Martingale transforms between Orlicz–Hardy spaces of predictable martingales

Using the technique of Burkholderʼs martingale transforms, the relations between “predictable” martingale Orlicz–Hardy spaces are investigated. Let Φ 1 and Φ 2 be two Young functions and Φ 1 ⋞ Φ 2 in some sense, a constructive proof is obtained of that the elements in Orlicz–Hardy space H Φ 1 are none other than the martingale transforms of those in Orlicz–Hardy space H Φ 2 , where H Φ ∈ { P Φ , Q Φ , H Φ s } . At the endpoint case, the space H ∞ must be replaced by a B M O space, it is also proved that a martingale is in H Φ ∈ { P Φ , Q Φ } if and only if it is the transform of a martingale from B M O ∈ { BMO 1 , BMO 2 } .