Martingale transforms between Orlicz-Hardy spaces of predictable martingales

Using the technique of Burkholder's martingale transforms, the relations between "predictable" martingale Orlicz-Hardy spaces are investigated. Let Phi(1) and Phi(2) be two Young functions and Phi(1) (sic) Phi(2) in some sense, a constructive proof is obtained of that the elements in Orlicz-Hardy space H-Phi 1 are none other than the martingale transforms of those in Orlicz-Hardy space H-Phi 2, where H-Phi is an element of {P-Phi, Q(Phi), H-Phi(s)}. At the endpoint case, the space H-infinity must be replaced by a BMO space, it is also proved that a martingale is in H-Phi is an element of {P-Phi, Q(Phi)}, if and only if it is the transform of a martingale from BMO is an element of {BMO1, BMO2}. (C) 2014 Elsevier Inc. All rights reserved.