Two-weight weak-type maximal inequalities for martingales

In this article, some necessary and sufficient conditions are shown in order that the inequality of the form Φ1(λ)Pu(f*>λ)≤Ev(Φ2(C|f∞|))Φ1(λ)Pu(f*>λ)≤Ev(Φ2(C|f∞|))holds with some constant C > 0 independent of martingale f = (fn)n≥o and λ > 0, where Φ1 and Φ2 are a pair of Young functions, f*=supn≥0|fn| and f∞=limn→∞fn a.e.