Boundedness of Cauchy singular integral operator under Hölder norm

Abstract

In this article, we will investigate the boundedness of the Cauchy singular integral operator under Hölder norm. This boundedness can be obtained by improving the method used in the early proof of the famous 2P theorems (Plemelj–Sokhotski formula and Privalov–Muskhelishvili theorem) by Jinyuan Du, which deepens the 2P theorems. Then, various Cauchy singular integral operators on the system of curves or open arcs are also proved to be bounded.