We prove that the existence of an accretive system in the sense of
M. Christ is equivalent to the boundedness of a
Calderón-Zygmund operator on L2(μ)$. 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\sp ∞. Thus we answer positively a question of
Christ as to whether the L\sp ∞-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 Calderón-Zygmund operators with
respect to very bad measures.