Boundedness of Some Marcinkiewicz Integral Operators Related to Higher Order Commutators on Hardy Spaces

In this paper, the authors study the boundedness properties of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mu ^{m}_{{\Omega ,b}} $$\end{document} generated by the function b ∈ Lipβ(ℝn)(0 < β ≤ 1/m) and the Marcinkiewicz integrals operator μΩ. The boundednesses are established on the Hardy type spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H^{p}_{{b^{m} ,s}} {\left( {\mathbb{R}^{n} } \right)} $$\end{document} and the Herz–Hardy type spaces \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ H_{{b^{m} }} \dot{K}^{{\alpha ,p}}_{q} {\left( {\mathbb{R}^{n} } \right)} $$\end{document}.