New weighted norm inequalities for Calderón–Zygmund operators with kernels of Dini’s type and their commutators

In this paper, we introduce certain classes of Calderón–Zygmund operators with kernels of Dini’s type including pseudodifferential operators with smooth symbols. Applying a class of new weight functions, we establish some weighted norm inequalities for certain classes of Calderón–Zygmund operators with kernels of Dini’s type. In addition, new BMO spaces with respect to the class of new weight functions are introduced. Naturally, the pointwise, weighted strong type and endpoint LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\,\mathrm {log}\,L$$\end{document} type estimates for the commutators with the new BMO functions are also obtained.