TY - JOUR
T1 - On Fundamental Solutions in Clifford Analysis
AU - Brackx, F.
AU - de Schepper, H.
AU - Luna-Elizarrarás, M. E.
AU - Shapiro, M.
PY - 2012/4
Y1 - 2012/4
N2 - Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ∂ called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dirac operators ∂ z and ∂ z† which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ∂E = δ. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.
AB - Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ∂ called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h-) monogenic functions, of two Hermitean Dirac operators ∂ z and ∂ z† which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford-Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford-Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ∂E = δ. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.
UR - http://www.scopus.com/inward/record.url?scp=84858797705&partnerID=8YFLogxK
U2 - 10.1007/s11785-010-0055-8
DO - 10.1007/s11785-010-0055-8
M3 - Artículo
SN - 1661-8254
VL - 6
SP - 325
EP - 339
JO - Complex Analysis and Operator Theory
JF - Complex Analysis and Operator Theory
IS - 2
ER -