Weighted Estimates for Multilinear Commutators of Marcinkiewicz Integrals with Bounded Kernel

Let μΩ,b→\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mu}_{\varOmega, \overrightarrow{b}} $$\end{document} be a multilinear commutator generalized by the n-dimensional Marcinkiewicz integral with bounded kernel μΏ and let bj∈OscexpLrj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {b}_j\ \in Os{c_{\exp}}_{L^{r_j}} $$\end{document} , 1 ≤ j ≤ m. We prove the following weighted inequalities for ω ∈ A∞ and 0 < p < ∞: μΩfLpω≤CMfLpω,μΩ,b→fLpω≤CMLlogL1/rfLpω.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\begin{array}{cc}\hfill {\left\Vert {\mu}_{\varOmega }(f)\right\Vert}_{L^p\left(\omega \right)}\le C{\left\Vert M(f)\right\Vert}_{L^p\left(\omega \right)},\hfill & \hfill \left\Vert {\mu}_{\varOmega, \overrightarrow{b}}(f)\right\Vert \hfill \end{array}}_{L^p\left(\omega \right)}\le C{\left\Vert {M}_{L{\left( \log L\right)}^{1/r}}(f)\right\Vert}_{L^p\left(\omega \right)}. $$\end{document}