Some New Φ-moment Martingale Inequalities Associated With Concave Functions

The atomic decomposition theorem plays an important role in martingale theory as well as in harmonic analysis. In martingale theory, atomic decomposition methods not only deal with martingale spaces with small index, but also unify the single index and multiple index. It is simple and effective for dealing with the duality and interpolation theory of martingale space.This paper introduces martingale Orlicz-Hardy spaces and summarizes the main work of T. Miyamoto, E. Nakai, G. Sadasue about the atomic decomposition theorem of martingale Orlicz-Hardy spaces in2012. The main innovation of this paper is the result of the third chapter. We establish the first atomic decomposition theorem without involving norm (even quasi-norm), by using the atomic decomposition theorem. We improve the results of Burkholder and Gundy about (?)-moment martingale inequalities.