Matrix Riemann-Hilbert problems with jumps across Carleson contours

Abstract

We develop a theory of $n \times n$-matrix Riemann-Hilbert problems for a class of jump contours and jump matrices of low regularity. Our basic assumption is that the contour $\Gamma$ is a finite union of simple closed Carleson curves in the Riemann sphere. In particular, contours with cusps, corners, and nontransversal intersections are allowed. We introduce a notion of $L^p$-Riemann-Hilbert problem and establish basic uniqueness results and a vanishing lemma. We also investigate the implications of Fredholmness for the unique solvability and prove a theorem on contour deformation.