Martingale Transformation And The Weak Hardy-Orlicz-Karamata Martingale Space

The analytic properties of the weak Hardy-Orlicz-Karamata martingale space are studied in this paper.The relationship between the weak Hardy-Orlicz-Karamata martingale Spaces is characterized by the use of martingale transformation.It majorly includes the following three aspects:Martingale transformation between space wH?1,bs and space wH?2,bs Martingale transformation between space wD?1,b and space wD?2,b;Martingale transformation between space wQ?1,b and space WQ?1,b.In the first part,The relationship between wH?1,bs and wH?2,bs in the weak martingale Hardy-Orlicz-Karamata space defined by the conditional mean square functions(f)is studied.It is proved that an arbitrary element f in the weak Hardy-Orlicz-Karamata martingale space is wH?1,bs martingale transformation of some martingale g in wH?2,bs.In the second part,Using the powerful tool of martingale transformation,the relationship between wD?1,b and wD?2,b in the weak Hardy-Orlicz-Karamata martingale space composed of wL?,b-predicable martingale is described respectively.And the characterization of the relationship between WQ?1,b and WQ?1,b of the weak Hardy-Orlicz-Karamata martingale space formed by the mean square function S(f)with wL?,b-controlled Martingale f.It is proved that any element of f in wQ?1,b in the weak Hardy-Orlicz-Karamata martingale space is wQ?1,b martingale transformation of some martingale g in wQ?2,b.