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 Φ1Φ1 and Φ2Φ2 be two Young functions and Φ1⋞Φ2Φ1⋞Φ2 in some sense, a constructive proof is obtained of that the elements in Orlicz–Hardy space HΦ1HΦ1 are none other than the martingale transforms of those in Orlicz–Hardy space HΦ2HΦ2, where HΦ∈{PΦ,QΦ,HΦs}. At the endpoint case, the space H∞H∞ must be replaced by a BMOBMO space, it is also proved that a martingale is in HΦ∈{PΦ,QΦ}HΦ∈{PΦ,QΦ} if and only if it is the transform of a martingale from BMO∈{BMO1,BMO2}BMO∈{BMO1,BMO2}.