Dichotomy property for maximal operators in non-doubling setting

We investigate a dichotomy property for Hardy–Littlewood maximal operators, non-centered $M$ and centered $M^c$, that was noticed by Bennett, DeVore and Sharpley. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^c$ in the context of non-doubling metric measure spaces $(X,\rho ,\mu )$. In addition, if $X = $\mathbb{R}$^d$,$d \geq 1$,and$\rho$ is the metric induced by an arbitrary norm on ${\mathbb{R}}^d$, then we give the exact characterization (in terms of $\mu$) of situations in which $M^c$ possesses the dichotomy property provided that $\mu$ satisfies some very mild assumptions.