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

中文翻译：

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矩阵 Riemann-Hilbert 问题与跨越 Carleson 等高线的跳跃

我们为一类低规则性的跳跃轮廓和跳跃矩阵开发了 $$n \times n$$n×n 矩阵黎曼-希尔伯特问题的理论。我们的基本假设是等高线 $$\Gamma $$Γ 是黎曼球体中简单闭合的卡尔森曲线的有限并集。特别是，允许​​具有尖点、拐角和非横向交叉点的无界轮廓。我们引入了 $$L^p$$Lp-Riemann-Hilbert 问题的概念，并建立了基本的唯一性结果和 Fredholm 性质。我们还研究了 Fredholmness 对唯一可解性的影响，并证明了一个关于轮廓变形的定理。