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

### Abstract

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
$$\mathcal{M}_2\; f(z) = \underset{z\in B}{\sup} \frac{1}{|B|} \int_B |f(z’)| dz’$$
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 $\mathcal{M}_2\; f(z)$ is taken over all balls containing the point $z$.

## Authors

Alexandru D. Ionescu