Let H1 (R 2 ), H1 (R+, rdr), H1 (T) and H1 (D) be the Hardy spaces on R, R+, T and D respectively . Classically relations between H1 (R 2 ) and H1 (R+, rdr), also H1 (T) and H1 (D) are well-known. We shall consider a relation between H1 (R+, rdr) and H1 (T). We give a theorem that the product of H1 (R+, rdr) and a Poisson transform of H1 (T) belongs to H1 (R 2 ). Similarly, we shall consider the case of SU(1, 1)/SO(2).
