Martingale transforms and weak Orlicz–Hardy spaces of predictable martingales

Abstract

Abstract Making use of the technique of Burkholder’s martingale transforms, the interrelationships between “predictable” martingale weak 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 weak Orlicz–Hardy space w H Φ 1 are none other than the martingale transforms of those in w H Φ 2 , where w H Φ ∈ { w P Φ , w Q Φ } . 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).