A remark on the boundedness of the Hardy-Littlewood maximal operator on Orlicz-Lorentz spaces

In this paper, we give an alternative proof of the main result in Hatano et al. (Tokyo J Math 46(1):125-160, 2023) that the Hardy-Littlewood maximal operator is bounded on the Orlicz-Lorentz space L Phi,q(Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L<^>{\Phi ,q}({\mathbb {R}}<^>n)$$\end{document} for a Young function Phi is an element of del 2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi \in \nabla _2$$\end{document} and 0<q<1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<q<1.$$\end{document}