Let $G$ be a real rank one connected semisimple Lie group with finite center. As well-known the radial, heat, and Poisson maximal operators satisfy the $L^p$-norm inequalities for any $p>1$ and a weak type $L^1$ estimate. The aim of this paper is to find a subspace of $L^1(G)$ from which they are bounded into $L^1(G)$. As an analogue of the atomic Hardy space on the real line, we introduce an atomic Hardy space on $G$ and prove that these maximal operators with suitable modifications are bounded from the atomic Hardy space on $G$ to $L^1(G)$.
"Hardy spaces and maximal operators on real rank one semisimple Lie groups, I." Tohoku Math. J. (2) 52 (1) 1 - 18, 2000. https://doi.org/10.2748/tmj/1178224654