BMO Inequalities Of Vector-valued Trigonometric Martingales And Convexity Of Complex Spaces

We study inequalities of trigonometric martingales and trigonometric martingales space with values in complex Banach spaces, using them give geometric characteriza-tions of complex Banach space. This paper contains threes chapters, the outline of the paper is arranged as follows.In chapter 1,we presents the background and give out some necessary notations as well as definitions of spaces.In chapter 2,we discuss the trigonometric martingale spaces and prove some analogous results to classical martingale spaces.In chapter 3, using BMO norm inequalities of trigonometric martingales with values in complex Banach space,we give its uniform convexity.