Vector valued commutators on non-homogeneous spaces

Let mu be a Borel measure on R-d which may be non doubling. The only condition that p must satisfy is mu(Q) <= c(0)l(Q)(n) for any cube Q subset of R-d with sides parallel to the coordinate axes and for some fixed n with 0 < n <= d. This paper is to develop the vector valued commutator theory in the context of the non-homogeneous spaces. As an application, the boundedness of the maximal commutator of any Calderon-Zygmund operator on the non-homogeneous space with a RBMO(mu) function introduced by Tolsa in [9] is obtained.