Weighted Inequalities for Vector Operators on Martingales

Sufficient conditions are given to get weighted inequalities between two maximal operators on Banach valued regular martingales. As an application we obtain generalizations with weights of the inequalities in the definitions of UMD- and MT-Banach spaces and weighted estimates for the vector valued square function operator. We also obtain results for Haar and Walsh series. In particular it is shown that LpX-convergence of the Walsh-Fourier series characterizes UMD spaces.