  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
$\bar\partial$ problem on product domains.
