Riemann-Hilbert problem in a family of weighted Hölder spaces (Q304642)

Let \(D\) be a simply connected domain bounded by a simple piecewise Lyapunov contour \(\Gamma\) which consists of the finite number of smooth arcs. The set of arc endpoints is denoted by \(F\). Let a given function \(G(t)\) be continuous on \(\Gamma \backslash F\) and the one-sided limits of \(G(t)\) exist at \(F\). The authors discuss the scalar Riemann-Hilbert problem \(\mathrm{Re }G(t) \phi(t) =f(t)\), \(t \in \Gamma \backslash F\) where \(f(t)\) is a given Hölder continuous function, \(\phi(z)\) is analytic in \(D\), Hölder continuous in its closure except \(F\) where it has a power-law behavior. The authors give the complete description of the solvability for this problem.