Spherical maximal operator on symmetric spaces, an end point estimate

Abstract

We establish an endpoint weak-type maximal inequality for the spherical maximal operator applied to radial functions on real rank-1 symmetric spaces of dimension n ≥ 2. More explicitly, we prove the Lorentz space estimate ∥Mƒ∥n′,∞ ≤ Cn∥ƒ∥n′,1,n′=nn−1 for every radial function in the Lorentz space Ln′,1' (X) associated with the natural isometry-invariant measure on X. The proof uses only geometric arguments and volume estimates and applies uniformly in every dimension.