c © 1997 Universitat de Barcelona The martingale Smirnov class

The Smirnov class N∗(δ) of dyadic martingales is studied. Continuous linear functionals on this class and its Fréchet envelope are described. It is proved that, in contrast to the case of Hp-spaces, the space N∗(δ) is not isomorphic to the Smirnov class of holomorphic functions on the unit disc. Finally, atomic decompositions of elements of N∗(δ) are obtained. This paper has been motivated by the following result [3]: For 0 < p <∞, the Hardy spaces Hp(D) on the unit disc D and Hp(T) on the unit circle T, and the Hardy space Hp(δ) of dyadic martingales on the interval [0, 1), are all isomorphic