We consider the centred and the uncentred Hardy–Littlewood maximal operators on metric measure spaces with exponential volume growth and “locally bounded geometry”. We discuss the problem of finding the optimal range of $p$’s such that either the centred or the uncentred HL maximal operator is bounded on $L^p$.

The prototypes of the metric measure spaces we consider are the symmetric spaces of the noncompact type and the homogeneous trees, where sharp $L^p$ estimates on the HL operators are available in the literature.

We shall show that interesting phenomena arise when homogeneous trees are replaced by non-homegeoeus ones, and noncompact symmetric spaces by Riemannian manifolds with pinched negative sectional curvature.