Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces

Recently, F. Nazarov, S. Treil and A. Volberg (and independently X. Tolsa) have extended the classical theory of Calderón-Zygmund operators to the context of a "non-homogeneous" space $(\mathbb {X},d,\mu )$, where, in particular, the measure $\mu $ may be non-doubling. In the present work we study weighted inequalities for these operators. Specifically, for $1 < p < \infty $, we identify sufficient conditions for the weight on one side, which guarantee the existence of another weight in the other side, so that the weighted $L^p$ inequality holds. We deal with this problem by developing a vector-valued theory for Calderón-Zygmund operators on non-homogeneous spaces which is interesting in its own right. For the case of the Cauchy integral operator, which is the most important example, we even prove that the conditions for the weights are also necessary.