Weighted estimates for the iterated commutators of multilinear maximal and fractional type operators (Q2848711)

\textit{C. Pérez} et al. [Bull. Lond. Math. Soc. 46, No. 1, 26--42 (2014; Zbl 1321.42032)] obtained weighted norm inequalities for iterated commutators of multilinear Calderón-Zygmund operators and pointwise multiplication with functions in \(\text{BMO}(\mathbb{R}^n)\). The author of this paper extends these results to maximal and fractional type operators.NEWLINENEWLINELet \(T\) be a multilinear Calderón-Zygmund operator that can be represented with a kernel \(K\) as NEWLINE\[NEWLINE T(f_1,\dots,f_m)(x) = \int_{(\mathbb{R}^n)^m}K(x,y_1,\dots,y_m)f_1(y_1)\cdots f_m(y_m)dy_1\cdots dy_m NEWLINE\]NEWLINE and let \(\mathbf{b}=(b_1,\dots,b_m)\in\text{BMO}^m(\mathbb{R}^n)\). Define the commutator of \(b_j\) and \(T\) in the \(j\)-th entry of \(T\) by NEWLINE\[NEWLINE [b_j,T]_j(f_1,\dots,f_m) = b_j T(f_1,\dots,f_j,\dots,f_m) - T(f_1,\dots,b_j f_j,\dots,f_m). NEWLINE\]NEWLINE The authors of [loc. cit.] studied mapping properties of the operator NEWLINE\[NEWLINE T_{\prod \mathbf{b}}(f_1,\dots,f_m) = [b_1,[b_2,\dots[b_{m-1},[b_m,T]_m]_{m-1}\dots]_2]_1(f_1,\dots,f_m) NEWLINE\]NEWLINE in the previously mentioned paper.NEWLINENEWLINEIn this paper, the author studies the following maximal and fractional type operators: NEWLINE\[NEWLINE T_{*,\prod \mathbf{b}}(f_1,\dots,f_m) = \sup_{\delta > 0}|[b_1,[b_2,\dots[b_{m-1},[b_m,T_{\delta}]_m]_{m-1}\dots]_2]_1(f_1,\dots,f_m)|,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE T_{\delta}(f_1,\dots,f_m)(x) = \int_{|x-y_1|^2+\dots+|x-y_m|^2>\delta^2}K(x,y_1,\dots,y_m)f_1(y_1)\cdots f_m(y_m)dy_1\cdots dy_m,NEWLINE\]NEWLINE and NEWLINE\[NEWLINE I_{\alpha,\prod \mathbf{b}}(f_1,\dots,f_m) = [b_1,[b_2,\dots[b_{m-1},[b_m,I_{\alpha}]_m]_{m-1}\dots]_2]_1(f_1,\dots,f_m),NEWLINE\]NEWLINE where NEWLINE\[NEWLINE I_{\alpha}(f_1,\dots,f_m)(x) = \int_{(\mathbb{R}^n)^m}\frac{1}{|(y_1,\dots,y_m)|^{mn-\alpha}}f_1(x-y_1)\cdots f_m(x-y_m)dy_1\cdots dy_m .NEWLINE\]NEWLINE In case of the operator \(T_{*,\prod \mathbf{b}}\), the author obtains the same Hölder type strong weighted estimates and \(L(\log L)\) weak-type weighted end-point estimates as in [loc. cit.] and in the case of \(I_{\alpha,\prod \mathbf{b}}\), the estimates are naturally modified to reflect the decay rate \(\alpha\) of the fractional integral operator \(I_{\alpha}\).