Martingale transforms and weak Orlicz–Hardy spaces of predictable martingales

Making use of the technique of Burkholder’s martingale transforms, the interrelationships between “predictable” martingale weak 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 weak Orlicz–Hardy space wHΦ1 are none other than the martingale transforms of those in wHΦ2, where wHΦ∈{wPΦ,wQΦ}. The results obtained here extend the corresponding results in the literature from strong-type spaces (normed space) to the setting of weak-type spaces (quasi-normed space).