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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2312.13982 |
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| _version_ | 1866914828724994048 |
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| author | Ghiloni, Riccardo Stoppato, Caterina |
| author_facet | Ghiloni, Riccardo Stoppato, Caterina |
| contents | After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular function, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function $f:Ω\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $Ω$ (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent $f$ better and better and allow to prove increasingly stronger results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_13982 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quaternionic slice regularity beyond slice domains Ghiloni, Riccardo Stoppato, Caterina Complex Variables 30G35 After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular function, the theory has focused on functions on the so-called slice domains. The present work defines the class of speared domains, which is a rather broad extension of the class of slice domains, and proves that the theory is extremely interesting on speared domains. A Semi-global Extension Theorem and a Semi-global Representation Formula are proven for slice regular functions on speared domains: they generalize and strengthen some known local properties of slice regular functions on slice domains. A proper subclass of speared domains, called hinged domains, is defined and studied in detail. For slice regular functions on a hinged domain, a Global Extension Theorem and a Global Representation Formula are proven. The new results are based on a novel approach: one can associate to each slice regular function $f:Ω\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $Ω$ (from an arbitrary quaternionic domain to a speared domain, to a hinged domain), these families represent $f$ better and better and allow to prove increasingly stronger results. |
| title | Quaternionic slice regularity beyond slice domains |
| topic | Complex Variables 30G35 |
| url | https://arxiv.org/abs/2312.13982 |