Potential Type Operators on Curves with Vorticity Points

Abstract

We study potential type operators on certain non-Lipschitz curves $\Gamma$. The curves under consideration are locally Lyapunov except for a finite set $F$ of singular points. The normal vector $v(y)$ to the curve $\Gamma$ does not have a limit at the singular points and, moreover, $v(y)$ may be an oscillating and rotating vector function in a neighborhood of the singular points. We establish a Fredholm theory of potential type operators in the spaces $L_{p,w}(\Gamma ,{\mathbb{C}}^n)$ where $p\in (1,\infty )$ and $w$ is a weight satisfying the Muckenhoupt condition.