Spherical maximal operator on symmetric spaces of constant curvature

We prove an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial funcions on symmetric spaces of constant curvature and dimension n greater than or equal to 2. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function f, parallel toMfparallel to(n', infinity) less than or equal to C(n)parallel tofparallel to(n'), n' = n/n-1. The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.