An endpoint estimate for the Kunze-Stein phenomenon and related maximal operators

One of the purposes of this paper is to prove that if G is a noncompact connected semisimple Lie group of real rank one with finite center, then L^2,1 (G) * L^2,1 (G) ⊆ L^2,∞(G). Let K be a maximal compact subgroup of G and X = G/K a symmetric space of real rank one. We will also prove that the noncentered maximal operator M₂f(z) = Sup_z∈B1/|B|∫_B|f(z′)\dz′ is bounded from L^2,1(X) to L^2,∞(X) and from L^p(X) to L^p(X) in the sharp range of exponents p ∈ (2, ∞]. The supremum in the definition of M₂f(z) is taken over all balls containing the point z.