DICHOTOMY PROPERTY FOR MAXIMAL OPERATORS IN A NONDOUBLING SETTING

We investigate a dichotomy property for Hardy–Littlewood maximal operators, noncentred $M$ and centred $M^{c}$ , that was noticed by Bennett et al. [‘Weak- $L^{\infty }$ and BMO’, Ann. of Math. (2) 113 (1981), 601–611]. 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 nondoubling metric measure spaces $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$ . In addition, if $X=\mathbb{R}^{d}$ , $d\geq 1$ , and $\unicode[STIX]{x1D70C}$ is the metric induced by an arbitrary norm on $\mathbb{R}^{d}$ , then we give the exact characterisation (in terms of $\unicode[STIX]{x1D707}$ ) of situations in which $M^{c}$ possesses the dichotomy property provided that $\unicode[STIX]{x1D707}$ satisfies some very mild assumptions.