Dirac and semi-Dirac pairs of differential operators
by
Mircea Martin
Department of Mathematics, Baker University, Baldwin City, KS 66006, USA

The standard Euclidean Dirac operators D = D_{euc, n} are first-order differential operators on Rⁿ, n ≥ 2, with coefficients in the real Clifford algebra A_n(R) associated with Rⁿ, that have the defining property D² = -D_{euc, n}, where D_{euc, n} stands for the Laplace operator on Rⁿ. As generalizations of this specific class of differential operators, we will investigate pairs (D, D^f) of first-order homogeneous differential operators on Rⁿ with coefficients in a real Banach algebra A, such that DD^f = m_LD_{euc, n} and \mathfrakD^fD = m_RD_{euc, n},  or     DD^f + \mathfrakD^fD = mD_{euc, n}, where m_L, m_R, or m are some elements of A. Every pair (D, D^f) that has the former property is called a Dirac pair of differential operators, and each pair (D, D^f) with the latter property is called a semi-Dirac pair. The two concepts have natural extensions in several complex variables, and in the setting of differential operators on a Clifford bundle over an oriented Riemannian manifold.

Our main goal is to prove that for any Dirac, or semi-Dirac pair, (D, D^f), we have two Cauchy-Pompeiu type, and two Bochner-Martinelli-Koppelman type integral representation formulas, one for D and another for D^f. In addition, we will show that the existence of such integral representation formulas characterizes the two classes of pairs of differential operators.

Date received: May 13, 2008