Slice regularity and harmonicity on Clifford algebras

Abstract

We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra ${\mathbb{R}}_n$ of signature $(0,n)$ and slice-regular functions on ${\mathbb{R}}_n$. The class of slice-regular functions, which comprises all polynomials with coefficients on one side, is the base of a recent function theory in several hypercomplex settings, including quaternions and Clifford algebras. In this paper we present formulas, relating the Cauchy-Riemann operator, the spherical Dirac operator, the differential operator characterizing slice regularity, and the {spherical derivative} of a slice function. The computation of the Laplacian of the spherical derivative of a slice regular function gives a result which implies, in particular, the Fueter-Sce Theorem. In the two four-dimensional cases represented by the paravectors of ${\mathbb{R}}_3$ and by the space of quaternions, these results are related to zonal harmonics on the three-dimensional sphere and to the Poisson kernel of the unit ball of ${\mathbb{R}}^4$.

17 pages. To appear in "Topics in Clifford Analysis - A Special Volume in Honor of Wolfgang Sprößig", Springer series Trends in Mathematics

References in corpus (3)

• Division algebras of slice functions
• Log-biharmonicity and a Jensen formula in the space of quaternions
• A four dimensional Jensen formula

Cited by in corpus (15)

• The harmonicity of slice regular functions
• Almansi-type theorems for slice-regular functions on Clifford algebras
• Generalized partial-slice monogenic functions
• On the Fueter-Sce theorem for generalized partial-slice monogenic functions
• Eigenvalue problems for slice functions
• Partial slice regularity and Fueter's Theorem in several quaternionic variables
• A four dimensional Jensen formula
• Some notions of subharmonicity over the quaternions
• Almansi-type decomposition for slice regular functions of several quaternionic variables
• Implementing zonal harmonics with the Fueter principle
• Slice regular holomorphic Cliffordian functions of order $k$
• Spherical coefficients of slice regular functions
• A four dimensional Bernstein Theorem
• Dunkl approach to slice regular functions
• A local Cauchy integral formula for slice-regular functions