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 wHΦ1,bs and space wHΦ2,bs Martingale transformation between space wDΦ1,b and space wDΦ2,b;Martingale transformation between space wQΦ1,b and space WQΦ1,b.In the first part,The relationship between wHΦ1,bs and wHΦ2,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 wHΦ1,bs martingale transformation of some martingale g in wHΦ2,bs.In the second part,Using the powerful tool of martingale transformation,the relationship between wDΦ1,b and wDΦ2,b in the weak Hardy-Orlicz-Karamata martingale space composed of wLΦ,b-predicable martingale is described respectively.And the characterization of the relationship between WQΦ1,b and WQΦ1,b of the weak Hardy-Orlicz-Karamata martingale space formed by the mean square function S(f)with wLΦ,b-controlled Martingale f.It is proved that any element of f in wQΦ1,b in the weak Hardy-Orlicz-Karamata martingale space is wQΦ1,b martingale transformation of some martingale g in wQΦ2,b.