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Characterizing higher-dimensional Hölder domains via spherical geometric functions

  • Emily Casey
  • Max Engelstein
  • Tatiana Toro
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Abstract

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a C1,β-manifold. This extends the analysis in \cite{Ca24} to dimensions ≥ 3 and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones β-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

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Cite

BibTeXRIS
BibTeX
@misc{casey2026characterizing,
  title = {{Characterizing higher-dimensional Hölder domains via spherical geometric functions}},
  author = {Emily Casey and Max Engelstein and Tatiana Toro},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13866},
  eprint = {2610.13866},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13866},
}
RIS
TY  - UNPB
TI  - Characterizing higher-dimensional Hölder domains via spherical geometric functions
AU  - Emily Casey
AU  - Max Engelstein
AU  - Tatiana Toro
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.13866
UR  - https://arxiv.org/abs/2610.13866
AB  - In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a C^1,β-manifold. This extends the analysis in \cite{Ca24} to dimensions ≥ 3 and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones β-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.
ER  -