The Density Of Finite Martingales In Weak Hardy-Orlicz Space Of Vector-Valued Martingales

This paper mainly studies the density of finite martingale in weak Hardy-Orlicz spaces of vector-valued martingale.In the first place,what we have paid attention to research is the density of finite martingale in predictable vector-valued martingale weak Hardy-Orlicz space wP?,B.And then,the sufficient and necessary property of its density is that Banach space has Radon-Nikodym property.Secondly,we will continue studying the other weak Hardy-Orlicz martingale of (?).The results show that the density of finite martingale is closely related to the geometric properties of Banach space.The main research tools and procedures are summarized as follows:In the first part,the concept of B-valued atomic martingale have been brought in.The decomposition theorem of "weak atom martingale" is established for B-value of weak Hardy-Orlicz martingale space without any constraint on the geometric properties of the Banach space,Then we use the generalized Davis decomposition theorem to decompose the vector-value martingale space WH?,B M,Finally,by combining the decomposition theorem of weak atom martingale and the generalized Davis decomposition theorem,we obtain the dense equivalent condition of finite martingale in weak Hardy-Orlicz space,which is that Banach space has Radon-Nikodym property.In the second part,the vector value "weak value atom martingale" is introduced,and the decomposition theorem of "weak atomic martingale" have been established on the value of weak Hardy-Orlicz martingale space without any limits on the geometric properties of Banach space.The Davis decomposition theorem is then extended to the weak Hardy-Orlicz martingale space,With the corresponding Davis decomposition,the space (?) have been well decomposed.Finally,through the decomposition theorem of weak atom martingale and the generalized Davis decomposition theorem,we get the result that the density of finite martingale in the weak Hardy-Orlicz space have closely connection with the geometrical properties of Banach space.These results obtained here extend the corresponding results in former literature.