Cauchy-green Type Formulae in Clifford Analysis

A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation (Dk+Yfcrn~JQ bmDm)f = 0 , where each bm is a complex number, D is the Dirac operator in R" , and f is defined on a domain in R" and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximation theorem. We also introduce a Bergman kernel for square integrable solutions to (D + X)f = 0 over bounded domains with piecewise C1 , or Lipschitz, boundary. Introduction It is well understood [1, 3, 4] that solutions to the Dirac equation, Df = 0, in R" can be described by a Cauchy integral formula. Here, D stands for the homogeneous Dirac operator _]"=x ejd/dXj , and the e, 's are the generators of a real Clifford algebra A„ . As D2 = -A, the negative Laplacian over 7?" , it may easily be shown (see [8]) that Green's formula for harmonic functions can be modified via Clifford algebras to more closely resemble a Cauchy integral formula. In [12], X. Zhenyuan shows that solutions to the inhomogeneous Dirac equation (D + k)f = 0, with k £ C, also possess a Cauchy integral formula, with a Cauchy kernel E_x(x y). A similar kernel for this equation is also produced for the case n = 3 by Giirlebeck and Sprossig [4, Chapter 4] and is used to describe boundary value problems for Helmholtz' equation An = k2h . Using the kernel Ex , produced in [12], these results automatically generalize to the cases n > 3. Clifford algebras have also been used by Mitrea [6] to study boundary value problems and associated Hp spaces for the Helmholtz equation over nonsmooth domains. In [11], Sommen and X. Zhenyuan construct a Cauchy kernel for D k as a power series _^=xamGm(x), where am £ R, and Gm(x) is a fundamental solution to the operator Dm . In this paper, we show that this method of construction generalizes to allow us to obtain Cauchy kernels and CauchyGreen type integral formulae for solutions to each polynomial equation (Dk + _lm~=obmDm)h = 0, with bm £ C. It automatically follows that many existing results in Clifford analysis extend to the context described here; for instance, the approximation theorems described in [ 1, Chapter 3] automatically Received by the editors October 14, 1993 and, in revised form, December 13, 1993; originally communicated to the Proceedings of the AMS by Palle E. T. Jorgensen. 1991 Mathematics Subject Classification. Primary 30G35. ©1995 American Mathematical Society 0002-9947/95 $1.00+ $.25 per page 1331 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use