Cauchy-Riemann operators and local slice analysis over real alternative algebras

Abstract

We prove some formulas relating Cauchy-Riemann operators defined on hypercomplex subspaces of an alternative *-algebra to a differential operator associated with the concept of slice-regularity and to the spherical Dirac operator. These results in particular allow to introduce a definition of locally slice-regular function and open the path for local slice analysis. Since Cauchy-Riemann operators factor the corresponding Laplacian operators, the proven formulas let us also obtain several results about the harmonicity and polyharmonicity properties of slice-regular functions.

34 pages

References in corpus (3)

• Geometric function theory over quaternionic slice domains
• Slice regular functions and orthogonal complex structures over ${\mathbb{R}}^8$
• Eigenvalue problems for slice functions

Cited by in corpus (5)

• A unified notion of regularity in one hypercomplex variable
• A unified theory of regular functions of a hypercomplex variable
• Quaternionic slice regularity beyond slice domains
• A manifold Fueter-Sce phenomenon in one hypercomplex variable
• Monogenic functions over real alternative *-algebras: the several hypercomplex variables case