Bilinear decompositions and commutators of singular integral operators

Abstract

Let b b be a B M O BMO -function. It is well known that the linear commutator [ b , T ] [b, T] of a Calderón-Zygmund operator T T does not, in general, map continuously H 1 ( R n ) H^1(\mathbb R^n) into L 1 ( R n ) L^1(\mathbb R^n) . However, Pérez showed that if H 1 ( R n ) H^1(\mathbb R^n) is replaced by a suitable atomic subspace H b 1 ( R n ) \mathcal H^1_b(\mathbb R^n) , then the commutator is continuous from H b 1 ( R n ) \mathcal H^1_b(\mathbb R^n) into L 1 ( R n ) L^1(\mathbb R^n) . In this paper, we find the largest subspace H b 1 ( R n ) H^1_b(\mathbb R^n) such that all commutators of Calderón-Zygmund operators are continuous from H b 1 ( R n ) H^1_b(\mathbb R^n) into L 1 ( R n ) L^1(\mathbb R^n) . Some equivalent characterizations of H b 1 ( R n ) H^1_b(\mathbb R^n) are also given. We also study the commutators [ b , T ] [b,T] for T T in a class K \mathcal K of sublinear operators containing almost all important operators in harmonic analysis. When T T is linear, we prove that there exists a bilinear operator R = R T \mathfrak R= \mathfrak R_T mapping continuously H 1 ( R n ) × B M O ( R n ) H^1(\mathbb R^n)\times BMO(\mathbb R^n) into L 1 ( R n ) L^1(\mathbb R^n) such that for all ( f , b ) ∈ H 1 ( R n ) × B M O ( R n ) (f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n) we have [ b , T ] ( f ) = R ( f , b ) + T ( S ( f , b ) ) , \begin{equation}[b,T](f)= \mathfrak R(f,b) + T(\mathfrak S(f,b)), \end{equation} where S \mathfrak S is a bounded bilinear operator from H 1 ( R n ) × B M O ( R n ) H^1(\mathbb R^n)\times BMO(\mathbb R^n) into L 1 ( R n ) L^1(\mathbb R^n) which does not depend on T T . In the particular case of T T a Calderón-Zygmund operator satisfying T 1 = T ∗ 1 = 0 T1=T^*1=0 and b b in B M O log ( R n ) BMO^\textrm {log}(\mathbb R^n) , the generalized B M O BMO type space that has been introduced by Nakai and Yabuta to characterize multipliers of B M O ( R n ) BMO(\mathbb {R}^n) , we prove that the commutator [ b , T ] [b,T] maps continuously H b 1 ( R n ) H^1_b(\mathbb R^n) into h 1 ( R n ) h^1(\mathbb R^n) . Also, if b b is in B M O ( R n ) BMO(\mathbb R^n) and T ∗ 1 = T ∗ b = 0 T^*1 = T^*b = 0 , then the commutator [ b , T ] [b, T] maps continuously H b 1 ( R n ) H^1_b (\mathbb R^n) into H 1 ( R n ) H^1(\mathbb R^n) . When T T is sublinear, we prove that there exists a bounded subbilinear operator R = R T : H 1 ( R n ) × B M O ( R n ) → L 1 ( R n ) \mathfrak R= \mathfrak R_T: H^1(\mathbb R^n)\times BMO(\mathbb R^n)\to L^1(\mathbb R^n) such that for all ( f , b ) ∈ H 1 ( R n ) × B M O ( R n ) (f,b)\in H^1(\mathbb R^n)\times BMO(\mathbb R^n) we have | T ( S ( f , b ) ) | − R ( f , b ) ≤ | [ b , T ] ( f ) | ≤ R ( f , b ) + | T ( S ( f , b ) ) | . \begin{equation}|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,T](f)|\leq \mathfrak R(f,b) + |T(\mathfrak S(f,b))|. \end{equation} The bilinear decomposition (1) and the subbilinear decomposition (2) allow us to give a general overview of all known weak and strong L 1 L^1 -estimates.