Real Hardy spaces on real rank 1 semisimple Lie groups

Let G be a real rank one connected semisimple Lie group with finite center. We introduce a real Hardy space H¹(G/K) on G as the space consisting of all K-bi-invariant functions f on G whose radial maximal functions Mϕf are integrable on G. We shall obtain a relation between H¹(G/K) and H¹(R), the real Hardy space on the real line R, via the Abel transform on G and give a characterization of H¹(G/K).