New weighted norm inequalities for multilinear singular integral operators with generalized kernels and their commutators

In this paper, we introduce a class of multilinear singular integral operators with generalized kernels, whose kernel is weaker than the kernel of certain Dini’s type. Then, we establish new weighted norm inequalities of multilinear singular integral operators with generalized kernels, multilinear commutators and multilinear iterative commutators, respectively. The weight function involved is Ap∞(φ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\textbf{p}}^\infty (\varphi )$$\end{document}. It contains the multiple weights class Ap\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\textbf{p}}$$\end{document}. Finally, as applications, we can obtain that our results generalize the results of a class of multilinear Calderón–Zygmund operators with kernels of Dini’s type and a class of multilinear singular integral operators with generalized kernels under certain conditions.