On the left linear Riemann problem in Clifford analysis

We consider a left-linear analogue to the classical Riemann problem: Dau = 0 in Rn\Γ u+ = H(x)u− + h(x) on Γ |u(x)| = O(|x|n2−1) as |x| → ∞. For this purpose, we state a Borel-Pompeiu formula for the disturbed Dirac operator Da = D+a with a paravector a and some functiontheoretical results. We reformulate the Riemann problem as an integral equation: Pau+ HQau = h on Γ, where Pa = 2(I + Sa) and Qa = I − Pa. We demonstrate that the essential part of the singular integral operator Sa which is constructed by the aid of a fundamental solution of D + a is just the singular integral operator S associated to D. In case Sa is simply S and Γ = Rn−1, then under the assumptions 1.H= ∑ β Hβeβ and allHβ are real-valued, measurable and essentially bounded; 2. (1 +H(x))(1 + H(x)) and H(x)H̄(x) are real numbers for all x ∈ Rn−1; 3. the scalar part H0 of H fulfils H0(x) > ε > 0 for all x ∈ Rn−1, the Riemann problem is uniquely solvable in L2,C(Rn−1) and the successive approximation un := 2(1 +H)−1h− (1 +H)(1−H)Sun−1, n = 1, 2, . . . , Received by the editors July 1995. Communicated by R. Delanghe. 1991 Mathematics Subject Classification : 35F15; 47B.