The authors develop a Hardy space theory for certain function spaces, among them the spaces
1 (formerly considered by A. Beurling [Ann. Inst. Fourier 14, No.2, 1-32 (1964; Zbl 0133.075)]), and the harmonic extension of their elements to the upper half-plane. Their results include a Burkholder-Gundy-Silverstein maximal function characterization of spaces related to the spaces above. Also considered are duality relations; for example, an analogue to the Fefferman-Stein theorem [C. Fefferman and E. M. Stein, Acta Math. 129, 137-193 (1972; Zbl 0257.46078)] on the duality between the classical Hardy space and BMO is proved.