  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)\ast L^{2,1}({G})\subseteq L^{2,\infty}({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 {\cal M}_2f(z) = \sup_{z\in B}
\frac{1}{|B|} \int_{B}|f(z')|\,dz' is bounded from L^{2,1}(X) to
L^{2,\infty}(X) and from L^p(X) to L^p(X) in the sharp range of exponents
p\in(2,\infty]. The supremum in the definition of {\cal M}_2f(z) is taken over
all balls containing the point z.
