Some Researches Of Properties On B-valued BMO Martingale Spaces

Several relevant analytical properties of vector-valued BMO martingale spaces are researched in this paper.First of all,BMOq?(X)martingales and BMOq?(X)martingales are described by different Carleson measures.And then,it not only proves the boundedness of mean-square operator in these martingale spaces,but also proves its boundedness in the weak BMO space.The main content of this article is summarized as the following three parts.The first part shows a characterization of BMOq?(X)martingales by fractional Carleson measure.Some properties of the vector-valued martingale spaces BMOq?(X)are investigated in this part.It is proved that a measure related to a X-valued martingale f?BMOq?(X)is a bounded(q,?)-Carleson measure and q-variation S(q)(f)on BMOq?(X)is bounded if and only if X has an equivalent norm which is q-uniformly convex.The results obtained here extend the corresponding results from real martingales to vector-valued martingales.And the second part shows a characterization of BMOq?(X)martingales by(q,?)-Carleson measure,The characterization of generalized Campanato spaces Lq,?(X)of vector-valued martingales by Carleson measures is investigated in this part.More precisely,the so-called(q,?)-Carleson measures are introduced and the relationship between vector-valued martingales in Lq,?(X)and(q,?)-Carleson measures is obtained.As a consequence,a new way to characterize the geometrical properties of a Banach space is found.As for the last part,the boundedness of mean-square operator on weak BMO martingale space is investigated.It is proved that X is isomorphic to a q-uniformly convex Banach spaces if and only if the q-mean-square operator for martingales S(q)(·)are of types(wBMOq?(X),wBMOq?(X))and(Lq(X),wBMOq?(X)).As well as,a new characterization of the q-uniformly convexity of Banach space is obtained.