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PreprintCC BY 4.0arXiv

BMO Domains and Quasicircles Associated with Zygmund Functions

  • Katsuhiko Matsuzaki
  • Fei Tao
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Abstract

We investigate graph domains and graph curves from the viewpoints of BMOA, Zygmund regularity, and asymptotic geometry. First, we prove that every graph domain is a BMO domain, which provides a proof of a statement of Astala and Zinsmeister. We then use a Weierstrass-type function to construct a Bishop-Jones quasicircle that is asymptotically conformal but not asymptotically smooth. This example yields the strict inclusions VMOA(ℍ) subsetneq BMOA(ℍ)∩ B0(ℍ), SS(ℝ) subsetneq SQS(ℝ)∩S(ℝ). Finally, by a localized lacunary construction, we produce a compactly supported locally absolutely continuous function f with f'∈BMO(ℝ)∖VMO(ℝ) whose graph is an asymptotically conformal chord-arc curve that is not asymptotically smooth.

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Cite

BibTeXRIS
BibTeX
@misc{matsuzaki2026domains,
  title = {{$\mathrmBMO$ Domains and Quasicircles Associated with Zygmund Functions}},
  author = {Katsuhiko Matsuzaki and Fei Tao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39567},
  eprint = {2610.39567},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39567},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - BMO Domains and Quasicircles Associated with Zygmund Functions
AU  - Katsuhiko Matsuzaki
AU  - Fei Tao
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39567
UR  - https://arxiv.org/abs/2610.39567
AB  - We investigate graph domains and graph curves from the viewpoints of BMOA, Zygmund regularity, and asymptotic geometry. First, we prove that every graph domain is a BMO domain, which provides a proof of a statement of Astala and Zinsmeister. We then use a Weierstrass-type function to construct a Bishop-Jones quasicircle that is asymptotically conformal but not asymptotically smooth. This example yields the strict inclusions VMOA(ℍ) subsetneq BMOA(ℍ)∩ B_0(ℍ), SS(ℝ) subsetneq SQS(ℝ)∩S(ℝ). Finally, by a localized lacunary construction, we produce a compactly supported locally absolutely continuous function f with f'∈BMO(ℝ)∖VMO(ℝ) whose graph is an asymptotically conformal chord-arc curve that is not asymptotically smooth.
ER  -