Let T be a Calderon-Zygmund operator in a "non-homogeneous" space (X,d,µ), where, in particular, the measure µ may be non- doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil and A. Vol- berg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T?, which is the supremum of the trun- cated operators associated with T. Specifically, for 1 < p < 1, we obtain sucient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L p weighted inequality holds for T?. The main tool to deal with this problem is the theory of vector-valued inequalities for T? and some re- lated operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderon-Zygmund operators in non-homogeneous spaces, developed in our previous pa- per (GM). For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C? to characterize the existence of principal values for functions in weighted L p .