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 L-1 estimate. The aim o f this paper is to find a subspace of L-1 (G) from which they are bounded i nto 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 t o L-1(G).