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,L-1(G)*L-2,L-1(G) subset of or equal to L-2,L-infinity(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 M(2)f(z) = sup(z epsilonB)1//B/ integral (B)/f(z')/dz' is bounded from L-2,L-1(X) to L-2,L-infinity(X) and from L-P(X) to L-P(X) in the sharp range of exponents p epsilon (2, infinity]. The supremum in the definition of M(2)f(z) is taken over all balls. containing the point z.