A differential criterium for regularity of quaternionic functions

Perotti, Alessandro

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

Soit $\Omega \subseteq {ℂ}^2$. Nous montrons l'existence de deux opérateurs différentiels T et N, à coefficients complexes, telle que une fonction $f:\Omega \to ℍ$ de classe C¹ est régulière si et seulement si (N−jT)f=0 sur $\partial \Omega$ (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 $\Omega$ ($\partial \Omega$ connexe) est holomorphe si et seulement si ${\overline{\partial }}_n\mathrm{h}=\overline{\mathrm{aL}\left(\mathrm{h}\right)}$ sur $\partial \Omega$, où ${\overline{\partial }}_n$ est la composante normale de $\overline{\partial }$, L est un opérateur différentiel tangentiel de Cauchy–Riemann et $\mathrm{a}\in ℂ$.

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

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