Cotlar-Stein lemma and the \(Tb\) theorem (Q2706583)

The Cotlar-Stein lemma provides an interesting approach to problems concerning singular integral operators of Calderón-Zygmund. Applications of this lemma can be found in several papers [\textit{A. P. Calderón} and \textit{R. Vaillancourt}, Proc. Natl. Acad. Sci. USA 69, 1185-1187 (1972; Zbl 0244.35074) and \textit{A. W. Knapp} and \textit{E. M. Stein}, Ann. Math., II. Ser. 93, 489-578 (1971; Zbl 0257.22015), amongst others].NEWLINENEWLINENEWLINEIn the proof of the \(T1\) theorem given by \textit{G. David} and \textit{J.-L. Journé} [Ann. Math., II. Ser. 120, 371-397 (1984; Zbl 0567.47025)] the Cotlar-Stein lemma was also used. However, it is not known how this lemma can be applied to prove the generalization of the \(T1\) theorem, the so-called \(Tb\) theorem, which was proved by \textit{G. David}, \textit{J.-L. Journé} and \textit{S. Semmes} [Rev. Mat. Iberoam. 1, No. 4, 1-56 (1985; Zbl 0604.42014)]. This fact motivates the aim of the paper: to give a generalization of the Cotlar-Stein lemma that allows to prove a special case of the \(Tb\) theorem.NEWLINENEWLINENEWLINEIn this theorem, the authors give sufficient conditions for certain Calderón-Zygmund operators \(T\) to be bounded on \(L^2(\mathbb{R}^n)\). These conditions are essential that the images of a certain positive real-valued function \(b\) under the actions of the operator \(T\) and its adjoint both be \(0\) and a certain weak boundedness property be satisfied by the operator \(M_b TM_b\), \(M_b\) being the multiplication operator by the function \(b\).NEWLINENEWLINENEWLINEThe generalized Cotlar-Stein lemma, which has a formulation similar to the known version but includes the multiplication operator \(M_b\), is then a successful tool in the proof of the theorem.