H p -spaces of martingales, 0

For 0 <p≦1 the dual Banach space to the space of H _p -martingales is characterized as a Lipschitz class of order α=l/p?1. When p=1 and α=0 this reduces to BMO being the dual of H ₁ . In the course of the investigation a hierarchy of martingale spaces _a L _p is introduced with the property that the index p is in the range ?∞ <1/p<∞. For l≦b≦q<∞ the space L _q is a subspace of _b L _q (equality when b=q) while for q<0 the space _b L _q corresponds to a Lipschitz class of order α=?1/q.