Conformal welding of chord-arc curves
Abstract
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 Ch u=u∘ 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 Ah denote the analytic projection of Ch. We prove that Ah 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 Ah 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.
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BibTeX
@misc{tailiang2026conformal,
title = {{Conformal welding of chord-arc curves}},
author = {Liu Tailiang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.24745},
eprint = {2608.24745},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.24745},
}
RIS
TY - UNPB
TI - Conformal welding of chord-arc curves
AU - Liu Tailiang
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2608.24745
UR - https://arxiv.org/abs/2608.24745
AB - 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∘ 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.
ER -