A unified version of weighted weak type inequality for martingale maximal operators

Let gamma and Phi(1) be nondecreasing and nonnegative functions defined on [0, infinity), and Phi(2) is an N-function, u, v and w are weights. A unified version of weighted weak type inequality of the form Phi(1)(lambda)P-u (f* > lambda) < CE (Phi(2) (C vertical bar f infinity vertical bar v/gamma(lambda)) w) for martingale maximal operators f* is considered, some necessary and sufficient conditions for it to hold are shown. In addition, we give a complete characterization of three-weight weak type maximal inequality of martingales. Our results generalize some known results on weighted theory of martingale maximal operators.