Let \(\mathcal{A}\) be the universal Clifford algebra constructed over a real \(n\)-dimensional quadratic vector space with an orthogonal basis \(e_1, e_2, \ldots , e_n\). A basis for \(\mathcal{A}\) is given by \[ \{e_A : A=(h_1, h_2, \ldots , h_r), \;1\leq h_1 < h_2 < \ldots < h_r \leq r, 1\leq r \leq n \}. \] Let \(\Omega \subset \mathbb{R}^m \times \mathbb{R}^k\). There are considered functions \(f: \Omega \rightarrow \mathcal{A}\), \(f(x,y)=\sum_A f_A(x,y)e_A.\) The generalized Cauchy-Riemann operators are defined as follows \[ D_x =\sum_{i=1}^m e_i \frac{\partial}{\partial x_i} , \;D_y =\sum_{j=1}^k e_{m+j}\frac{\partial}{\partial y_j}. \] Let \((g,h)\) be a pair of \(\mathcal{A}\)-valued functions. The authors study the inhomogeneous system of equations \(D_xf=g, fD_y=h.\) At first they study the case of the algebra of quaternions. Then they extend their results to the case of Clifford algebra.
Earlier Le Hung Son [Complex Var. Theory Appl. 20, No.1–4, 255–263 (1992; Zbl 0768.30038)] investigated analogous questions. The authors weaken some assumptions of Le Hung Son.
The article is a continuation of the works of the same authors [Far East J. Math. Sci. (FJMS) 16, No. 3, 313–349 (2005; Zbl 1095.30041), ibid. 20, No. 3, 309–229 (2006; Zbl 1158.30037)].
For the entire collection see [Zbl 1134.30002].