Estimates of commutators on Herz-type spaces with variable exponent and applications

Let [b, T] be the commutator generated by b and T, where b∈BMO(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b\in \mathrm {BMO}({\mathbb {R}}^{n})$$\end{document} and T is a Calderón–Zygmund singular integral operator. In this paper, the authors establish some strong type and weak type boundedness estimates including the LlogL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L\log L$$\end{document} type inequality for [b, T] on the Herz-type spaces with variable exponent. Meanwhile, the similar results for the commutators [b,Il]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[b,I_l]$$\end{document} of fractional integral operator are also obtained. As applications, we consider the regularity in the Herz-type spaces with variable exponent of strong solutions to nondivergence elliptic equations with VMO\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {VMO}$$\end{document} coefficients.