Characterization for boundedness of some commutators of the multilinear fractional Calderón-Zygmund operators with Dini type kernel

Let T alpha\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{\alpha}$$\end{document} be an m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m$$\end{document}-linear fractional Calder & oacute;n-Zygmund operator with kernel of mild regularity, and b ->=(b1,b2,& mldr;,bm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec{b} =(b_{1},b_{2} ,\ldots,b_{m})$$\end{document} be a collection of locally integrable functions. In this paper, the main purpose is to establish some estimates for the mapping property of the multilinear commutators T alpha,Sigma b ->\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ T_{{\alpha,\Sigma \vec{b}}}$$\end{document} in the context of the variable exponent function spaces. The key tools used are the Fourier series and the pointwise estimates involving the sharp maximal operator of the multilinear commutator and certain associated maximal operators.