On the existence of principal values for the Cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X, d, mu ), where, in particular the measure mu may be non-doubling. Much of the cla ssical theory of singular integrals has been recently extended to this cont ext by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T-star, which is the supremum of the truncated operators associated with T. Specifically, for 1 < p < infinity, we obtain sufficient conditions for the weight in on e side, which guarantee that another weight exists in the other side, so th at the corresponding L-P weighted inequality holds for T-star. The main too l to deal with this problem is the theory of vector-valued inequalities for T-star and some related operators. We discuss it first by showing how thes e operators are connected to the general theory of vector-valued Calderon-Z ygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we app ly the two-weight inequalities for C-star to characterize the existence of principal values for functions in weighted L-P.