Spherical maximal operator on symmetric spaces of constant curvature

Abstract

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 > 2. More explicitly, in the Lorentz space associated with the natural isometry-invariant measure, we show that, for every radial function f, ||Mf|| n', ∞≤C n ||f|| n',1 , n'=. The proof uses only geometric arguments and volume estimates, and applies uniformly in every dimension.