Atomic Decompositions Of Weak And Strong Martingale Spaces And Their Applications

Atomic decompositions is an important method for studying martingale spaces theory.Because of its simplicity and effectiveness in dealing with problems,it has attracted much attention in recent years.This paper uses atomic decompositions as a tool to study the prop-erties and application of weak and strong dyadic martingale spaces with variable exponents and B-value weak quasi-martingale spaces.Firstly,we study the weak and strong dyadic martingale spaces with variable exponents.We establishe the atomic decompositions of the weak and strong dyadic martingale spaces with variable exponents.By atomic decompositions,we prove that sublinear operator T is bounded from wHp(·)? to wLp(·);Cesaro operator is bounded from Hp(·)M to Lp(·).and from lp(·)to Lp(·).Secondly,we study B-valued weak Orlicz quasi-martingale spaces.We combine the geometric properties of B anach spaces to establish the atomic decompositions for B-valued weak Orlicz quasi-martingale spaces.By atomic decompositions,the embedding relation-ship between the B-valued weak Orlicz quasi-martingale spaces and the boundedness of the sublinear operators are proved.Finally,we make a general summary of the research contents of this paper,and propose the next research direction.