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 \(L^{\Phi ,q}({\mathbb {R}}^n)\) $L^{\mathrm{\Phi },q}({\mathbb{R}}^n)$ for a Young function \(\Phi \in \nabla _2\) $\mathrm{\Phi }\in {\mathrm{\nabla }}_2$ and \(0<q<1.\) $0<q<1.$