Atomic decompositions and inequalities for vector-valued discrete-time martingales

We consider martingales with discrete time taking values in a Banach lattice $X$ that has UMD-property (UMD means unconditionality of martingale differences). We suppose that the UMD-lattice $X$ consists of real-valued functions. Two notions of maximal value for such martingales are introduced (in the case of real-valued martingales these notions are the same and also coincide with the notion of usual maximal value). We also introduce the notion of quadratic variation and both usual and predictable classes of martingale spaces corresponding to maximal values and quadratic variation. The equivalence of these classes is established. In particular, Davis inequalities are proved with the help of atomic decompositions. The case of a regular stochastic basis is considered separately.