Bilinear decompositions and commutators of singular integral operators

Let b be aBMO-function. It is well-known that the linear commutator [b, T ] of a Calderón-Zygmund operator T does not, in general, map continuously H(R) into L(R). However, Pérez showed that if H(R) is replaced by a suitable atomic subspace H b(R ) then the commutator is continuous from H b(R ) into L(R). In this paper, we find the largest subspace H b (R ) such that all commutators of Calderón-Zygmund operators are continuous from H b (R ) into L(R). Some equivalent characterizations of H b (R ) are also given. We also study the commutators [b, T ] for T in a class K of sublinear operators containing almost all important operators in harmonic analysis. When T is linear, we prove that there exists a bilinear operators R = RT mapping continuously H (R) × BMO(R) into L(R) such that for all (f, b) ∈ H(R)×BMO(R), we have (1) [b, T ](f) = R(f, b) + T (S(f, b)), where S is a bounded bilinear operator from H(R) × BMO(R) into L(R) which does not depend on T . In the particular case of T a CalderónZygmund operator satisfying T 1 = T ∗1 = 0 and b in BMO(R)– the generalized BMO type space that has been introduced by Nakai and Yabuta to characterize multipliers of BMO(R) –we prove that the commutator [b, T ] maps continuously H b (R ) into h(R). Also, if b is in BMO(R) and T ∗1 = T ∗b = 0, then the commutator [b, T ] maps continuously H b (R ) into H(R). When T is sublinear, we prove that there exists a bounded subbilinear operator R = RT : H (R) × BMO(R) → L(R) such that for all (f, b) ∈ H(R)×BMO(R), we have (2) |T (S(f, b))| −R(f, b) ≤ |[b, T ](f)| ≤ R(f, b) + |T (S(f, b))|. The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong L-estimates.