Cauchy singular integral operator with parameters in Log-H\"older spaces

This paper is motivated by a claim in the classical textbook of Muskhelishvili concerning the Cauchy singular integral operator $S$ on H"older functions with parameters. To the contrary of the claim, a counter example was constructed by Tumanov which shows that $S$ with parameters fails to maintain the same H"older regularity with respect to the parameters. In view of the example, the behavior of the Cauchy singular integral operator with parameters between a type of Log-H"older spaces is investigated to obtain the sharp norm estimates. At the end of the paper, we discuss its application to the $\bar{\mathrm{\partial }}$ problem on product domains.