Accretive system Tb-theorems on nonhomogeneous spaces

We prove that the existence of an accretive system in the sense of M. Christ is equivalent to the boundedness of a Calderon-Zygmund operator on L-2(mu). We do not assume any kind of doubling condition on the measure mu, so we are in the nonhomogeneous situation. Another interesting difference from the theorem of Christ is that we allow the operator to send the functions of our accretive system into the space bounded mean oscillation (BMO) rather than L-infinity. Thus we answer positively a question of Christ as to whether the L-infinity-assumption can be replaced by a BMO assumption. We believe that nonhomogeneous analysis is useful in many questions at the junction of analysis and geometry In fact, it allows one to get rid of all superfluous regularity conditions for rectifiable sets. The nonhomogeneous accretive system theorem represents a flexible tool for dealing with Calderon-Zygmund operators with respect to very bad measures.