A local result for systems of Riemann-Hilbert barrier problems

Abstract:

The Riemann-Hilbert barrier problem (for $n$ pairs of functions) \[ G{\Phi ^ + } = {\Phi ^ - } + g\] is investigated for the square integrable functions on a union of analytic Jordan curves $C$ bounding a domain in the complex plane. In the special case, where at each point ${t_0}$ of $C$ the symbol $G$ has at most two essential cluster values ${G_1}({t_0}),{G_2}({t_0})$, then the condition $\det [(1 - \lambda ){G_1}({t_0}) + \lambda {G_2}({t_0})] \ne 0$, for all ${t_0}$ in $C$ and all $\lambda (0 \leqslant \lambda \leqslant 1)$, implies the Riemann-Hilbert operator is Fredholm. In the case, where for some ${t_0}$ in $C$ and some ${\lambda _0}(0 \leqslant {\lambda _0} \leqslant 1),\det [(1 - {\lambda _0}){G_1}({t_0}) + {\lambda _0}{G_2}({t_0})] = 0$, the Riemann-Hilbert operator is not Fredholm. An application is given to systems of singular integral equation on ${L^2}(E)$, where $E$ is a measurable subset of $C$.