A differential criterium for regularity of quaternionic functions

Alessandro Perotti

doi:10.1016/S1631-073X(03)00284-X

Complex Analysis

A differential criterium for regularity of quaternionic functions
[Une condition différentielle de régularité pour les fonctions quaternioniennes]

Présenté par : Jean-Pierre Demailly

Alessandro Perotti ¹

¹ Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy

Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 89-92.

Résumé
Abstract

Soit $Ω \subseteq ℂ^{2}$ . Nous montrons l'existence de deux opérateurs différentiels T et N, à coefficients complexes, telle que une fonction $f : Ω \to ℍ$ de classe C¹ est régulière si et seulement si (N−jT)f=0 sur $\partial Ω$ (j un quaternion de base de $ℍ$ ) et f est harmonique. Nous obtenons aussi une généralisation d'un résultat de Kytmanov et Aizenberg. Nous montrons qu'une fonction harmonique complexe h sur $Ω$ ( $\partial Ω$ connexe) est holomorphe si et seulement si ${\bar{\partial}}_{n} h = \bar{aL (h)}$ sur $\partial Ω$ , où ${\bar{\partial}}_{n}$ est la composante normale de $\bar{\partial}$ , L est un opérateur différentiel tangentiel de Cauchy–Riemann et $a \in ℂ$ .

Let $Ω \subseteq ℂ^{2}$ . We prove that there exist differential operators T and N, with complex coefficients, such that a function $f : \bar{Ω} \to ℍ$ of class C¹ is regular if and only if (N−jT)f=0 on $\partial Ω$ (j a basic quaternion) and f is harmonic on $Ω$ . At the same time we generalize a result of Kytmanov and Aizenberg. We show that a complex harmonic function h on $Ω$ ( $\partial Ω$ connected) is holomorphic if and only if ${\bar{\partial}}_{n} h = \bar{aL (h)}$ on $\partial Ω$ , where ${\bar{\partial}}_{n}$ is the normal part of $\bar{\partial}$ , L is a tangential Cauchy–Riemann operator and $a \in ℂ$ .

Reçu le : 2003-03-14
Accepté le : 2003-06-11
Publié le : 2003-07-09

DOI : 10.1016/S1631-073X(03)00284-X

Affiliations des auteurs :

Alessandro Perotti ¹

¹ Dipartimento di Matematica, Università degli Studi di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy

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     title = {A differential criterium for regularity of quaternionic functions},
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     pages = {89--92},
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Alessandro Perotti. A differential criterium for regularity of quaternionic functions. Comptes Rendus. Mathématique, Volume 337 (2003) no. 2, pp. 89-92. doi : 10.1016/S1631-073X(03)00284-X. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00284-X/

Bibliographie
Cité par

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[8] M.V. Shapiro; N.L. Vasilevski Quaternionic ψ-hyperholomorphic functions, singular integral operators and boundary value problems. I. ψ-hyperholomorphic function theory, Complex Variables Theory Appl., Volume 27 (1995) no. 1, pp. 17-46

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