Characterization for commutators of n-dimensional fractional Hardy operators

In this paper, it was proved that the commutator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{H}_{\beta ,b} $$ \end{document} generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from Lp1 (ℝn) to Lp2 (ℝn) if and only if b is a CṀO(ℝn) function, where 1/p1 − 1/p2 = β/n, 1 < p1 < ∞, 0 ⩽ β < n. Furthermore, the characterization of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathcal{H}_{\beta ,b} $$ \end{document} on the homogenous Herz space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\dot K_q^{\alpha ,p} $$ \end{document}(ℝn) was obtained.