The topic of study are weighted norm inequalities for iterated commutators of multilinear Calderón-Zygmund operators and pointwise multiplication with functions in \(\text{BMO}(\mathbb{R}^n)\). Let \(T\) be a multilinear Calderón-Zygmund operator and \(\mathbf{b}=(b_1,\dots,b_m)\in\text{BMO}^m(\mathbb{R}^n)\). Define the commutator of \(b_j\) and \(T\) in the \(j\)-th entry of \(T\) by \[ [b_j,T]_j(f_1,\dots,f_m) = b_j T(f_1,\dots,f_j,\dots,f_m) - T(f_1,\dots,b_j f_j,\dots,f_m). \] The operators studied in this paper are iterated commutators of the form \[ T_{\prod \mathbf{b}}(f_1,\dots,f_m) = [b_1,[b_2,\dots[b_{m-1},[b_m,T]_m]_{m-1}\dots]_2]_1(f_1,\dots,f_m) . \] For \(\vec{p}=(p_1,\dots,p_m)\), \(1\leq p_1,\dots,p_m <\infty\), a multiple weight \(\vec{w}=(w_1,\dots,w_m)\) is said to satisfy the multilinear \(A_{\vec{p}}\) condition if for \[ \nu_{\vec{w}}=\prod_{j=1}^{m}w_j^{p/p_j} \] it holds that \[ \sup_{Q}\left(\frac{1}{|Q|}\int_Q \nu_{\vec{w}} \right)^{1/p}\prod_{j=1}^{m}\left(\frac{1}{|Q|}\int_Q w_j^{1-p_j^{'}} \right)^{1/p_j^{'}} < \infty, \] where the \(Q\) are cubes and \(p_j^{'} = 1 - \frac{1}{p_j}\).
The authors obtain the following strong-type weighted bounds: \[ \| T_{\prod \mathbf{b}}(f_1,\dots,f_m) \|_{L^{p}(\nu_{\vec{w}})} \leq C\prod_{j=1}^{m}\|b_j \|_{BMO} \prod_{j=1}^{m}\|f_j \|_{L^{p_j}(w_j)}, \] where \(C\) is a constant and \(\vec{w}\in A_{\vec{p}}\), where \(1<p_j<\infty\) for \(j=1,\dots,m\) and \[ \frac{1}{p}=\frac{1}{p_1}+\dots+\frac{1}{p_m}. \] At the end-point they obtain the following weak-type weighted bounds: \[ \nu_{\vec{w}}(\{ x\in\mathbb{R}^n : |T_{\prod \mathbf{b}}(\mathbf{f})(x)| > t^m \}) \leq C\prod_{j=1}^{m}\left(\int_{\mathbb{R}^n}\Phi^{(m)}\left( \frac{|f_j(x)|}{t} \right)w_j(x)dx \right)^{1/m}, \] where \(\Phi(t) = t(1+\log^{+}t)\) and \(\vec{w}\in A_{\vec{1}}\).
In the case of Lebesgue measure weights and a linear operator, \(m=1\), these results recover a classic result from [R. R. Coifman et al., Ann. Math. (2) 103, 611–635 (1976; Zbl 0326.32011)]. These results also compliment and extend previous results by C. Pérez and R. H. Torres [Contemp. Math. 320, 323–331 (2003; Zbl 1045.42011)] and A. K. Lerner et al. [Adv. Math. 220, No. 4, 1222–1264 (2009; Zbl 1160.42009)] on commutators of the type \[ T_{\sum\mathbf{b}}(f_1,\dots,f_m) = \sum_{j=1}^{m}[b_j,T]_j(f_1,\dots,f_m). \] The difference is that these commutators are not iterated.