Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spaces

We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form ∫{x:Mf(x)>λ}φ(λω1(x))ω2(x)dμ≤c∫Xφ(cf(x)ω3(x))ω4(x)dμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _{\{ {x: \mathcal {M} f(x) > \lambda } \}} {\varphi (\lambda {\omega _1}(x))} {\omega _2}(x) \,d\mu \le {c} \int _X \varphi ({c}f(x){\omega _3}(x)){\omega _4}(x) \,d\mu \end{aligned}$$\end{document}in Orlicz classes, where Mf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} f$$\end{document} is a Hardy–Littlewood maximal function defined on homogeneous type spaces. Our main result extends some known results.