The endpoints $p_-=1$ and $p_+=\infty$ for the maximal operator on $L^{p(\cdot)}$ over spaces of homogeneous type

We investigate the boundedness of the Hardy-Littlewood maximal operator $M$ on variable Lebesgue spaces $L^{p(\cdot)}(Ω,d,μ)$ over spaces of homogeneous type, focusing on the endpoints $p_-=1$ and $p_+=\infty$. At the lower endpoint, we prove that the boundedness of $M$ forces $p_->1$ for all expone