Hardy spaces and maximal operators on real rank one semisimple lie groups I

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 Ll estimate. The aim of this paper is to find a subspace of L¹ (G) from which they are bounded into L (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 L¹ (G).