converse is true. We answer this question affirmatively, establishing that Ph- is compact if and only if log h'∈VMO(𝕋)), which completes the VMO theory of the universal Teichmüller space.
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BibTeX
@misc{tailiang2026compactness,
title = {{A Compactness Characterization of Strongly Symmetric Homeomorphisms}},
author = {Liu Tailiang and Shen Yuliang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.28074},
eprint = {2608.28074},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28074},
}
RIS
TY - UNPB
TI - A Compactness Characterization of Strongly Symmetric Homeomorphisms
AU - Liu Tailiang
AU - Shen Yuliang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.28074
UR - https://arxiv.org/abs/2608.28074
AB - A self-homeomorphism h of the unit circle 𝕋 is strongly symmetric if it is absolutely continuous and log h'∈VMO(𝕋). Let P_h^- be the anti-analytic component of the pullback operator P_h: F↦ F∘ h on BMOA(𝔻), where 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'∈VMO(𝕋)), which completes the VMO theory of the universal Teichmüller space.
ER -