Generalized Riemann problem on a Riemann surface (Q1594180)

The author considers the Riemann-Hilbert type problem with the matrix boundary condition \[ \text{Re } A(t)\phi_\gamma^+(t) =f(t), \quad 0<t<1, \] where \(A(t)\) is a given complex-valued \(m\times m\) matrix-function and \(\phi_\gamma^+(t)\) stands for the parameterized boundary values of a function analytic in a domain on a Riemann surface. Namely, let \(D\) be an open relatively compact set on a Riemann surface. It is assumed that \(D\) has one-sided piece-wise smooth boundary, which means that there exists a family \(\Gamma=(\Gamma_j)_1^m\) of smooth arcs covering the boundary of \(D\) and there exists the corresponding family \(D_j\) (\(j=1,\dots ,m\)) of pair-wise disjoint subdomains of \(D\) such that the boundary \(\partial D_j\) is a simple piece-wise smooth contour and contains \(\Gamma_j\). In such a general setting, the author's goal is to avoid the specific nature of the Riemann surface and to reduce the problem by triangulation to the plane case. The boundary problem is considered under the assumption that the boundary values are in the weighted Hölder class on the boundary. The weight function is of power type with the powers ``fixed'' to the end-points of the arcs \(\Gamma_j\). The author gives some statements on the solvability of the problem and gives a formula for the index, generalizing that one of the theory of boundary problems in weighted Hölder classes on the plane.