L¹Estimates for Maximal Functions and Riesz Transform on Real Rank 1 Semisimple Lie Groups

LetGbe a real rank one semisimple Lie group andKa maximal compact subgroup ofG. Radial maximal operators for suitable dilations, the heat and Poisson maximal operators, and the Riesz transform, which act onK-bi-invariant functions onG, satisfy theL^p-norm inequalities forp 1 and a weak typeL¹estimate. In this paper, through the Fourier theories onRandGwe shall duplicate the Hardy spaceH¹(R) to a subspaceH¹_s(G) (s≥0) ofL¹(G) and show that these operators are bounded fromH¹_s(G) toL¹(G).