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A local representation formula for quaternionic slice regular functions


Authors: Graziano Gentili and Caterina Stoppato
Journal: Proc. Amer. Math. Soc. 149 (2021), 2025-2034
MSC (2020): Primary 30G35; Secondary 32D05
DOI: https://doi.org/10.1090/proc/15339
Published electronically: March 2, 2021
MathSciNet review: 4232195
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Abstract:

After their introduction in 2006, quaternionic slice regular functions have mostly been studied over domains that are symmetric with respect to the real axis. This choice was motivated by some foundational results published in 2009, such as the Representation Formula for axially symmetric domains.

The present work studies slice regular functions over domains that are not axially symmetric, partly correcting the hypotheses of some previously published results. In particular, this work includes a Local Representation Formula valid without the symmetry hypothesis. Moreover, it determines a class of domains, called simple, having the following property: every slice regular function on a simple domain can be uniquely extended to the symmetric completion of its domain.


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Graziano Gentili
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy
MR Author ID: 189767
ORCID: 0000-0002-5001-2187
Email: graziano.gentili@unifi.it

Caterina Stoppato
Affiliation: Dipartimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, I-50134 Firenze, Italy
MR Author ID: 862712
ORCID: 0000-0001-9859-6559
Email: caterina.stoppato@unifi.it

Received by editor(s): July 21, 2020
Received by editor(s) in revised form: August 18, 2020
Published electronically: March 2, 2021
Additional Notes: This work was partly supported by INdAM, through: GNSAGA; INdAM project “Hypercomplex function theory and applications”. It was also partly supported by MIUR, through the projects: Finanziamento Premiale FOE 2014 “Splines for accUrate NumeRics: adaptIve models for Simulation Environments”; PRIN 2017 “Real and complex manifolds: topology, geometry and holomorphic dynamics”.
Communicated by: Filippo Bracci
Article copyright: © Copyright 2021 American Mathematical Society