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

Abstract

We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form $$\begin{aligned} \int _{\{ {x: \mathcal {M} f(x) > \lambda } \}} {\varphi (\lambda {\omega _1}(x))} {\omega _2}(x) \,d\mu \le {c} \int _X \varphi ({c}f(x){\omega _3}(x)){\omega _4}(x) \,d\mu \end{aligned}$$ in Orlicz classes, where $$\mathcal {M} f$$ is a Hardy–Littlewood maximal function defined on homogeneous type spaces. Our main result extends some known results.