Abstract

Let Sn−1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{S}^{n-1}$\\end{document} denote unit sphere in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\mathbb{R}^{n}$\\end{document} equipped with the normalized Lebesgue measure. Let Φ∈Ls(Sn−1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\Phi \\in L^{s}(\\mathbb{S}^{n-1})$\\end{document} be a homogeneous function of degree zero such that ∫Sn−1Φ(y′)dσ(y′)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\int _{\\mathbb{S}^{n-1}}\\Phi (y^{\\prime})d \\sigma (y^{\\prime})=0$\\end{document}, where y′=y/|y|\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$y^{\\prime}=y/|y|$\\end{document} for any y≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$y\ eq 0$\\end{document}. The commutators of variable Marcinkiewicz fractional integral operator is defined as [b,μΦ]βm(f)(x)=(∫0∞|∫|x−y|≤sΦ(x−y)[b(x)−b(y)]m|x−y|n−1−β(x)f(y)dy|2dss3)12.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ [b,\\mu _{\\Phi}]^{m}_{\\beta }(f)(x )= \\left ( \\int \\limits _{0} ^{ \\infty }\\left |\\int \\limits _{|x -y | \\leq s} \\frac{\\Phi (x -y )[b(x )-b(y )]^{m}}{|x -y |^{n-1-\\beta (x )}}f(y )dy \\right |^{2} \\frac{ds}{s^{3}}\\right )^{\\frac{1}{2}}. $$\\end{document}In this paper, we obtain the boundedness of the commutators of the variable Marcinkiewicz fractional integral operator on grand variable Herz spaces K˙p(⋅)α(⋅),q),θ(Rn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}${\\dot{K} ^{\\alpha (\\cdot ), q),\ heta}_{ p(\\cdot )}(\\mathbb{R}^{n})}$\\end{document}.

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