The Application Of Some Geometric Properties Of Orlicz Space In The Martingale Theory

The theory of Orlicz space and the martingale theory in their respective have had certain development and perfection, but the application of some geometric properties of Orlicz space in the martingale theory research has just begun, especially the application of some geometric properties of Orlicz space equipped with the p-Amemiya norm in the martingale theory research has not begun. The p-Amemiya norm inequalities of B-valued martingales and the boundedness of some operators of Martingale space and p uniform convexity of Orlicz space will be studied. This thesis consists of four parts. The main results are as follows:It is reviewed that the application of theory of Orlicz space in the martingale theory and developing process of the p-uniform convexity and q-uniform smoothness of Banach space. Moreover, main research results relative to propewrties mentioned above that have been already summaried and the main contents of this thesis are shown.The p-Amemiya norm inequalities of the maximal functions and the p-mean functions of B-valued martingales are proved, meaning while criteria for a Banach space are the p-uniform convexity and q-uniform smoothness are presented. This will help us to study the Orlicz space and Martingale theory better.The boundedness of pingxiao operator and variable operator of Martingale space are proved. Moreover, the criterion of the p-uniform convexity and q-uniform smoothness of Banach space are presented.P-uniform convexity is an important geometric property in Banach space. According to N functionΦ, we define another N function M. So the criterion of the p-uniform convexity of Orlicz space L_M equipped with the Luxemburg norm and the Orlicz norm and the p-Amemiya norm are presented.