Singular integral operators on Riemann surfaces and elliptic differential systems
On some closed Riemann surface \(M\) of genus \(\geq 0\), a first-order elliptic differential system for two unknown functions \(u,v\) is considered which, by \(w=u+iv\), is rewritten in the complex form \[ \overline\partial w=q_1 \partial w+q_2 \overline{\partial w}+Aw+ B\overline w;\tag{1} \] here is assumed that \(A,B\) belong to \(L_p(M)\), \(p>2\), and \(q_1\) and \(q_2\) are measurable functions satisfying an ellipticity condition for (1). For the investigation of solutions to (1), a representation of the form \(w=\exp T_0\varphi\) with \(T_0 \varphi= -{1\over\pi} \iint{\varphi(t) d\sigma_t\over t-z}\) is utilized; hence, \[ \varphi= q_1S\varphi+ q_2{\overline {\exp T_0\varphi} \over\exp T_0 \varphi} \overline {\partial T_0\varphi} +A+B{\overline {\exp T_0\varphi}\over \exp T_0 \varphi}. \tag{2} \] First, as auxiliary tools some period relations for Abelian differentials are derived in Section 2. Then the operator \(S\) (introduced in ansatz (2)) is defined and studied in Sec. 3. Finally, in Sec. 4, the existence of a regular solution (a generalized solution without singularities) to (1) is proved. In view of (2) this involves the consideration of an integral equation and an application of a fixed-point argument.
scientific article; zbMATH DE number 1463799