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 theLp-norm inequalities forp>1 and a weak typeL1estimate. In this paper, through the Fourier theories onRandGwe shall duplicate the Hardy spaceH1(R) to a subspaceH1s(G) (s⩾0) ofL1(G) and show that these operators are bounded fromH1s(G) toL1(G).