A singular integral equation with a multiple-valued kernel on a hyperelliptic Riemann surface

Abstract

In this note, we generalize to the case of a singular integral equation with the multiplevalued analog of the Cauchy kernel [4, 5] on a hyperelliptic Riemann surface. We give a complete treatment of this equation in-the case of a composite piecewise-smooth contour. The solution is found in a class of Holder functions. As an application, we consider a homogeneous singular integral equation of polynomial type on the torus. The principal result is a characterization of the number of linearly independent solutions. N I. Let R be a double-sheeted Riemann surface whose algebraic equation is u 2 = ~ (z--aj) j=l (z -bj) and whose genus is p = N -I. Let L Be a composite piecewise-smooth contour on R consisting of a finite number of smooth oriented (open or closed) curves which can have a