Martingale transforms between Hardy-Orlicz spaces QΦ1 and QΦ2 of martingales

Using the technique of martingale transforms, the relation between Hardy-Orlicz spaces of the martingales with predictable quadratic variations is investigated. Let Φ1 and Φ2 be two Young functions and Φ1≺Φ2 in some sense, a constructive proof is obtained that the elements in the Hardy-Orlicz space QΦ1 are none other than the martingale transforms of those in the Hardy-Orlicz space QΦ2. The results obtained here extend the corresponding results in the former literature.