The vector-valued nonhomogeneous Tb theorem (Q2878692)

The article is devoted to obtaining a Banach space-valued extension of the Tb Theorem of Nazarov-Treil-Volberg [\textit{F. Nazarov} et al., Int. Math. Res. Not. 1997, No. 15, 703--726 (1997; Zbl 0889.42013)] on the boundedness of singular integral operators with respect to a measure \(\mu\) satisfying only an upper control on the size of balls. As the author notes, the aim of the present paper is ``to bring together two so-far distinct lines along which the classical Calderón-Zygmund theory has been generalized: one of them related to the domain, the other to the range of the functions under consideration''.NEWLINENEWLINELet us introduce some definitions. An operator \(T\) satisfies the \textit{rectangular weak boundedness property} if for any rectangle \(R\subset{\mathbb R}^N\), NEWLINE\[NEWLINE \left|\int_{{\mathbb R}^N}1_R\cdot T1_R\,d\mu\right|\leq\mu(R). NEWLINE\]NEWLINENEWLINENEWLINEA function \(b\in L^1_{loc}(\mu)\) is called \textit{weakly accretive} if NEWLINE\[NEWLINE \frac{1}{\mu(Q)}\left|\int_Q b\,d\mu\right|\geq\deltaNEWLINE\]NEWLINE for all cubes \(Q\) and some fixed positive \(\delta\). \(M_b\) denotes the operator of pointwise multiplication by \(b\), i.e. \(M_b:\,f\mapsto b\cdot f\).NEWLINENEWLINE\(\mathrm{BMO}_\lambda^p(\mu)\), \(\lambda,p\in[1,\infty)\), is the space of functions \(h\in L^1_{loc}(\mu)\) satisfying NEWLINE\[NEWLINE \left\|h\right\|_{\mathrm{BMO}_\lambda^p(\mu)}:= \sup_Q\left(\frac{1}{\mu(\lambda Q)}\int_Q\left|h-\left<h\right>_Q\right|^p\,d\mu\right)^{1/p}<\infty, NEWLINE\]NEWLINE where the supremum is taken over all cubes \(Q\subset{\mathbb R}^N\), and \(\left<h\right>_Q:=\frac{1}{\mu(Q)}\int_Q h\,d\mu\).NEWLINENEWLINEA Banach space \(X\) has the \textit{UMD property} if NEWLINE\[NEWLINE \left\|\sum_{k=1}^n\varepsilon_k d_k\right\|_{L^p\left(\mu;X\right)}\leq C \left\|\sum_{k=1}^nd_k\right\|_{L^p\left(\mu;X\right)} NEWLINE\]NEWLINE for any \(\left(d_k\right)_{k=1}^n\) -- a martingale difference sequence in \(L^p\left(\mu;X\right)\), and \(\varepsilon_k=\pm 1\).NEWLINENEWLINEUMD is a strong requirement. As the author notes, the UMD property implies reflexivity, but the converse is not true. For example, reflexive \(L^p\)-spaces, Sobolev spaces, and Besov spaces do not satisfy the UMD property.NEWLINENEWLINEThe main results of the article are four Tb theorems. Let us cite two of them.NEWLINENEWLINE``{Tb Theorem 1.} Let \(X\) be a UMD space and \(1<p<\infty\). Let \(T\) be a Calderón-Zygmund operator for which \(M_{b_2}TM_{b_1}\) satisfies the rectangular weak boundedness property and NEWLINE\[NEWLINE \left\|Tb_1\right\|_{\mathrm{BMO}_\lambda^1(\mu)}\leq 1,\quad \left\|T^*b_2\right\|_{\mathrm{BMO}_\lambda^1(\mu)}\leq 1. NEWLINE\]NEWLINE Then \(\left\|T\right\|_{{\mathcal L}\left(L^p\left(\mu;X\right)\right)}\leq C\).'' Here the constant \(C\) depends on some explicitly specified parameters.NEWLINENEWLINE``{Tb Theorem 4.} Let \(X\) be a UMD space and \(1<p<\infty\). Let \(T\) be an \({\mathcal L}(X)\)-valued Rademacher-Calderón-Zygmund operator for which \(M_{b_2}TM_{b_1}\) satisfies the rectangular weak Rademacher boundedness property. Let \(Y\subset{\mathcal L}(X)\) and \(Z\subset{\mathcal L}(X^*)\) be subspaces with the UMD property, and NEWLINE\[NEWLINE \left\|Tb_1\right\|_{\mathrm{BMO}_\lambda^p(\mu;Y)}\leq 1,\quad \left\|T^*b_2\right\|_{\mathrm{BMO}_\lambda^{p'}(\mu;Z)}\leq 1. NEWLINE\]NEWLINE Then \(\left\|T\right\|_{{\mathcal L}\left(L^p\left(\mu;X\right)\right)}\leq C\), where \(C\) is allowed to depend on \(Y\) and \(Z\), in addition to the usual parameters.''NEWLINENEWLINEIt is impossible to outline the ideas of the proofs in one page. This is done very well in Section 2 of the article which is called ''Strategy of the Proof with Historical Remarks''. Let us only note that the author has systematically developed several interesting methods and approaches to deal with the vector-valued case. The article is a serious investigation, which should be interesting for specialists in Harmonic Analysis, Functional Analysis, Operator Theory, and related areas.