Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss

Let C f (g) = T (f • g)f • T (g) where T is the Calderón-Zygmund singular integral operator.It is a well known result of Coifman, Rochberg and Weiss that the boundedness of this operator on L p (R n ) is equivalent to f being in the space BMO.In this paper we introduce a version of this operator which allows f to be in the Besov-Lipschitz class Λβ , with 0 < β < n.For 1 < p < ∞, p is the dual, 1/p + 1/p = 1, and we use the notation