After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular functions, 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 Semiglobal 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 $f : \Omega\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $\Omega$ (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.

Quaternionic slice regularity beyond slice domains / Ghiloni, Riccardo; Stoppato, Caterina. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - STAMPA. - 306:3(2024), pp. 5501-5542. [10.1007/s00209-024-03434-7]

### Quaternionic slice regularity beyond slice domains

#### Abstract

After Gentili and Struppa introduced in 2006 the theory of quaternionic slice regular functions, 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 Semiglobal 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 $f : \Omega\to\mathbb{H}$ a family of holomorphic stem functions and a family of induced slice regular functions. As we tighten the hypotheses on $\Omega$ (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.
##### Scheda breve Scheda completa Scheda completa (DC)
2024
3
Ghiloni, Riccardo; Stoppato, Caterina
Quaternionic slice regularity beyond slice domains / Ghiloni, Riccardo; Stoppato, Caterina. - In: MATHEMATISCHE ZEITSCHRIFT. - ISSN 0025-5874. - STAMPA. - 306:3(2024), pp. 5501-5542. [10.1007/s00209-024-03434-7]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/410433