Slice Regularity and Harmonicity on Clifford Algebras

Abstract

We present some new relations between the Cauchy-Riemann operator on the real Clifford algebra Rn of signature (0, n) and slice-regular functions on Rn. 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 R3 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 R 4 .

Log-biharmonicity and a Jensen formula in the space of quaternions

“…As pointed out by Perotti in [ [32], Section 6], for any class C 1 slice function f , the following two formulas hold true:…”

“…45 ([32]). Let Ω D be any circular domain and let f :Ω D ⊂ H → H be a slice function of class C 1 (Ω D ), then f is regular if and only if D CF f = −2∂ s f .…”

Spherical Coefficients of Slice Regular Functions

“…By [25,Theorem 6.3], if f is a slice regular function, then ∂ c ∂ s f equals −1/4 the four dimensional real Laplacian of f . From this point of view the formula in the previous lemma is related to the standard formula for the Laplacian of a product.…”

“…It was proven in [25] that the spherical derivative operator is indeed a differential operator when applied to slice functions. It is in fact equal to some other differential operator related to the operators∂ c , ∂ c and to the Cauchy-Dirac operator.…”

Some Notions of Subharmonicity over the Quaternions

“…In the present work, we consider several notions of subharmonicity related to the class of regular quaternionic functions. In contrast with the work [9], which studies the relation between quaternionvalued (or Clifford-valued) regular functions and real harmonicity, we consider real-valued functions of a quaternionic variable and look for new notions of subharmonicity compatible with composition with regular functions.…”