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PreprintCC BY 4.0arXiv

The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem

  • Fabrizio Colombo
  • Elodie Pozzi
  • Irene Sabadini
  • Brett D. Wick
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Abstract

In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra ℝn for any n≥ 2. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.

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Cite

BibTeXRIS
BibTeX
@misc{colombo2026corona,
  title = {{The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem}},
  author = {Fabrizio Colombo and Elodie Pozzi and Irene Sabadini and Brett D. Wick},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.12258},
  eprint = {2608.12258},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.12258},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Corona Problem for Slice Holomorphic Functions via the Matrix Corona Problem
AU  - Fabrizio Colombo
AU  - Elodie Pozzi
AU  - Irene Sabadini
AU  - Brett D. Wick
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2608.12258
UR  - https://arxiv.org/abs/2608.12258
AB  - In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra ℝ_n for any n≥ 2. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.
ER  -