Z,-boundedness of the Cauchy Transform on Smooth Non-lipschitz Curves

If A is a Lipschitz function, i.e., || A |L < °°, then %A makes a very significant example of non-convolution type singular integral operators. The problem of L -boundedness of the Cauchy transform was raised and solved when || A |L is small by A. P. Calderόn in relation to the Dirichlet problem on Lipschitz domains [Call, Cal2]. Since then, it has been a central problem in the theory of singular integral operators and several significant techniques has been developed to deal with this problem. Among them are the 7X1) -Theorem of David and Journe, the technique of Coifman, Mclntosh, and Meyer, and the technique of Coifman, Jones, and Semmes [DJ, C.M.M, CJ.S]. We refer to [Chi, Mur] for a history of development in the last decades on the theory of the Cauchy transform. If IIA'IL = °°, then the Cauchy kernel K(x, y) given in (1.1) is not a standard kernel. An integral kernel on the line is called a standard kernel if it satisfies \K(x, y ) \ < C\χ-y\~ a n d \VXΛKb9y)\ < C\χ-y\~\ If | | J 4 ' | L = ° ° , then the Cauchy kernel does not satisfy both estimates. So, the theory of the singular integral operators may not be applied directly. Nevertheless, the question of L -boundedness of <βA is still an interesting one. In this paper, we deal with L -boundedness of cβA when A is smooth and \\A |L = °°.